diff --git a/MSetFoldWithAbort.v b/MSetFoldWithAbort.v
index 18dbc49..145290f 100644
--- a/MSetFoldWithAbort.v
+++ b/MSetFoldWithAbort.v
@@ -29,7 +29,7 @@
 
 
 Require Export MSetInterface.
-Require Import ssreflect.
+Require Import mathcomp.ssreflect.ssreflect.
 Require Import MSetWithDups.
 Require Import Int.
 Require Import MSetGenTree MSetAVL MSetRBT. 
@@ -861,7 +861,7 @@ End SetsWithFoldA.
 (** *** GenTree implementation 
     Finally, provide such a fold with abort operation for generic trees. *)
 Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp) 
-  (Import Raw:Ops E I)
+  (Import Raw: MSetGenTree.Ops E I)
   (M : MSetGenTree.Props E I Raw).
 
   Fixpoint foldWithAbort_Raw {A B: Type} (f_pre : E.t -> B) f_lt (f: E.t -> B -> A -> A) f_gt t (base: A) : A :=
@@ -926,7 +926,7 @@ Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
          M.bst l /\ M.bst r /\ M.lt_tree e l /\ M.gt_tree e r. {
          inversion H_bst. done.
        }
-       
+
        have H_sub_l : (forall e0 : E.t, M.In e0 l -> M.In e0 s /\ E.lt e0 e). {
          intros e0 H_in_l.
          split; last by apply H_lt_tree_l. 
@@ -934,7 +934,7 @@ Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
          rewrite /M.In M.In_node_iff.
          tauto.
        }
-       move : (IHl H_bst_l) => {IHl} IHl {H_bst_l} {H_lt_tree_l}.
+       move : (IHl H_bst_l) => {} IHl {H_bst_l} {H_lt_tree_l}.
        have H_sub_r : (forall e0 : E.t, M.In e0 r -> M.In e0 s /\ E.lt e e0). {
          intros e0 H_in_r.
          split; last by apply H_gt_tree_r. 
@@ -942,7 +942,7 @@ Module MakeGenTreeFoldA (Import E : OrderedType) (Import I:InfoTyp)
          rewrite /M.In M.In_node_iff.
          tauto.
        }
-       move : (IHr H_bst_r) => {IHr} IHr {H_bst_r} {H_gt_tree_r}.
+       move : (IHr H_bst_r) => {} IHr {H_bst_r} {H_gt_tree_r}.
        have H_in_e : M.In e s. {
          eapply H_sub.
          rewrite /M.In M.In_node_iff.
diff --git a/MSetIntervals.v b/MSetIntervals.v
index 03ec64e..4a28c26 100644
--- a/MSetIntervals.v
+++ b/MSetIntervals.v
@@ -19,10 +19,18 @@
  * If not, see <http://www.gnu.org/licenses/>.
  *)
 
+(* CHANGES
+ *
+ * 25-09-2016 Thomas Tuerk <thomas@tuerk-brechen.de>
+ *   implement union, inter, diff, subset, choose efficiently
+ * 03-10-2016 Thomas Tuerk <thomas@tuerk-brechen.de>
+ *   implement fold, exists_, for_all, filter, partition efficiently
+ *)
+
 (** * Weak sets implemented by interval lists 
 
-    This file contains an implementation of the weak set interface
-    [WSetsOn] which uses internally intervals of Z. This allows some
+    This file contains an implementation of the set interface
+    [SetsOn] which uses internally intervals of Z. This allows some
     large sets, which naturally map to intervals of integers to be
     represented very efficiently.
 
@@ -34,10 +42,11 @@
 
 Require Import MSetInterface OrdersFacts OrdersLists.
 Require Import BinNat.
-Require Import ssreflect.
+Require Import mathcomp.ssreflect.ssreflect.
 Require Import NArith.
 Require Import ZArith.
 Require Import NOrder.
+Require Import Lia.
 Require Import DecidableTypeEx.
 Module Import NOP := NOrderProp N.
 Open Scope Z_scope.
@@ -56,6 +65,44 @@ Proof.
   apply N2Z.is_nonneg.
 Qed.
 
+Lemma Z_lt_add_r : forall (z : Z) (n : N),
+  (n <> 0)%N ->
+  z < z + Z.of_N n.
+Proof.
+  move => z n H_neq_0.
+  suff : (z + Z.of_N 0 < z + Z.of_N n). {
+    rewrite Z.add_0_r //.
+  }
+  apply Z.add_lt_mono_l, N2Z.inj_lt.
+  by apply N.neq_0_lt_0.
+Qed.
+
+Lemma Z_lt_le_add_r : forall y1 y2 c, 
+  y1 < y2 ->
+  y1 <= y2 + Z.of_N c.
+Proof.
+  intros y1 y2 c H.
+  apply Z.le_trans with (m := y2). {
+    by apply Z.lt_le_incl.
+  } {
+    apply Z_le_add_r.
+  }
+Qed.
+
+Lemma Z_to_N_minus_neq_0 : forall (x y : Z),
+    y < x ->
+    Z.to_N (x - y) <> 0%N.
+Proof.
+  intros x y H_y_lt.
+  apply N.neq_0_lt_0.
+  apply N2Z.inj_lt.
+  suff H :  0 < x - y. {
+    rewrite Z2N.id => //.
+    by apply Z.lt_le_incl.
+  }
+  by apply Z.lt_0_sub.
+Qed.
+
 Lemma add_add_sub_eq : forall (x y : Z), (x + (y - x) = y). 
 Proof.
   intros x y.
@@ -142,7 +189,7 @@ Qed.
     provides encode and decode function. *)
 
 Module Type ElementEncode.
-  Declare Module E : DecidableType.
+  Declare Module E : OrderedType.
 
   Parameter encode : E.t -> Z.
   Parameter decode : Z -> E.t.
@@ -158,6 +205,10 @@ Module Type ElementEncode.
   Axiom encode_eq : forall (e1 e2 : E.t),
     (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2.
 
+  (** Encoding is compatible with the order of elements. *)
+  Axiom encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+                                      
 End ElementEncode.
 
 
@@ -182,7 +233,7 @@ End ElementEncode.
     the decode function when checking it. This allows for sets with other element types
     than Z.
  *)
-Module Ops (Enc : ElementEncode) <: WOps Enc.E.
+Module Ops (Enc : ElementEncode) <: Ops Enc.E.
   Definition elt := Enc.E.t.
   Definition t := list (Z * N).
 
@@ -190,7 +241,7 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
   Definition empty : t := nil.
   Definition is_empty (l : t) := match l with nil => true | _ => false end.
 
-  
+ 
   (** Defining the list of elements, is much more tricky, especially, 
       if it needs to be executable. *) 
   Lemma acc_pred : forall n p, n = Npos p -> Acc N.lt n -> Acc N.lt (N.pred n).
@@ -201,31 +252,34 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
     apply N.lt_pred_l.
     discriminate.
   Defined.
-
-  Fixpoint elementsZ_aux'' (acc : list Z) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : list Z :=
-    match c as c0 return c = c0 -> list Z with
-    | N0 => fun _ => acc
-    | c => fun Heq => elementsZ_aux'' (x::acc) (Z.succ x) (N.pred c) (acc_pred _ _ Heq H)
+ 
+  Fixpoint fold_elementsZ_aux {A} (f : A -> Z -> option A) (acc : A) (x : Z) (c : N) (H : Acc N.lt c) { struct H } : (bool * A) :=
+    match c as c0 return c = c0 -> (bool * A) with
+    | N0 => fun _ => (false, acc)
+    | c => fun Heq => match (f acc x) with
+        | None => (true, acc)
+        | Some acc' =>             
+          fold_elementsZ_aux f acc' (Z.succ x) (N.pred c) (acc_pred _ _ Heq H) end
     end (refl_equal _).
 
-  Extraction Inline elementsZ_aux''.
-
-  Definition elementsZ_aux' acc x c := elementsZ_aux'' acc x c (lt_wf_0 _).
+  Definition fold_elementsZ_single {A} f (acc : A) x c := fold_elementsZ_aux f acc x c (lt_wf_0 _).
 
-  Fixpoint elementsZ_aux acc (s : t) : list Z := 
+  Fixpoint fold_elementsZ {A} f (acc : A) (s : t) : (bool * A) := 
     match s with 
-    | nil => acc
-    | (x, c) :: s' => 
-        elementsZ_aux (elementsZ_aux' acc x c) s'
+    | nil => (false, acc)
+    | (x, c) :: s' =>
+      match fold_elementsZ_single f acc x c with
+          (false, acc') => fold_elementsZ f acc' s'
+        | (true, acc') => (true, acc')
+      end
     end.
 
   Definition elementsZ (s : t) : list Z := 
-    elementsZ_aux nil s.
+    snd (fold_elementsZ (fun l x => Some (x :: l)) nil s).
 
   Definition elements (s : t) : list elt := 
     rev_map Enc.decode (elementsZ s).   
-
-
+  
   (** membership is easily defined *)
   Fixpoint memZ (x : Z) (s : t) :=
     match s with
@@ -238,34 +292,108 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
 
   Definition mem (x : elt) (s : t) := memZ (Enc.encode x) s.
 
+  (** Comparing intervals *)
+  Inductive interval_compare_result := 
+      ICR_before
+    | ICR_before_touch
+    | ICR_overlap_before 
+    | ICR_overlap_after
+    | ICR_equal
+    | ICR_subsume_1
+    | ICR_subsume_2
+    | ICR_after
+    | ICR_after_touch.
+
+  Definition interval_compare (i1 i2 : (Z * N)) : interval_compare_result := 
+    match (i1, i2) with ((y1, c1), (y2, c2)) => 
+      let yc2 := (y2+Z.of_N c2) in
+      match (Z.compare yc2 y1) with
+        | Lt => ICR_after
+        | Eq => ICR_after_touch
+        | Gt => let yc1 := (y1+Z.of_N c1) in
+                match (Z.compare yc1 y2) with
+                | Lt => ICR_before
+                | Eq => ICR_before_touch
+                | Gt => (* y2 < y1+c1 /\ y1 < y2 + c2 *)
+                        match (Z.compare y1 y2, Z.compare yc1 yc2) with
+                        | (Lt, Lt) => ICR_overlap_before
+                        | (Lt, _) => ICR_subsume_2
+                        | (Eq, Lt) => ICR_subsume_1
+                        | (Eq, Gt) => ICR_subsume_2
+                        | (Eq, Eq) => ICR_equal
+                        | (Gt, Gt) => ICR_overlap_after
+                        | (Gt, _) => ICR_subsume_1
+                        end
+                end
+      end
+    end.
+
+  Definition interval_1_compare (y1 : Z) (i : (Z * N)) : interval_compare_result := 
+    match i with (y2, c2) => 
+      let yc2 := (y2+Z.of_N c2) in
+      match (Z.compare yc2 y1) with
+        | Lt => ICR_after
+        | Eq => ICR_after_touch
+        | Gt => match (Z.compare (Z.succ y1) y2) with
+                | Lt => ICR_before 
+                | Eq => ICR_before_touch
+                | Gt => ICR_subsume_1
+                end
+      end
+    end.
+
+  Fixpoint compare (s1 s2 : t) :=
+    match (s1, s2) with
+      | (nil, nil) => Eq
+      | (nil, _ :: _) => Lt
+      | (_ :: _, nil) => Gt
+      | ((y1, c1)::s1', (y2, c2)::s2') =>
+        match (Z.compare y1 y2) with
+          | Lt => Lt
+          | Gt => Gt
+          | Eq => match N.compare c1 c2 with
+                    | Lt => Lt
+                    | Gt => Gt
+                    | Eq => compare s1' s2'
+                  end
+        end
+    end.
+  
+  (** Auxiliary functions for inserting at front and merging intervals *)
+  Definition merge_interval_size (x1 : Z) (c1 : N) (x2 : Z) (c2 : N) : N :=
+    (N.max c1 (Z.to_N (x2 + Z.of_N c2 - x1))).
+
+  Fixpoint insert_interval_begin (x : Z) (c : N) (l : t) := 
+    match l with
+    | nil => (x,c)::nil
+    | (y, c')::l' => 
+         match (Z.compare (x + Z.of_N c) y) with
+         | Lt => (x, c) :: l
+         | Eq => (x, (c+c')%N) :: l'
+         | Gt => insert_interval_begin x (merge_interval_size x c y c') l'
+         end
+    end.
+
   (** adding an element needs to be defined carefully again in order to
       generate efficient code *)
   Fixpoint addZ_aux (acc : list (Z * N)) (x : Z) (s : t) :=
     match s with
     | nil => List.rev' ((x, (1%N))::acc)
     | (y, c) :: l =>
-        match (Z.compare (Z.succ x) y) with
-        | Lt => List.rev_append ((x, (1%N))::acc) s
-        | Eq => List.rev_append ((x, N.succ c)::acc) l
-        | Gt => match (Z.compare x (y+Z.of_N c)) with
-                 | Lt => List.rev_append acc s
-                 | Gt => addZ_aux ((y,c) :: acc) x l
-                 | Eq => match l with
-                           | nil => List.rev' ((y, N.succ c)::acc) 
-                           | (z, c') :: l' => if (Z.eqb z (Z.succ x)) then
-                                List.rev_append ((y,N.succ (c+c')) :: acc) l'
-                             else
-                                List.rev_append ((y,N.succ c) :: acc) l
-                         end
-                end                 
+        match (interval_1_compare x (y,c)) with
+          | ICR_before       => List.rev_append ((x, (1%N))::acc) s
+          | ICR_before_touch => List.rev_append ((x, N.succ c)::acc) l
+          | ICR_after        => addZ_aux ((y,c) :: acc) x l
+          | ICR_after_touch  => List.rev_append acc (insert_interval_begin y (N.succ c) l)
+          | _  => List.rev_append ((y, c)::acc) l
         end
     end.
 
   Definition addZ x s := addZ_aux nil x s.
   Definition add x s := addZ (Enc.encode x) s.
 
-  (** [add_list] simple extension to add many elements, which then allows to
-      define from_elements. *)
+  (** [add_list] is a simple extension to add many elements.
+      This is used to define the function [from_elements]. *)
   Definition add_list (l : list elt) (s : t) : t :=
      List.fold_left (fun s x => add x s) l s.
 
@@ -279,7 +407,7 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
 
 
   (** removing needs to be done with code extraction in mind again. *)
-  Definition removeZ_aux_insert_guarded (x : Z) (c : N) s :=
+  Definition insert_intervalZ_guarded (x : Z) (c : N) s :=
      if (N.eqb c 0) then s else (x, c) :: s.
 
   Fixpoint removeZ_aux (acc : list (Z * N)) (x : Z) (s : t) : t :=
@@ -288,9 +416,9 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
     | (y, c) :: l =>
         if (Z.ltb x y) then List.rev_append acc s else
         if (Z.ltb x (y+Z.of_N c)) then (
-           List.rev_append (removeZ_aux_insert_guarded (Z.succ x) 
+           List.rev_append (insert_intervalZ_guarded (Z.succ x) 
               (Z.to_N ((y+Z.of_N c)- (Z.succ x))) 
-             (removeZ_aux_insert_guarded y (Z.to_N (x-y)) acc)) l
+             (insert_intervalZ_guarded y (Z.to_N (x-y)) acc)) l
         ) else removeZ_aux ((y,c)::acc) x l
     end.
 
@@ -300,42 +428,222 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
   Definition remove_list (l : list elt) (s : t) : t :=
      List.fold_left (fun s x => remove x s) l s.
 
-  (** all other operations are defined trivially (if not always efficiently)
-      in terms of already defined ones. In the future it might be worth implementing
-      some of them more efficiently. *)
-  Definition union (s1 s2 : t) :=
-    add_list (elements s1) s2.
+  (** union *)
+  Fixpoint union_aux (s1 : t) :=
+    fix aux (s2 : t) (acc : list (Z * N)) :=
+    match (s1, s2) with
+    | (nil, _) => List.rev_append acc s2
+    | (_, nil) => List.rev_append acc s1
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => union_aux l1 s2 ((y1, c1)::acc)
+          | ICR_before_touch => 
+              union_aux l1 (
+               insert_interval_begin y1 ((c1+c2)%N) l2) acc
+          | ICR_after        => aux l2 ((y2, c2)::acc)
+          | ICR_after_touch  => union_aux l1 (
+              insert_interval_begin y2 ((c1+c2)%N) l2) acc
+          | ICR_overlap_before => 
+              union_aux l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2) acc
+          | ICR_overlap_after => 
+              union_aux l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2) acc
+          | ICR_equal => union_aux l1 s2 acc
+          | ICR_subsume_1 => union_aux l1 s2 acc
+          | ICR_subsume_2 => aux l2 acc
+        end
+    end.
+
+  Definition union s1 s2 := union_aux s1 s2 nil.
+
+  (** diff *)
 
-  Definition filter (f : elt -> bool) (s : t) : t :=
-    from_elements (List.filter f (elements s)).
 
-  Definition inter (s1 s2 : t) : t :=
-    filter (fun x => mem x s2) s1.
+  Fixpoint diff_aux (y2 : Z) (c2 : N) (acc : list (Z * N)) (s : t) : (list (Z * N) * t) :=
+    match s with
+    | nil => (acc, nil)
+    | ((y1, c1) :: l1) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => diff_aux y2 c2 ((y1, c1)::acc) l1
+          | ICR_before_touch => diff_aux y2 c2 ((y1, c1)::acc) l1
+          | ICR_after        => (acc, s)
+          | ICR_after_touch  => (acc, s)
+          | ICR_overlap_before => diff_aux y2 c2 ((y1, Z.to_N (y2 - y1))::acc) l1
+          | ICR_overlap_after => (acc, (y2+Z.of_N c2, Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) :: l1)
+          | ICR_equal => (acc, l1)
+          | ICR_subsume_1 => diff_aux y2 c2 acc l1
+          | ICR_subsume_2 => ((insert_intervalZ_guarded y1
+                (Z.to_N (y2 - y1)) acc), 
+              insert_intervalZ_guarded (y2+Z.of_N c2) (Z.to_N ((y1 + Z.of_N c1) - (y2 + Z.of_N c2))) l1) 
+        end
+    end.
 
-  Definition diff (s1 s2 : t) : t :=
-    remove_list (elements s2) s1.
+  Fixpoint diff_aux2 (acc : list (Z * N)) (s1 s2 : t) : (list (Z * N)) :=
+    match (s1, s2) with
+    | (nil, _) => rev_append acc s1
+    | (_, nil) => rev_append acc s1
+    | (_, (y2, c2) :: l2) =>
+      match diff_aux y2 c2 acc s1 with
+        (acc', s1') => diff_aux2 acc' s1' l2
+      end
+    end.
 
-  Definition subset s s' :=
-    List.forallb (fun x => mem x s') (elements s).
+  Definition diff s1 s2 := diff_aux2 nil s1 s2.
+  
+  (** subset *)
+  Fixpoint subset (s1 : t) :=
+    fix aux (s2 : t) :=
+    match (s1, s2) with
+    | (nil, _) => true
+    | (_ :: _, nil) => false
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => false
+          | ICR_before_touch => false
+          | ICR_after        => aux l2 
+          | ICR_after_touch  => false
+          | ICR_overlap_before => false
+          | ICR_overlap_after => false
+          | ICR_equal => subset l1 l2
+          | ICR_subsume_1 => subset l1 s2
+          | ICR_subsume_2 => false
+        end
+    end.
 
+  (** equal *)
   Fixpoint equal (s s' : t) : bool := match s, s' with
     | nil, nil => true
     | ((x,cx)::xs), ((y,cy)::ys) => andb (Z.eqb x y) (andb (N.eqb cx cy) (equal xs ys))
     | _, _ => false
   end. 
 
+  (** inter *)
+  Fixpoint inter_aux (y2 : Z) (c2 : N) (acc : list (Z * N)) (s : t) : (list (Z * N) * t) :=
+    match s with
+    | nil => (acc, nil)
+    | ((y1, c1) :: l1) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => inter_aux y2 c2 acc l1
+          | ICR_before_touch => inter_aux y2 c2 acc l1
+          | ICR_after        => (acc, s)
+          | ICR_after_touch  => (acc, s)
+          | ICR_overlap_before => inter_aux y2 c2 ((y2, Z.to_N (y1 + Z.of_N c1 - y2))::acc) l1
+          | ICR_overlap_after => ((y1, Z.to_N (y2 + Z.of_N c2 - y1))::acc, s)
+          | ICR_equal => ((y1,c1)::acc, l1)
+          | ICR_subsume_1 => inter_aux y2 c2 ((y1, c1)::acc) l1
+          | ICR_subsume_2 => ((y2, c2)::acc, s)
+        end
+    end.
+
+  Fixpoint inter_aux2 (acc : list (Z * N)) (s1 s2 : t) : (list (Z * N)) :=
+    match (s1, s2) with
+    | (nil, _) => List.rev' acc
+    | (_, nil) => List.rev' acc
+    | (_, (y2, c2) :: l2) =>
+      match inter_aux y2 c2 acc s1 with
+        (acc', s1') => inter_aux2 acc' s1' l2
+      end
+    end.
+
+  Definition inter s1 s2 := inter_aux2 nil s1 s2.
+
+
+  (** Partition and filter *)
+
+  Definition partitionZ_fold_insert
+             (cur : option (Z * N)) (x : Z) :=
+    match cur with
+       None => (x, 1%N)
+     | Some (y, c) => (y, N.succ c)
+    end.
+
+  Definition partitionZ_fold_skip (acc : list (Z * N))
+             (cur : option (Z * N)) : (list (Z * N)) :=
+    match cur with
+       None => acc
+     | Some yc => yc::acc
+    end.
+  
+  Definition partitionZ_fold_fun f st (x : Z) :=
+    match st with ((acc_t, c_t), (acc_f, c_f)) =>
+      if (f x) then
+        ((acc_t, Some (partitionZ_fold_insert c_t x)),
+         (partitionZ_fold_skip acc_f c_f, None))
+      else
+        ((partitionZ_fold_skip acc_t c_t, None),
+         (acc_f, Some (partitionZ_fold_insert c_f x)))
+    end.
+  
+  Definition partitionZ_single_aux f st (x : Z) (c : N) :=
+    snd (fold_elementsZ_single (fun st x => Some (partitionZ_fold_fun f st x)) st x c).
+
+  Definition partitionZ_single f acc_t acc_f x c :=
+    match partitionZ_single_aux f ((acc_t, None), (acc_f, None)) x c with
+    | ((acc_t, c_t), (acc_f, c_f)) =>
+        (partitionZ_fold_skip acc_t c_t,
+         partitionZ_fold_skip acc_f c_f)
+    end.
+
+  Fixpoint partitionZ_aux acc_t acc_f f s :=
+    match s with
+    | nil => (List.rev acc_t, List.rev acc_f)
+    | (y, c) :: s' =>
+      match partitionZ_single f acc_t acc_f y c with
+      | (acc_t', acc_f') => partitionZ_aux acc_t' acc_f' f s'
+      end
+    end.
+
+  Definition partitionZ := partitionZ_aux nil nil.
+
+  Definition partition (f : elt -> bool) : t -> (t * t) :=
+    partitionZ (fun z => f (Enc.decode z)).
+
+
+  Definition filterZ_fold_fun f st (x : Z) :=
+    match st with (acc_t, c_t) =>
+      if (f x) then
+        (acc_t, Some (partitionZ_fold_insert c_t x))
+      else
+        (partitionZ_fold_skip acc_t c_t, None)
+    end.
+  
+  Definition filterZ_single_aux f st (x : Z) (c : N) :=
+    snd (fold_elementsZ_single (fun st x => Some (filterZ_fold_fun f st x)) st x c).
+
+  Definition filterZ_single f acc x c :=
+    match filterZ_single_aux f (acc, None) x c with
+    | (acc, c) =>
+        (partitionZ_fold_skip acc c)
+    end.
+
+  Fixpoint filterZ_aux acc f s :=
+    match s with
+    | nil => (List.rev acc)
+    | (y, c) :: s' =>
+      filterZ_aux (filterZ_single f acc y c) f s'
+    end.
+  
+  Definition filterZ := filterZ_aux nil.
+
+  Definition filter (f : elt -> bool) : t -> t :=
+    filterZ (fun z => f (Enc.decode z)).
+
+  
+  (** Simple wrappers *)
+
   Definition fold {B : Type} (f : elt -> B -> B) (s : t) (i : B) : B :=
-    List.fold_left (flip f) (elements s) i.
+    snd (fold_elementsZ (fun b z => Some (f (Enc.decode z) b)) i s).
 
   Definition for_all (f : elt -> bool) (s : t) : bool :=
-    List.forallb f (elements s).
+    snd (fold_elementsZ (fun b z =>
+      if b then
+        Some (f (Enc.decode z))
+      else None) true s).
 
   Definition exists_ (f : elt -> bool) (s : t) : bool :=
-    List.existsb f (elements s).
-
-  Definition partition (f : elt -> bool) (s : t) : t * t :=
-    (filter f s, filter (fun x => negb (f x)) s).
-
+    snd (fold_elementsZ (fun b z =>
+      if b then
+        None
+      else Some (f (Enc.decode z))) false s).
 
   Fixpoint cardinalN c (s : t) : N := match s with
     | nil => c
@@ -344,18 +652,33 @@ Module Ops (Enc : ElementEncode) <: WOps Enc.E.
 
   Definition cardinal (s : t) : nat := N.to_nat (cardinalN (0%N) s).
 
-  Definition chooseZ (s : t) : option Z :=
-    match List.rev' (elementsZ s) with
+  Definition min_eltZ (s : t) : option Z :=
+    match s with
     | nil => None
-    | x :: _ => Some x
+    | (x, _) :: _ => Some x
+    end.
+
+  Definition min_elt (s : t) : option elt :=
+    match (min_eltZ s) with
+    | None => None
+    | Some x => Some (Enc.decode x)
     end.
 
-  Definition choose (s : t) : option elt :=
-    match elements s with
-      | nil => None
-      | e :: _ => Some e
+  Definition choose := min_elt.
+
+  Fixpoint max_eltZ (s : t) : option Z :=
+    match s with
+    | nil => None
+    | (x, c) :: nil => Some (Z.pred (x + Z.of_N c))
+    | (x, _) :: s' => max_eltZ s'
     end.
 
+  Definition max_elt (s : t) : option elt :=
+    match (max_eltZ s) with
+    | None => None
+    | Some x => Some (Enc.decode x)
+    end.
+  
 End Ops.
 
 
@@ -370,31 +693,29 @@ Module Raw (Enc : ElementEncode).
   Include (Ops Enc).
 
   (** *** Defining invariant [IsOk] *)
+  Definition is_encoded_elems_list (l : list Z) : Prop :=
+    (forall x, List.In x l -> exists e, Enc.encode e = x).
 
-  Definition inf (x:Z) (l: t) :=
-   match l with
-   | nil => true
-   | (y,_)::_ => Z.ltb x y
-   end.
+  Definition interval_list_elements_greater (x : Z) (l : t) : bool :=
+    match l with 
+      | nil => true
+      | (y, _)::_ => Z.ltb x y 
+    end.
 
-  Fixpoint isok (l : t) :=
+  Fixpoint interval_list_invariant (l : t) :=
    match l with
    | nil => true
-   | (x, c) ::l => inf (x+(Z.of_N c)) l && negb (N.eqb c 0) && isok l
+   | (x, c) :: l' => 
+       interval_list_elements_greater (x + (Z.of_N c)) l' && negb (N.eqb c 0) && interval_list_invariant l'
    end.
 
-  Definition is_encoded_elems_list (l : list Z) : Prop :=
-    (forall x, List.In x l -> exists e, Enc.encode e = x).
-
-  Definition IsOk s := (isok s = true /\ is_encoded_elems_list (elementsZ s)).
+  Definition IsOk s := (interval_list_invariant s = true /\ is_encoded_elems_list (elementsZ s)).
 
 
   (** *** Defining notations *)
   Section ForNotations.
 
     Class Ok (s:t) : Prop := ok : IsOk s.
-    Hint Resolve @ok.
-    Hint Unfold Ok.
     Instance IsOk_Ok s `(Hs : IsOk s) : Ok s := { ok := Hs }.
 
     Definition In x s := (SetoidList.InA Enc.E.eq x (elements s)).
@@ -453,7 +774,6 @@ Module Raw (Enc : ElementEncode).
     }
   Qed.
 
-
   Lemma elementsZ_single_succ_front : forall x c,
     elementsZ_single x (N.succ c) = 
     x :: elementsZ_single (Z.succ x) c.
@@ -471,32 +791,23 @@ Module Raw (Enc : ElementEncode).
     induction c as [| c' IH] using N.peano_ind. {
       intros y x.
       rewrite elementsZ_single_base Z.add_0_r /=.
-      split; first done.
-      move => [H_x_le H_y_lt].     
-      have := Z.le_lt_trans _ _ _ H_x_le H_y_lt.
-      apply Z.lt_irrefl.
+      lia.
     } {
       intros y x.
       rewrite elementsZ_single_succ in_app_iff IH /= N2Z.inj_succ Z.add_succ_r Z.lt_succ_r.
       split. {
-        move => []. {
-          move => [H_x_le H_y_le].
-          split; first done.
-          by apply Z.lt_le_incl.
-        }
-        move => [] // <-.
-        split. {
-          apply Z_le_add_r.
+        move => [ | []] //. { 
+          move => [H_x_le H_y_le]. 
+          lia.
         } {
-          by apply Z.le_refl.
+          move => <-.
+          split.
+            - by apply Z_le_add_r.
+            - by apply Z.le_refl.
         }
       } {
         move => [H_x_le] H_y_lt.
-        apply Z.lt_eq_cases in H_y_lt as [H_y_lt | <-]. {
-          by left. 
-        } {
-          by right; left.
-        }
+        lia.
       }
     }
   Qed.
@@ -506,13 +817,8 @@ Module Raw (Enc : ElementEncode).
     (x = y).
   Proof.
     intros y x.
-    rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r.
-    split. {
-      move => [? ?]. by apply Z.le_antisymm.
-    } {
-      move => ->.
-      split; apply Z.le_refl.
-    }
+    rewrite In_elementsZ_single /= Z.add_1_r Z.lt_succ_r. 
+    lia.
   Qed.
 
   Lemma length_elementsZ_single : forall cx x,
@@ -527,85 +833,181 @@ Module Raw (Enc : ElementEncode).
     }
   Qed.
 
-  Lemma elementsZ_aux''_irrel : forall c acc x H1 H2,
-      elementsZ_aux'' acc x c H1 = elementsZ_aux'' acc x c H2.
+  Lemma fold_elementsZ_aux_irrel {A} :
+    forall f c (acc : A) x H1 H2,
+      fold_elementsZ_aux f acc x c H1 =
+      fold_elementsZ_aux f acc x c H2.
   Proof.
-    intro c.
-    induction c using (well_founded_ind lt_wf_0).
+    intros f c.
+    induction c as [c IH] using (well_founded_ind lt_wf_0).
     case_eq c. {
-      intros H_c acc x; case; intro; case; intro.
+      intros H_c acc x; case; intro H_H1; case; intro H_H2.
       reflexivity.
     } {
-      intros p H_c acc x; case; intro; case; intro.
-      simpl.
-      apply H.
+      intros p H_c acc x; case; intro H_H1; case; intro H_H2.
+      unfold fold_elementsZ_aux; fold (@fold_elementsZ_aux A).
+      case (f acc x) => // acc'.
+      apply IH.
       rewrite H_c.
-      rewrite N.pos_pred_spec.
       apply N.lt_pred_l.
       discriminate.
     }
   Qed.
 
 
-  Lemma elementsZ_aux'_pos : forall s x p,
-      elementsZ_aux' s x (N.pos p) = elementsZ_aux' (x::s) (Z.succ x) (Pos.pred_N p).
+  Lemma fold_elementsZ_single_pos {A} : forall f (acc : A) x p,
+    fold_elementsZ_single f acc x (N.pos p) =
+    match f acc x with
+    | Some acc' =>
+        fold_elementsZ_single f acc' (Z.succ x)
+         (N.pred (N.pos p))
+    | None => (true, acc)
+    end.
   Proof.
-    intros s x p.
-    unfold elementsZ_aux'.
-    unfold elementsZ_aux''.
+    intros f acc x p.        
+    unfold fold_elementsZ_single.
+    unfold fold_elementsZ_aux.
     case: (lt_wf_0 _).
+    fold (@fold_elementsZ_aux A).
     intro.
-    apply elementsZ_aux''_irrel.
+    case (f acc x) => // acc'.
+    apply fold_elementsZ_aux_irrel.
   Qed.
 
 
-  Lemma elementsZ_aux'_zero : forall s x,
-      elementsZ_aux' s x (0%N) = s.
+  Lemma fold_elementsZ_single_zero {A} : forall f (acc : A) x,
+      fold_elementsZ_single f acc x (0%N) = (false, acc).
   Proof.
-    intros s x.
-    unfold elementsZ_aux'.
-    by case (lt_wf_0 (0%N)); intro.
+    intros f acc x.
+    unfold fold_elementsZ_single.
+    case (lt_wf_0 (0%N)); intro.
+    unfold fold_elementsZ_aux.
+    reflexivity.
   Qed.
 
 
-  Lemma elementsZ_aux'_succ : forall s x c,
-      elementsZ_aux' s x (N.succ c) = elementsZ_aux' (x::s) (Z.succ x) c.
+  Lemma fold_elementsZ_single_succ {A} : forall f (acc : A) x c,
+    fold_elementsZ_single f acc x (N.succ c) =
+    match f acc x with
+      | Some acc' =>
+          fold_elementsZ_single f acc' (Z.succ x) c
+      | None => (true, acc)
+    end.
+  Proof.
+    intros f acc x c.
+    case c. {
+      by rewrite fold_elementsZ_single_pos.
+    } {
+      intro p; simpl.
+      rewrite fold_elementsZ_single_pos.
+      case (f acc x) => // acc' /=.
+      by rewrite Pos.pred_N_succ.
+    }
+  Qed.
+
+  Fixpoint fold_opt {A B} f (acc : A) (bs : list B) : (bool * A) :=
+    match bs with
+      | nil => (false, acc)
+      | (b :: bs') =>
+        match f acc b with
+        | Some acc' => fold_opt f acc' bs'
+        | None => (true, acc)
+        end
+    end.
+
+  Lemma fold_opt_list_cons : forall {A} (bs : list A) (acc : list A),
+    fold_opt (fun l x => Some (x :: l)) acc bs =
+    (false, List.rev bs ++ acc).
   Proof.
-    intros s x c.
-    case c.
-    - by rewrite elementsZ_aux'_pos.
-    - intro p; simpl.
-      rewrite elementsZ_aux'_pos.
-        by rewrite Pos.pred_N_succ.
+    induction bs as [| b bs' IH] => acc. {
+      by simpl.
+    } {
+      rewrite /= IH -app_assoc //.
+    }
   Qed.
 
+  Lemma fold_opt_app {A B} : forall f (acc : A) (l1 l2 : list B),
+    fold_opt f acc (l1 ++ l2) =
+    (let (ab, acc') := fold_opt f acc l1 in
+     if ab then (true, acc') else fold_opt f acc' l2).
+  Proof.
+    intros f acc l1 l2.
+    move : acc.
+    induction l1 as [| b l1' IH] => acc. {
+      rewrite app_nil_l //.
+    } {
+      rewrite /=.
+      case (f acc b); last done.
+      intro acc'.
+      rewrite IH //.
+    }
+  Qed.
 
-  Lemma elementsZ_single_intro : forall c s x,
-     elementsZ_aux' s x c =
-     (List.rev (elementsZ_single x c)) ++ s.
+  
+  Lemma fold_elementsZ_single_alt_def {A} : forall f c (acc : A) x,
+     fold_elementsZ_single f acc x c =
+     fold_opt f acc (elementsZ_single x c).
   Proof.
+    intro f.
     induction c as [| c' IH] using N.peano_ind. {
-      intros s x.
-      rewrite elementsZ_aux'_zero.
-      rewrite elementsZ_single_base //.
+      intros acc x.
+      rewrite fold_elementsZ_single_zero
+              elementsZ_single_base /fold_opt //.
     } {
-      intros s x.
-      rewrite elementsZ_aux'_succ elementsZ_single_succ_front IH.
-      rewrite -app_assoc /= //.
+      intros acc x.
+      rewrite fold_elementsZ_single_succ
+              elementsZ_single_succ_front /=.
+      case (f acc x); last reflexivity.
+      intro acc'.
+      apply IH.
     }
   Qed.
 
+  Lemma fold_elementsZ_nil {A} : forall f (acc : A),
+     fold_elementsZ f acc nil = (false, acc).
+  Proof. done. Qed.
 
-  Lemma elementsZ_aux_alt_def : forall s acc,
-    elementsZ_aux acc s = elementsZ s ++ acc. 
+  Lemma fold_elementsZ_cons {A} : forall f (acc : A) y c s,
+    fold_elementsZ f acc ((y, c)::s) =
+    (let (ab, acc') := fold_elementsZ_single f acc y c in
+     if ab then (true, acc') else fold_elementsZ f acc' s).
   Proof.
-    induction s as [| [x c] s' IH]. {
-      move => acc.
-      rewrite /elementsZ_aux elementsZ_nil app_nil_l //.
+    intros f acc y c s.
+    done.
+  Qed.
+
+
+  Lemma fold_elementsZ_alt_def_aux : forall (s : t) base,
+    (snd (fold_elementsZ
+      (fun (l : list Z) (x : Z) => Some (x :: l)) base s)) =
+    elementsZ s ++ base.
+  Proof.
+    induction s as [| [y1 c1] s' IH] => base. {
+      rewrite elementsZ_nil /fold_elementsZ /fold_opt /snd
+        app_nil_l //.
+    } {
+      rewrite /elementsZ !fold_elementsZ_cons.
+      rewrite !fold_elementsZ_single_alt_def !fold_opt_list_cons.
+      rewrite !IH app_nil_r app_assoc //.
+    }
+  Qed.
+
+  
+  Lemma fold_elementsZ_alt_def {A} : forall f s (acc : A),
+     fold_elementsZ f acc s =
+     fold_opt f acc (rev (elementsZ s)).
+  Proof.
+    intro f.
+    induction s as [| [y1 c1] s' IH] => acc. {
+      by simpl.
     } {
-      move => acc.
-      rewrite /elementsZ /elementsZ_aux -/elementsZ_aux !IH !elementsZ_single_intro.
-      rewrite -!app_assoc app_nil_l //.
+      rewrite /elementsZ !fold_elementsZ_cons.
+      rewrite !fold_elementsZ_single_alt_def
+              fold_opt_list_cons app_nil_r
+              fold_elementsZ_alt_def_aux rev_app_distr
+              rev_involutive fold_opt_app.
+      case (fold_opt f acc (elementsZ_single y1 c1)).
+      move => [] //.
     }
   Qed.
 
@@ -613,8 +1015,12 @@ Module Raw (Enc : ElementEncode).
      ((elementsZ s) ++ (List.rev (elementsZ_single x c))). 
   Proof. 
     intros x c s.
-    rewrite /elementsZ /elementsZ_aux -/elementsZ_aux.
-    rewrite !elementsZ_aux_alt_def elementsZ_single_intro !app_nil_r //.
+    rewrite /elementsZ fold_elementsZ_cons
+            !fold_elementsZ_alt_def 
+            fold_elementsZ_single_alt_def.
+    rewrite !fold_opt_list_cons.
+    rewrite !app_nil_r !rev_involutive /=.
+    rewrite fold_elementsZ_alt_def_aux //.
   Qed.
 
   Lemma elements_cons : forall x c s, elements (((x, c) :: s) : t) = 
@@ -625,6 +1031,83 @@ Module Raw (Enc : ElementEncode).
     rewrite !rev_map_alt_def map_app rev_app_distr map_rev rev_involutive //. 
   Qed.
 
+  Lemma elementsZ_app : forall (s1 s2 : t), elementsZ (s1 ++ s2) = 
+     ((elementsZ s2) ++ (elementsZ s1)). 
+  Proof. 
+    induction s1 as [| [x1 c1] s1 IH1]. {
+      move => s2.
+      rewrite elementsZ_nil app_nil_r //.
+    } 
+    move => s2.
+    rewrite -app_comm_cons !elementsZ_cons IH1 -app_assoc //.
+  Qed.
+
+  Lemma InZ_nil : forall y, InZ y nil <-> False. 
+  Proof. 
+    intro y.
+    done.
+  Qed.
+
+  Lemma InZ_cons : forall y x c s, InZ y (((x, c) :: s) : t) <-> 
+     List.In y (elementsZ_single x c) \/ InZ y s. 
+  Proof. 
+    intros y x c s.
+    rewrite /InZ elementsZ_cons in_app_iff -in_rev.
+    firstorder.
+  Qed.
+
+  Lemma InZ_app : forall s1 s2 y, 
+     InZ y (s1 ++ s2) <-> InZ y s1 \/ InZ y s2.
+  Proof. 
+    intros s1 s2 y.   
+    rewrite /InZ elementsZ_app in_app_iff.
+    tauto.
+  Qed.
+
+  Lemma InZ_rev : forall s y, 
+     InZ y (List.rev s) <-> InZ y s.
+  Proof. 
+    intros s x.
+    rewrite /InZ.
+    induction s as [| [y c] s' IH]; first done.
+    simpl.
+    rewrite elementsZ_app in_app_iff IH.
+    rewrite !elementsZ_cons !in_app_iff elementsZ_nil
+            -!in_rev /=.
+    tauto.
+  Qed.
+
+  Lemma In_elementsZ_single_dec : forall y x c,
+    {List.In y (elementsZ_single x c)} +
+    {~ List.In y (elementsZ_single x c)}.
+  Proof.  
+    intros y x c.
+    case (Z_le_dec x y); last first. {
+      right; rewrite In_elementsZ_single; tauto.
+    }
+    case (Z_lt_dec y (x + Z.of_N c)); last first. {
+      right; rewrite In_elementsZ_single; tauto.
+    }
+    left; rewrite In_elementsZ_single; tauto.
+  Qed.
+
+  Lemma InZ_dec : forall y s, 
+     {InZ y s} + {~InZ y s}.
+  Proof. 
+    intros y.
+    induction s as [| [x c] s IH]. {
+      by right.
+    }
+    move : IH => [] IH. {
+      by left; rewrite InZ_cons; right.
+    }
+    case (In_elementsZ_single_dec y x c). {
+      by left; rewrite InZ_cons; left.
+    } {
+      by right; rewrite InZ_cons; tauto.
+    }
+  Qed.
+
   Lemma In_elementsZ_single_hd : forall (c : N) x, (c <> 0)%N -> List.In x (elementsZ_single x c).
   Proof.
     intros c x H_c_neq.
@@ -639,275 +1122,249 @@ Module Raw (Enc : ElementEncode).
     }
   Qed.
 
+ 
+  (** *** comparing intervals *)
 
-  (** *** Alternative definition of addZ *)
-
-  (** [addZ] is defined with efficient execution in mind.
-      We derive first an alternative definition that demonstrates
-      the intention better and is better suited for proofs. *)
-  Lemma addZ_ind : 
-    forall (P : Z -> list (Z * N) -> Prop),
-       (* case s = nil *)
-       (forall (x : Z), P x nil) ->
-
-       (* case s = [y, c] :: l, compare (x+1) y = Eq *)
-       (forall (x : Z)  (l : list (Z * N)) (c : N),
-        P x ((x + 1, c) :: l)) ->
-
-       (* case s = [y, c] :: l, compare (x+1) y = Lt *)       
-       (forall (x : Z) (l : list (Z * N)) (y : Z) (c : N),
-        (x + 1 ?= y) = Lt ->
-        P x ((y, c) :: l)) ->
-
-       (* case s = [y, c] :: l, compare (x+1) y = Gt, compare x (y+c) = Eq,
-           l = nil *)       
-       (forall (y : Z) (c : N),
-        ((y + Z.of_N c) + 1 ?= y) = Gt ->
-        P (y + Z.of_N c) ((y, c) :: nil)) ->
-
-       (* case s = (y, c) :: (z, c') :: l, compare (x+1) y = Gt, compare x (y+c) = Eq,
-          (z = x + 1) *)       
-       (forall (l : list (Z * N)) (y : Z) (c c' : N),
-        ((y + Z.of_N c) + 1 ?= y) = Gt ->
-        (P (y+Z.of_N c) l) ->
-        P (y+Z.of_N c) ((y, c) :: (((y+Z.of_N c) + 1, c') :: l))) ->
-
-       (* case s = (y, c) :: (z, c') :: l, compare (x+1) y = Gt, compare x (y+c) = Eq,
-          (z <> x + 1) *)       
-       (forall (l : list (Z * N)) (y : Z) (c : N) (z : Z) (c' : N),
-        ((y + Z.of_N c) + 1 ?= y) = Gt ->
-        (z =? (y+Z.of_N c) + 1) = false ->
-        (P (y+Z.of_N c) ((y, c) :: (z, c') :: l))) ->
-
-
-       (* case s = (y, c) :: l, compare (x+1) y = Gt, compare x (y+c) = Lt *)   
-       (forall (x : Z) (l : list (Z * N)) (y : Z) (c : N),
-        (x + 1 ?= y) = Gt ->
-        (x ?= y + Z.of_N c) = Lt -> 
-        P x ((y, c) :: l)) ->
-
-       (* case s = (y, c) :: l, compare (x+1) y = Gt, compare x (y+c) = Lt *)   
-       (forall (x : Z)(l : list (Z * N)) (y : Z) (c : N),
-        (x + 1 ?= y) = Gt ->
-        (x ?= y + (Z.of_N c)) = Gt ->
-        (P x l) ->
-        P x ((y, c) :: l)) ->
-
-
-       forall (x : Z) (s : list (Z * N)),
-       P x s.
-  Proof.
-    intros P H1 H2 H3 H4 H5 H6 H7 H8.
-    move => x s.
-    remember (length s) as n.
-    revert s Heqn. 
-    induction n as [n IH] using Wf_nat.lt_wf_ind. 
-    case. {
-      move => _. apply H1.
-    } {
-      move => [y c] s' H_n.
-      simpl in H_n.
-      have IH_s': (P x s'). {
-        apply (IH (length s')) => //.
-        rewrite H_n //.
-      }
-      case_eq (x + 1 ?= y). {
-        move => /Z.compare_eq_iff <-.
-        apply H2.
-      } {
-        move => H_x1y_comp.
-        by apply H3.
-      } {
-        move => H_x1y_comp.
-        case_eq (x ?= y + Z.of_N c). {            
-          move => H_x_eqb.
-          move : (H_x_eqb) => /Z.compare_eq_iff H_x_eq.
-          case_eq s'. {
-            move => _.
-            rewrite H_x_eq.
-            apply H4.
-            rewrite -H_x1y_comp H_x_eq //.
-          } {
-            move => [z c'] l H_s'.
-            have IH_l: (P x l). {
-              apply (IH (length l)) => //.
-              rewrite H_n H_s' /=.
-              apply Lt.lt_trans with (m := S (length l)) => //.
-            }
-            case_eq (z =? x + 1). { 
-              move => /Z.eqb_eq ->.
-              rewrite H_x_eq.
-              apply H5. {
-                rewrite -H_x1y_comp H_x_eq //.
-              } {
-                rewrite -H_x_eq.
-                apply IH_l.
-              }
-            } {
-              move => H_z_neq.
-              rewrite H_x_eq.
-              eapply H6 => //; rewrite -H_x_eq //.
-            }
-          }
-        } {
-          by apply H7.
-        } {
-          move => H_x_gt.
-          apply H8 => //.
+  Ltac Z_named_compare_cases H := match goal with
+    | [ |- context [Z.compare ?z1 ?z2] ] =>
+      case_eq (Z.compare z1 z2); [move => /Z.compare_eq_iff | move => /Z.compare_lt_iff | move => /Z.compare_gt_iff]; move => H //
+  end.
+
+  Ltac Z_compare_cases := let H := fresh "H" in Z_named_compare_cases H.
+  
+  Lemma interval_compare_elim : forall (y1 : Z) (c1 : N) (y2 : Z) (c2 : N), 
+    match (interval_compare (y1, c1) (y2, c2)) with
+      | ICR_before         => (y1 + Z.of_N c1) < y2
+      | ICR_before_touch   => (y1 + Z.of_N c1) = y2
+      | ICR_after          => (y2 + Z.of_N c2) < y1
+      | ICR_after_touch    => (y2 + Z.of_N c2) = y1
+      | ICR_equal          => (y1 = y2) /\ (c1 = c2)
+      | ICR_overlap_before => (y1 < y2) /\ (y2 < y1 + Z.of_N c1) /\ (y1 + Z.of_N c1 < y2 + Z.of_N c2)
+      | ICR_overlap_after  => (y2 < y1) /\ (y1 < y2 + Z.of_N c2) /\ (y2 + Z.of_N c2 < y1 + Z.of_N c1)
+      | ICR_subsume_1      => (y2 <= y1) /\ (y1 + Z.of_N c1 <= y2 + Z.of_N c2) /\ (y2 < y1 \/ y1 + Z.of_N c1 < y2 + Z.of_N c2)
+      | ICR_subsume_2      => (y1 <= y2) /\ (y2 + Z.of_N c2 <= y1 + Z.of_N c1) /\ (y1 < y2 \/ y2 + Z.of_N c2 < y1 + Z.of_N c1)
+    end.
+  Proof.
+    intros y1 c1 y2 c2.
+    rewrite /interval_compare.
+    (repeat Z_compare_cases); subst; repeat split;
+       try (by apply Z.eq_le_incl); 
+       try (by apply Z.lt_le_incl);
+       try (by left); try (by right).
+
+    apply Z.add_reg_l in H2.
+    by apply N2Z.inj.
+  Qed.
+
+  Lemma interval_compare_swap : forall (y1 : Z) (c1 : N) (y2 : Z) (c2 : N),
+    (c1 <> 0%N) \/ (c2 <> 0%N) -> 
+    interval_compare (y2, c2) (y1, c1) =
+    match (interval_compare (y1, c1) (y2, c2)) with
+      | ICR_before         => ICR_after
+      | ICR_before_touch   => ICR_after_touch
+      | ICR_after          => ICR_before
+      | ICR_after_touch    => ICR_before_touch
+      | ICR_equal          => ICR_equal
+      | ICR_overlap_before => ICR_overlap_after
+      | ICR_overlap_after  => ICR_overlap_before
+      | ICR_subsume_1      => ICR_subsume_2
+      | ICR_subsume_2      => ICR_subsume_1
+    end.
+  Proof.
+    intros y1 c1 y2 c2 H_c12_neq_0.
+    rewrite /interval_compare.
+    move : (Z.compare_antisym y1 y2) => ->.
+    move : (Z.compare_antisym (y1 + Z.of_N c1) (y2 + Z.of_N c2)) => ->.
+    have H_suff : y1 + Z.of_N c1 <= y2 -> y2 + Z.of_N c2 <= y1 -> False. {
+      move => H1 H2.
+      case H_c12_neq_0 => H_c_neq_0. {
+        suff : (y1 + Z.of_N c1 <= y1). {
+          apply Z.nle_gt.
+          by apply Z_lt_add_r.
         }
+        eapply Z.le_trans; eauto.
+        eapply Z.le_trans; eauto.
+        apply Z_le_add_r.
+      } {
+        suff : (y2 + Z.of_N c2 <= y2). {
+          apply Z.nle_gt.
+          by apply Z_lt_add_r.
+        }
+        eapply Z.le_trans; eauto.
+        eapply Z.le_trans; eauto.
+        apply Z_le_add_r.
       }
     }
+    repeat Z_compare_cases. {
+      exfalso; apply H_suff.
+        - by rewrite H; apply Z.le_refl.
+        - by rewrite H0; apply Z.le_refl.
+    } {
+      exfalso; apply H_suff.
+        - by rewrite H; apply Z.le_refl.
+        - by apply Z.lt_le_incl.
+    } {
+      exfalso; apply H_suff.
+        - by apply Z.lt_le_incl.
+        - by rewrite H0; apply Z.le_refl.
+    } {
+      exfalso; apply H_suff.
+        - by apply Z.lt_le_incl.
+        - by apply Z.lt_le_incl.
+    }
   Qed.
-
-  Lemma addZ_aux_alt_def : forall x s acc,
-    addZ_aux acc x s = (List.rev acc) ++ addZ x s. 
+  
+  Lemma interval_1_compare_alt_def : forall (y : Z) (i : (Z * N)), 
+    interval_1_compare y i = match (interval_compare (y, (1%N)) i) with
+      | ICR_equal => ICR_subsume_1
+      | ICR_subsume_1 => ICR_subsume_1
+      | ICR_subsume_2 => ICR_subsume_1
+      | r => r
+    end.
   Proof.
-    move => ? ?.
-    eapply addZ_ind with (P := (fun x s => forall acc, addZ_aux acc x s = rev acc ++ addZ x s)); clear. {
-      intros x acc.
-      rewrite /addZ_aux /rev' -rev_alt //.
-    } {
-      intros x l c acc.
-      rewrite /addZ /= Z.compare_refl rev_append_rev //.
+    move => y1 [y2 c2].
+    rewrite /interval_1_compare /interval_compare.
+    replace (y1 + Z.of_N 1) with (Z.succ y1); last done.
+    repeat Z_compare_cases. {
+      contradict H1.
+      by apply Zle_not_lt, Zlt_succ_le.
     } {
-      intros x l y c H_y_lt acc.
-      rewrite /addZ /= H_y_lt rev_append_rev //.
+      contradict H.
+      by apply Zle_not_lt, Zlt_succ_le.
+    }
+  Qed.
+
+  Lemma interval_1_compare_elim : forall (y1 : Z) (y2 : Z) (c2 : N), 
+    match (interval_1_compare y1 (y2, c2)) with
+      | ICR_before         => Z.succ y1 < y2
+      | ICR_before_touch   => y2 = Z.succ y1 
+      | ICR_after          => (y2 + Z.of_N c2) < y1
+      | ICR_after_touch    => (y2 + Z.of_N c2) = y1
+      | ICR_equal          => False
+      | ICR_overlap_before => False
+      | ICR_overlap_after  => False
+      | ICR_subsume_1      => (c2 = 0%N) \/ ((y2 <= y1) /\ (y1 < y2 + Z.of_N c2))
+      | ICR_subsume_2      => False
+    end.
+  Proof.
+    intros y1 y2 c2.
+    move : (interval_compare_elim y1 (1%N) y2 c2).
+    rewrite interval_1_compare_alt_def.
+    have H_succ: forall z, z + Z.of_N 1 = Z.succ z by done.
+
+    case_eq (interval_compare (y1, 1%N) (y2, c2)) => H_comp;
+      rewrite ?H_succ ?Z.lt_succ_r ?Z.le_succ_l //. {
+      move => [H_lt] [H_le] _.
+      contradict H_lt.
+      by apply Zle_not_lt.
     } {
-      intros y c H_y_comp acc.
-      rewrite /addZ /= H_y_comp Z.compare_refl /rev' -rev_alt //.
+      move => [_] [H_lt] H_le.
+      contradict H_lt.
+      by apply Zle_not_lt.
     } {
-      intros l y c c' H_y_comp IH acc.
-      rewrite /addZ /= H_y_comp Z.compare_refl Z.eqb_refl.
-      rewrite rev_append_rev //.
+      move => [->] <-.
+      rewrite ?Z.lt_succ_r.
+      right.
+      split; apply Z.le_refl.
     } {
-      intros l y c z c' H_y_comp H_z_comp acc.
-      rewrite /addZ /= H_y_comp Z.compare_refl H_z_comp.
-      rewrite rev_append_rev /= //.
+      tauto.
     } {
-      intros x l y c H_y_comp H_x_comp acc.
-      rewrite /addZ /= H_x_comp H_y_comp rev_append_rev //.
+      case (N.zero_or_succ c2). { 
+        move => -> _; by left.
+      } {
+        move => [c2'] ->.
+        rewrite !N2Z.inj_succ Z.add_succ_r -Z.succ_le_mono Z.le_succ_l.
+        move => [H_y1_le] [H_le_y1].        
+        suff -> : y1 = y2. {
+          move => [] H_pre; contradict H_pre. {
+            apply Z.lt_irrefl.
+          } {
+            apply Zle_not_lt, Z_le_add_r.
+          }
+        }
+        apply Z.le_antisymm => //.
+        eapply Z.le_trans; last apply H_le_y1.
+        apply Z_le_add_r.
+      }
+    }
+  Qed.
+      
+   (** *** Alternative definition of addZ *)
+  Lemma addZ_aux_alt_def : forall x s acc,
+    addZ_aux acc x s = (List.rev acc) ++ addZ x s. 
+  Proof. 
+    intros y1 s.
+    unfold addZ.
+    induction s as [| [y2 c2] s' IH] => acc. {
+      rewrite /addZ_aux /addZ /= /rev' !rev_append_rev /= app_nil_r //.
     } {
-      intros x l y c H_y_comp H_x_comp IH acc.      
-      rewrite /addZ /= H_x_comp H_y_comp.
-      rewrite !IH /= -app_assoc //.
+      unfold addZ_aux.
+      case (interval_1_compare y1 (y2, c2)); fold addZ_aux;
+        rewrite ?rev_append_rev /= ?app_assoc_reverse //.
+      rewrite (IH ((y2,c2)::acc)) (IH ((y2,c2)::nil)).
+      rewrite /= app_assoc_reverse //.
     }
   Qed.
        
   Lemma addZ_alt_def : forall x s, 
     addZ x s =
     match s with
-    | nil => (x, 1%N)::nil
+    | nil => (x, (1%N))::nil
     | (y, c) :: l =>
-        match (Z.compare (x+1) y) with
-        | Lt => (x, 1%N)::s
-        | Eq => (x, (c+1)%N)::l
-        | Gt => match (Z.compare x (y+Z.of_N c)) with
-                 | Lt => s
-                 | Gt => (y,c) :: addZ x l
-                 | Eq => match l with
-                           | nil => (y, (c+1)%N)::nil 
-                           | (z, c') :: l' => if (Z.eqb z (x + 1)) then
-                                (y, (c + c' + 1)%N) :: l'
-                             else
-                                (y,(c+1)%N) :: (z, c') :: l'
-                         end
-                end                 
-        end  
+        match (interval_1_compare x (y,c)) with
+          | ICR_before       => (x, (1%N))::s
+          | ICR_before_touch => (x, N.succ c)::l
+          | ICR_after        => (y,c)::(addZ x l)
+          | ICR_after_touch  => insert_interval_begin y (N.succ c) l
+          | _  => (y, c)::l
+        end
     end.
   Proof.
     intros x s.
-    rewrite /addZ Z.add_1_r. 
+    rewrite /addZ.
     case s => //.
-    move => [y c] s' /=.
-    case (Z.succ x ?= y). {
-      rewrite N.add_1_r //.
-    } { 
-      reflexivity.
-    } { 
-      case (x ?= y+ Z.of_N c) => //. {
-        rewrite !N.add_1_r.
-        case s' => //.
-        move => [z c'] s'' //.
-        rewrite !N.add_1_r //.
-      } {
-        rewrite addZ_aux_alt_def //.
-      }
-    }
+    move => [y c] s'.
+    unfold addZ_aux.
+    case (interval_1_compare x (y, c)); fold addZ_aux;
+      rewrite ?rev_append_rev /= ?app_assoc_reverse //.
+    rewrite addZ_aux_alt_def //.
   Qed.
    
 
-  (** *** Alternative definition of removeZ *)
-
-  (** [removeZ] is defined with efficient execution in mind.
-      We derive first an alternative definition that demonstrates
-      the intention better and is better suited for proofs. *)
+    
+  (** *** Auxiliary Lemmata about Invariant *)
 
-  Lemma removeZ_aux_alt_def : forall s x acc,
-    removeZ_aux acc x s = (List.rev acc) ++ removeZ x s. 
+  Lemma interval_list_elements_greater_cons : forall z x c s,
+    interval_list_elements_greater z ((x, c) :: s) = true <-> 
+    (z < x).  
   Proof.
-    induction s as [| [y c] s' IH]. {
-      intros x acc.
-      rewrite /removeZ /removeZ_aux /= app_nil_r /rev' -rev_alt //.
-    } {
-      intros x acc.
-      rewrite /removeZ /removeZ_aux -/removeZ_aux.
-      case (x <? y). {
-        rewrite !rev_append_rev //=.
-      } {
-        case (x <? y + Z.of_N c). {
-          rewrite /removeZ_aux_insert_guarded.
-          case (Z.to_N (y + Z.of_N c - (Z.succ x)) =? 0)%N, (Z.to_N (x - y) =? 0)%N;
-            rewrite !rev_append_rev /= // -!app_assoc /= //.
-        } {
-          rewrite !IH /= -app_assoc /= //.
-        }
-      }
-    }
+    intros z x c s.
+    rewrite /=.
+    apply Z.ltb_lt.
   Qed.
 
-  Lemma removeZ_alt_def : forall x s,
-    removeZ x s =
-    match s with
-    | nil => nil
-    | (y, c) :: l =>
-        if (Z.ltb x y) then s else
-        if (Z.ltb x (y+Z.of_N c)) then (
-           (removeZ_aux_insert_guarded y (Z.to_N (x-y)) 
-             (removeZ_aux_insert_guarded (Z.succ x) (Z.to_N ((y+Z.of_N c)- (Z.succ x))) l))
-        ) else (y, c) :: removeZ x l
-    end.
-  Proof.
-    intros x s.
-    rewrite /removeZ /=.
-    case s => //.
-    move => [y c] s' /=.
-    case(x <? y) => //.
-    case(x <? y + Z.of_N c). {
-      rewrite /removeZ_aux_insert_guarded.
-      case (Z.to_N (y + Z.of_N c - (Z.succ x)) =? 0)%N, (Z.to_N (x - y) =? 0)%N => //. 
-    } {
-      rewrite removeZ_aux_alt_def //.
-    }
-  Qed.
-            
-
-  (** *** Auxiliary Lemmata about Invariant *)
-
-  Lemma inf_impl : forall x y s,
-    (y <= x) -> inf x s = true -> inf y s = true.
+  Lemma interval_list_elements_greater_impl : forall x y s,
+    (y <= x) -> 
+    interval_list_elements_greater x s = true -> 
+    interval_list_elements_greater y s = true.
   Proof.
     intros x y s.
     case s => //.
     move => [z c] s'.
-    rewrite /inf.
+    rewrite /interval_list_elements_greater.
     move => H_y_leq /Z.ltb_lt H_x_lt.
     apply Z.ltb_lt.
     eapply Z.le_lt_trans; eauto.
   Qed.
 
+  Lemma interval_list_invariant_nil : interval_list_invariant nil = true.
+  Proof.
+    by [].
+  Qed.
+ 
   Lemma Ok_nil : Ok nil <-> True. 
   Proof.
-    rewrite /Ok /IsOk /isok /is_encoded_elems_list //.
+    rewrite /Ok /IsOk /interval_list_invariant /is_encoded_elems_list //.
   Qed.
 
   Lemma is_encoded_elems_list_app : forall l1 l2,
@@ -934,10 +1391,12 @@ Module Raw (Enc : ElementEncode).
     ).
   Qed.
 
-  Lemma isok_cons : forall y c s', isok ((y, c) :: s') = true <-> 
-    (inf (y+Z.of_N c) s' = true /\ ((c <> 0)%N) /\ isok s' = true). 
+  Lemma interval_list_invariant_cons : forall y c s', 
+    interval_list_invariant ((y, c) :: s') = true <-> 
+    (interval_list_elements_greater (y+Z.of_N c) s' = true /\ 
+      ((c <> 0)%N) /\ interval_list_invariant s' = true). 
   Proof.
-    rewrite /isok -/isok.
+    rewrite /interval_list_invariant -/interval_list_invariant.
     intros y c s'.
     rewrite !Bool.andb_true_iff negb_true_iff.
     split. {
@@ -947,71 +1406,245 @@ Module Raw (Enc : ElementEncode).
     }
   Qed.
 
+  Lemma interval_list_invariant_sing : forall x c,
+    interval_list_invariant ((x, c)::nil) = true <-> (c <> 0)%N.
+  Proof.
+    intros x c.
+    rewrite interval_list_invariant_cons.
+    split; tauto.
+  Qed.
+
   Lemma Ok_cons : forall y c s', Ok ((y, c) :: s') <-> 
-    (inf (y+Z.of_N c) s' = true /\ ((c <> 0)%N) /\ 
+    (interval_list_elements_greater (y+Z.of_N c) s' = true /\ ((c <> 0)%N) /\ 
      is_encoded_elems_list (elementsZ_single y c) /\ Ok s'). 
   Proof.
     intros y c s'.
-    rewrite /Ok /IsOk isok_cons elementsZ_cons is_encoded_elems_list_app
+    rewrite /Ok /IsOk interval_list_invariant_cons elementsZ_cons is_encoded_elems_list_app
        is_encoded_elems_list_rev.
     tauto.
   Qed.
 
   Lemma Nin_elements_greater : forall s y,
-     inf y s = true ->
-     isok s = true ->
-     forall x, x <= y ->
-     ~(InZ x s).
+     interval_list_elements_greater y s = true ->
+     interval_list_invariant s = true ->
+     forall x, x <= y -> ~(InZ x s).
   Proof.
     induction s as [| [z c] s' IH]. {
       intros y _ _ x _.
-      done.
+      by simpl.
     } {
-      rewrite /inf.
-      move => y /Z.ltb_lt H_y_lt /isok_cons [H_inf'] [H_c] H_s' x H_x_le.
-      rewrite /InZ elementsZ_cons in_app_iff -!in_rev In_elementsZ_single. 
-      move => []. {
-        eapply IH; eauto.
-        apply Z.le_trans with (m := z). {
-          eapply Z.lt_le_incl.
-          eapply Z.le_lt_trans; eauto.
-        } {
-          apply Z_le_add_r.
-        }
-      } {
+      move => y /interval_list_elements_greater_cons H_y_lt
+        /interval_list_invariant_cons [H_gr] [H_c] H_s'
+        x H_x_le.
+      rewrite InZ_cons In_elementsZ_single. 
+      have H_x_lt : x < z by eapply Z.le_lt_trans; eauto.
+
+      move => []. {              
         move => [H_z_leq] _; contradict H_z_leq.
-        apply Z.nle_gt.
-        eapply Z.le_lt_trans; eauto.
+        by apply Z.nle_gt.
+      } {
+        eapply IH; eauto.
+        by apply Z_lt_le_add_r.
       }
     }
   Qed.
 
-  Lemma isok_inf_nin :
+  Lemma Nin_elements_greater_equal :
      forall x s,
-       isok s = true ->
-       inf x s = true ->
+       interval_list_elements_greater x s = true ->
+       interval_list_invariant s = true ->
        ~ (InZ x s). 
   Proof.
-    move => x.
-    induction s as [| [y c] s' IH]. {
-      move => _ _.
-      rewrite /InZ elementsZ_nil //.
+    move => x s H_inv H_gr. 
+    apply (Nin_elements_greater s x) => //.
+    apply Z.le_refl.
+  Qed.
+
+  Lemma interval_list_elements_greater_alt_def : forall s y,
+     interval_list_invariant s = true ->
+
+     (interval_list_elements_greater y s = true <->
+      (forall x, x <= y -> ~(InZ x s))).
+  Proof.
+    intros s y H_inv.
+    split. {
+      move => H_gr.
+      apply Nin_elements_greater => //.
+    } {
+      move : H_inv.
+      case s as [| [x2 c] s'] => //.
+      rewrite interval_list_invariant_cons interval_list_elements_greater_cons.
+      move => [_] [H_c_neq] _ H.
+      apply Z.nle_gt => H_ge.
+      apply (H x2) => //.
+      rewrite InZ_cons; left.
+      apply In_elementsZ_single_hd => //.
+    }
+  Qed.
+
+  Lemma interval_list_elements_greater_alt2_def : forall s y,
+     interval_list_invariant s = true ->
+
+     (interval_list_elements_greater y s = true <->
+      (forall x, InZ x s -> y < x)).
+  Proof.
+    intros s y H.
+
+    rewrite interval_list_elements_greater_alt_def //.
+    split. {
+      move => H_notInZ x H_inZ.
+      apply Z.nle_gt.
+      move => H_lt.
+      move : (H_notInZ x H_lt H_inZ) => //.
+    } {
+      move => H_lt x H_le H_inZ.
+      move : (H_lt x H_inZ).
+      lia.
+    }
+  Qed.
+
+  Lemma interval_list_elements_greater_intro : forall s y,
+     interval_list_invariant s = true ->
+     (forall x, InZ x s -> y < x) ->
+     interval_list_elements_greater y s = true.
+  Proof.
+    intros s y H1 H2.
+    rewrite interval_list_elements_greater_alt2_def //.
+  Qed.
+               
+  Lemma interval_list_elements_greater_app_elim_1 : forall s1 s2 y,
+    interval_list_elements_greater y (s1 ++ s2) = true ->
+    interval_list_elements_greater y s1 = true.
+  Proof.
+    intros s1 s2 y.
+    case s1 => //.
+  Qed.
+
+  Lemma interval_list_invariant_app_intro : forall s1 s2,
+      interval_list_invariant s1 = true ->
+      interval_list_invariant s2 = true ->
+      (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2) ->
+      interval_list_invariant (s1 ++ s2) = true.
+  Proof.
+    induction s1 as [| [y1 c1] s1' IH]. {
+      move => s2 _ //.
     } {
-      rewrite isok_cons.
-      move => [H_inf] [_] H_ok_s' /Z.ltb_lt H_x_lt_y.
-      rewrite /InZ elementsZ_cons in_app_iff -in_rev In_elementsZ_single Z.le_ngt.
-      suff : inf x s' = true by tauto.
-      eapply inf_impl; last apply H_inf.
-      apply Z.le_trans with (m := y). {
-        by apply Z.lt_le_incl.
+      move => s2.
+      rewrite -app_comm_cons !interval_list_invariant_cons.
+      move => [H_gr] [H_c1_neq] H_inv_s1' H_inv_s2 H_inz_s2.
+      split; last split. {
+        move : H_gr H_inz_s2.
+        case s1' as [| [y1' c1'] s1'']; last done.
+        move => _ H_inz_s2.
+        rewrite app_nil_l.
+        apply interval_list_elements_greater_intro => //.
+        move => x H_x_in_s2.
+        suff H_inz : InZ (Z.pred (y1 + Z.of_N c1)) ((y1, c1) :: nil). {
+          move : (H_inz_s2 _ _ H_inz H_x_in_s2).
+          by rewrite Z.succ_pred.
+        }
+        rewrite InZ_cons In_elementsZ_single -Z.lt_le_pred; left.
+        split. {
+          by apply Z_lt_add_r.
+        } {
+          apply Z.lt_pred_l.
+        }
       } {
-        by apply Z_le_add_r.
+        assumption.
+      } {
+        apply IH => //.
+        intros x1 x2 H_in_x1 H_in_x2.
+        apply H_inz_s2 => //.
+        rewrite InZ_cons; by right.
       }
-    } 
+    }
+  Qed.
+
+
+  Lemma interval_list_invariant_app_elim : forall s1 s2,
+      interval_list_invariant (s1 ++ s2) = true ->
+      interval_list_invariant s1 = true /\
+      interval_list_invariant s2 = true /\
+      (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2).
+  Proof.
+    move => s1 s2.
+    induction s1 as [| [y1 c1] s1' IH]; first done.
+    rewrite -app_comm_cons !interval_list_invariant_cons.
+
+    move => [H_gr] [H_c1_neq_0] /IH [H_inv_s1'] [H_inv_s2] H_in_s1'_s2.
+    repeat split; try assumption. {
+      move : H_gr.
+      case s1'; first done.
+      move => [y2 c2] s1''.
+      rewrite interval_list_elements_greater_cons //.
+    } {
+      move => x1 x2.
+      rewrite InZ_cons In_elementsZ_single.
+      move => []; last by apply H_in_s1'_s2.
+      move => [] H_y1_le H_x1_lt H_x2_in.
+      move : H_gr.
+      rewrite interval_list_elements_greater_alt2_def; last first. {
+          by apply interval_list_invariant_app_intro.
+      }
+      move => H_in_s12'.
+      have : (y1 + Z.of_N c1 < x2). {
+        apply H_in_s12'.
+        rewrite InZ_app.
+        by right.
+      }
+      move => H_lt_x2.
+      apply Z.le_lt_trans with (m := y1 + Z.of_N c1) => //.
+      by apply Zlt_le_succ.
+    }
+  Qed.
+
+  Lemma interval_list_invariant_app_iff : forall s1 s2,
+      interval_list_invariant (s1 ++ s2) = true <->
+      (interval_list_invariant s1 = true /\
+      interval_list_invariant s2 = true /\
+      (forall (x1 x2 : Z), InZ x1 s1 -> InZ x2 s2 -> Z.succ x1 < x2)).
+  Proof.
+    intros s1 s2.
+    split. {
+      by apply interval_list_invariant_app_elim.
+    } {
+      move => [] H_inv_s1 [].
+      by apply interval_list_invariant_app_intro.
+    }
+  Qed.
+
+  Lemma interval_list_invariant_snoc_intro : forall s1 y2 c2,
+      interval_list_invariant s1 = true ->
+      (c2 <> 0)%N ->
+      (forall x, InZ x s1 -> Z.succ x < y2) ->
+      interval_list_invariant (s1 ++ ((y2, c2)::nil)) = true.
+  Proof.
+    intros s1 y2 c2 H_inv_s1 H_c2_neq H_in_s1.
+    apply interval_list_invariant_app_intro => //. {
+      rewrite interval_list_invariant_cons; done.
+    } {
+      intros x1 x2 H_in_x1.
+      rewrite InZ_cons.
+      move => [] //.
+      rewrite In_elementsZ_single.
+      move => [H_y2_le] _.
+      eapply Z.lt_le_trans; eauto.
+    }
   Qed.
 
+    
   (** *** Properties of In and InZ *)
 
+  Lemma encode_decode_eq : forall x s, Ok s -> InZ x s ->
+    (Enc.encode (Enc.decode x) = x).
+  Proof.
+    intros x s.
+    rewrite /Ok /IsOk /InZ.
+    move => [_] H_enc H_in_x.
+    move : (H_enc _ H_in_x) => [x'] <-.
+    rewrite Enc.decode_encode_ok //.
+  Qed.
+
   Lemma In_alt_def : forall x s, Ok s -> 
     (In x s <-> List.In x (elements s)).
   Proof.
@@ -1019,7 +1652,7 @@ Module Raw (Enc : ElementEncode).
     rewrite /In InA_alt /elements rev_map_alt_def.
     split. {
       move => [y] [H_y_eq].
-      rewrite -!in_rev !in_map_iff.
+      rewrite -!in_rev !in_map_iff.      
       move => [x'] [H_y_eq'] H_x'_in.
       suff H_x'_eq :  (Enc.encode x = x'). {
         exists x'.
@@ -1065,30 +1698,43 @@ Module Raw (Enc : ElementEncode).
     apply H_ok.
   Qed.
 
+  Lemma InZ_In : forall x s, Ok s -> 
+    (InZ x s -> In (Enc.decode x) s).
+  Proof.
+    intros x s H_ok.
+    rewrite In_InZ /InZ.
+    move : H_ok.
+    rewrite /Ok /IsOk /is_encoded_elems_list.
+    move => [_] H_enc.
+    move => H_in.
+    move : (H_enc _ H_in) => [e] H_x.
+    subst.
+    by rewrite Enc.decode_encode_ok.
+  Qed.
 
+  
   (** *** Membership specification *)
           
   Lemma memZ_spec :
-   forall (s : t) (x : Z) (Hs : Ok s), memZ x s = true <-> InZ x s.
+    forall (s : t) (x : Z) (Hs : Ok s), memZ x s = true <-> InZ x s.
   Proof.
     induction s as [| [y c] s' IH]. {
       intros x _.
       rewrite /InZ elementsZ_nil //.
     } {    
       move => x /Ok_cons [H_inf] [H_c] [H_is_enc] H_s'.  
-      rewrite /InZ /memZ elementsZ_cons -/mem.
-      rewrite /isok -/isok in_app_iff -!in_rev In_elementsZ_single.
+      rewrite /InZ /memZ elementsZ_cons -/memZ.
+      rewrite in_app_iff -!in_rev In_elementsZ_single.
 
       case_eq (x <? y). {
         move => /Z.ltb_lt H_x_lt.
         split; first done.
         move => []. {
           move => H_x_in; contradict H_x_in.
-          eapply Nin_elements_greater; eauto; first apply H_s'.
-          apply Z.le_trans with (m := y). {
-            by apply Z.lt_le_incl.
+          apply Nin_elements_greater with (y := (y + Z.of_N c)) => //. {
+            apply H_s'.
           } {
-            apply Z_le_add_r.
+            apply Z_lt_le_add_r => //.
           }
         } {
           move => [H_y_le]; contradict H_y_le.
@@ -1120,201 +1766,345 @@ Module Raw (Enc : ElementEncode).
     rewrite /mem memZ_spec In_InZ //.
   Qed.
 
+  Lemma merge_interval_size_neq_0 : forall x1 c1 x2 c2,
+     (c1 <> 0%N) ->
+     (merge_interval_size x1 c1 x2 c2 <> 0)%N.
+  Proof.
+    intros x1 c1 x2 c2.
+    rewrite /merge_interval_size !N.neq_0_lt_0 N.max_lt_iff.
+    by left.
+  Qed.
+
 
-  (** *** add specification *)
-  Lemma addZ_spec :
-   forall (s : t) (x y : Z) (Hs : Ok s),
-    InZ y (addZ x s) <-> Z.eq y x \/ InZ y s.
-  Proof.
-    intros s x y.
-    eapply addZ_ind with (P := (fun (x:Z) s => Ok s -> (InZ y (addZ x s) <-> Z.eq y x \/ InZ y s))); clear s x. {
-      move => x _.
-      rewrite /InZ addZ_alt_def elementsZ_cons elementsZ_nil app_nil_l -in_rev
-              In_elementsZ_single1 /=.
-      firstorder.
+  (** *** insert if length not 0 *)
+
+  Lemma interval_list_invariant_insert_intervalZ_guarded : forall x c s,
+    interval_list_invariant s = true -> 
+    interval_list_elements_greater (x + Z.of_N c) s = true ->
+    interval_list_invariant (insert_intervalZ_guarded x c s) = true.
+  Proof.
+    intros x c s.
+    rewrite /insert_intervalZ_guarded.
+    case_eq (c =? 0)%N => //.
+    move => /N.eqb_neq.
+    rewrite interval_list_invariant_cons.
+    tauto.
+  Qed.
+
+  Lemma interval_list_elements_greater_insert_intervalZ_guarded : forall x c y s,
+    interval_list_elements_greater y (insert_intervalZ_guarded x c s) = true <->
+    (if (c =? 0)%N then (interval_list_elements_greater y s = true) else (y < x)).
+  Proof.
+    intros x c y s.
+    rewrite /insert_intervalZ_guarded.
+    case (c =? 0)%N => //.       
+    rewrite /interval_list_elements_greater Z.ltb_lt //.
+  Qed.
+
+  Lemma insert_intervalZ_guarded_app : forall x c s1 s2,
+    (insert_intervalZ_guarded x c s1) ++ s2 =
+    insert_intervalZ_guarded x c (s1 ++ s2).
+  Proof.
+    intros x c s1 s2.
+    rewrite /insert_intervalZ_guarded.
+    case (N.eqb c 0) => //.
+  Qed.
+
+  Lemma insert_intervalZ_guarded_rev_nil_app : forall x c s,
+    rev (insert_intervalZ_guarded x c nil) ++ s =
+    insert_intervalZ_guarded x c s.
+  Proof.
+    intros x c s.
+    rewrite /insert_intervalZ_guarded.
+    case (N.eqb c 0) => //.
+  Qed.
+
+Lemma elementsZ_insert_intervalZ_guarded : forall x c s,
+    elementsZ (insert_intervalZ_guarded x c s) = elementsZ ((x, c) :: s). 
+  Proof.
+    intros x c s.
+    rewrite /insert_intervalZ_guarded.
+    case_eq (c =? 0)%N => //.
+    move => /N.eqb_eq ->.
+    rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //.
+  Qed.
+
+  Lemma InZ_insert_intervalZ_guarded : forall y x c s,
+    InZ y (insert_intervalZ_guarded x c s) = InZ y ((x, c) :: s). 
+  Proof.
+    intros y x c s.
+    rewrite /InZ elementsZ_insert_intervalZ_guarded //.
+  Qed.
+  
+  (** *** Merging intervals *)
+
+  Lemma merge_interval_size_add : forall x c1 c2,
+     (merge_interval_size x c1 (x + Z.of_N c1) c2 = (c1 + c2))%N.
+  Proof.
+    intros x c1 c2.
+    rewrite /merge_interval_size.
+    replace (x + Z.of_N c1 + Z.of_N c2 - x) with
+            (Z.of_N c1 + Z.of_N c2) by lia.
+    rewrite -N2Z.inj_add N2Z.id.
+    apply N.max_r, N.le_add_r.
+  Qed.
+
+  Lemma merge_interval_size_eq_max : forall y1 c1 y2 c2,
+     y1 <= y2 + Z.of_N c2 ->
+     y1 + Z.of_N (merge_interval_size y1 c1 y2 c2) = 
+     Z.max (y1 + Z.of_N c1) (y2 + Z.of_N c2).
+  Proof.
+    intros y1 c1 y2 c2 H_y1_le.
+    rewrite /merge_interval_size N2Z.inj_max Z2N.id; last first. {
+      by apply Zle_minus_le_0.
+    }
+    rewrite -Z.add_max_distr_l.
+    replace (y1 + (y2 + Z.of_N c2 - y1)) with (y2 + Z.of_N c2) by lia.
+    done.
+  Qed.
+
+  Lemma merge_interval_size_invariant : forall y1 c1 y2 c2 z s,
+    interval_list_invariant s = true ->
+    y1 + Z.of_N c1 <= y2 + Z.of_N c2 ->
+    y2 + Z.of_N c2 <= z ->
+    interval_list_elements_greater z s = true ->
+    (c1 <> 0)%N ->
+    interval_list_invariant ((y1, merge_interval_size y1 c1 y2 c2) :: s) =
+   true.
+  Proof.
+    intros y1 c1 y2 c2 z s H_inv H_le H_le_z H_gr H_c1_neq_0.
+    rewrite interval_list_invariant_cons.
+    split; last split. {
+      rewrite merge_interval_size_eq_max; last first. {
+        eapply Z.le_trans; last apply H_le.
+        apply Z_le_add_r.
+      } {
+        rewrite Z.max_r => //.
+        eapply interval_list_elements_greater_impl; first apply H_le_z.
+        done.
+      }
     } {
-      intros x l c _.
-      rewrite /InZ addZ_alt_def Z.compare_refl !elementsZ_cons !in_app_iff -!in_rev N.add_1_r
-              Z.add_1_r !In_elementsZ_single N2Z.inj_succ Z.add_succ_l 
-              Z.add_succ_r Z.le_succ_l Z.lt_succ_r.   
-      split. {
-        move => []; first by tauto.
-        move => [] /Z.lt_eq_cases. 
-        firstorder.
+      apply merge_interval_size_neq_0.
+      assumption.
+    } {
+      assumption.
+    }
+  Qed.
+
+
+  Lemma In_merge_interval : forall x1 c1 x2 c2 y,
+    x1 <= x2 -> 
+    x2 <= x1 + Z.of_N c1 -> (
+    List.In y (elementsZ_single x1 (merge_interval_size x1 c1 x2 c2)) <->
+    List.In y (elementsZ_single x1 c1) \/ List.In y (elementsZ_single x2 c2)).
+  Proof.
+    intros x1 c1 x2 c2 y H_x1_le H_x2_le.
+    rewrite !In_elementsZ_single merge_interval_size_eq_max; 
+      last first. {
+      eapply Z.le_trans; eauto.
+      by apply Z_le_add_r.
+    }
+    rewrite Z.max_lt_iff.
+    split. {
+      move => [H_x_le] [] H_y_lt. {
+        by left.
       } {
-        move => [-> | ]. {
-          right; split. 
-            - by apply Z.le_refl.
-            - by apply Z_le_add_r.
+        case_eq (Z.leb x2 y). {
+          move => /Z.leb_le H_y'_le.
+          by right.
         } {
-          move => []; first by tauto.
-          move => [H_x_lt H_y_le].
-          right; split => //.
-          by apply Z.lt_le_incl.
+          move => /Z.leb_gt H_y_lt_x2.
+          left.
+          split => //.
+          eapply Z.lt_le_trans; eauto.
         }
-      }  
-    } {
-      intros x l y0 c H_y_comp _.
-      rewrite /InZ addZ_alt_def H_y_comp elementsZ_cons in_app_iff
-          In_elementsZ_single1.
-      firstorder.
-    } {
-      move => y0 c H_y_comp H_ok.
-      rewrite addZ_alt_def H_y_comp Z.compare_refl.   
-      rewrite /InZ !elementsZ_cons elementsZ_nil !app_nil_l N.add_1_r.
-      rewrite elementsZ_single_succ -!in_rev in_app_iff /=.
-      firstorder.
-    } {
-      move => l y0 c c' H_y_comp _ _.
-      rewrite addZ_alt_def H_y_comp Z.compare_refl Z.eqb_refl.
-      rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev.   
-      have -> : (c + c' + 1 = N.succ c + c')%N. {
-        rewrite -N.add_1_r. ring.
       }
-      rewrite Z.add_1_r elementsZ_single_add elementsZ_single_succ !in_app_iff
-              N2Z.inj_succ Z.add_succ_r.
-      firstorder.
-    } {
-      intros l y0 c z c'.
-      move => H_y_comp H_z_comp _.
-      rewrite addZ_alt_def H_y_comp Z.compare_refl H_z_comp.
-      rewrite /InZ !elementsZ_cons N.add_1_r !in_app_iff -!in_rev.
-      rewrite elementsZ_single_succ in_app_iff /=.   
-      firstorder.
     } {
-      intros x l y0 c H_y_comp H_x_comp _.
-      rewrite addZ_alt_def H_y_comp H_x_comp.   
-      split. {
-        by right.
+      move => []. {
+        tauto.
       } {
-        move => [] // ->.
-        rewrite /InZ elementsZ_cons in_app_iff -in_rev In_elementsZ_single.
-        right. split. {
-          move : H_y_comp => /Z.compare_gt_iff.
-          rewrite Z.add_1_r Z.lt_succ_r //.
+        move => [H_x2_le'] H_y_lt.
+        split. {
+          eapply Z.le_trans; eauto.
         } {
-          move : H_x_comp => /Z.compare_lt_iff //.
+          by right.
         }
       }
+    }
+  Qed.
+
+  Lemma insert_interval_begin_spec : forall y s x c,
+     interval_list_invariant s = true -> 
+     interval_list_elements_greater x s = true ->
+     (c <> 0)%N ->
+      (
+     interval_list_invariant (insert_interval_begin x c s) = true /\
+     (InZ y (insert_interval_begin x c s) <->
+     (List.In y (elementsZ_single x c) \/ InZ y s))).
+  Proof.
+    intros y.
+    induction s as [| [y' c'] s' IH]. {
+      intros x c _ H_c_neq H_z_lt.
+      rewrite /insert_interval_begin InZ_cons interval_list_invariant_cons //.
     } {
-      intros x l y0 c H_y_comp H_x_comp IH H_ok.
-      have H_ok' : Ok l. {
-        move : H_ok. 
-        rewrite Ok_cons.
+      intros x c.
+      rewrite interval_list_invariant_cons
+       interval_list_elements_greater_cons.
+      move => [H_gr] [H_c'_neq_0] H_inv_s' H_x_lt H_c_neq_0.
+      unfold insert_interval_begin.
+      Z_named_compare_cases H_y'; fold insert_interval_begin. {
+        subst.
+        rewrite !InZ_cons elementsZ_single_add in_app_iff.
+        split; last tauto.
+        rewrite interval_list_invariant_cons N2Z.inj_add
+          Z.add_assoc N.eq_add_0.
         tauto.
+      } {
+        rewrite !InZ_cons !interval_list_invariant_cons
+          interval_list_elements_greater_cons.
+        repeat split => //.
+      } {
+        set c'' := merge_interval_size x c y' c'.
+        have H_x_lt' : x < y' + Z.of_N c'. {
+          eapply Z.lt_le_trans with (m := y') => //. 
+          by apply Z_le_add_r.
+        } 
+
+        have H_pre : interval_list_elements_greater x s' = true. {
+          eapply interval_list_elements_greater_impl; eauto.
+          by apply Z.lt_le_incl.
+        }
+        have H_pre2 : c'' <> 0%N. {
+          by apply merge_interval_size_neq_0.
+        }
+        move : (IH x c'' H_inv_s' H_pre H_pre2) => {IH} {H_pre} {H_pre2} [->] ->.
+
+        split; first reflexivity.
+        unfold c''; clear c''.
+        rewrite In_merge_interval. {
+          rewrite InZ_cons. 
+          tauto.
+        } {
+          by apply Z.lt_le_incl.
+        } {
+          by apply Z.lt_le_incl.
+        }
       }
-      rewrite addZ_alt_def H_y_comp H_x_comp.   
-      unfold InZ in IH.
-      rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev (IH H_ok').
+    }
+  Qed.
+
+
+  (** *** add specification *)
+  Lemma addZ_InZ :
+   forall (s : t) (x y : Z),
+    interval_list_invariant s = true ->
+    (InZ y (addZ x s) <-> x = y \/ InZ y s).
+  Proof.
+    move => s x y.
+    induction s as [| [z c] s' IH]. {
+      move => _.
+      rewrite /InZ addZ_alt_def
+              elementsZ_cons elementsZ_nil app_nil_l -in_rev
+              In_elementsZ_single1 /=.
       firstorder.
+    } {
+      move => /interval_list_invariant_cons [H_greater] [H_c_neq_0] H_inv_c'.
+      move : (IH H_inv_c') => {} IH.
+      rewrite addZ_alt_def.
+      have H_succ : forall z, z + Z.of_N 1 = Z.succ z by done.
+      move : (interval_1_compare_elim x z c).      
+      case (interval_1_compare x (z, c));
+        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
+        move => ->.
+        rewrite elementsZ_single_succ_front /=.
+        tauto.
+      } {
+        move => [] // H_x_in.
+        split; first tauto.
+        move => [] // <-.
+        left.
+        by rewrite In_elementsZ_single.
+      } {
+        rewrite IH.
+        tauto.
+      } {
+        move => H_x_eq.
+        have -> : (InZ y (insert_interval_begin z (N.succ c) s') <->
+                   List.In y (elementsZ_single z (N.succ c)) \/ InZ y s'). {
+          eapply insert_interval_begin_spec. {
+            by apply H_inv_c'.
+          } {
+            eapply interval_list_elements_greater_impl; eauto.
+            apply Z_le_add_r.
+          } {
+            by apply N.neq_succ_0.
+          }
+        }
+        rewrite -H_x_eq elementsZ_single_succ in_app_iff /=.
+        tauto.
+      }
     }
   Qed.
 
-  Lemma addZ_isok : forall s x, isok s = true -> isok (addZ x s) = true.
+  Lemma addZ_invariant : forall s x, 
+    interval_list_invariant s = true -> 
+    interval_list_invariant (addZ x s) = true.
   Proof. 
-    intros s x.
-    have H_add_1_neq : (forall x, x + 1 <> 0)%N. {
-      move => x'.
-      rewrite N.eq_add_0. 
-      have : (1 <> 0)%N by done.
-      tauto.
-    }
-    
-    eapply addZ_ind with (P := (fun (x : Z) (s : t) => isok s = true -> 
-          isok (addZ x s) = true)); clear s x. {
-      move => x _ //.
-    } {    
-      intros x l c.
-      rewrite addZ_alt_def Z.compare_refl.   
-      rewrite !isok_cons.
-      move => [H_inf] [H_c_neq] H_ok_l.
-      split; last split => //. {
-        suff -> : (x + Z.of_N (c + 1) = x + 1 + Z.of_N c) by done.
-        rewrite N2Z.inj_add /=.
-        ring.
-      }
-    } {
-      intros x l y c H_y_comp.
-      rewrite addZ_alt_def H_y_comp.   
-      rewrite !isok_cons => [[H_inf] [H_c_neq] H_ok_l].
-      split; last split; last split; last split; try done. 
-      by apply Z.ltb_lt.      
-    } {
-      move => y c H_y_comp _.
-      rewrite addZ_alt_def H_y_comp Z.compare_refl isok_cons.   
-      done.      
-    } {
-      move => l y c c' H_y_comp IH.
-      rewrite addZ_alt_def H_y_comp Z.compare_refl Z.eqb_refl.
-      rewrite !isok_cons.   
-      move => [H_inf1] [H_c_neq] [H_inf2] [H_c'_neq] H_ok_l.
-      split; last split => //. {
-        suff -> : (y + Z.of_N (c + c' + 1) =  y + Z.of_N c + 1 + Z.of_N c') by assumption.
-        rewrite !N2Z.inj_add /=.
-        ring.
-      }
-    } {
-      intros l y c z c'.
-      move => H_y_comp H_z_comp.
-      rewrite addZ_alt_def H_y_comp Z.compare_refl H_z_comp.
-      rewrite isok_cons => [] [] H_inf [] H_c_neq H_ok_l.
-      rewrite isok_cons.
-      split; last split; try done.
-      move : H_inf.
-      rewrite /inf. 
-      move : H_z_comp => /Z.eqb_neq H_z_neq /Z.ltb_lt H_z_lt.
-      apply Z.ltb_lt.
-
-      have -> : Z.of_N (c + 1) = Z.of_N c + 1. {
-        rewrite N2Z.inj_add //.
-      }
-
-      suff : y + (Z.of_N c + 1) <= z. {
-        rewrite Z.lt_eq_cases.
-        move => [] // H_z_eq.
-        contradict H_z_neq.
-        rewrite -H_z_eq.
-        ring.
-      }
-      suff H_suff : (Z.pred (y + (Z.of_N c + 1)) = y + Z.of_N c). {
-        apply Z.lt_pred_le.
-        by rewrite H_suff.
-      }
-      rewrite Z.add_1_r Z.add_succ_r Z.pred_succ //.
-    } {
-      intros x l y c H_y_comp H_x_comp.
-      rewrite addZ_alt_def H_y_comp H_x_comp //.
-    } {
-      intros x l y c H_y_comp H_x_comp IH.
-      rewrite addZ_alt_def H_y_comp H_x_comp.   
-      rewrite !isok_cons => [] [] H_inf [] H_c_neq H_ok_l.
-      split; last split; try done. {
-        suff : exists x0 c0 l0, addZ x l = (x0, c0)::l0 /\ (y+Z.of_N c < x0). {
-           move => [x0] [c0] [l0] [->].
-           rewrite /inf.
-           apply Z.ltb_lt.
-        }
-        apply Z.compare_gt_iff in H_x_comp.
-        move : H_inf.
-        case l. {
-          exists x, 1%N, nil.
-          by rewrite addZ_alt_def.
-        } {
-          move =>  [z c'] l0 /Z.ltb_lt H_z_gt.
-          rewrite addZ_alt_def.
-          case (x + 1 ?= z); try by eauto.
-          case (x ?= z + Z.of_N c'); try by eauto.
-          case l0; try by eauto.
-          move => [z' c''] ?.
-          case (z' =? x + 1); by eauto.
+    move => s x.
+    induction s as [| [z c] s' IH]. {
+      move => _.
+      by simpl.
+    } {
+      move => /interval_list_invariant_cons [H_greater] [H_c_neq_0]
+              H_inv_c'.
+      move : (IH H_inv_c') => {} IH.
+      rewrite addZ_alt_def.
+      have H_succ : forall z, z + Z.of_N 1 = Z.succ z by done.
+      move : (interval_1_compare_elim x z c).      
+      case_eq (interval_1_compare x (z, c)) => H_comp;
+        rewrite ?InZ_cons ?In_elementsZ_single1 ?H_succ ?Z.lt_succ_r //. {
+        move => H_z_gt.
+        rewrite interval_list_invariant_cons /= !andb_true_iff !H_succ.
+        repeat split => //. {
+          by apply Z.ltb_lt.
+        } {
+          apply negb_true_iff, N.eqb_neq => //.
         }
       } {
-        apply IH => //.
+        move => ?; subst.
+        rewrite /= !andb_true_iff.
+        repeat split => //. {
+          move : H_greater.
+          rewrite Z.add_succ_l -Z.add_succ_r N2Z.inj_succ //.
+        } {
+          apply negb_true_iff, N.eqb_neq => //.
+          apply N.neq_succ_0.
+        }
+      } {
+        move => [] // _.
+        rewrite interval_list_invariant_cons /=.
+        tauto.
+      } {
+        rewrite interval_list_invariant_cons.
+        move => H_lt_x.
+        repeat split => //.
+        apply interval_list_elements_greater_intro => //.
+        move => xx.
+        rewrite addZ_InZ => //.
+        move => [<- //|]. 
+        apply interval_list_elements_greater_alt2_def => //.
+      } {
+        move => H_x_eq.
+        apply insert_interval_begin_spec => //. {
+          eapply interval_list_elements_greater_impl; eauto.
+          apply Z_le_add_r.
+        } {
+          by apply N.neq_succ_0.
+        }
       }
     }
   Qed.
-
-
+  
   Global Instance add_ok s x : forall `(Ok s), Ok (add x s).
   Proof. 
     move => H_ok_s.
@@ -1322,13 +2112,13 @@ Module Raw (Enc : ElementEncode).
     rewrite /Ok /IsOk /is_encoded_elems_list /add.
     move => [H_isok_s] H_pre.
     split. {
-      apply addZ_isok => //.
+      apply addZ_invariant => //.
     } {
       intros y.
-      move : (addZ_spec s (Enc.encode x) y H_ok_s).
+      move : (addZ_InZ s (Enc.encode x) y H_isok_s).
       rewrite /InZ => ->.
       move => []. {
-        move => ->.
+        move => <-.
         by exists x.
       } {
         move => /H_pre //.
@@ -1343,264 +2133,20 @@ Module Raw (Enc : ElementEncode).
     intros s x y Hs.
     have Hs' := (add_ok s x Hs).
     rewrite !In_InZ.
-    rewrite /add addZ_spec Enc.encode_eq //.
+    rewrite /add addZ_InZ. {
+      rewrite -Enc.encode_eq /Z.eq.
+      firstorder. 
+    } {
+      apply Hs.
+    }
   Qed.
 
 
-  (** *** remove specification *)
-
-  Lemma isok_removeZ_aux_insert_guarded : forall x c s,
-    isok s = true -> inf (x + Z.of_N c) s = true ->
-    isok (removeZ_aux_insert_guarded x c s) = true.
-  Proof.
-    intros x c s.
-    rewrite /removeZ_aux_insert_guarded.
-    case_eq (c =? 0)%N => //.
-    move => /N.eqb_neq.
-    rewrite isok_cons.
-    tauto.
-  Qed.
+  (** *** empty specification *)
 
-  Lemma inf_removeZ_aux_insert_guarded : forall x c y s,
-    inf y (removeZ_aux_insert_guarded x c s) = true <->
-    (if (c =? 0)%N then (inf y s = true) else (y < x)).
+  Global Instance empty_ok : Ok empty.
   Proof.
-    intros x c y s.
-    rewrite /removeZ_aux_insert_guarded.
-    case (c =? 0)%N => //.       
-    rewrite /inf Z.ltb_lt //.
-  Qed.
-
-
-  Lemma removeZ_counter_pos_aux : forall y c x,
-     x < y + Z.of_N c ->
-     0 <= y + Z.of_N c - Z.succ x.
-  Proof.
-    intros y c x H_x_lt.
-    rewrite Z.le_0_sub Z.le_succ_l.
-    assumption.
-  Qed.
-
-  Lemma removeZ_isok : forall s x, isok s = true -> isok (removeZ x s) = true.
-  Proof.
-    intros s x.
-    induction s as [| [y c] s' IH]. {
-      rewrite removeZ_alt_def //.
-    } {
-      rewrite removeZ_alt_def.
-      case_eq (x <? y) => //.
-      move => /Z.ltb_ge H_y_le_x.
-      case_eq (x <? y + Z.of_N c). {
-        move => /Z.ltb_lt H_x_lt /isok_cons [H_inf] [H_c_neq] H_ok_s'.
-        apply isok_removeZ_aux_insert_guarded; first apply isok_removeZ_aux_insert_guarded. {
-          assumption.
-        } {
-          rewrite Z2N.id; last by apply removeZ_counter_pos_aux.
-          rewrite add_add_sub_eq.
-          assumption.
-        } {
-          rewrite /removeZ_aux_insert_guarded Z2N.id; last first. {
-            by rewrite Z.le_0_sub.
-          }
-          rewrite add_add_sub_eq.
-          case  (Z.to_N (y + Z.of_N c - Z.succ x) =? 0)%N. {
-            eapply inf_impl; eauto.
-            by apply Z.lt_le_incl.
-          } {
-            rewrite /inf.
-            apply Z.ltb_lt, Z.lt_succ_diag_r.
-          }
-        }
-      } {
-        move => /Z.ltb_ge H_yc_le_x.
-        rewrite !isok_cons.
-        move => [H_inf] [H_c_neq] H_ok_s'.
-        split; last split => //; last by apply IH. 
-        move : H_inf.
-        case_eq s' => //.
-        move => [z c'] l' H_s'_eq.
-        rewrite removeZ_alt_def.
-        case (x <? z); first by eauto.
-        case (x <? z + Z.of_N c'); last by eauto.
-        move => /Z.ltb_lt H_lt_z.
-        rewrite inf_removeZ_aux_insert_guarded.
-        case (Z.to_N (x - z) =? 0)%N => //.
-        rewrite inf_removeZ_aux_insert_guarded.
-        case (Z.to_N (z + Z.of_N c' - Z.succ x) =? 0)%N. {
-          move : H_ok_s'.
-          rewrite H_s'_eq !isok_cons.
-          move => [H_inf] _.
-          eapply inf_impl; last apply H_inf.
-          apply Z.le_trans with (m := z). {
-            by apply Z.lt_le_incl.
-          } {
-            by apply Z_le_add_r.
-          }
-        } {
-          rewrite Z.lt_succ_r //.
-        }
-      }
-    }
-  Qed.
-
-
-  Lemma elementsZ_removeZ_aux_insert_guarded : forall x c s,
-    elementsZ (removeZ_aux_insert_guarded x c s) = elementsZ ((x, c) :: s). 
-  Proof.
-    intros x c s.
-    rewrite /removeZ_aux_insert_guarded.
-    case_eq (c =? 0)%N => //.
-    move => /N.eqb_eq ->.
-    rewrite elementsZ_cons elementsZ_single_base /= app_nil_r //.
-  Qed.
- 
-  Lemma removeZ_spec :
-   forall (s : t) (x y : Z) (Hs : isok s = true),
-    InZ y (removeZ x s) <-> InZ y s /\ ~Z.eq y x.
-  Proof.
-    intros s x y Hs.
-    move : Hs x y.
-    induction s as [| [y c] s' IH] . {
-      move => _ x y.
-      rewrite /=.
-      firstorder.
-    } {
-      intros H_ok x y0.
-      rewrite removeZ_alt_def.
-      case_eq (x <? y). {
-        move => /Z.ltb_lt H_x_lt.
-        split; last tauto.
-        move => H_in.
-        split => // H_y0_eq.
-        apply (isok_inf_nin y0 ((y, c)::s')) => //.
-        rewrite /inf H_y0_eq. 
-        by apply Z.ltb_lt.        
-      } {
-        move => /Z.ltb_ge H_y_le.
-        have H_ok' : isok s' = true. {
-          move : H_ok.
-          rewrite isok_cons; tauto.
-        } 
-        move : (IH H_ok') => {IH} IH.
-        case_eq (x <? y + Z.of_N c). {
-          move => /Z.ltb_lt H_x_lt /=.
-          rewrite /InZ elementsZ_removeZ_aux_insert_guarded !elementsZ_cons.
-          rewrite elementsZ_removeZ_aux_insert_guarded elementsZ_cons !in_app_iff -!in_rev.
-          rewrite !In_elementsZ_single !Z2N.id; last first. {
-            by apply removeZ_counter_pos_aux.
-          } {
-            by rewrite Z.le_0_sub.
-          }
-          rewrite !add_add_sub_eq Z.le_succ_l.
-          case_eq (y0 =? x). {
-            move => /Z.eqb_eq ->.
-            split; last first. {
-              move => [_] H_neq.
-              contradict H_neq => //.
-            } {
-              move => [[] |]. {
-                move => H_in. exfalso.
-                apply (isok_inf_nin x s') => //.
-                move : H_ok.
-                case s' => //.
-                move => [z c'] s''.
-                rewrite !isok_cons /inf.
-                move => [] /Z.ltb_lt H_lt_x' _.
-                apply Z.ltb_lt.
-                eapply Z.lt_trans; eauto.
-              } {
-                move => [] /Z.lt_irrefl //.
-              } {
-                move => [_] /Z.lt_irrefl //.
-              }
-            } 
-          }
-          move => /Z.eqb_neq H_y0_neq.
-          split. {
-            move => H.
-            split; last by apply H_y0_neq.
-            move : H => [[] |]. {
-              by left.
-            } {
-              move => [H_x_lt'] H_y0_lt.
-              right; split => //.
-              apply Z.le_trans with (m := x) => //.
-              by apply Z.lt_le_incl.
-            } {
-              move => [H_y_le'] H_y0_lt.
-              right; split => //.
-              eapply Z.lt_trans; eauto.
-            }
-          } {
-            move => [[]]; first by tauto.
-            move => [H_y_le'] H_y0_lt _.
-            case_eq (y0 <? x). {
-              move => /Z.ltb_lt H_y0_lt_x.
-              right. done.
-            } {
-              move => /Z.ltb_ge /Z.lt_eq_cases []. {
-                move => H_x_lt'.
-                left; right.
-                done.
-              } {
-                move => H_x_eq.
-                contradict H_y0_neq => //.
-              }
-            }
-          } 
-        } {
-          move => /Z.ltb_ge H_yc_le.
-          unfold InZ in IH.
-          rewrite /InZ !elementsZ_cons !in_app_iff !IH.
-          split; last tauto.
-          move => []; first tauto.
-          rewrite -!in_rev In_elementsZ_single.
-          move => [H_y_le'] H_y0_lt.
-          split; first by right.        
-          move => H_y0_eq.
-          move : H_y0_lt.
-          apply Z.nlt_ge.
-          by rewrite H_y0_eq.
-        }
-      }
-    }
-  Qed.
-
-
-  Global Instance remove_ok s x : forall `(Ok s), Ok (remove x s).
-  Proof.
-    rewrite /Ok /IsOk /remove.
-    move => [H_is_ok_s H_enc_s].
-    split. {
-      by apply removeZ_isok.
-    } {
-      rewrite /is_encoded_elems_list => y.
-      move : (removeZ_spec s (Enc.encode x) y H_is_ok_s).
-      rewrite /InZ => -> [] H_y_in _.
-      apply H_enc_s => //.
-    }
-  Qed.
-
-  Lemma remove_spec :
-   forall (s : t) (x y : elt) (Hs : Ok s),
-    In y (remove x s) <-> In y s /\ ~Enc.E.eq y x.
-  Proof.
-    intros s x y Hs.
-    have H_rs := (remove_ok s x Hs).
-    rewrite /remove !In_InZ.
-    rewrite removeZ_spec. {
-      rewrite Enc.encode_eq //.
-    } {
-      apply Hs.
-    }
-  Qed.
-
-
-  (** *** empty specification *)
-
-  Global Instance empty_ok : Ok empty.
-  Proof.
-    rewrite /empty Ok_nil //.
+    rewrite /empty Ok_nil //.
   Qed.
 
   Lemma empty_spec' : forall x, (In x empty <-> False).
@@ -1689,148 +2235,2079 @@ Module Raw (Enc : ElementEncode).
     }
   Qed.
 
-  (** *** remove_list specification *)
+  (** *** union specification *)
 
-  Lemma remove_list_ok : forall l s, Ok s -> Ok (remove_list l s).
+  Lemma union_aux_flatten_alt_def : forall (s1 s2 : t) acc, 
+    union_aux s1 s2 acc =
+    match (s1, s2) with
+    | (nil, _) => List.rev_append acc s2
+    | (_, nil) => List.rev_append acc s1
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => union_aux l1 s2 ((y1, c1)::acc)
+          | ICR_before_touch => 
+              union_aux l1 (
+                insert_interval_begin y1 ((c1+c2)%N) l2) acc
+          | ICR_after        => union_aux s1 l2 ((y2, c2)::acc)
+          | ICR_after_touch  => union_aux l1 (
+              insert_interval_begin y2 ((c1+c2)%N) l2) acc
+          | ICR_overlap_before => 
+              union_aux l1 (
+                insert_interval_begin y1 
+                  (merge_interval_size y1 c1 y2 c2) l2) acc
+          | ICR_overlap_after => 
+              union_aux l1 (
+                insert_interval_begin y2 
+                  (merge_interval_size y2 c2 y1 c1) l2) acc
+          | ICR_equal => union_aux l1 s2 acc
+          | ICR_subsume_1 => union_aux l1 s2 acc
+          | ICR_subsume_2 => union_aux s1 l2 acc
+        end
+    end.
   Proof.
-    induction l as [| x l' IH]. {
-      done.
-    } {
-      move => s H_s_ok /=.
-      apply IH.
-      by apply remove_ok.
+    intros s1 s2 acc.    
+    case s1, s2 => //. 
+  Qed.
+
+  Lemma union_aux_alt_def : forall (s1 s2 : t) acc, 
+    union_aux s1 s2 acc =
+    List.rev_append acc (union s1 s2).
+  Proof.
+    rewrite /union.
+    intros s1 s2 acc.
+    move : acc s2.
+    induction s1 as [| [y1 c1] l1 IH1]. {
+      intros acc s2.
+      rewrite !union_aux_flatten_alt_def.
+      rewrite !rev_append_rev //.
     }
+    intros acc s2; move : acc.
+    induction s2 as [| [y2 c2] l2 IH2]; first by simpl.
+    move => acc.
+    rewrite !(union_aux_flatten_alt_def ((y1, c1)::l1) ((y2, c2)::l2)).
+    case (interval_compare (y1, c1) (y2, c2));
+      rewrite ?(IH1 ((y1, c1) :: acc)) ?(IH1 ((y1, c1) :: nil))
+              ?(IH2 ((y2, c2) :: acc)) ?(IH2 ((y2, c2) :: nil))
+              ?(IH1 acc) //.
   Qed.
 
-  Lemma remove_list_spec : forall x l s, Ok s -> 
-     (In x (remove_list l s) <-> ~(InA Enc.E.eq x l) /\ In x s).
+  Lemma union_alt_def : forall (s1 s2 : t), 
+    union s1 s2 =
+    match (s1, s2) with
+    | (nil, _) => s2
+    | (_, nil) => s1
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => (y1, c1) :: (union l1 s2)
+          | ICR_before_touch => 
+              union l1 (insert_interval_begin y1 ((c1+c2)%N) l2)
+          | ICR_after        => (y2, c2) :: union s1 l2
+          | ICR_after_touch  => union l1 
+              (insert_interval_begin y2 ((c1+c2)%N) l2)
+          | ICR_overlap_before => 
+              union l1 (insert_interval_begin y1 (merge_interval_size y1 c1 y2 c2) l2)
+          | ICR_overlap_after => 
+              union l1 (insert_interval_begin y2 (merge_interval_size y2 c2 y1 c1) l2)
+          | ICR_equal => union l1 s2 
+          | ICR_subsume_1 => union l1 s2 
+          | ICR_subsume_2 => union s1 l2
+        end
+     end.
   Proof.
-    move => x.
-    induction l as [| y l' IH]. {
-      intros s H.
-      rewrite /= InA_nil.
+    intros s1 s2.
+    rewrite /union union_aux_flatten_alt_def.
+    case s1 as [| [y1 c1] l1] => //.
+    case s2 as [| [y2 c2] l2] => //.
+    case (interval_compare (y1, c1) (y2, c2));
+      rewrite union_aux_alt_def //.
+  Qed.
+
+  Lemma union_InZ :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    forall y, (InZ y (union s1 s2) <-> InZ y s1 \/ InZ y s2).
+  Proof.
+    intro s1.
+    induction s1 as [| [y1 c1] l1 IH1]. {
+      intros s2 _ _ y.
+      rewrite union_alt_def /InZ /=.
       tauto.
     } {
-      move => s H_ok /=.
-      rewrite IH remove_spec InA_cons.
-      tauto.
+      induction s2 as [| [y2 c2] l2 IH2]. {
+        intros _ _ y.
+        rewrite union_alt_def /InZ /=.
+        tauto.
+      } {
+        move => H_inv_s1 H_inv_s2.
+        move : (H_inv_s1) (H_inv_s2).
+        rewrite !interval_list_invariant_cons.
+        move => [H_gr_l1] [H_c1_neq_0] H_inv_l1.
+        move => [H_gr_l2] [H_c2_neq_0] H_inv_l2.
+        move : (IH2 H_inv_s1 H_inv_l2) => {} IH2.
+        have :  forall s2 : t,
+          interval_list_invariant s2 = true ->
+          forall y : Z, InZ y (union l1 s2) <-> InZ y l1 \/ InZ y s2. {
+          intros. by apply IH1.
+        }
+        move => {} IH1 y.
+        rewrite union_alt_def.
+        move : (interval_compare_elim y1 c1 y2 c2).
+        case (interval_compare (y1, c1) (y2, c2)). {
+          rewrite !InZ_cons IH1 // InZ_cons.
+          tauto.
+        } {
+          move => H_y2_eq.
+          replace (c1 + c2)%N with (merge_interval_size y1 c1 y2 c2);
+            last first. {
+            rewrite -H_y2_eq merge_interval_size_add //.
+          }
+          set c'' := merge_interval_size y1 c1 y2 c2.
+          have [H_inv_insert H_InZ_insert] : 
+               interval_list_invariant (insert_interval_begin y1 c'' l2) = true /\
+                (InZ y (insert_interval_begin y1 c'' l2) <->
+                List.In y (elementsZ_single y1 c'') \/ InZ y l2). {
+            apply insert_interval_begin_spec => //. {
+              eapply interval_list_elements_greater_impl; eauto.
+              rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
+              apply Z_le_add_r.
+            } {
+              by apply merge_interval_size_neq_0.
+            } 
+          }
+  
+          rewrite IH1 => //.
+          rewrite H_InZ_insert !InZ_cons /c''.
+          rewrite -H_y2_eq In_merge_interval. {
+            tauto.
+          } {
+            apply Z_le_add_r.
+          } {
+            by apply Z.le_refl.
+          }
+        } {
+          move => [H_y1_lt] [H_y2_lt] H_y1_c1_lt.
+          set c'' := merge_interval_size y1 c1 y2 c2.
+          have [H_inv_insert H_InZ_insert] : 
+               interval_list_invariant (insert_interval_begin y1 c'' l2) = true /\
+                (InZ y (insert_interval_begin y1 c'' l2) <->
+                List.In y (elementsZ_single y1 c'') \/ InZ y l2). {
+            apply insert_interval_begin_spec => //. {
+              eapply interval_list_elements_greater_impl; eauto.
+              apply Z_lt_le_add_r => //.
+            } {
+              by apply merge_interval_size_neq_0.
+            } 
+          }
+          rewrite IH1 => //.
+          rewrite H_InZ_insert !InZ_cons /c''.
+          rewrite In_merge_interval. {
+            tauto.
+          } {
+            by apply Z.lt_le_incl.
+          } {
+            by apply Z.lt_le_incl.
+          }
+        } {
+          move => [H_y2_lt] [H_y1_lt] H_y2_c_lt.
+          set c'' := merge_interval_size y2 c2 y1 c1.
+          have [H_inv_insert H_InZ_insert] : 
+               interval_list_invariant (insert_interval_begin y2 c'' l2) = true /\
+                (InZ y (insert_interval_begin y2 c'' l2) <->
+                List.In y (elementsZ_single y2 c'') \/ InZ y l2). {
+            apply insert_interval_begin_spec => //. {
+              eapply interval_list_elements_greater_impl; eauto.
+              apply Z_le_add_r.
+            } {
+              by apply merge_interval_size_neq_0.
+            } 
+          }
+  
+          rewrite IH1 => //.
+          rewrite H_InZ_insert !InZ_cons /c''.
+          rewrite In_merge_interval. {
+            tauto.
+          } {
+            by apply Z.lt_le_incl.
+          } {
+            by apply Z.lt_le_incl.
+          }
+        } {
+          move => [? ?]; subst.
+          rewrite IH1 => //.
+          rewrite !InZ_cons. 
+          tauto.
+        } {
+          move => [H_y2_le] [H_y1_c1_le] _.
+          rewrite IH1 => //.
+          rewrite !InZ_cons. 
+          suff : (List.In y (elementsZ_single y1 c1) ->
+                  List.In y (elementsZ_single y2 c2)). {
+            tauto.
+          }
+          rewrite !In_elementsZ_single.
+          move => [H_y1_le H_y_lt].
+          split; lia.
+        } {
+          move => [H_y1_le] [H_y2_c2_le] _.
+          rewrite IH2.
+          rewrite !InZ_cons. 
+          suff : (List.In y (elementsZ_single y2 c2) ->
+                  List.In y (elementsZ_single y1 c1)). {
+            tauto.
+          }
+          rewrite !In_elementsZ_single.
+          move => [H_y2_le H_y_lt].
+          split; lia.
+        } {
+          rewrite !InZ_cons IH2 !InZ_cons.
+          tauto.
+        } {
+          move => H_y1_eq.
+          replace (c1 + c2)%N with (merge_interval_size y2 c2 y1 c1);
+            last first. {
+            rewrite -H_y1_eq merge_interval_size_add N.add_comm //.
+          }
+          set c'' := merge_interval_size y2 c2 y1 c1.
+          have [H_inv_insert H_InZ_insert] : 
+               interval_list_invariant (insert_interval_begin y2 c'' l2) = true /\
+                (InZ y (insert_interval_begin y2 c'' l2) <->
+                List.In y (elementsZ_single y2 c'') \/ InZ y l2). {
+            apply insert_interval_begin_spec => //. {
+              eapply interval_list_elements_greater_impl; eauto.
+              apply Z_le_add_r.
+            } {
+              by apply merge_interval_size_neq_0.
+            } 
+          }
+  
+          rewrite IH1 => //.
+          rewrite H_InZ_insert !InZ_cons /c''.
+          rewrite -H_y1_eq In_merge_interval. {
+            tauto.
+          } {
+            apply Z_le_add_r.
+          } {
+            by apply Z.le_refl.
+          }
+        }
+      }
     }
   Qed.
 
-  (** *** union specification *)
+  Lemma union_invariant :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    interval_list_invariant (union s1 s2) = true.
+  Proof.
+    intro s1.
+    induction s1 as [| [y1 c1] l1 IH1]. {
+      intros s2 _ H_inv_s2.
+      rewrite union_alt_def /InZ //.
+    } {
+      induction s2 as [| [y2 c2] l2 IH2]. {
+        intros H_inv_s1 _.
+        rewrite union_alt_def /InZ //.
+      } {
+        move => H_inv_s1 H_inv_s2.
+        move : (H_inv_s1) (H_inv_s2).
+        rewrite !interval_list_invariant_cons.
+        move => [H_gr_l1] [H_c1_neq_0] H_inv_l1.
+        move => [H_gr_l2] [H_c2_neq_0] H_inv_l2.
+        move : (IH2 H_inv_s1 H_inv_l2) => {} IH2.
+        have :  forall s2 : t,
+          interval_list_invariant s2 = true ->
+          interval_list_invariant (union l1 s2) = true. {
+          intros. by apply IH1.
+        }
+        move => {} IH1.
+
+        rewrite union_alt_def.
+        move : (interval_compare_elim y1 c1 y2 c2).
+        case (interval_compare (y1, c1) (y2, c2)). {
+          move => H_lt_y2.
+          have H_inv' : interval_list_invariant (union l1 ((y2, c2) :: l2)) = true. {
+            by apply IH1.
+          }
+
+          rewrite interval_list_invariant_cons.
+          repeat split => //. 
+          apply interval_list_elements_greater_intro => // x.
+          rewrite union_InZ => //.
+          move => []. {
+            apply interval_list_elements_greater_alt2_def => //.
+          } {
+            apply interval_list_elements_greater_alt2_def => //.
+            rewrite interval_list_elements_greater_cons //.
+          }
+        } {
+          move => H_y2_eq.
+          apply IH1.
+          apply insert_interval_begin_spec => //. {
+            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
+            rewrite -H_y2_eq -Z.add_assoc -N2Z.inj_add.
+            apply Z_le_add_r.
+          } {
+            rewrite N.eq_add_0.
+            tauto.
+          }
+        } {
+          move => [H_y1_lt] _.
+          apply IH1. 
+          apply insert_interval_begin_spec => //. {
+            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
+            apply Z_lt_le_add_r => //.
+          } {
+            apply merge_interval_size_neq_0 => //.
+          }
+        } {
+          move => [H_y2_lt] _.
+          apply IH1.
+          apply insert_interval_begin_spec => //. {
+            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
+            apply Z_le_add_r => //.
+          } {
+            apply merge_interval_size_neq_0 => //.
+          }
+        } {
+          move => [? ?]; subst.
+          apply IH1 => //. 
+        } {
+          move => _.
+          apply IH1 => //. 
+        } {
+          move => _.
+          apply IH2 => //. 
+        } {
+          move => H_lt_y1.
+          rewrite interval_list_invariant_cons => //.
+          repeat split => //.
+          apply interval_list_elements_greater_intro => // x.
+          rewrite union_InZ => //.
+          move => []. {
+            apply interval_list_elements_greater_alt2_def => //.
+            rewrite interval_list_elements_greater_cons //.
+          } {
+            apply interval_list_elements_greater_alt2_def => //.
+          }
+        } {
+          move => H_y1_eq.
+          apply IH1 => //.
+          apply insert_interval_begin_spec => //. {
+            eapply interval_list_elements_greater_impl; last apply H_gr_l2.
+            apply Z_le_add_r.
+          } {
+            rewrite N.eq_add_0.
+            tauto.
+          }
+        }
+      }
+    }
+  Qed.
 
-  Global Instance union_ok s s' : forall `(Ok s, Ok s'), Ok (union s s').
+ 
+  Global Instance union_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (union s1 s2).
   Proof.
-    move => H_ok H_ok'.
-    rewrite /union.
-    by apply add_list_ok.
+    move => H_ok_s1 H_ok_s2.
+    move : (H_ok_s1) (H_ok_s2).
+    rewrite /Ok /IsOk /is_encoded_elems_list /add.
+    move => [H_inv_s1] H_pre1.
+    move => [H_inv_s2] H_pre2.
+    split. {
+      apply union_invariant => //.
+    } {
+      intros y.
+      move : (union_InZ s1 s2 H_inv_s1 H_inv_s2).    
+      rewrite /InZ => ->.      
+      move => []. {
+        apply H_pre1.
+      } {
+        apply H_pre2.
+      }
+    }
   Qed.
-    
+
   Lemma union_spec :
    forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
    In x (union s s') <-> In x s \/ In x s'.
   Proof.
     intros s s' x H_ok H_ok'.
-    rewrite /union add_list_spec //.
+    rewrite !In_InZ.
+    rewrite union_InZ => //. {
+      apply H_ok.
+    } {
+      apply H_ok'.
+    }
   Qed.
 
 
-  (** *** filter specification *)
-
-  Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s).
-  Proof.
-    move => _.
-    rewrite /filter /from_elements.
-    apply add_list_ok, empty_ok.
-  Qed.
-
-  Lemma filter_spec :
-   forall (s : t) (x : elt) (f : elt -> bool),
-   Proper (Enc.E.eq==>eq) f ->
-   (In x (filter f s) <-> In x s /\ f x = true).
+  (** *** inter specification *)
+ 
+  Lemma inter_aux_alt_def :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      inter_aux y2 c2 acc s = match inter_aux y2 c2 nil s with
+                               (acc', s') => (acc' ++ acc, s')
+                             end.
   Proof.
-    move => s x f H_prop.
-    rewrite /filter /from_elements add_list_spec  empty_spec' /In !InA_alt.
-    setoid_rewrite filter_In.
-    split. {
-      move => [] //.
-      move => [y] [H_y_eq] [H_y_in] H_fy.
-      split; first by exists y.
-      rewrite -H_fy.
-      by f_equiv.
+    intros y2 c2.
+    induction s as [| [y1 c1] s' IH] => acc. {
+      rewrite /inter_aux app_nil_l //.
     } {
-      move => [] [y] [H_y_eq] H_y_in H_fx.
-      left. 
-      exists y.
-      split; last split; try assumption.
-      apply Logic.eq_sym.
-      rewrite -H_fx.
-      by f_equiv.
+      simpl.
+      case_eq (inter_aux y2 c2 nil s') => acc'' s'' H_eq.
+      case (interval_compare (y1, c1) (y2, c2));
+        rewrite ?(IH acc)
+                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: acc))
+                ?(IH ((y2, Z.to_N (y1 + Z.of_N c1 - y2)) :: nil))
+                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: acc))
+                ?(IH ((y1, Z.to_N (y2 + Z.of_N c2 - y1)) :: nil))
+                ?(IH ((y1, c1) :: acc))
+                ?(IH ((y1, c1) :: nil))
+                
+                ?H_eq -?app_assoc -?app_comm_cons //.     
     }
   Qed.
 
+  Lemma inter_aux_props :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      interval_list_invariant (rev acc) = true ->
+      interval_list_invariant s = true ->
+      (forall x1 x2, InZ x1 acc -> InZ x2 s ->
+                     List.In x2 (elementsZ_single y2 c2) ->
+                     Z.succ x1 < x2) ->
+      (c2 <> 0%N) ->
+      match (inter_aux y2 c2 acc s) with (acc', s') =>
+        (forall y, (InZ y acc' <->
+                   (InZ y acc \/ (InZ y s /\ (List.In y (elementsZ_single y2 c2)))))) /\
+        (forall y, InZ y s' -> InZ y s) /\
+        (forall y, InZ y s -> y2 + Z.of_N c2 < y -> InZ y s') /\
+        interval_list_invariant (rev acc') = true /\
+        interval_list_invariant s' = true
+      end.
+  Proof.
+    intros y2 c2.     
+    induction s as [| [y1 c1] s1' IH] => acc. {
+      rewrite /inter_aux.       
+      move => H_inv_acc _ _ _.
+      split; last split; try done.
+      move => y. rewrite InZ_nil.
+      tauto.
+    } {
+      rewrite interval_list_invariant_cons.       
+      move => H_inv_acc [H_gr_s1'] [H_c1_neq_0] H_inv_s1'.
+      move => H_in_acc_lt H_c2_neq_0.
+       
+      rewrite inter_aux_alt_def.
+      case_eq (inter_aux y2 c2 nil ((y1, c1) :: s1')).
+      move => acc' s' H_inter_aux_eq.
+
+      set P1 := forall y : Z,
+        (InZ y acc' <->
+        ((InZ y ((y1, c1) :: s1') /\ List.In y (elementsZ_single y2 c2)))).
+      set P2 := (forall y,
+                  (InZ y s' -> InZ y ((y1, c1) :: s1')) /\
+                  (InZ y ((y1, c1) :: s1') ->
+                     y2 + Z.of_N c2 < y -> InZ y s')).
+      set P3 := interval_list_invariant (rev acc') = true.
+      set P4 := interval_list_invariant s' = true.
+
+      suff : (P1 /\ P2 /\ P3 /\ P4). {
+        move => [H_P1] [H_P2] [H_P3] H_P4.
+        split; last split; last split; last split. {
+          move => y.
+          move : (H_P1 y).
+          rewrite !InZ_app InZ_cons !In_elementsZ_single.
+          move => <-.
+          tauto.
+        } {
+          move => y H_y_in.
+          by apply H_P2.
+        } {
+          move => y H_y_in.
+          by apply H_P2.
+        } {
+          rewrite rev_app_distr.
+          apply interval_list_invariant_app_intro => //.
+          move => x1 x2.
+          rewrite !InZ_rev.
+          move => H_x1_in /H_P1 [H_x2_in1] H_x2_in2.
+          apply H_in_acc_lt => //.
+        } {
+          apply H_P4.
+        }
+      }
+
+      move : (H_gr_s1').
+      rewrite interval_list_elements_greater_alt2_def => // => H_gr_s1'_alt.
+
+      have : forall (acc : list (Z * N)),
+        interval_list_invariant (rev acc) = true ->
+        (forall y, InZ y acc <-> (
+           y1 <= y < y1 + Z.of_N c1 /\
+           y2 <= y < y2 + Z.of_N c2)) ->
+        (y1 + Z.of_N c1 <= y2 + Z.of_N c2) ->        
+        (inter_aux y2 c2 acc s1' = (acc', s')) ->
+        P1 /\ P2 /\ P3 /\ P4. {
+        
+        intros acc0 H_inv_acc0 H_in_acc0 H_y2c_lt H_inter_aux_eq0.
+        have H_in_acc0_lt : (forall x1 x2 : Z,
+          InZ x1 acc0 ->
+          InZ x2 s1' ->
+          List.In x2 (elementsZ_single y2 c2) ->
+          Z.succ x1 < x2). {
+
+          intros x1 x2 H_x1_in_acc0 H_x2_in_s1' H_x2_in_yc2.
+
+          suff H_yc1_lt_x2 : Z.succ x1 <= y1 + Z.of_N c1. {
+            apply Z.le_lt_trans with (m := y1 + Z.of_N c1) => //.
+            by apply H_gr_s1'_alt.
+          }
+          move : (H_x1_in_acc0).
+          rewrite H_in_acc0 Z.le_succ_l.
+          tauto.
+        }          
+        
+        move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_acc0_lt H_c2_neq_0).
+        rewrite H_inter_aux_eq0.
+        move => [H1] [H2] [H3] [H4] H5.
+        split; last split => //. {
+          move => y.
+          rewrite (H1 y).
+          rewrite InZ_cons !In_elementsZ_single
+                  H_in_acc0.
+          tauto.
+        } {
+          move => y.
+          split. {
+            move => /H2.
+            rewrite InZ_cons.
+            by right.
+          } {
+            rewrite InZ_cons In_elementsZ_single.
+            move => []. {
+              move => [_] H_y_lt H_lt_y.
+              exfalso.
+              suff : (y < y) by apply Z.lt_irrefl.
+              apply Z.lt_le_trans with (m := y1 + Z.of_N c1) => //.
+              apply Z.le_trans with (m := y2 + Z.of_N c2) => //.
+
+              by apply Z.lt_le_incl.
+            } {
+              apply H3.
+            }
+          }
+        } 
+      }
+      move => {} IH.
+
+      clear H_inv_acc H_in_acc_lt acc.
+      move : (interval_compare_elim y1 c1 y2 c2) H_inter_aux_eq.
+      unfold inter_aux.
+      case_eq (interval_compare (y1, c1) (y2, c2)) => H_comp;
+         fold inter_aux. {
+        move => H_lt_y2.
+        apply IH. {
+          done.
+        } {
+          move => x.
+          rewrite InZ_nil.
+          split => //.
+          lia.
+        } {
+          apply Z.le_trans with (m := y2). {
+            by apply Z.lt_le_incl.
+          } {
+            apply Z_le_add_r.
+          }
+        }          
+      } {
+        move => H_eq_y2. 
+        apply IH. {
+          done.
+        } {
+          move => x.
+          rewrite InZ_nil.
+          split => //.
+          lia.
+        } {
+          rewrite H_eq_y2.
+          apply Z_le_add_r.
+        }          
+      } {
+        move => [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2.
+        apply IH. {
+          rewrite interval_list_invariant_sing.
+          by apply Z_to_N_minus_neq_0.
+        } {
+          move => x.
+          rewrite InZ_cons InZ_nil In_elementsZ_single Z2N.id; last lia.
+          replace (y1 + (y2 - y1)) with y2 by lia.
+          split; lia.
+        } {
+          by apply Z.lt_le_incl.
+        }
+      } {
+        rewrite /P1 /P2 /P3 /P4.
+        move => [H_y2_lt] [H_y1_lt] H_yc1_lt.
+        move => [] H_acc' H_s'.
+        clear IH P1 P2 P3 P4 H_comp.
+        subst s' acc'.
+        split; last split; last split. {          
+          move => y.
+          have H_yc2_intro : y1 + Z.of_N (Z.to_N (y2 + Z.of_N c2 - y1)) =
+                             y2 + Z.of_N c2. {
+            rewrite Z2N.id; lia.
+          }       
+          
+          rewrite !InZ_cons !In_elementsZ_single InZ_nil H_yc2_intro.
+          split. {
+            move => [] //.
+            move => [H_y1_le] H_y_lt.
+            split; last by lia.
+            left. lia.
+          } {
+            move => [H_in_s] [H_y2_le] H_y_lt.
+            left.
+            split; last assumption.
+            move : H_in_s => []. {
+              tauto.
+            } {              
+              move => /H_gr_s1'_alt H_lt_y.
+              apply Z.le_trans with (m := y1 + Z.of_N c1). {
+                by apply Z_le_add_r.
+              } {
+                by apply Z.lt_le_incl.
+              }
+            }
+          }
+        } {
+          move => y.
+          split; done.
+        } {
+          rewrite interval_list_invariant_sing.
+          by apply Z_to_N_minus_neq_0.
+        } {
+          by rewrite interval_list_invariant_cons.
+        }
+      } {
+        rewrite /P1 /P2 /P3 /P4.
+        move => [H_y12_eq] H_c12_eq [] H_acc' H_s'.
+        clear IH P1 P2 P3 P4 H_comp.
+        subst.
+        split; last split; last split. {
+          move => y.
+          rewrite !InZ_cons InZ_nil In_elementsZ_single.
+          split; last by tauto. {         
+            move => [] //.
+            tauto.
+          }
+        } {
+          move => y.
+          rewrite InZ_cons In_elementsZ_single.
+          split; first by right.
+          move => [] //.
+          move => [_] H_y_lt H_lt_y.
+          exfalso.
+          suff : (y2 + Z.of_N c2 < y2 + Z.of_N c2) by
+              apply Z.lt_irrefl.
+          apply Z.lt_trans with (m := y) => //.
+        } {
+          rewrite interval_list_invariant_sing //.          
+        } {
+          assumption.
+        }        
+      } {
+        move => [H_y2_le_y1] [H_yc1_le_yc2] _.
+        apply IH. {
+          by rewrite interval_list_invariant_sing.
+        } {
+          move => y.
+          rewrite InZ_cons InZ_nil In_elementsZ_single.
+          split. {
+            move => [] //.
+            move => [H_y1_le] H_y_lt.
+            split; first done.
+            split; lia.
+          } {
+            move => [?] _.
+            by left.
+          }
+        } {
+          assumption.
+        }          
+      } {
+        rewrite /P1 /P2 /P3 /P4.
+        move => [H_y1_le] [H_yc2_le] _.
+        move => [] H_acc' H_s'.
+        clear IH P1 P2 P3 P4 H_comp.
+        subst.
+        split; last split; last split. {          
+          move => y.
+          rewrite !InZ_cons !In_elementsZ_single InZ_nil.
+          split. {
+            move => [] //.
+            move => [H_y2_le] H_y_lt.
+            split; last by lia.
+            left. lia.
+          } {
+            move => [H_in_s] [H_y2_le] H_y_lt.
+            by left.
+          }
+        } {
+          tauto.
+        } {
+          by rewrite interval_list_invariant_sing.
+        } {
+          by rewrite interval_list_invariant_cons.
+        }
+      } {
+        rewrite /P1 /P2 /P3 /P4.
+        move => H_yc2_lt [] H_acc' H_s'.
+        clear IH P1 P2 P3 P4 H_comp.
+        subst.
+        split; last split; last split. {
+          move => y.
+          rewrite InZ_cons !In_elementsZ_single InZ_nil.
+          split; first done.
+          move => [] []. {
+            move => [H_y1_le_y] H_y_lt_yc1.
+            move => [H_y2_le_y] H_y_lt_yc2.
+            lia.
+          } {
+            move => /H_gr_s1'_alt H_lt_y [_] H_y_lt.
+            suff : (y1 + Z.of_N c1 < y1). {
+              apply Z.nlt_ge.
+              apply Z_le_add_r.
+            }
+            lia.
+          }
+        } {
+          tauto.
+        } {
+          done.
+        } {
+          by rewrite interval_list_invariant_cons.
+        }         
+      } {
+        rewrite /P1 /P2 /P3 /P4.
+        move => H_y1_eq [] H_acc' H_s'.
+        clear IH P1 P2 P3 P4 H_comp.
+        subst acc' s'.
+        split; last split; last split. {
+          move => y.
+          rewrite InZ_cons !In_elementsZ_single InZ_nil.
+          split; first done.
+          move => [] []. {
+            move => [H_y1_le_y] H_y_lt_yc1.
+            move => [H_y2_le_y] H_y_lt_yc2.
+            lia.
+          } {
+            move => /H_gr_s1'_alt H_lt_y [_] H_y_lt.
+            suff : (y1 + Z.of_N c1 < y1). {
+              apply Z.nlt_ge.
+              apply Z_le_add_r.
+            }
+            lia.
+          }
+        } {
+          tauto.
+        } {
+          done.
+        } {
+          by rewrite interval_list_invariant_cons.
+        }
+      }
+    }
+  Qed.
 
-  (** *** inter specification *)
-
-  Global Instance inter_ok s s' : forall `(Ok s, Ok s'), Ok (inter s s').
+  Lemma inter_aux2_props :
+   forall (s2 s1 acc : t),
+    interval_list_invariant (rev acc) = true ->
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    (forall x1 x2, InZ x1 acc -> InZ x2 s1 -> InZ x2 s2 -> Z.succ x1 < x2) ->
+    ((forall y, (InZ y (inter_aux2 acc s1 s2) <->
+               (InZ y acc) \/ ((InZ y s1) /\ InZ y s2))) /\
+    (interval_list_invariant (inter_aux2 acc s1 s2) = true)).
   Proof.
-    intros H_ok H_ok'.
+    induction s2 as [| [y2 c2] s2' IH]. {
+      move => s1 acc.
+      move => H_inv_acc _ _ _.
+      unfold inter_aux2.
+      replace (match s1 with
+        | nil => rev' acc
+        | _ :: _ => rev' acc
+               end) with (rev' acc); last by case s1.
+      rewrite /rev' rev_append_rev app_nil_r.
+      split; last done.
+      move => y.
+      rewrite InZ_nil InZ_rev.
+      tauto.
+    } {
+      intros s1 acc H_inv_acc H_inv_s1.
+      rewrite interval_list_invariant_cons.
+      move => [H_gr_s2'] [H_c2_neq_0] H_inv_s2'.
+      move => H_acc_s12.
+
+      move : H_gr_s2'.
+      rewrite interval_list_elements_greater_alt2_def //.
+      move => H_gr_s2'.
+      
+      rewrite /inter_aux2; fold inter_aux2.
+      case_eq s1. {
+        move => H_s1_eq.
+        split. {
+          move => y.
+          rewrite /rev' rev_append_rev app_nil_r InZ_nil
+                  InZ_rev.
+          tauto.
+        } {
+          rewrite /rev' rev_append_rev app_nil_r //.
+        }
+      } {
+        move => [_ _] _ <-.
+        case_eq (inter_aux y2 c2 acc s1).
+        move => acc' s1' H_inter_aux_eq.
+
+        have H_acc_s1_yc2 : forall x1 x2 : Z,
+          InZ x1 acc ->
+          InZ x2 s1 ->
+          List.In x2 (elementsZ_single y2 c2) ->
+          Z.succ x1 < x2. {
+          intros x1 x2 H_x1_in H_x2_in1 H_x2_in2.
+          apply H_acc_s12 => //.
+          rewrite InZ_cons; by left.
+        }
+
+        move : (inter_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1_yc2 H_c2_neq_0).
+        rewrite H_inter_aux_eq.
+        move => [H_in_acc'] [H_in_s1'_elim] [H_in_s1'_intro]
+                [H_inv_acc'] H_inv_s1'.
+
+        have H_in_acc'_s2' : forall x1 x2 : Z,
+            InZ x1 acc' -> InZ x2 s1' -> InZ x2 s2' -> Z.succ x1 < x2. {
+          move => x1 x2 /H_in_acc'.
+          move => []. {
+            move => H_in_acc H_in_s1' H_in_s2'.
+            apply H_acc_s12 => //. {
+              by apply H_in_s1'_elim.
+            } {
+              rewrite InZ_cons; by right.
+            }
+          } {
+            rewrite In_elementsZ_single.
+            move => [H_in_s1] [_] H_x1_lt _.
+            move => /H_gr_s2' H_lt_x2.
+            apply Z.le_lt_trans with (m := y2 + Z.of_N c2) => //.
+            by apply Z.le_succ_l.
+          }
+        }
+       
+        move : (IH s1' acc' H_inv_acc' H_inv_s1' H_inv_s2' H_in_acc'_s2').
+        move => [H_inZ_res] H_inv_res.
+        split; last assumption.
+        move => y.
+        rewrite H_inZ_res H_in_acc' InZ_cons
+                In_elementsZ_single.
+        split. {
+          move => []; first by tauto.
+          move => [H_y_in_s1' H_y_in_s2'].
+          right.
+          split; last by right.
+          by apply H_in_s1'_elim.
+        } {
+          move => []. {
+            move => H_y_in_acc.
+            by left; left.
+          } {
+            move => [H_y_in_s1].
+            move => []. {
+              move => H_in_yc2.
+              by left; right.
+            } {
+              right.
+              split; last assumption.
+              apply H_in_s1'_intro => //.
+              by apply H_gr_s2'.
+            }
+          }
+        }
+      }
+    }
+  Qed.
+ 
+  Lemma inter_InZ :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    forall y, (InZ y (inter s1 s2) <-> InZ y s1 /\ InZ y s2).
+  Proof.
+    intros s1 s2 H_inv_s1 H_inv_s2 y.
     rewrite /inter.
-    by apply filter_ok.
+
+    move : (inter_aux2_props s2 s1 nil).
+    move => [] //.
+    move => H_in_inter _.
+    rewrite H_in_inter InZ_nil.
+    tauto.
+  Qed.
+
+  Lemma inter_invariant :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    interval_list_invariant (inter s1 s2) = true.
+  Proof.
+    intros s1 s2 H_inv_s1 H_inv_s2.
+    rewrite /inter.
+
+    move : (inter_aux2_props s2 s1 nil).
+    move => [] //.
+  Qed.
+  
+ 
+  Global Instance inter_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (inter s1 s2).
+  Proof.
+    move => H_ok_s1 H_ok_s2.
+    move : (H_ok_s1) (H_ok_s2).
+    rewrite /Ok /IsOk /is_encoded_elems_list /add.
+    move => [H_inv_s1] H_pre1.
+    move => [H_inv_s2] H_pre2.
+    split. {
+      apply inter_invariant => //.
+    } {
+      intros y.
+      move : (inter_InZ s1 s2 H_inv_s1 H_inv_s2).    
+      rewrite /InZ => ->.      
+      move => [].
+      move => /H_pre1 //.
+    }
+  Qed.
+
+  Lemma inter_spec :
+   forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
+   In x (inter s s') <-> In x s /\ In x s'.
+  Proof.
+    intros s s' x H_ok H_ok'.
+    rewrite !In_InZ.
+    rewrite inter_InZ => //. {
+      apply H_ok.
+    } {
+      apply H_ok'.
+    }
+  Qed.
+
+  
+  (** *** diff specification *)
+ 
+  Lemma diff_aux_alt_def :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      diff_aux y2 c2 acc s = match diff_aux y2 c2 nil s with
+                               (acc', s') => (acc' ++ acc, s')
+                             end.
+  Proof.
+    intros y2 c2.
+    induction s as [| [y1 c1] acc' IH] => acc. {
+      rewrite /diff_aux app_nil_l //.
+    } {
+      simpl.
+      case_eq (diff_aux y2 c2 nil acc') => acc'' s'' H_eq.
+      case (interval_compare (y1, c1) (y2, c2));
+        rewrite ?(IH ((y1, c1)::acc)) ?(IH ((y1, c1)::nil))
+                ?(IH acc) ?(IH ((y1, Z.to_N (y2 - y1)) :: acc))
+                ?(IH ((y1, Z.to_N (y2 - y1)) :: nil)) ?H_eq;
+        rewrite ?insert_intervalZ_guarded_app -?app_assoc -?app_comm_cons //.
+    }
+  Qed.
+
+  
+  Lemma diff_aux_props :
+    forall (y2 : Z) (c2 : N) (s : t) acc,
+      interval_list_invariant (List.rev acc) = true ->
+      interval_list_invariant s = true ->
+      (forall x1 x2, InZ x1 acc -> InZ x2 s -> Z.succ x1 < x2) ->
+      (forall x, InZ x acc -> x < y2) ->
+      (c2 <> 0%N) ->
+      match (diff_aux y2 c2 acc s) with
+        (acc', s') => (forall y, InZ y (List.rev_append acc' s') <->
+                                 InZ y (List.rev_append acc s) /\ ~(List.In y (elementsZ_single y2 c2))) /\
+                      (interval_list_invariant (List.rev_append acc' s') = true) /\
+                      (forall x, InZ x acc' -> x < y2 + Z.of_N c2)
+      end.
+  Proof.
+    intros y2 c2.     
+    induction s as [| [y1 c1] s1' IH] => acc. {
+      rewrite /diff_aux -rev_alt.       
+      move => H_inv_acc _ _ H_in_acc H_c2_neq.
+      split; last split. {
+        move => y; split; last by move => [] //.
+        rewrite InZ_rev.         
+        move => H_in. split => //.
+        move => /In_elementsZ_single => [] [] /Z.nlt_ge H_neq.
+        contradict H_neq.
+        by apply H_in_acc.
+      } {
+        assumption.
+      } {
+        intros x H_in_acc'.
+        apply Z.lt_le_trans with (m := y2). {
+          by apply H_in_acc.
+        } {
+          by apply Z_le_add_r.
+        }
+      }
+    } {
+      rewrite interval_list_invariant_cons.       
+      move => H_inv_acc [H_gr_s1'] [H_c1_neq_0] H_inv_s1'.
+      move => H_in_s1 H_in_acc H_c2_neq_0.
+       
+      rewrite diff_aux_alt_def.
+      case_eq (diff_aux y2 c2 nil ((y1, c1) :: s1')).
+      move => acc' s' H_diff_aux_eq.
+
+      set P1 := forall y : Z,
+        (InZ y acc' \/ InZ y s') <->
+        InZ y ((y1, c1) :: s1') /\ ~ List.In y (elementsZ_single y2 c2).
+      set P2 := interval_list_invariant (rev acc' ++ s') = true.
+      set P3 := forall x : Z, InZ x acc' -> (x < y2 + Z.of_N c2).
+
+      suff : (P1 /\ P2 /\ P3). {
+        move => [H_P1] [H_P2] H_P3.
+        split; last split. {
+          move => y.
+          move : (H_P1 y).
+          rewrite !rev_append_rev rev_app_distr !InZ_app
+                  !InZ_rev In_elementsZ_single.
+          suff : (InZ y acc -> ~ y2 <= y < y2 + Z.of_N c2). {
+            tauto.
+          }
+          move => /H_in_acc H_y_lt [H_y_ge] _.
+          contradict H_y_ge.
+          by apply Zlt_not_le.
+        } {
+          rewrite rev_append_rev rev_app_distr -app_assoc.
+          apply interval_list_invariant_app_intro => //.
+          move => x1 x2.
+          rewrite InZ_app !InZ_rev.
+          move => H_in_acc' H_x2_in_s'.
+          suff : (InZ x2 ((y1, c1)::s1')). {
+            by apply H_in_s1.
+          }
+          move : (H_P1 x2).
+          tauto.
+        } {
+          move => x.
+          rewrite InZ_app.
+          move => []. {
+            apply H_P3.
+          } {
+            move => /H_in_acc H_x_lt.
+            eapply Z.lt_trans; eauto.
+            by apply Z_lt_add_r.
+          }
+        }
+      }
+
+      move : (H_gr_s1').
+      rewrite interval_list_elements_greater_alt2_def => // => H_gr_s1'_alt.
+
+      have : forall (acc : list (Z * N)),
+        interval_list_invariant (rev acc) = true ->
+        (forall x : Z,
+            InZ x acc <->
+            ((y1 <= x < y1 + Z.of_N c1) /\ (x < y2))) ->
+        (y1 + Z.of_N c1 <= y2 + Z.of_N c2) ->
+        (diff_aux y2 c2 acc s1' = (acc', s')) ->
+        P1 /\ P2 /\ P3. {
+        
+        intros acc0 H_inv_acc0 H_in_acc0 H_c1_before H_diff_aux_eq0.
+        have H_in_s1' : (forall x1 x2 : Z,
+                            InZ x1 acc0 -> InZ x2 s1' -> Z.succ x1 < x2). {
+          intros x1 x2 H_x1_in_acc0.
+          move => /H_gr_s1'_alt.
+
+          eapply Z.le_lt_trans.
+          move : H_x1_in_acc0.
+          rewrite Z.le_succ_l H_in_acc0.
+          tauto.
+        }
+        have H_in_acc0' : (forall x : Z, InZ x acc0 -> x < y2). {
+          move => x.
+          rewrite H_in_acc0.
+          move => [_] //.
+        }
+        move : (IH acc0 H_inv_acc0 H_inv_s1' H_in_s1' H_in_acc0' H_c2_neq_0).
+        rewrite H_diff_aux_eq0 !rev_append_rev.
+        move => [H1] [H2] H3.
+        split; last split => //. {
+          move => y.
+          move : (H1 y).
+          rewrite !InZ_app !InZ_rev In_elementsZ_single.
+          move => ->.
+          rewrite InZ_cons In_elementsZ_single.
+          split. {
+            rewrite H_in_acc0 -(Z.nle_gt y2 y).
+            tauto.
+          } {
+            rewrite H_in_acc0 -(Z.nle_gt y2 y).
+            move => [] H_in H_nin_i2.
+            split; last by assumption.
+            move : H_in => [] H_in; last by right.
+            left.
+            lia.
+          }
+        }
+      }
+      move => {} IH.
+       
+      clear H_inv_acc H_in_s1 H_in_acc acc.
+      move : (interval_compare_elim y1 c1 y2 c2) H_diff_aux_eq.
+      unfold diff_aux.
+      case_eq (interval_compare (y1, c1) (y2, c2)) => H_comp;
+                                                        fold diff_aux. {
+        move => H_lt_y2.
+        apply IH. {
+          by rewrite interval_list_invariant_sing.
+        } {
+          move => x.
+          rewrite InZ_cons In_elementsZ_single.
+          split; last by tauto.
+          move => []; last done.
+          move => [H_y1_le H_x_lt].         
+          split; first done.       
+          eapply Z.lt_trans; eauto.
+        } {
+          apply Z.le_trans with (m := y2). 
+            - by apply Z.lt_le_incl.
+            - by apply Z_le_add_r.
+        }
+      } {
+        move => H_eq_y2. 
+        apply IH. {
+          by rewrite interval_list_invariant_sing.
+        } {
+          move => x.
+          rewrite InZ_cons In_elementsZ_single -H_eq_y2.
+          split; last by tauto.
+          move => []; last done.
+          move => []. done.
+        } {
+          rewrite H_eq_y2.
+          by apply Z_le_add_r.
+        }
+      } {
+        move => [H_y1_lt_y2] [H_y2_lt_yc1] H_yc1_lt_yc2.
+        apply IH. {
+          rewrite interval_list_invariant_sing.
+          by apply Z_to_N_minus_neq_0.
+        } {
+          move => x.
+          rewrite InZ_cons In_elementsZ_single Z2N.id; last lia.
+          replace (y1 + (y2 - y1)) with y2 by lia.
+          split; last tauto.
+          move => [] //.
+          move => [H_y1_le] H_x_lt.
+          repeat split => //.
+          apply Z.lt_trans with (m := y2) => //.
+        } {
+          by apply Z.lt_le_incl.
+        }
+      } {
+        rewrite /P1 /P2 /P3.
+        move => [H_y2_lt] [H_y1_lt] H_yc1_lt [H_acc'] H_s'.
+        clear IH P1 P2 P3 H_comp.
+        subst.
+        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
+          rewrite Z2N.id; lia.
+        }
+        have H_nin_yc2 : forall y,
+            InZ y s1' -> ~ y2 <= y < y2 + Z.of_N c2. {
+          move => y /H_gr_s1'_alt H_lt_y.
+          move => [H_y2_le_y].
+          apply Z.le_ngt, Z.lt_le_incl.           
+          by apply Z.lt_trans with (m := y1 + Z.of_N c1).
+        }
+        split; last split. {
+          move => y.
+          rewrite !InZ_cons !In_elementsZ_single H_yc1_intro.
+          split. {             
+            move => [] //.
+            move => []. {
+              move => [H_le_y] H_y_lt.
+              split. {
+                left; lia.
+              } {
+                move => [_].
+                by apply Z.nlt_ge.
+              }
+            } {
+              move : (H_nin_yc2 y). tauto.
+            }
+          } {
+            move => [] []; last by right; right.
+            move => [H_y_ge] H_y_lt_yc1 H_nin_yc2'.
+            right; left. lia.
+          }
+        } {
+          rewrite interval_list_invariant_cons H_yc1_intro.
+          split => //.
+          split => //.
+          by apply Z_to_N_minus_neq_0.
+        } {
+          move => [] //.
+        }
+      } {
+        rewrite /P1 /P2 /P3.
+        move => [H_y12_eq] H_c12_eq [] H_acc' H_s'.
+        clear IH P1 P2 P3 H_comp.
+        subst.
+        split; last split. {
+          move => y.
+          rewrite InZ_cons In_elementsZ_single.
+          split; last by tauto. {         
+            move => [] //.
+            move => H_y_in.
+            split; first by right.
+            move => [] _.
+            by apply Z.nlt_ge, Z.lt_le_incl, H_gr_s1'_alt.
+          }
+        } {
+          apply H_inv_s1'.
+        } {
+          move => x [] //.
+        }
+      } {
+        move => [H_y2_le_y1] [H_yc1_le_yc2] _.
+        apply IH. {
+          done.
+        } {
+          move => x.
+          split; first done.
+          lia.
+        } {
+          assumption.
+        }
+      } {
+        rewrite /P1 /P2 /P3.
+        move => [H_y1_le] [H_yc2_le_yc1] _ [] H_acc' H_s'.
+        clear IH P1 P2 P3 H_comp.
+        subst.
+        have H_yc1_intro : y2 + Z.of_N c2 + Z.of_N (Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2))) = y1 + Z.of_N c1. {
+          rewrite Z2N.id; lia.
+        }       
+        have H_y1_intro : y1 + Z.of_N (Z.to_N (y2 - y1)) = y2. {
+          rewrite Z2N.id; lia.
+        }              
+        split; last split. {
+          move => y.
+          rewrite !InZ_insert_intervalZ_guarded
+                  !InZ_cons !In_elementsZ_single
+                  H_yc1_intro H_y1_intro InZ_nil.
+          split. {
+            rewrite -!or_assoc.
+            move => [[[]|]|] //. {
+              move => [H_y1_le_y] H_y_lt.
+              split. {
+                left.
+                split => //.
+                apply Z.lt_le_trans with (m := y2 + Z.of_N c2) => //.
+                apply Z.lt_trans with (m := y2) => //.
+                by apply Z_lt_add_r.
+              } {
+                move => [] /Z.le_ngt //.
+              }
+            } {
+              move => [H_y2c_le_y] H_y_lt_yc1.
+              split. {
+                left.
+                split => //.
+                apply Z.le_trans with (m := y2 + Z.of_N c2) => //.
+                apply Z.le_trans with (m := y2) => //.
+                apply Z_le_add_r.
+              } {
+                move => [] _ /Z.lt_nge //.
+              }
+            } {
+              move => H_y_in_s1'.
+              split; first by right.
+              suff H_suff : y2 + Z.of_N c2 <= y. {
+                move => [] _ /Z.lt_nge //.
+              }
+              apply Z.le_trans with (m := y1 + Z.of_N c1) => //.
+              apply Z.lt_le_incl.
+              by apply H_gr_s1'_alt.
+            }
+          } {
+            move => [] []; last by tauto.
+            move => [H_y1_le_y] H_y_lt H_neq_y2.
+            apply not_and in H_neq_y2; last by apply Z.le_decidable.          
+            case H_neq_y2. {
+              move => /Z.nle_gt H_y_lt'.
+              left; left; done.
+            } {
+              move => /Z.nlt_ge H_le_y.
+              right; left; done.               
+            }
+          }
+        } {
+          rewrite insert_intervalZ_guarded_rev_nil_app.
+          rewrite !interval_list_invariant_insert_intervalZ_guarded => //. {
+            rewrite H_yc1_intro => //.
+          } {
+            rewrite /insert_intervalZ_guarded.
+            case_eq ((Z.to_N (y1 + Z.of_N c1 - (y2 + Z.of_N c2)) =? 0)%N). {
+              rewrite H_y1_intro.
+              move => /N.eqb_eq /N2Z.inj_iff.
+              rewrite Z2N.id; last first. {
+                by apply Z.le_0_sub.
+              }
+              move => /Zminus_eq H_yc1_eq.
+              eapply interval_list_elements_greater_impl;
+                last apply H_gr_s1'.
+              rewrite H_yc1_eq.
+              apply Z_le_add_r.
+            } {
+              move => _.
+              rewrite interval_list_elements_greater_cons
+                      H_y1_intro.
+              by apply Z_lt_add_r.
+            }
+          }
+        } {
+          move => x.
+          rewrite InZ_insert_intervalZ_guarded InZ_cons In_elementsZ_single H_y1_intro InZ_nil.
+          move => [] //.
+          move => [_] H_x_lt.
+          apply Z.lt_le_trans with (m := y2) => //.
+          apply Z_le_add_r.
+        }
+      } {
+        rewrite /P1 /P2 /P3.
+        move => H_yc2_lt [] H_acc' H_s'.
+        clear IH P1 P2 P3 H_comp.
+        subst.
+        split; last split. {
+          move => y.
+          rewrite InZ_cons In_elementsZ_single.
+          split; last by tauto. {         
+            move => [] //.
+            move => H_y_in.
+            split; first assumption.
+            rewrite In_elementsZ_single.
+            move => [] H_y2_le H_y_lt.
+            case H_y_in; first by lia.
+            move => /H_gr_s1'_alt H_lt_y.
+            suff : y2 + Z.of_N c2 < y. {
+              move => ?. lia.
+            }
+            apply Z.lt_trans with (m := y1 + Z.of_N c1) => //.
+            apply Z.lt_le_trans with (m := y1) => //.
+            apply Z_le_add_r.
+          }
+        } {
+          by rewrite interval_list_invariant_cons.
+        } {
+          done.
+        }
+      } {
+        rewrite /P1 /P2 /P3.
+        move => H_yc2_eq [] H_acc' H_s'.
+        clear IH P1 P2 P3 H_comp.
+        subst.
+        split; last split. {
+          move => y.
+          rewrite InZ_cons In_elementsZ_single.
+          split; last by tauto. {         
+            move => [] //.
+            move => H_y_in.
+            split; first assumption.
+            rewrite In_elementsZ_single.
+            move => [] H_y2_le H_y_lt.
+            case H_y_in; first by lia.
+            move => /H_gr_s1'_alt H_lt_y.
+            suff : y2 + Z.of_N c2 < y. {
+              move => ?. lia.
+            }
+            apply Z.lt_trans with (m := (y2 + Z.of_N c2) + Z.of_N c1) => //. 
+            by apply Z_lt_add_r.
+          }
+        } {
+          by rewrite interval_list_invariant_cons.
+        } {
+          done.
+        }
+      }
+    }
+  Qed.
+
+
+  Lemma diff_aux2_props :
+   forall (s2 s1 acc : t),
+    interval_list_invariant (rev_append acc s1) = true ->
+    interval_list_invariant s2 = true ->
+    (forall x1 x2, InZ x1 acc -> InZ x2 s2 -> Z.succ x1 < x2) ->
+    ((forall y, (InZ y (diff_aux2 acc s1 s2) <->
+               ((InZ y acc) \/ (InZ y s1)) /\ ~InZ y s2)) /\
+    (interval_list_invariant (diff_aux2 acc s1 s2) = true)).
+  Proof.
+    induction s2 as [| [y2 c2] s2' IH]. {
+      move => s1 acc H_inv_acc_s1 _ _. 
+      rewrite /diff_aux2.
+      replace (match s1 with
+        | nil => rev_append acc s1
+        | _ :: _ => rev_append acc s1
+       end) with (rev_append acc s1); last by case s1.
+      split. {
+        move => y.
+        rewrite rev_append_rev InZ_app InZ_rev InZ_nil.
+        tauto.
+      } {
+        assumption.
+      }        
+    } {
+      intros s1 acc H_inv_acc_s1.
+      rewrite interval_list_invariant_cons.
+      move => [H_gr_s2'] [H_c2_neq_0] H_inv_s2'.
+      move => H_acc_s2.
+      rewrite /diff_aux2; fold diff_aux2.
+      case_eq s1. {
+        move => H_s1_eq.
+        split. {
+          move => y.
+          rewrite rev_append_rev InZ_app InZ_nil InZ_rev.
+          split; last tauto.
+          move => [] // H_y_in.
+          split; first by left.
+          move => H_y_in'.
+          move : (H_acc_s2 _ _ H_y_in H_y_in').
+          apply Z.nlt_succ_diag_l.
+        } {
+          move : H_inv_acc_s1.
+           by rewrite H_s1_eq.
+        }
+      } {
+        move => [_ _] _ <-.
+        case_eq (diff_aux y2 c2 acc s1).
+        move => acc' s1' H_diff_aux_eq.
+
+        have H_acc_lt_y2 : (forall x : Z, InZ x acc -> x < y2). {
+          move => x H_x_in.
+          have H_y2_in: (InZ y2 ((y2,c2) :: s2')). {
+            rewrite InZ_cons.
+            left.
+            by apply In_elementsZ_single_hd.
+          }
+          move : (H_acc_s2 _ _ H_x_in H_y2_in).
+          apply Z.lt_trans, Z.lt_succ_diag_r.
+        }
+
+        have [H_inv_acc [H_inv_s1 H_acc_s1]] :
+          interval_list_invariant (rev acc) = true /\
+          interval_list_invariant s1 = true /\
+          (forall x1 x2 : Z,
+             InZ x1 acc -> InZ x2 s1 -> Z.succ x1 < x2). {
+
+          move : H_inv_acc_s1.
+          rewrite rev_append_rev.
+          move => /interval_list_invariant_app_elim.
+          move => [?] [?] H_x.
+          split; first assumption.
+          split; first assumption.
+          move => x1 x2 H_in_x1.
+          apply H_x.
+          by rewrite InZ_rev.
+        }
+
+        move : (diff_aux_props y2 c2 s1 acc H_inv_acc H_inv_s1 H_acc_s1 H_acc_lt_y2 H_c2_neq_0).
+        rewrite !H_diff_aux_eq.
+        move => [H_inZ_res] [H_inv_res] H_inZ_acc'.
+
+        have H_acc'_s2' : (forall x1 x2 : Z,
+                              InZ x1 acc' -> InZ x2 s2' -> Z.succ x1 < x2). {
+          move => x1 x2 H_x1_in H_x2_in.
+          apply Z.le_lt_trans with (m := y2 + Z.of_N c2). {
+            apply Z.le_succ_l.
+            by apply H_inZ_acc'.
+          } {
+            move : H_gr_s2'.
+            rewrite interval_list_elements_greater_alt2_def //.
+            move => H_gr_s2'.
+            by apply H_gr_s2'.
+          }
+        }
+
+        move : (IH s1' acc' H_inv_res H_inv_s2' H_acc'_s2').
+        move => [] H_inZ_diff_res ->.
+        split; last done.
+        move => y.
+        rewrite H_inZ_diff_res.
+        move : (H_inZ_res y).
+        rewrite !rev_append_rev !InZ_app !InZ_rev InZ_cons.
+        move => ->.
+        tauto.
+      }
+    }
+  Qed.
+
+
+  Lemma diff_InZ :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    forall y, (InZ y (diff s1 s2) <-> InZ y s1 /\ ~InZ y s2).
+  Proof.
+    intros s1 s2 H_inv_s1 H_inv_s2 y.
+    rewrite /diff.
+
+    move : (diff_aux2_props s2 s1 nil).
+    move => [] //.
+    move => H_in_diff _.
+    rewrite H_in_diff InZ_nil.
+    tauto.
+  Qed.
+
+  Lemma diff_invariant :
+   forall (s1 s2 : t),
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    interval_list_invariant (diff s1 s2) = true.
+  Proof.
+    intros s1 s2 H_inv_s1 H_inv_s2.
+    rewrite /diff.
+
+    move : (diff_aux2_props s2 s1 nil).
+    move => [] //.
+  Qed.
+  
+ 
+  Global Instance diff_ok s1 s2 : forall `(Ok s1, Ok s2), Ok (diff s1 s2).
+  Proof.
+    move => H_ok_s1 H_ok_s2.
+    move : (H_ok_s1) (H_ok_s2).
+    rewrite /Ok /IsOk /is_encoded_elems_list /add.
+    move => [H_inv_s1] H_pre1.
+    move => [H_inv_s2] H_pre2.
+    split. {
+      apply diff_invariant => //.
+    } {
+      intros y.
+      move : (diff_InZ s1 s2 H_inv_s1 H_inv_s2).    
+      rewrite /InZ => ->.      
+      move => [].
+      move => /H_pre1 //.
+    }
+  Qed.
+
+  Lemma diff_spec :
+   forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
+   In x (diff s s') <-> In x s /\ ~In x s'.
+  Proof.
+    intros s s' x H_ok H_ok'.
+    rewrite !In_InZ.
+    rewrite diff_InZ => //. {
+      apply H_ok.
+    } {
+      apply H_ok'.
+    }
   Qed.
 
-  Lemma inter_spec :
-   forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
-   In x (inter s s') <-> In x s /\ In x s'.
+
+  (** *** remove specification *)
+
+  Lemma removeZ_alt_def : forall x s acc,
+    interval_list_invariant s = true ->  
+    removeZ_aux acc x s = match diff_aux x (1%N) acc s with
+                    (acc', s') => rev_append acc' s'
+                  end.
   Proof.
-    intros s s' x Hs Hs'.
-    rewrite /inter.
-    setoid_rewrite filter_spec. {
-      by rewrite mem_spec.
-    } {
-      repeat red.
-      intros y y' H_y_y'_eq.
-      apply eq_true_iff_eq.
-      rewrite !mem_spec /In.
-      setoid_rewrite H_y_y'_eq.
+    intros y2.
+    induction s as [| [y1 c1] s' IH]; first done.
+
+    move => acc.
+    rewrite interval_list_invariant_cons /=. 
+    move => [H_gr] [H_c1_neq_0] H_inv_s'.
+    move : (interval_1_compare_elim y2 y1 c1).
+    rewrite interval_1_compare_alt_def.
+    rewrite !(interval_compare_swap y1 c1 y2); last first. {
+      right. done.
+    }
+    move : (interval_compare_elim y1 c1 y2 (1%N)).    
+    case_eq (interval_compare (y1, c1) (y2, (1%N))) => H_eq. {
+      move => H_lt_y2 _.
+      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
+        by apply Z.nlt_ge, Z.lt_le_incl.
+      }
+      have H_y2_nlt : ~(y2 < y1). {
+        move => H_y2_y1.
+        apply H_yc2_nlt.
+        apply Z.lt_le_trans with (m := y1) => //.
+        apply Z_le_add_r.
+      }
+      move : (H_y2_nlt) (H_yc2_nlt) => /Z.ltb_nlt -> /Z.ltb_nlt ->.
+      rewrite IH //.
+    } {
+      move => H_lt_y2 _.
+      have H_yc2_nlt : ~(y2 < y1 + Z.of_N c1). {
+        apply Z.nlt_ge.
+        rewrite H_lt_y2.
+        apply Z.le_refl.
+      }
+      have H_y2_nlt : ~(y2 < y1). {
+        move => H_y2_y1.
+        apply H_yc2_nlt.
+        apply Z.lt_le_trans with (m := y1) => //.
+        apply Z_le_add_r.
+      }
+      move : (H_y2_nlt) (H_yc2_nlt) => /Z.ltb_nlt -> /Z.ltb_nlt ->.
+      rewrite IH //.
+    } {
       done.
+    } {
+      done.
+    } {
+      move => [H_y1_eq] H_c1_eq.
+      move => [] // .
+      move => [H_lt_y2] H_y2_lt.
+      have H_y2_nlt : ~(y2 < y1). {
+        apply Z.nlt_ge => //.
+      }
+      move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->.
+      rewrite /insert_intervalZ_guarded.
+
+      have -> : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
+        rewrite H_y1_eq H_c1_eq Z.add_1_r Z.sub_diag //.
+      }
+
+      have -> : (Z.to_N (y2 - y1) =? 0 = true)%N. {
+        rewrite H_y1_eq Z.sub_diag //.
+      }
+      done.       
+    } {
+      move => [H_y2_le] [H_yc1_le] _.
+      move => [] //.
+      move => [H_y1_le] H_y2_lt.
+      have H_y2_nlt : ~(y2 < y1). {
+        apply Z.nlt_ge => //.
+      }
+      move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->.
+
+      have H_y1_eq : (y1 = y2) by lia.
+      have H_yc1_eq : (y1 + Z.of_N c1 = Z.succ y2). {
+        apply Z.le_antisymm. {
+          move : H_yc1_le.
+          rewrite Z.add_1_r //.
+        } {
+          by apply Z.le_succ_l.
+        }
+      }
+
+      rewrite /insert_intervalZ_guarded.
+      have -> : (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0 = true)%N. {
+        rewrite H_yc1_eq Z.sub_diag //.
+      }
+
+      have -> : (Z.to_N (y2 - y1) =? 0 = true)%N. {
+        rewrite H_y1_eq Z.sub_diag //.
+      }
+
+      suff -> : diff_aux y2 (1%N) acc s' = (acc, s') by done.
+
+      move : H_gr.
+      rewrite H_yc1_eq.
+      case s' as [| [y' c'] s'']. {
+        done.
+      } {
+        rewrite interval_list_elements_greater_cons /=
+                /interval_compare.
+        move => H_lt_y'.
+        have -> : y2 + Z.of_N 1 ?= y' = Lt. {
+          apply Z.compare_lt_iff.
+          by rewrite Z.add_1_r.
+        }
+        done.
+      }
+    } { 
+      move => [H_y1_le] [H_yc2_le] _.
+      move => [] //.
+      move => [_] H_y2_lt.
+      have H_y2_nlt : ~(y2 < y1). {
+        apply Z.nlt_ge => //.
+      }
+      move : (H_y2_nlt) (H_y2_lt) => /Z.ltb_nlt -> /Z.ltb_lt ->.
+      rewrite !rev_append_rev /insert_intervalZ_guarded Z.add_1_r.
+      case ((Z.to_N (y2 - y1) =? 0)%N),  (Z.to_N (y1 + Z.of_N c1 - Z.succ y2) =? 0)%N. {
+        reflexivity.
+      } {
+        rewrite /= -!app_assoc //.
+      } {
+        reflexivity.
+      } {
+        rewrite /= -!app_assoc //.
+      }
+    } {
+      move => _ H_y2_lt'.
+      have H_y2_lt : (y2 < y1). {
+        apply Z.lt_trans with (m := Z.succ y2) => //.
+        apply Z.lt_succ_diag_r.
+      }
+      move : (H_y2_lt) => /Z.ltb_lt -> //.
+    } {
+      move => _ H_y1_eq.
+      have H_y2_lt : (y2 < y1). {
+        rewrite H_y1_eq.
+        apply Z.lt_succ_diag_r.
+      }
+      move : (H_y2_lt) => /Z.ltb_lt -> //.
     }
   Qed.
 
+  Lemma removeZ_interval_list_invariant : forall s x, interval_list_invariant s = true -> interval_list_invariant (removeZ x s) = true.
+  Proof.
+    intros s x H_inv.
+    rewrite /removeZ removeZ_alt_def //.
+    move : (diff_aux_props x (1%N) s nil).
+    case_eq (diff_aux x 1%N nil s).
+    move => acc' s' H_eq [] //.
+    move => _ [] //.
+  Qed.
 
-  (** *** diff specification *)
+  Lemma removeZ_spec :
+   forall (s : t) (x y : Z) (Hs : interval_list_invariant s = true),
+    InZ y (removeZ x s) <-> InZ y s /\ ~Z.eq y x.
+  Proof.
+    intros s x y H_inv.
+    rewrite /removeZ removeZ_alt_def //.
+    move : (diff_aux_props x (1%N) s nil).
+    case_eq (diff_aux x 1%N nil s).
+    move => acc' s' H_eq [] //.
+    move => -> _.
+    rewrite rev_append_rev InZ_app InZ_rev InZ_nil
+            In_elementsZ_single1.
+    split; move => [H1 H2]; split => //;
+      move => H3; apply H2; by rewrite H3.
+  Qed.
+
+  Global Instance remove_ok s x : forall `(Ok s), Ok (remove x s).
+  Proof.
+    rewrite /Ok /interval_list_invariant /remove.
+    move => [H_is_ok_s H_enc_s].
+    split. {
+      by apply removeZ_interval_list_invariant.
+    } {
+      rewrite /is_encoded_elems_list => y.
+      move : (removeZ_spec s (Enc.encode x) y H_is_ok_s).
+      rewrite /InZ => -> [] H_y_in _.
+      apply H_enc_s => //.
+    }
+  Qed.
 
-  Global Instance diff_ok s s' : forall `(Ok s, Ok s'), Ok (diff s s').
+  Lemma remove_spec :
+   forall (s : t) (x y : elt) (Hs : Ok s),
+    In y (remove x s) <-> In y s /\ ~Enc.E.eq y x.
   Proof.
-    intros ? ?.
-    by apply remove_list_ok.
+    intros s x y Hs.
+    have H_rs := (remove_ok s x Hs).
+    rewrite /remove !In_InZ.
+    rewrite removeZ_spec. {
+      rewrite Enc.encode_eq //.
+    } {
+      apply Hs.
+    }
   Qed.
+  
 
-  Lemma diff_spec :
-   forall (s s' : t) (x : elt) (Hs : Ok s) (Hs' : Ok s'),
-   In x (diff s s') <-> In x s /\ ~In x s'.
+  (** *** remove_list specification *)
+
+  Lemma remove_list_ok : forall l s, Ok s -> Ok (remove_list l s).
   Proof.
-    intros s s' x Hs Hs'.
-    rewrite /diff remove_list_spec.
-    tauto.
+    induction l as [| x l' IH]. {
+      done.
+    } {
+      move => s H_s_ok /=.
+      apply IH.
+      by apply remove_ok.
+    }
   Qed.
 
+  Lemma remove_list_spec : forall x l s, Ok s -> 
+     (In x (remove_list l s) <-> ~(InA Enc.E.eq x l) /\ In x s).
+  Proof.
+    move => x.
+    induction l as [| y l' IH]. {
+      intros s H.
+      rewrite /= InA_nil.
+      tauto.
+    } {
+      move => s H_ok /=.
+      rewrite IH remove_spec InA_cons.
+      tauto.
+    }
+  Qed.
 
+  
   (** *** subset specification *)
+ 
+  Lemma subset_flatten_alt_def : forall (s1 s2 : t), 
+    subset s1 s2 =
+    match (s1, s2) with
+    | (nil, _) => true
+    | (_ :: _, nil) => false
+    | ((y1, c1) :: l1, (y2, c2) :: l2) =>
+        match (interval_compare (y1, c1) (y2,c2)) with
+          | ICR_before       => false
+          | ICR_before_touch => false
+          | ICR_after        => subset s1 l2 
+          | ICR_after_touch  => false
+          | ICR_overlap_before => false
+          | ICR_overlap_after => false
+          | ICR_equal => subset l1 l2
+          | ICR_subsume_1 => subset l1 s2
+          | ICR_subsume_2 => false
+        end
+    end.
+  Proof.
+    intros s1 s2.    
+    case s1, s2 => //. 
+  Qed.
+
+  Lemma subset_props_aux : forall y1 c1 l1 y2 c2 l2,
+    (exists y, InZ y ((y1, c1) :: l1) /\ ~InZ y ((y2, c2) :: l2)) ->
+    (false = true <->
+    (forall y : Z,
+        InZ y ((y1, c1) :: l1) -> InZ y ((y2, c2) :: l2))).
+  Proof.
+    intros y1 c1 l1 y2 c2 l2.
+    move => [y] [H_y_in H_y_nin].
+    split; first done.
+    move => H.
+    contradict H_y_nin.
+    by apply H.
+  Qed.
+      
+  Lemma subset_props_aux_before : forall y1 c1 l1 y2 c2 l2,
+    (c1 <> 0%N) ->
+    interval_list_invariant ((y2, c2) :: l2) = true ->
+    (y1 < y2) ->
+    (false = true <->
+    (forall y : Z,
+        InZ y ((y1, c1) :: l1) -> InZ y ((y2, c2) :: l2))).
+  Proof.
+    intros y1 c1 l1 y2 c2 l2.
+    rewrite interval_list_invariant_cons.
+    move => H_c1_neq_0 [H_gr] [H_inv_l2] H_c2_neq_0 H_y1_lt.
+    apply subset_props_aux.
+    exists y1.
+    split. {
+      rewrite InZ_cons.
+      left.
+      by apply In_elementsZ_single_hd.
+    } {
+      rewrite InZ_cons.
+      suff : ~ (List.In y1 (elementsZ_single y2 c2)) /\ ~ InZ y1 l2 by tauto.
+      split. {
+        rewrite In_elementsZ_single.
+        move => [] /Z.le_ngt //.
+      } {
+        eapply Nin_elements_greater; eauto.
+        apply Z.le_trans with (m := y2). {
+          by apply Z.lt_le_incl.
+        } {
+          apply Z_le_add_r.
+        }
+      }
+    }
+  Qed.
+
+  
+  Lemma subset_props : forall s1 s2 : t,
+    interval_list_invariant s1 = true ->
+    interval_list_invariant s2 = true ->
+    (subset s1 s2 = true <->
+     (forall y, InZ y s1 -> InZ y s2)).
+  Proof.
+    induction s1 as [| [y1 c1] l1 IH1]. {
+      move => s2 _ _.
+      rewrite subset_flatten_alt_def.
+      split; done.
+    } {
+      induction s2 as [| [y2 c2] l2 IH2]. {
+        rewrite interval_list_invariant_cons
+                subset_flatten_alt_def.
+        move => [_] [H_c1_neq_0] _ _.        
+        split => //.
+        move => H; move : (H y1).
+        rewrite InZ_nil => {} H.
+        contradict H.
+        rewrite InZ_cons; left.
+        by apply In_elementsZ_single_hd.
+      } {
+        move => H_inv_s1 H_inv_s2.
+        move : (H_inv_s1) (H_inv_s2).
+        rewrite !interval_list_invariant_cons.
+        move => [H_gr_l1] [H_c1_neq_0] H_inv_l1.
+        move => [H_gr_l2] [H_c2_neq_0] H_inv_l2.
+        move : (IH2 H_inv_s1 H_inv_l2) => {} IH2.
+        have :  forall s2 : t,
+          interval_list_invariant s2 = true ->
+          (subset l1 s2 = true <->
+          (forall y : Z, InZ y l1 -> InZ y s2)). {
+          intros. by apply IH1.
+        }
+        move => {} IH1.
 
+        have H_yc2_nin : ~ InZ (y2 + Z.of_N c2) ((y2, c2) :: l2). {
+          rewrite !InZ_cons !In_elementsZ_single.
+          move => []. {
+            move => [_] /Z.lt_irrefl //.
+          } {
+            eapply Nin_elements_greater; eauto.
+            apply Z.le_refl.
+          }
+        }
+          
+        rewrite subset_flatten_alt_def.
+        move : (interval_compare_elim y1 c1 y2 c2).
+        case (interval_compare (y1, c1) (y2, c2)). {
+          move => H_lt_y2.
+          apply subset_props_aux_before => //.
+          apply Z.le_lt_trans with (m := y1 + Z.of_N c1) => //.
+          apply Z_le_add_r.          
+        } {
+          move => H_y2_eq.
+          apply subset_props_aux_before => //.
+          rewrite -H_y2_eq.
+          by apply Z_lt_add_r.
+        } {
+          move => [H_y1_lt] _.
+          apply subset_props_aux_before => //.
+        } {
+          move => [H_y2_lt] [H_y1_lt] H_yc2_lt.          
+          apply subset_props_aux.
+          exists (y2 + Z.of_N c2).
+          split => //.
+          rewrite !InZ_cons !In_elementsZ_single.
+          left.
+          split => //.
+          by apply Z.lt_le_incl.
+        } {
+          move => [H_y1_eq] H_c1_eq; subst.
+          rewrite IH1 => //.
+          split; move => H_pre y; move : (H_pre y) => {H_pre};
+            rewrite !InZ_cons. {
+            tauto.
+          } {
+            move => H_pre H_y_in_l1.
+            suff : ~(List.In y (elementsZ_single y2 c2)). {
+              tauto.
+            }
+            move : H_gr_l1.
+            rewrite interval_list_elements_greater_alt2_def
+              // In_elementsZ_single.
+            move => H; move : (H y H_y_in_l1) => {H}.
+            move => /Z.lt_ngt H_neq [_] //.
+          }
+        } {
+          move => [H_y2_lt_y1] [H_yc1_le] _.
+          rewrite IH1.
+          split; move => H_pre y; move : (H_pre y) => {H_pre};
+            rewrite !InZ_cons. {
+            move => H []; last apply H. move => {H}.
+            rewrite !In_elementsZ_single.
+            move => [H_y1_le] H_y_lt.
+            left.
+            lia.
+          } {
+            move => H_pre H_y_in_l1.
+            apply H_pre.
+            by right.
+          } {
+            assumption.
+          }
+        } {
+          move => [H_y1_le] [_] []. {
+            apply subset_props_aux_before => //.
+          } {
+            move => H_yc2_lt.
+            apply subset_props_aux.
+            exists (y2 + Z.of_N c2).
+            split => //.
+            rewrite !InZ_cons !In_elementsZ_single.
+            left.
+            split => //.
+            apply Z.le_trans with (m := y2) => //.
+            apply Z_le_add_r.
+          }
+        } {
+          move => H_yc2_lt_y1.
+          rewrite IH2.
+          split; move => H_pre y; move : (H_pre y) => {H_pre};
+                                                        rewrite !InZ_cons. {
+            tauto.
+          } {
+            rewrite !In_elementsZ_single.
+            move => H_pre H_y_in.
+            suff : ~(y2 <= y < y2 + Z.of_N c2). {
+              tauto.
+            }
+            move => [H_y2_le H_y_lt].
+            move : H_y_in => []. {
+              move => [H_y1_le] H_y_lt'.
+              lia.
+            } {
+              eapply Nin_elements_greater; eauto.
+              apply Z.le_trans with (m := y1); last first. {
+                apply Z_le_add_r.
+              }
+              apply Z.lt_le_incl.
+              apply Z.lt_trans with (m := y2 + Z.of_N c2) => //.
+            }
+          }
+        } {
+          move => H_y1_eq.
+          apply subset_props_aux.
+          exists y1.
+          rewrite !InZ_cons.
+          split. {
+            left.
+            by apply In_elementsZ_single_hd.
+          } {
+            rewrite !In_elementsZ_single H_y1_eq. 
+            move => []. {
+              move => [_] /Z.lt_irrefl //.
+            } {
+              eapply Nin_elements_greater; eauto.
+              rewrite H_y1_eq.
+              apply Z.le_refl.
+            }
+          }
+        }
+      }
+    }
+  Qed. 
+     
   Lemma subset_spec :
    forall (s s' : t) (Hs : Ok s) (Hs' : Ok s'),
    subset s s' = true <-> Subset s s'.
   Proof.
     intros s s' Hs Hs'.
-    rewrite /subset forallb_forall /Subset.
-    split; (
-      move => H x /In_alt_def H_x_in;
-      move : (H x H_x_in) => {H};
-      rewrite mem_spec //
-    ).
+    move : (Hs) (Hs').
+    rewrite /Ok /IsOk.
+    move => [H_inv_s H_enc_s] [H_inv_s' H_enc_s'].
+    rewrite (subset_props s s' H_inv_s H_inv_s').
+    rewrite /Subset.
+    split. {
+      move => H_pre enc_y.
+      rewrite !In_InZ.
+      apply H_pre.
+    } {
+      move => H_pre y H_y_in.
+      move : (H_enc_s _ H_y_in) => [e] H_e. subst.
+      move : (H_pre e) H_y_in.
+      rewrite !In_InZ //.
+    }
   Qed.
-
+  
 
   (** *** elements and elementsZ specification *)
 
@@ -1879,7 +4356,7 @@ Module Raw (Enc : ElementEncode).
       apply NoDupA_nil.
     } {
       move => H_ok_s.
-      move : (H_ok_s) => /Ok_cons [H_inf] [H_c] [H_enc] H_s'.
+      move : (H_ok_s) => /Ok_cons [H_interval_list_elements_greater] [H_c] [H_enc] H_s'.
       rewrite elementsZ_cons.
       apply NoDupA_app. {
         apply Z.eq_equiv.
@@ -1897,285 +4374,91 @@ Module Raw (Enc : ElementEncode).
         move => [_] [<-] H_y_in.
         move => [_] [<-] H_y_in'.
         move : H_y_in'.
-        rewrite -in_rev In_elementsZ_single /=.
-        move => [H_x_le] H_y_lt.
-        eapply Nin_elements_greater; eauto; first apply H_s'.
-        by apply Z.lt_le_incl.
-      }
-    }
-  Qed.
-
-  Lemma elements_spec2w : forall (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
-  Proof.
-    intros s Hs.
-    rewrite /elements rev_map_alt_def.
-    apply NoDupA_rev. {
-      apply Enc.E.eq_equiv.
-    } {
-      eapply NoDupA_map; first by apply elementsZ_spec2w.
-
-      intros x1 x2 H_x1_in H_x2_in H_dec_eq.
-      have H_is_enc: is_encoded_elems_list (elementsZ s). {
-        apply Hs.
-      }
-      move : (H_is_enc _ H_x1_in) => [y1 H_x1_eq].
-      move : (H_is_enc _ H_x2_in) => [y2 H_x2_eq].
-      move : H_dec_eq.
-      rewrite -H_x1_eq -H_x2_eq !Enc.decode_encode_ok Enc.encode_eq //.
-    }
-  Qed.
-
-
-
-  (** *** equal specification *)
-
-  Lemma equal_alt_def : forall s1 s2,
-    equal s1 s2 = true <-> (s1 = s2).
-  Proof.
-    induction s1 as [| [x cx] xs IH]. {
-      move => [] //.
-    } {
-      move => [] //=.
-      move => [y cy] ys.
-      rewrite !andb_true_iff IH N.eqb_eq Z.eqb_eq.
-      split. {
-        move => [->] [->] -> //.
-      } {
-        move => [->] -> -> //.
-      }
-    }
-  Qed.
-
-  Lemma elementsZ_cons_le_start : forall x cx xs y cy ys,
-     isok ((x, cx) :: xs) = true ->
-     isok ((y, cy) :: ys) = true ->
-     (forall z, List.In z (elementsZ ((y, cy) :: ys)) ->
-                List.In z (elementsZ ((x, cx) :: xs))) ->
-     (x <= y). 
-  Proof.
-    intros x cx xs y cy ys.
-    rewrite !isok_cons.
-    move => [H_inf_xs] [H_cx] H_xs [H_inf_ys] [H_cy] H_ys H.
-
-    case_eq (x <=? y). {
-      move => /Z.leb_le //.
-    } {
-      move => /Z.leb_gt H_y_lt.
-      exfalso.
-      move : (H y).
-      have -> : (List.In y (elementsZ ((x, cx) :: xs)) <-> False). {
-        split => //.
-        rewrite elementsZ_cons in_app_iff -in_rev In_elementsZ_single.
-        move => []. {
-          suff : (~ InZ y xs). {
-            rewrite /InZ //.
-          }
-          eapply Nin_elements_greater; eauto.
-          eapply Z.le_trans with (m := x). {
-            by apply Z.lt_le_incl.
-          } {
-            apply Z_le_add_r.
-          }
-        } {
-          apply Z.lt_nge in H_y_lt.
-          tauto.
-        }
-      } 
-      have -> : (List.In y (elementsZ ((y, cy) :: ys)) <-> True). {
-        split => // _.
-        rewrite elementsZ_cons in_app_iff -in_rev.
-        right.
-        apply In_elementsZ_single_hd => //.
-      } 
-      tauto.
-    }
-  Qed.
-
-  Lemma elementsZ_cons_le_end : forall x cx xs y cy ys,
-     isok ((x, cx) :: xs) = true ->
-     isok ((y, cy) :: ys) = true ->
-     (x <= y + Z.of_N cy) ->
-     (forall z, List.In z (elementsZ ((x, cx) :: xs)) ->
-                List.In z (elementsZ ((y, cy) :: ys))) ->
-     (x + Z.of_N cx <= y + Z.of_N cy). 
-  Proof.
-    intros x cx xs y cy ys.
-    rewrite !isok_cons.
-    move => [H_inf_xs] [H_cx] H_xs [H_inf_ys] [H_cy] H_ys H_x_le H.
-
-    case_eq (x + Z.of_N cx <=? y + Z.of_N cy). {
-      move => /Z.leb_le //.
-    } {
-      move => /Z.leb_gt H_y_lt.
-      exfalso.
-      move : (H (y + Z.of_N cy)).
-      have -> : ((List.In (y + Z.of_N cy)) (elementsZ ((y, cy) :: ys)) <-> False). {
-        split => //.
-        rewrite elementsZ_cons in_app_iff -in_rev In_elementsZ_single.
-        move => []. {
-          suff : (~ InZ (y + Z.of_N cy) ys). {
-            rewrite /InZ //.
-          }
-          eapply Nin_elements_greater; eauto.
-          apply Z.le_refl.
+        rewrite -in_rev In_elementsZ_single /=.
+        move => [H_x_le] H_y_lt.
+        eapply (Nin_elements_greater s' (x + Z.of_N c)) => //. {
+          apply H_s'.
         } {
-          move => [_] /Z.lt_irrefl //.
+          apply Z.lt_le_incl, H_y_lt.
+        } {
+          apply H_y_in.
         }
-      } 
-      have -> : (List.In (y + Z.of_N cy) (elementsZ ((x, cx) :: xs)) <-> True). {
-        split => // _.
-        rewrite elementsZ_cons in_app_iff -in_rev.
-        right.
-        rewrite In_elementsZ_single => //.
-      } 
-      tauto.
+      }
     }
   Qed.
 
-
-  Lemma elementsZ_cons_equiv_hd : forall x cx xs y cy ys,
-     isok ((x, cx) :: xs) = true ->
-     isok ((y, cy) :: ys) = true ->
-     (forall z, List.In z (elementsZ ((x, cx) :: xs)) <->
-                List.In z (elementsZ ((y, cy) :: ys))) ->
-     (x = y) /\ (cx = cy). 
+  Lemma elements_spec2w : forall (s : t) (Hs : Ok s), NoDupA Enc.E.eq (elements s).
   Proof.
-    intros x cx xs y cy ys H_ok_xs H_ok_ys H.
-    have H_xy_eq : x = y. {
-      have H_x_le : (x <= y). {
-        eapply (elementsZ_cons_le_start); eauto.
-        intro z.
-        rewrite H => //.
-      } 
-      have H_y_le : (y <= x). {
-        eapply (elementsZ_cons_le_start); eauto.
-        intro z.
-        rewrite H => //.
-      } 
-      apply Z.le_antisymm => //.
-    }
-    subst.
-    split => //.
+    intros s Hs.
+    rewrite /elements rev_map_alt_def.
+    apply NoDupA_rev. {
+      apply Enc.E.eq_equiv.
+    } {
+      eapply NoDupA_map; first by apply elementsZ_spec2w.
 
-    have : (y + Z.of_N cx = y + Z.of_N cy). {
-      have H_cx_le : (y + Z.of_N cx <= y + Z.of_N cy). {
-        eapply elementsZ_cons_le_end; eauto. {       
-          apply Z_le_add_r.
-        } {
-          intro Z.
-          rewrite H => //.
-        }
-      }
-      have H_cy_le : (y + Z.of_N cy <= y + Z.of_N cx). {
-        eapply elementsZ_cons_le_end; eauto. {       
-          apply Z_le_add_r.
-        } {
-          intro Z.
-          rewrite H => //.
-        }
+      intros x1 x2 H_x1_in H_x2_in H_dec_eq.
+      have H_is_enc: is_encoded_elems_list (elementsZ s). {
+        apply Hs.
       }
-      apply Z.le_antisymm => //.
+      move : (H_is_enc _ H_x1_in) => [y1 H_x1_eq].
+      move : (H_is_enc _ H_x2_in) => [y2 H_x2_eq].
+      move : H_dec_eq.
+      rewrite -H_x1_eq -H_x2_eq !Enc.decode_encode_ok Enc.encode_eq //.
     }
-    rewrite Z.add_cancel_l.
-    apply N2Z.inj.
   Qed.
 
 
-  Lemma elementsZ_single_equiv : forall x cx y cy,
-     (cx <> 0)%N ->
-     (cy <> 0)%N ->
-     (forall z, List.In z (elementsZ_single x cx) <->
-                List.In z (elementsZ_single y cy)) ->
-     (x = y) /\ (cx = cy). 
+
+  (** *** equal specification *)
+
+  Lemma equal_alt_def : forall s1 s2,
+    equal s1 s2 = true <-> (s1 = s2).
   Proof.
-    intros x cx y cy H_cx H_cy H.
-    apply elementsZ_cons_equiv_hd with (xs := nil) (ys := nil). {
-      rewrite isok_cons //.
-    } {
-      rewrite isok_cons //.
+    induction s1 as [| [x cx] xs IH]. {
+      move => [] //.
     } {
-      intro z. 
-      move : (H z) => {H}.
-      rewrite !elementsZ_cons !in_app_iff !elementsZ_nil -!in_rev /= => -> //.
+      move => [] //=.
+      move => [y cy] ys.
+      rewrite !andb_true_iff IH N.eqb_eq Z.eqb_eq.
+      split. {
+        move => [->] [->] -> //.
+      } {
+        move => [->] -> -> //.
+      }
     }
   Qed.
-   
+
 
   Lemma equal_elementsZ :
     forall (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
     (forall x, (InZ x s <-> InZ x s')) -> (s = s').
   Proof.
-    induction s as [| [x cx] xs IH]. {
-      move => [] //.
-      move => [y cy ys] _ /Ok_cons [_] [H_cy] _.
-      rewrite /InZ elementsZ_nil /= => H.
-      exfalso.
-      rewrite (H y) elementsZ_cons in_app_iff -in_rev.
-      right.
-      apply In_elementsZ_single_hd => //.
-    } {  
-      move => []. {
-        move => /Ok_cons [_] [H_xc] _ _.
-        rewrite /InZ elementsZ_nil /= => H.
-        exfalso.
-        rewrite -(H x) elementsZ_cons in_app_iff -in_rev.
-        right.
-        apply In_elementsZ_single_hd => //.
-      } {
-        move => [y cy ys] H_ok_xxs H_ok_yys. 
-        rewrite /InZ.
-        move => H_in.
-
-        have [H_ok_xs H_inf_xs] : Ok xs /\ inf (x + Z.of_N cx) xs = true. {
-          move : H_ok_xxs => /Ok_cons; tauto.
-        }
-        have [H_ok_ys H_inf_ys] : Ok ys /\ inf (y + Z.of_N cy) ys = true. {
-          move : H_ok_yys => /Ok_cons. tauto.
-        }
-        have H_isok_xs : isok xs=true by apply H_ok_xs.
-        have H_isok_ys : isok ys=true by apply H_ok_ys.
-
-        have [H_x_eq H_cx_eq] : (x = y) /\ (cx = cy). {
-          eapply elementsZ_cons_equiv_hd. 
-            - by apply H_ok_xxs.
-            - by apply H_ok_yys.
-            - by apply H_in.
-        }
-        subst.
-
-        suff -> : (xs = ys) by [].
-        suff H_suff : (forall x : Z, InZ x xs <-> InZ x ys) by apply IH.
-
-        move => z.
-        move : (H_in z) => {H_in}.
-        move : (Nin_elements_greater _ _ H_inf_xs H_isok_xs z).
-        move : (Nin_elements_greater _ _ H_inf_ys H_isok_ys z).
-        rewrite /InZ !elementsZ_cons !in_app_iff -!in_rev !In_elementsZ_single.
-        move => H_nin_ys H_nin_xs H_equiv.
-        split. {
-          move => H_z_in.
-          have : List.In z (elementsZ ys) \/ y <= z < y + Z.of_N cy. {
-            rewrite -H_equiv.
-            by left.
-          }
-          move => [] //.
-          move => [_] /Z.lt_le_incl H_z_le.
-          contradict H_z_in.
-          by apply H_nin_xs.
-        } {
-          move => H_z_in.
-          have : List.In z (elementsZ xs) \/ y <= z < y + Z.of_N cy. {
-            rewrite H_equiv.
-            by left.
-          }
-          move => [] //.
-          move => [_] /Z.lt_le_incl H_z_le.
-          contradict H_z_in.
-          by apply H_nin_ys.
-        }
-      }
+    intros s s'.
+    move => H_ok_s H_ok_s' H_InZ_eq.
+    have [] : ((subset s s' = true) /\ (subset s' s = true)). {
+      rewrite !subset_spec /Subset.
+      split => x; rewrite !In_InZ H_InZ_eq //.
     }
+    have : interval_list_invariant s' = true by apply H_ok_s'.
+    have : interval_list_invariant s = true by apply H_ok_s.
+    clear H_ok_s H_ok_s' H_InZ_eq.
+    move : s s'.
+    induction s as [| [x1 c1] s1 IH];
+      case s' as [| [x2 c2] s2] => //.
+    rewrite !interval_list_invariant_cons.
+    move => [H_gr_s1] [H_c1_neq_0] H_inv_s1.
+    move => [H_gr_s2] [H_c2_neq_0] H_inv_s2.
+    rewrite subset_flatten_alt_def
+            (subset_flatten_alt_def ((x2, c2)::s2)).
+    rewrite (interval_compare_swap x1 c1); last by left.
+    move : (interval_compare_elim x1 c1 x2 c2).
+    case (interval_compare (x1, c1) (x2, c2)) => //.
+    move => [->] -> H_sub_s12 H_sub_s21.
+
+    suff -> : s1 = s2 by done.
+    by apply IH.
   Qed.
+ 
 
   Lemma equal_spec :
     forall (s s' : t) {Hs : Ok s} {Hs' : Ok s'},
@@ -2215,86 +4498,538 @@ Module Raw (Enc : ElementEncode).
     }
   Qed.
 
+  (** *** compare *)
+  Definition lt (s1 s2 : t) : Prop := (compare s1 s2 = Lt).
+
+  Lemma compare_eq_Eq : forall s1 s2,
+    (compare s1 s2 = Eq <-> equal s1 s2 = true).
+  Proof.
+    induction s1 as [| [y1 c1] s1' IH];
+      case s2 as [| [y2 c2] s2'] => //.
+    rewrite /= !andb_true_iff -IH Z.eqb_eq N.eqb_eq.
+    move : (Z.compare_eq_iff y1 y2).
+    case (Z.compare y1 y2). {
+      move => H.
+      have -> : y1 = y2. by apply H.
+      clear H.
+
+      move : (N.compare_eq_iff c1 c2).
+      case (N.compare c1 c2). {
+        move => H.
+        have -> : c1 = c2. by apply H.
+        tauto.
+      } {
+        move => H.
+        have H_neq : ~(c1 = c2). by rewrite -H => {H}.
+        tauto.
+      } {        
+        move => H.
+        have H_neq : ~(c1 = c2). by rewrite -H => {H}.
+        tauto.
+      }
+    } {
+      move => H.
+      have H_neq : ~(y1 = y2). by rewrite -H => {H}.
+      tauto.
+    } {
+      move => H.
+      have H_neq : ~(y1 = y2). by rewrite -H => {H}.
+      tauto.
+    }
+  Qed.
+
+  Lemma compare_eq_Lt_nil_l : forall s,
+    compare nil s = Lt <-> s <> nil.
+  Proof.
+    intros s.
+    case s => //=.
+  Qed.
+
+  Lemma compare_eq_Lt_nil_r : forall s,
+    ~(compare s nil = Lt).
+  Proof.
+    intros s.
+    case s as [| [y1 c1] s'] => //=.
+  Qed.
+  
+  Lemma compare_eq_Lt_cons : forall y1 y2 c1 c2 s1 s2,
+    compare ((y1, c1)::s1) ((y2, c2)::s2) = Lt <->
+    (y1 < y2) \/ ((y1 = y2) /\ (c1 < c2)%N) \/
+    ((y1 = y2) /\ (c1 = c2) /\ compare s1 s2 = Lt).
+  Proof.
+    intros y1 y2 c1 c2 s1 s2.
+    rewrite /=.
+    case_eq (Z.compare y1 y2). {
+      move => /Z.compare_eq_iff ->.
+      case_eq (N.compare c1 c2). {
+        move => /N.compare_eq_iff ->.
+        split. {
+          move => H.
+          right; right.
+          done.
+        } {
+          move => [| []]. {
+            move => /Z.lt_irrefl //.
+          } {
+            move => [_] /N.lt_irrefl //.
+          } {
+            move => [_] [_] -> //.
+          }
+        }
+      } {
+        move => /N.compare_lt_iff H_c1_lt.
+        split => //.
+        move => _.
+        right; left. done.
+      } {
+        move => /N.compare_gt_iff H_c2_lt.
+        split => //.
+        move => [| []]. {
+          move => /Z.lt_irrefl //.
+        } {
+          move => [_] /N.lt_asymm //.
+        } {
+          move => [_] [] H_c1_eq.
+          contradict H_c2_lt.
+          subst c1.
+          by apply N.lt_irrefl.
+        }
+      }
+    } {
+      move => /Z.compare_lt_iff.
+      tauto.
+    } {
+      move => /Z.compare_gt_iff H_y2_lt.
+      split => //.
+      move => [| []]. {
+        move => /Z.lt_asymm //.
+      } {
+        move => [] H_y1_eq.
+        exfalso. lia.
+      } {
+        move => [] H_y1_eq.
+        exfalso. lia.
+      }
+    }
+  Qed.
+  
+  
+  Lemma compare_antisym: forall (s1 s2 : t),
+    (compare s1 s2) = CompOpp (compare s2 s1).
+  Proof.
+    induction s1 as [| [y1 c1] s1' IH];
+      case s2 as [| [y2 c2] s2'] => //.
+    rewrite /= (Z.compare_antisym y1 y2) (N.compare_antisym c1 c2).
+    case (Z.compare y1 y2) => //=.
+    case (N.compare c1 c2) => //=.
+  Qed.
+
+
+  Lemma compare_spec : forall s1 s2,
+    CompSpec eq lt s1 s2 (compare s1 s2).
+  Proof.
+    intros s1 s2.
+    rewrite /CompSpec /lt (compare_antisym s2 s1).
+    case_eq (compare s1 s2). {
+      rewrite compare_eq_Eq equal_alt_def => ->.
+      by apply CompEq.
+    } {
+      move => _.
+      by apply CompLt.
+    } {
+      move => _.
+      by apply CompGt.
+    }
+  Qed.
+
+  Lemma lt_Irreflexive : Irreflexive lt.
+  Proof.
+    rewrite /Irreflexive /Reflexive /complement /lt.
+    intros x.
+    suff -> : compare x x = Eq by done.
+    rewrite compare_eq_Eq equal_alt_def //.
+  Qed.
+
+  Lemma lt_Transitive : Transitive lt.
+  Proof.
+    rewrite /Transitive /lt.
+    induction x as [| [y1 c1] s1' IH];
+      case y as [| [y2 c2] s2'];
+      case z as [| [y3 c3] s3'] => //.
+
+    rewrite !compare_eq_Lt_cons.
+    move => [H_y1_lt | [[->] H_c1_lt | [->] [->] H_comp]]
+            [H_y2_lt | [[<-] H_c2_lt | [<-] [<-] H_comp']]. {
+      left.
+      by apply Z.lt_trans with (m := y2).
+    } {
+      by left.
+    } {
+      by left.
+    } {
+      by left.
+    } {
+      right; left.
+      split => //.
+      by apply N.lt_trans with (m := c2).
+    } {
+      by right; left.
+    } {
+      by left.
+    } {
+      by right; left.
+    } {
+      right; right.
+      split => //.
+      split => //.
+      by apply (IH s2').
+    }
+  Qed.
+             
+  (** *** elements is sorted *)
+  
+  Lemma elementsZ_single_sorted : forall c x,
+    sort Z.lt (elementsZ_single x c).
+  Proof.
+    induction c as [| c' IH] using N.peano_ind. {
+      intro x.
+      rewrite elementsZ_single_base.
+      apply Sorted_nil.
+    } {
+      intro x.
+      rewrite elementsZ_single_succ_front.
+      apply Sorted_cons. {
+        apply IH.
+      } {
+        case (N.zero_or_succ c'). {
+          move => ->.
+          rewrite elementsZ_single_base //.
+        } {
+          move => [c''] ->.
+          rewrite elementsZ_single_succ_front.
+          constructor.
+          apply Z.lt_succ_diag_r.
+        }
+      }
+    }
+  Qed.
+
+  Lemma elementsZ_sorted : forall s,
+    interval_list_invariant s = true ->
+    sort Z.lt (rev (elementsZ s)).
+  Proof.
+    induction s as [| [y c] s' IH]. {
+      move => _.
+      rewrite elementsZ_nil.
+      apply Sorted_nil.
+    } {
+      rewrite interval_list_invariant_cons elementsZ_cons
+              rev_app_distr rev_involutive.
+      move => [H_gr] [H_c_neq_0] H_inv_s'.
+      apply SortA_app with (eqA := Logic.eq). {
+        apply eq_equivalence.
+      } {
+        apply elementsZ_single_sorted.
+      } {
+        by apply IH.
+      } {
+        intros x1 x2.
+        move => /InA_alt [_] [<-] /In_elementsZ_single [_ H_x1_lt].
+        move => /InA_alt [_] [<-].
+        rewrite -In_rev => H_x2_in.
+
+        apply Z.lt_trans with (m := (y + Z.of_N c)) => //.
+        eapply interval_list_elements_greater_alt2_def;
+          eauto.
+      }
+    }
+  Qed.
 
+  
+  Lemma elements_sorted : forall s,
+    Ok s ->
+    sort Enc.E.lt (elements s).
+  Proof.
+    move => s [H_inv] H_enc.
+    rewrite /elements rev_map_alt_def -map_rev.
+    have : (forall x : Z, List.In x (rev (elementsZ s)) ->
+           exists e : Enc.E.t, Enc.encode e = x). {
+      move => x.
+      move : (H_enc x).
+      rewrite In_rev //.
+    }
+    move : (elementsZ_sorted s H_inv) => {H_enc}.
+    generalize (rev (elementsZ s)).
+    induction l as [| x xs IH]. {
+      rewrite /= => _ _.
+      apply Sorted_nil.
+    } {
+      move => H_sort H_enc.
+      apply Sorted_inv in H_sort as [H_sort H_hd_rel].
+      simpl.
+      apply Sorted_cons. {
+        apply IH => //.
+        move => xx H_xx_in.
+        apply H_enc.
+        by apply in_cons.
+      } {
+        move : H_hd_rel H_enc.
+        case xs => //=.
+        move => x' xs' H_hd_rel H_enc.
+        apply HdRel_inv in H_hd_rel.
+        apply HdRel_cons.
+        rewrite -Enc.encode_lt.
+        have [y H_y] : (exists y, Enc.encode y = x). {
+          apply H_enc. by left.
+        }
+        have [y' H_y'] : (exists y', Enc.encode y' = x'). {
+          apply H_enc. by right; left.
+        }
+        move : H_hd_rel.
+        rewrite -!H_y -!H_y' !Enc.decode_encode_ok //.
+      }
+    }
+  Qed.   
+        
+  
   (** *** choose specification *)
 
-  Lemma choose_alt_def : forall s, 
-    choose s = match chooseZ s with
-      | None => None
-      | Some e => Some (Enc.decode e)
-    end.
+  Definition min_eltZ_spec1 :
+    forall (s : t) (x : Z),
+      interval_list_invariant s = true ->
+      min_eltZ s = Some x -> InZ x s. 
+  Proof.   
+    intros s x.
+    case s as [| [x' c] s']. {
+      rewrite /min_eltZ //.
+    } {
+      rewrite /min_eltZ InZ_cons interval_list_invariant_cons.
+      move => [_] [H_c_neq] _ [->].
+      left.
+      by apply In_elementsZ_single_hd.
+    }
+  Qed.
+
+  Lemma min_eltZ_spec2 :
+    forall (s : t) (x y : Z) (Hs : Ok s),
+    min_eltZ s = Some x -> InZ y s -> ~ Z.lt y x.
   Proof.
+    intros s x y H_ok H_min H_in H_y_lt_x.
+    eapply (Nin_elements_greater s (Z.pred x)) => //; last apply H_in. {
+      move : H_ok H_min.
+      case s => //.
+      move => [z c] s' _ [<-].
+      rewrite interval_list_elements_greater_cons.
+      apply Z.lt_pred_l.
+    } {
+      apply H_ok.
+    } {
+      by apply Z.lt_le_pred.
+    }
+  Qed.
+
+  Definition min_eltZ_spec3 :
+    forall (s : t),
+      min_eltZ s = None -> forall x, ~InZ x s. 
+  Proof.   
     intros s.
-    rewrite /choose /chooseZ /elements /rev' rev_append_rev app_nil_r 
-            rev_map_alt_def -map_rev.
-    case (rev (elementsZ s)) => //.
+    case s as [| [x' c] s'];
+      rewrite /min_eltZ //.
+    move => _ x //.
+  Qed.
+
+  Definition min_elt_spec1 :
+    forall (s : t) (x : elt) (Hs : Ok s), min_elt s = Some x -> In x s. 
+  Proof.   
+    rewrite /min_elt.
+    move => s x H_ok.
+    case_eq (min_eltZ s) => //.
+    move => z H_min_elt [<-].
+    apply InZ_In => //.
+    apply min_eltZ_spec1 => //.
+    apply H_ok.
+  Qed.
+
+  Definition min_elt_spec2 :
+    forall (s : t) (x y : elt) (Hs : Ok s), min_elt s = Some x -> In y s -> ~(Enc.E.lt y x). 
+  Proof.   
+    rewrite /min_elt.
+    move => s x y H_ok.
+    case_eq (min_eltZ s) => //.
+    move => z H_min_elt [<-].
+    rewrite In_InZ => H_inZ.    
+    have H_y_eq : y = Enc.decode (Enc.encode y). {
+      by rewrite Enc.decode_encode_ok.
+    }
+    rewrite H_y_eq -Enc.encode_lt.
+    apply (min_eltZ_spec2 _ _ _ H_ok); last first. {
+      by rewrite Enc.decode_encode_ok.
+    }
+    suff -> : Enc.encode (Enc.decode z) = z by assumption.
+    apply encode_decode_eq with (s := s) => //.
+    apply min_eltZ_spec1 => //.
+    apply H_ok.
+  Qed.
+
+  Definition min_elt_spec3 :
+    forall s : t, min_elt s = None -> Empty s.
+  Proof.   
+    rewrite /min_elt /min_eltZ /Empty /In.
+    case s as [| [x' c] s'] => //.
+    move => _ e.
+    rewrite elements_nil InA_nil //.
+  Qed.
+
+
+  Definition choose_spec1 :
+    forall (s : t) (x : elt) (Hs : Ok s), choose s = Some x -> In x s. 
+  Proof.   
+    rewrite /choose.
+    apply min_elt_spec1.
+  Qed.
+
+  Definition choose_spec2 :
+    forall s : t, choose s = None -> Empty s.
+  Proof.   
+    rewrite /choose.
+    apply min_elt_spec3.
+  Qed.
+
+  Lemma choose_spec3: forall s s' x x', Ok s -> Ok s' ->
+    choose s = Some x -> choose s' = Some x' -> Equal s s' -> x = x'.
+  Proof.
+    intros s s' x x' Hs Hs' Hx Hx'.
+    rewrite -equal_spec equal_alt_def => H_s_eq.
+    move : Hx Hx'.
+    rewrite H_s_eq => -> [] //.
   Qed.
 
-  Definition choose_spec1 :
-    forall (s : t) (x : elt), choose s = Some x -> In x s. 
+    
+  Definition max_eltZ_spec1 :
+    forall (s : t) (x : Z),
+      interval_list_invariant s = true ->
+      max_eltZ s = Some x -> InZ x s. 
   Proof.   
     intros s x.
-    rewrite /choose /In.
-    case (elements s) => //.
-    move => z l [<-].
-    apply InA_cons_hd.
-    apply Enc.E.eq_equiv.
+    induction s as [| [x' c] s' IH]. {
+      rewrite /max_eltZ //.
+    } {
+      rewrite InZ_cons interval_list_invariant_cons /=.
+      move => [_] [H_c_neq].
+      case s' as [| [y' c'] s'']. {
+        move => _ [<-].
+        left. {
+          rewrite In_elementsZ_single.
+          split. {
+            rewrite -Z.lt_le_pred.
+            by apply Z_lt_add_r.
+          } {
+            apply Z.lt_pred_l.
+          }
+        }
+      } {
+        move => H_inv H_max_eq.
+        right.
+        by apply IH.
+      }
+    }
   Qed.
 
-  Definition choose_spec2 :
-    forall s : t, choose s = None -> Empty s.
-  Proof.   
-    rewrite /choose /Empty /In.
-    intro s.
-    case (elements s) => //.
-    move => _ a.
-    rewrite InA_nil //.
+  Lemma max_eltZ_spec2 :
+    forall (s : t) (x y : Z),
+    interval_list_invariant s = true ->
+    max_eltZ s = Some x -> InZ y s -> ~ Z.lt x y.
+  Proof.
+    induction s as [| [y c] s' IH]. {
+      done.
+    } {
+      move => x x'.
+      rewrite interval_list_invariant_cons.
+      move => [H_gr] [H_c_neq_0] H_inv_s'.
+      have H_gr' : (forall xx : Z, InZ xx (s') -> y + Z.of_N c < xx). {
+        apply interval_list_elements_greater_alt2_def => //.
+      }
+
+      case s' as [| [y' c'] s'']. {
+        move => [<-].
+        rewrite InZ_cons InZ_nil In_elementsZ_single.
+        lia.
+      } {
+        move => H_max_eq.
+        rewrite InZ_cons.
+        move => []; last by apply IH.
+        rewrite In_elementsZ_single.
+        move => [_] H_x'_lt H_lt_x'.
+
+        have H_x_in : InZ x ((y', c')::s''). {
+          by apply max_eltZ_spec1.
+        }
+        
+        move : (H_gr' _ H_x_in).
+        apply Z.nlt_ge, Z.lt_le_incl.
+        by apply Z.lt_trans with (m := x').
+      }
+    }
   Qed.
 
+  Lemma max_eltZ_eq_None :
+    forall (s : t),
+      max_eltZ s = None -> s = nil.
+  Proof.
+    induction s as [| [x' c] s' IH] => //.
+    move : IH.
+    case s' as [| [y' c'] s''] => //=.
+    move => H H_pre.
+    move : (H H_pre) => //.
+  Qed.   
+    
 
-  Lemma chooseZ_min :
-    forall (s : t) (x y : Z) (Hs : Ok s),
-    chooseZ s = Some x -> InZ y s -> ~ Z.lt y x.
+  Definition max_eltZ_spec3 :
+    forall (s : t),
+      max_eltZ s = None -> forall x, ~InZ x s. 
   Proof.
-    intros s x y H_ok H_min H_in H_y_lt_x.
-    eapply (Nin_elements_greater s (Z.pred x)) => //; last apply H_in. {
-      move : H_ok H_min.
-      case s => //.
-      move => [z c] s'.
-      rewrite /chooseZ Ok_cons elementsZ_cons /rev' -rev_alt rev_app_distr
-              rev_involutive.
-      move => [_] [/N.neq_0_r] [c'] -> _.
-      rewrite elementsZ_single_succ_front /=. 
-      move => [->].
-      apply Z.ltb_lt, Z.lt_pred_l.
-    } {
-      apply H_ok.
-    } {
-      by apply Z.lt_le_pred.
-    }
+    move => s /max_eltZ_eq_None -> x /InZ_nil //.
   Qed.
 
-  Lemma chooseZ_InZ :
-    forall (s : t) (x : Z),
-    chooseZ s = Some x -> InZ x s.
+  Definition max_elt_spec1 :
+    forall (s : t) (x : elt) (Hs : Ok s), max_elt s = Some x -> In x s. 
   Proof.   
-    intros s x.
-    rewrite /chooseZ /InZ /rev' -rev_alt in_rev.
-    case (rev (elementsZ s)) => //.
-    move => z l [<-] /=.
-    by left.
+    rewrite /max_elt.
+    move => s x H_ok.
+    case_eq (max_eltZ s) => //.
+    move => z H_max_elt [<-].
+    apply InZ_In => //.
+    apply max_eltZ_spec1 => //.
+    apply H_ok.
   Qed.
 
+  Definition max_elt_spec2 :
+    forall (s : t) (x y : elt) (Hs : Ok s), max_elt s = Some x -> In y s -> ~(Enc.E.lt x y). 
+  Proof.   
+    rewrite /max_elt.
+    move => s x y H_ok.
+    move : (H_ok) => [H_inv] _.
+    case_eq (max_eltZ s) => //.
+    move => z H_max_elt [<-].
+    rewrite In_InZ => H_inZ.    
+    rewrite -Enc.encode_lt.
+    apply (max_eltZ_spec2 _ _ _ H_inv) => //.
+    suff -> : Enc.encode (Enc.decode z) = z => //.
+    apply encode_decode_eq with (s := s) => //.
+    apply max_eltZ_spec1 => //.
+  Qed.
 
-  Lemma chooseZ_spec3: forall s s' x x', Ok s -> Ok s' ->
-   chooseZ s = Some x -> chooseZ s' = Some x' -> Equal s s' -> x = x'.
+  Definition max_elt_spec3 :
+    forall s : t, max_elt s = None -> Empty s.
   Proof.
-    intros s s' x x' Hs Hs' Hx Hx'.
-    rewrite -equal_spec equal_alt_def => H_s_eq.
-    move : Hx Hx'.
-    rewrite H_s_eq => -> [] //.
+    intro s.
+    rewrite /max_elt /Empty.
+    case_eq (max_eltZ s) => //.
+    move => /max_eltZ_eq_None -> _ x.
+    rewrite /In elements_nil InA_nil //.
   Qed.
 
+
   (** *** fold specification *)
 
   Lemma fold_spec :
@@ -2302,7 +5037,16 @@ Module Raw (Enc : ElementEncode).
    fold f s i = fold_left (flip f) (elements s) i.
   Proof.
     intros s A i f.
-    rewrite /fold //.
+    rewrite /fold fold_elementsZ_alt_def /elements
+            rev_map_alt_def -map_rev.
+    move : i.
+    generalize (rev (elementsZ s)).
+    induction l as [| x xs IH]. {
+      done.
+    } {
+      move => i.
+      rewrite /= IH //.
+    }
   Qed.
 
  
@@ -2341,11 +5085,43 @@ Module Raw (Enc : ElementEncode).
    (for_all f s = true <-> For_all (fun x => f x = true) s).
   Proof.
     intros s f Hs H.
-    rewrite /for_all forallb_forall /For_all //.
-    split; (
-      move => HH x; move : {HH} (HH x);
-      rewrite In_alt_def //
-    ).
+    rewrite /for_all /For_all /In fold_elementsZ_alt_def
+            /elements rev_map_alt_def -map_rev.
+    generalize (rev (elementsZ s)).
+    induction l as [| x xs IH]. {      
+      split => // _ x /= /InA_nil //.
+    } {
+      rewrite /=.
+      case_eq (f (Enc.decode x)) => H_f_eq. {
+        rewrite IH.
+        split. {
+          move => HH x' /InA_cons []. {
+            by move => ->.
+          } {
+            apply HH.
+          }
+        } {
+          move => HH x' H_in.
+          apply HH.
+          apply InA_cons.
+          by right.
+        }
+      } {
+        split; move => HH. {
+          contradict HH.
+          case xs => //.
+        } {
+          exfalso.
+          have H_in: (InA Enc.E.eq (Enc.decode x) (Enc.decode x :: map Enc.decode xs)). {
+            apply InA_cons.
+            left.
+            apply Enc.E.eq_equiv.
+          }
+          move : (HH _ H_in).
+          rewrite H_f_eq => //.
+        }
+      }
+    }
   Qed.
 
   (** *** exists specification *)
@@ -2356,46 +5132,526 @@ Module Raw (Enc : ElementEncode).
    (exists_ f s = true <-> Exists (fun x => f x = true) s).
   Proof.
     intros s f Hs H.
-    rewrite /exists_ /Exists existsb_exists //.
-    split; (
-      move => [x] [HH1 HH2]; exists x; move : HH1 HH2;
-      rewrite In_alt_def //
-    ).
+    rewrite /exists_ /Exists /In fold_elementsZ_alt_def
+            /elements rev_map_alt_def -map_rev.
+    generalize (rev (elementsZ s)).
+    induction l as [| x xs IH]. {      
+      split => //.
+      move => [x] /= [] /InA_nil //.
+    } {
+      rewrite /=.
+      case_eq (f (Enc.decode x)) => H_f_eq. {
+        split => _. {
+          exists (Enc.decode x).
+          split => //.
+          apply InA_cons.
+          left.
+          apply Enc.E.eq_equiv.
+        } {
+          case xs => //.
+        }
+      } {
+        rewrite IH.
+        split. {
+          move => [x0] [H_in] H_f_x0.
+          exists x0.
+          split => //.
+          apply InA_cons.
+          by right.
+        } {
+          move => [x0] [] /InA_cons H_in H_f_x0.
+          exists x0.
+          split => //.
+          move : H_in => [] // H_in.
+          contradict H_f_x0.
+          rewrite H_in H_f_eq //.
+        }
+      }
+    }
+  Qed.
+
+
+  (** *** filter specification *)    
+    
+  Definition partitionZ_aux_invariant (x : Z) acc c :=
+    interval_list_invariant (List.rev (partitionZ_fold_skip acc c)) = true /\
+    match c with
+      None => (forall y', InZ y' acc -> Z.succ y' < x)
+    | Some (y, c') => (x = y + Z.of_N c')
+    end.
+
+  Lemma partitionZ_aux_invariant_insert : forall x acc c,
+    partitionZ_aux_invariant x acc c ->
+    partitionZ_aux_invariant (Z.succ x) acc
+      (Some (partitionZ_fold_insert c x)).
+  Proof.
+    intros x acc c.
+    rewrite /partitionZ_fold_insert /partitionZ_aux_invariant
+            /partitionZ_fold_skip.
+    case c; last first. {
+      move => [H_inv] H_in.
+      rewrite /= interval_list_invariant_app_iff Z.add_1_r.
+      split; last done.
+      split; first done.
+      split; first done.
+      move => x1 x2.
+      rewrite InZ_rev InZ_cons InZ_nil In_elementsZ_single1.
+      move => H_x1_in [] // <-.
+      by apply H_in.
+    } {
+      move => [y c'].
+      rewrite /= !interval_list_invariant_app_iff
+        N2Z.inj_succ Z.add_succ_r .
+      rewrite !interval_list_invariant_cons !interval_list_invariant_nil.
+      move => [] [H_inv_acc] [] [] _ [H_c_neq_0] _
+        H_in_c ->.      
+      split; last done.
+      split; first done.
+      split. {
+        split; first done.
+        split; last done.
+        apply N.neq_succ_0.
+      } {
+        move => x1 x2.
+        rewrite InZ_cons InZ_nil In_elementsZ_single.
+        move => H_x1_in [] // [H_y_le] H_x2_lt.
+        apply Z.lt_le_trans with (m := y) => //. 
+        apply H_in_c => //.
+        rewrite InZ_cons In_elementsZ_single.
+        left.
+        split. {
+          apply Z.le_refl.
+        } {
+          by apply Z_lt_add_r.
+        }
+      }
+    }
+  Qed.
+
+  Lemma partitionZ_aux_invariant_skip : forall x acc c,
+    partitionZ_aux_invariant x acc c ->
+    partitionZ_aux_invariant (Z.succ x) (partitionZ_fold_skip acc c) None.
+  Proof.
+    intros x acc c.
+    rewrite /partitionZ_fold_skip /partitionZ_aux_invariant
+            /partitionZ_fold_skip.
+    case c; last first. {
+      move => [H_inv] H_in.
+      split; first done.
+      move => y' H_y'_in.
+      apply Z.lt_trans with (m := x). {
+        by apply H_in.
+      } {
+        apply Z.lt_succ_diag_r.
+      }
+    } {
+      move => [y c'] [H_inv] ->.
+      split => //.
+      move => y'.
+      rewrite InZ_cons In_elementsZ_single.
+      move => []. {
+        move => [_].
+        rewrite -Z.succ_lt_mono //.
+      } {
+        move : H_inv.
+        rewrite /= !interval_list_invariant_app_iff interval_list_invariant_cons.
+        move => [_] [] [_] [H_c'_neq] _ H_pre H_y'_in.
+        apply Z.lt_trans with (m := y). {
+          apply H_pre. {
+            by rewrite InZ_rev.
+          } {
+            rewrite InZ_cons.
+            left.
+            by apply In_elementsZ_single_hd.
+          }
+        }
+        apply Z.lt_succ_r, Z_le_add_r.
+      }
+    }
+  Qed.
+
+  Definition partitionZ_fold_current (c : option (Z * N)) :=
+    match c with
+       None => nil
+     | Some yc => yc::nil
+    end.
+  
+  Lemma InZ_partitionZ_fold_current_Some : forall yc y,
+     InZ y (partitionZ_fold_current (Some yc)) <->
+     InZ y (yc :: nil).
+  Proof. done. Qed.
+
+  Lemma InZ_partitionZ_fold_insert : forall c x y l,
+   match c with
+   | Some (y, c') => x = y + Z.of_N c'
+   | None => True
+   end -> (
+   InZ y (partitionZ_fold_insert c x :: l) <->
+    ((x = y) \/ InZ y (partitionZ_fold_current c) \/
+       InZ y l)).
+  Proof.
+    intros c x y l.
+    rewrite /partitionZ_fold_insert /partitionZ_fold_current
+            /partitionZ_fold_skip.
+    case c. {
+      move => [y' c'] ->.
+      rewrite !InZ_cons elementsZ_single_succ in_app_iff
+              InZ_nil /=.
+      tauto.
+    } {
+      rewrite InZ_cons InZ_nil In_elementsZ_single1.
+      tauto.
+    }
+  Qed.  
+
+  Lemma InZ_partitionZ_fold_skip : forall c acc y,
+    InZ y (partitionZ_fold_skip acc c) <->
+    (InZ y (partitionZ_fold_current c) \/ InZ y acc).
+  Proof.
+    intros c acc y.
+    rewrite /partitionZ_fold_skip /partitionZ_fold_current
+            /partitionZ_fold_skip.
+    case c. {
+      move => [y' c'].
+      rewrite !InZ_cons InZ_nil /=.
+      tauto.
+    } {
+      rewrite InZ_nil.
+      tauto.
+    }
+  Qed.  
+  
+  Lemma filterZ_single_aux_props :
+    forall f c x acc cur,
+      partitionZ_aux_invariant x acc cur ->
+      match (filterZ_single_aux f (acc, cur) x c) with
+        (acc', c') =>
+        let r := partitionZ_fold_skip acc' c' in
+        interval_list_invariant (List.rev r) = true /\
+        (forall y', InZ y' r <-> (InZ y' (partitionZ_fold_skip acc cur) \/
+                                    (f y' = true /\ List.In y' (elementsZ_single x c))))
+
+      end.
+  Proof.
+    intro f.
+    induction c as [| c' IH] using N.peano_ind. {
+      intros x acc cur.
+      rewrite /partitionZ_aux_invariant.
+      move => [H_inv] _.
+      rewrite /filterZ_single_aux fold_elementsZ_single_zero /=.
+      tauto.
+    }
+    intros x acc cur H_inv.
+    have -> :  filterZ_single_aux f (acc, cur) x (N.succ c') =
+               filterZ_single_aux f (filterZ_fold_fun f (acc, cur) x) (Z.succ x) c'. {
+        by rewrite /filterZ_single_aux fold_elementsZ_single_succ.
+    }
+    case_eq (filterZ_fold_fun f (acc, cur) x). 
+    move => acc' cur' H_fold_eq.
+
+    case_eq (filterZ_single_aux f (acc', cur') (Z.succ x) c').
+    move => acc'' cur'' H_succ_eq.
+
+    have H_inv' : partitionZ_aux_invariant (Z.succ x) acc' cur'. {
+      move : H_fold_eq H_inv.
+      rewrite /filterZ_fold_fun.
+      case (f x); move => [<-] <-. {
+        apply partitionZ_aux_invariant_insert.
+      } {
+        apply partitionZ_aux_invariant_skip.
+      }
+    }
+   
+    move : (IH (Z.succ x) acc' cur' H_inv') => {IH}.
+    rewrite H_succ_eq /=.
+    set r := partitionZ_fold_skip acc'' cur''.
+    move => [H_inv_r] H_in_r.
+    split; first assumption.
+
+    move => y'.
+    move : H_fold_eq.
+    rewrite H_in_r /filterZ_fold_fun.
+    case_eq (f x) => H_fx [<-] <-. {
+      rewrite InZ_partitionZ_fold_skip InZ_partitionZ_fold_current_Some InZ_partitionZ_fold_skip elementsZ_single_succ_front.
+      rewrite InZ_partitionZ_fold_insert; last first. {
+        move : H_inv.
+        rewrite /partitionZ_aux_invariant => [[_]].
+        case cur => //.
+      }
+      rewrite InZ_nil /=.
+      split; last by tauto.
+      move => []; last by tauto.
+      move => []; last by tauto.
+      move => []. {
+        move => <-.
+        tauto.
+      } {
+        tauto.
+      }
+    } {
+      rewrite InZ_partitionZ_fold_skip /partitionZ_fold_current InZ_partitionZ_fold_skip elementsZ_single_succ_front !InZ_nil /=.
+      split; first by tauto.
+      move => []; first by tauto.
+      move => [] H_fy' []. {
+        move => H_x_eq; subst y'.
+        contradict H_fy'.
+        by rewrite H_fx.
+      } {
+        tauto.
+      }
+    }
+  Qed.
+        
+
+  Lemma filterZ_single_props :
+    forall f c x acc,
+      interval_list_invariant (rev acc) = true ->
+      (forall y' : Z, InZ y' acc -> Z.succ y' < x) ->     
+      match (filterZ_single f acc x c) with
+        r =>
+        interval_list_invariant (List.rev r) = true /\
+        (forall y', InZ y' r <-> (InZ y' acc \/
+                                 (f y' = true /\ List.In y' (elementsZ_single x c))))
+
+      end.
+  Proof.
+    intros f c x acc.
+    move => H_inv H_acc.
+    rewrite /filterZ_single.
+
+    have H_inv' : partitionZ_aux_invariant x acc None. {
+      by rewrite /partitionZ_aux_invariant /=.
+    }
+    move : (filterZ_single_aux_props f c x acc None H_inv').
+    case_eq (filterZ_single_aux f (acc, None) x c).
+    move => acc' cur' /= H_res.
+    tauto.
+  Qed.
+
+  
+  Lemma filterZ_aux_props :
+    forall f s acc,
+      interval_list_invariant s = true ->
+      interval_list_invariant (rev acc) = true ->
+      (forall x1 x2 : Z, InZ x1 acc -> InZ x2 s -> Z.succ x1 < x2) ->   
+      match (filterZ_aux acc f s) with
+        r =>
+        interval_list_invariant r = true /\
+        (forall y', InZ y' r <-> (InZ y' acc \/
+                                 (f y' = true /\ InZ y' s)))
+      end.
+  Proof.
+    intro f.
+    induction s as [| [y c] s' IH]. {
+      intros acc.
+      move => _ H_inv _.
+      rewrite /filterZ_aux.
+      split; first assumption.
+      move => y'; rewrite InZ_rev InZ_nil; tauto.
+    } {
+      intros acc.
+      rewrite interval_list_invariant_cons.
+      move => [H_gr] [H_c_neq_0] H_inv_s' H_inv H_in_acc /=.
+      move : H_gr.
+      rewrite interval_list_elements_greater_alt2_def => // H_gr.
+
+      have H_pre : (forall y' : Z, InZ y' acc -> Z.succ y' < y). {
+        move => x1 H_x1_in.
+        apply H_in_acc => //.
+        rewrite InZ_cons.
+        by left; apply In_elementsZ_single_hd.
+      }
+      
+      move : (filterZ_single_props f c y acc H_inv H_pre) => {H_pre}.
+      set acc' := filterZ_single f acc y c.
+      move => [H_inv'] H_in_acc'.
+
+      have H_pre : (forall x1 x2 : Z,
+                      InZ x1 acc' -> InZ x2 s' -> Z.succ x1 < x2). {
+        move => x1 x2.
+        rewrite H_in_acc' In_elementsZ_single.
+        move => []. {
+          move => H_x1_in H_x2_in.
+          apply H_in_acc => //.
+          rewrite InZ_cons.
+          by right.
+        } {
+          move => [_] [_] H_x1_lt H_x2_in.
+          apply Z.le_lt_trans with (m := y + Z.of_N c).
+            - by apply Z.le_succ_l.
+            - by apply H_gr.
+        }
+      }
+      move : (IH acc' H_inv_s' H_inv' H_pre) => {H_pre}.
+
+      move => [H_inv_r] H_in_r.
+      split; first assumption.
+      move => y'.
+      rewrite H_in_r H_in_acc' InZ_cons.
+      tauto.
+    }
+  Qed.
+
+  Lemma filterZ_props :
+    forall f s,
+      interval_list_invariant s = true ->
+      match (filterZ f s) with r =>
+        interval_list_invariant r = true /\
+        (forall y', InZ y' r <-> (f y' = true /\ InZ y' s))
+      end.
+  Proof.
+    intros f s H_inv_s.
+    rewrite /filterZ.
+
+    have H_pre_1 : interval_list_invariant (rev nil) = true by done.
+    have H_pre_2 : (forall x1 x2 : Z, InZ x1 nil -> InZ x2 s -> Z.succ x1 < x2) by done.
+    
+    move : (filterZ_aux_props f s nil H_inv_s H_pre_1 H_pre_2) => {H_pre_1} {H_pre_2}.
+    move => [H_inv'] H_in_r.
+    split; first assumption.
+    move => y'.
+    rewrite H_in_r InZ_nil.
+    tauto.
+  Qed.
+
+  Global Instance filter_ok s f : forall `(Ok s), Ok (filter f s).
+  Proof.
+    move => [H_inv H_enc].
+    rewrite /filter.
+    set f' := (fun z : Z => f (Enc.decode z)).
+    move : (filterZ_props f' s H_inv).
+    move => [H_inv'] H_in_r.
+    rewrite /Ok /IsOk /is_encoded_elems_list.
+    split; first assumption.
+    move => x /H_in_r [_] H_x_in.
+    by apply H_enc.
+  Qed.
+
+  Lemma filter_spec :
+   forall (s : t) (x : elt) (f : elt -> bool),
+   Ok s ->      
+   (In x (filter f s) <-> In x s /\ f x = true).
+  Proof.
+    move => s x f H_ok.
+    suff H_suff :
+      (forall x, (InZ x (filter f s)) <->
+                 InZ x s /\ f (Enc.decode x) = true). {
+      rewrite !In_alt_def /elements !rev_map_alt_def
+              -!in_rev !in_map_iff.
+      setoid_rewrite H_suff.
+      rewrite /InZ.
+      split. {
+        move => [y] [<-] [?] ?.
+        split => //.
+        by exists y.
+      } {
+        move => [] [y] [<-] ? ?.
+        by exists y.
+      }
+    }
+    rewrite /filter.
+    set f' := (fun z : Z => f (Enc.decode z)).
+    move : (H_ok) => [H_inv _].
+    move : (filterZ_props f' s H_inv).
+    move => [H_inv'] H_in_r.
+    move => y; rewrite H_in_r; tauto.
+  Qed.
+  
+
+  (** *** partition specification *)    
+
+  Lemma partitionZ_single_aux_alt_def : forall f c y acc_t c_t acc_f c_f,
+    partitionZ_single_aux f ((acc_t, c_t), (acc_f, c_f)) y c =
+    (filterZ_single_aux f (acc_t, c_t) y c,
+     filterZ_single_aux (fun x : Z => negb (f x)) (acc_f, c_f) y c).
+  Proof.
+    intros f.
+    rewrite /partitionZ_single_aux /filterZ_single_aux.
+    induction c as [| c' IH] using N.peano_ind. {
+      intros y acc_t c_t acc_f c_f.
+      rewrite !fold_elementsZ_single_zero //.                       } {
+      intros y acc_t c_t acc_f c_f.
+      rewrite !fold_elementsZ_single_succ.
+      case_eq (partitionZ_fold_fun f (acc_t, c_t, (acc_f, c_f)) y) => [] [acc_t' c_t'] [acc_f' c_f'] H_fold_eq.
+      rewrite IH => {IH}.
+      suff : (filterZ_fold_fun f (acc_t, c_t) y = (acc_t', c_t')) /\
+             (filterZ_fold_fun (fun x : Z => negb (f x)) (acc_f, c_f) y = (acc_f', c_f')). {
+        move => [->] -> //.
+      }
+      move : H_fold_eq.
+      rewrite /partitionZ_fold_fun /filterZ_fold_fun.
+      case (f y); move => [<-] <- <- <- //.
+    }
+  Qed.
+                      
+  Lemma partitionZ_aux_alt_def : forall f s acc_t acc_f,
+   partitionZ_aux acc_t acc_f f s =
+   (filterZ_aux acc_t f s,
+    filterZ_aux acc_f (fun x : Z => negb (f x)) s).
+  Proof.
+    intros f.
+    induction s as [| [y c] s' IH]. {
+      done.
+    } {
+      intros acc_t acc_f.
+      rewrite /= /partitionZ_single /filterZ_single
+              partitionZ_single_aux_alt_def.
+      case (filterZ_single_aux f (acc_t, None) y c) => acc_t' c_t'.
+      case (filterZ_single_aux (fun x : Z => negb (f x)) (acc_f, None) y c) => acc_f' c_f'.
+      rewrite IH //.
+    }
+  Qed.
+      
+  Lemma partitionZ_alt_def : forall f s,
+    partitionZ f s = (filterZ f s,
+                      filterZ (fun x => negb (f x)) s).
+  Proof.
+    intros f s.
+    rewrite /partitionZ /filterZ
+            partitionZ_aux_alt_def //.    
+  Qed.
+
+  Lemma partition_alt_def : forall f s,
+    partition f s = (filter f s,
+                     filter (fun x => negb (f x)) s).
+  Proof.
+    intros f s.
+    rewrite /partition /filter partitionZ_alt_def.
+    done.
   Qed.
 
-  (** *** partition specification *)
 
   Global Instance partition_ok1 s f : forall `(Ok s), Ok (fst (partition f s)).
   Proof.
-    intros H_ok.
-    rewrite /partition /fst.
+    move => H_ok.
+    rewrite partition_alt_def /fst.
     by apply filter_ok.
-  Qed.
+  Qed.  
 
   Global Instance partition_ok2 s f : forall `(Ok s), Ok (snd (partition f s)).
   Proof.
-    intros H_ok.
-    rewrite /partition /snd.
+    move => H_ok.
+    rewrite partition_alt_def /snd.
     by apply filter_ok.
   Qed.
 
   Lemma partition_spec1 :
    forall (s : t) (f : elt -> bool),
-   Proper (Enc.E.eq==>eq) f -> Equal (fst (partition f s)) (filter f s).
+   Equal (fst (partition f s)) (filter f s).
   Proof.
-    intros s f H_ok.
-    rewrite /partition /fst /Equal //.
+    intros s f.
+    rewrite partition_alt_def /fst /Equal //.
   Qed.
 
   Lemma partition_spec2 :
    forall (s : t) (f : elt -> bool),
-   Proper (Enc.E.eq==>eq) f ->
+   Ok s ->    
    Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
   Proof.
-    intros s f H_ok.
-    rewrite /partition /snd /Equal //.
+    intros s f.
+    rewrite partition_alt_def /snd /Equal //.
   Qed.
-
+  
 End Raw.
 
 
@@ -2404,7 +5660,7 @@ End Raw.
     We can now build the invariant into the set type to obtain an instantiation
     of module type [WSetsOn]. *)
 
-Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
+Module MSetIntervals (Enc : ElementEncode) <: SetsOn Enc.E.
   Module E := Enc.E.
   Module Raw := Raw Enc.
 
@@ -2416,7 +5672,7 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
  Record t_ := Mkt {this :> Raw.t; is_ok : Raw.Ok this}.
  Definition t := t_.
  Arguments Mkt this {is_ok}.
- Hint Resolve is_ok : typeclass_instances.
+ #[local] Hint Resolve is_ok : typeclass_instances.
 
  Definition In (x : elt)(s : t) := Raw.In x s.(this).
  Definition Equal (s s' : t) := forall a : elt, In a s <-> In a s'.
@@ -2437,7 +5693,10 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
  Definition empty : t := Mkt Raw.empty.
  Definition is_empty (s : t) := Raw.is_empty s.
  Definition elements (s : t) : list elt := Raw.elements s.
+ Definition min_elt (s : t) : option elt := Raw.min_elt s.
+ Definition max_elt (s : t) : option elt := Raw.max_elt s.
  Definition choose (s : t) : option elt := Raw.choose s.
+ Definition compare (s1 s2 : t) : comparison := Raw.compare s1 s2.
  Definition fold {A : Type}(f : elt -> A -> A)(s : t) : A -> A := Raw.fold f s.
  Definition cardinal (s : t) := Raw.cardinal s.
  Definition filter (f : elt -> bool)(s : t) : t := Mkt (Raw.filter f s).
@@ -2446,7 +5705,7 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
  Definition partition (f : elt -> bool)(s : t) : t * t :=
    let p := Raw.partition f s in (Mkt (fst p), Mkt (snd p)).
 
- Instance In_compat : Proper (E.eq==>eq==>iff) In.
+ #[local] Instance In_compat : Proper (E.eq==>eq==>iff) In.
  Proof. 
    repeat red.
    move => x y H_eq_xy x' y' ->.
@@ -2457,10 +5716,9 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
 
  Definition eq : t -> t -> Prop := Equal.
 
- Instance eq_equiv : Equivalence eq.
+ #[local] Instance eq_equiv : Equivalence eq.
  Proof. firstorder. Qed.
 
-
  Definition eq_dec : forall (s s':t), { eq s s' }+{ ~eq s s' }.
  Proof.
   intros (s,Hs) (s',Hs').
@@ -2469,7 +5727,39 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
    rewrite <- Raw.equal_spec; congruence.
  Defined.
 
+ Definition lt : t -> t -> Prop := Raw.lt.
+
+ #[local] Instance lt_strorder : StrictOrder lt.
+ Proof.
+   unfold lt.
+   constructor. {
+     move : Raw.lt_Irreflexive.
+     rewrite /Irreflexive /complement /Reflexive.
+     move => H x.
+     apply H.
+   } {
+     move : Raw.lt_Transitive.
+     rewrite /Transitive.
+     move => H x y z.
+     apply H.
+   }
+ Qed.
 
+ #[local] Instance lt_compat : Proper (eq==>eq==>iff) lt.
+ Proof.
+   repeat red.
+   move => [x1 H_x1_ok] [y1 H_y1_ok] H_eq.
+   move => [x2 H_x2_ok] [y2 H_y2_ok].
+   move : H_eq.
+   rewrite /eq /lt /Equal /In /=.
+   replace (forall a : elt, Raw.In a x1 <-> Raw.In a y1) with
+     (Raw.Equal x1 y1) by done.
+   replace (forall a : elt, Raw.In a x2 <-> Raw.In a y2) with
+     (Raw.Equal x2 y2) by done.
+   rewrite -!Raw.equal_spec !Raw.equal_alt_def.
+   move => -> -> //.
+ Qed.
+ 
  Section Spec.
   Variable s s' : t.
   Variable x y : elt.
@@ -2505,7 +5795,7 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
   Proof. exact (@Raw.cardinal_spec s). Qed.
   Lemma filter_spec : compatb f ->
     (In x (filter f s) <-> In x s /\ f x = true).
-  Proof. exact (@Raw.filter_spec _ _ _). Qed.
+  Proof. move => _; exact (@Raw.filter_spec _ _ _ _). Qed.
   Lemma for_all_spec : compatb f ->
     (for_all f s = true <-> For_all (fun x => f x = true) s).
   Proof. exact (@Raw.for_all_spec _ _ _). Qed.
@@ -2513,19 +5803,60 @@ Module MSetIntervals (Enc : ElementEncode) <: WSetsOn Enc.E.
     (exists_ f s = true <-> Exists (fun x => f x = true) s).
   Proof. exact (@Raw.exists_spec _ _ _). Qed.
   Lemma partition_spec1 : compatb f -> Equal (fst (partition f s)) (filter f s).
-  Proof. exact (@Raw.partition_spec1 _ _). Qed.
+  Proof. move => _; exact (@Raw.partition_spec1 _ _). Qed.
   Lemma partition_spec2 : compatb f ->
       Equal (snd (partition f s)) (filter (fun x => negb (f x)) s).
-  Proof. exact (@Raw.partition_spec2 _ _). Qed.
+  Proof. move => _; exact (@Raw.partition_spec2 _ _ _). Qed.
   Lemma elements_spec1 : InA E.eq x (elements s) <-> In x s.
   Proof. rewrite /In /Raw.In /elements //. Qed.
   Lemma elements_spec2w : NoDupA E.eq (elements s).
   Proof. exact (Raw.elements_spec2w _ _). Qed.
+  Lemma elements_spec2 : sort E.lt (elements s).  
+  Proof. exact (Raw.elements_sorted _ _). Qed.
   Lemma choose_spec1 : choose s = Some x -> In x s.
-  Proof. exact (Raw.choose_spec1 _ _). Qed.
+  Proof. exact (Raw.choose_spec1 _ _ _). Qed.
   Lemma choose_spec2 : choose s = None -> Empty s.
   Proof. exact (Raw.choose_spec2 _). Qed.
+  Lemma choose_spec3 : choose s = Some x -> choose s' = Some y ->
+    Equal s s' -> E.eq x y.
+  Proof.
+    intros H1 H2 H3.
+    suff -> : x = y. {
+      apply E.eq_equiv.
+    }
+    move : H1 H2 H3.
+    exact (Raw.choose_spec3 _ _ _ _ _ _).
+  Qed.
 
+  Lemma min_elt_spec1 : choose s = Some x -> In x s.
+  Proof. exact (Raw.min_elt_spec1 _ _ _). Qed.
+  Lemma min_elt_spec2 : min_elt s = Some x -> In y s -> ~ E.lt y x.
+  Proof. exact (Raw.min_elt_spec2 _ _ _ _). Qed.
+  Lemma min_elt_spec3 : choose s = None -> Empty s.
+  Proof. exact (Raw.min_elt_spec3 _). Qed.
+  
+  Lemma max_elt_spec1 : max_elt s = Some x -> In x s.
+  Proof. exact (Raw.max_elt_spec1 _ _ _). Qed.
+  Lemma max_elt_spec2 : max_elt s = Some x -> In y s -> ~ E.lt x y.
+  Proof. exact (Raw.max_elt_spec2 _ _ _ _). Qed.
+  Lemma max_elt_spec3 : max_elt s = None -> Empty s.
+  Proof. exact (Raw.max_elt_spec3 _). Qed.
+
+  Lemma compare_spec : CompSpec eq lt s s' (compare s s').
+  Proof.
+    generalize s s'.
+    move => [s1 H_ok_s1] [s2 H_ok_s2].
+    move : (Raw.compare_spec s1 s2).
+    rewrite /CompSpec /eq /Equal /In /lt /compare /=.
+    replace (forall a : elt, Raw.In a s1 <-> Raw.In a s2) with
+    (Raw.Equal s1 s2) by done.
+    suff H_eq : (Raw.Equal s1 s2) <-> (s1 = s2). {
+      move => [] H; constructor => //.
+      by rewrite H_eq.
+    }
+    rewrite -Raw.equal_spec Raw.equal_alt_def //.
+  Qed.
+  
  End Spec.
 
 End MSetIntervals.
@@ -2552,10 +5883,14 @@ Module ElementEncodeZ <: ElementEncode.
     (Z.eq (encode e1) (encode e2)) <-> E.eq e1 e2.
   Proof. by []. Qed.
 
+  Lemma encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+  Proof. by []. Qed.
+  
 End ElementEncodeZ.
 
-Module MSetIntervalsZ <: WSetsOn Z := MSetIntervals ElementEncodeZ.
 
+Module MSetIntervalsZ <: SetsOn Z := MSetIntervals ElementEncodeZ.
 
 (** *** N *)
 
@@ -2578,14 +5913,22 @@ Module ElementEncodeN <: ElementEncode.
     intros e1 e2.
     rewrite /encode /Z.eq N2Z.inj_iff /E.eq //.
   Qed.
+
+  Lemma encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+  Proof.
+    intros e1 e2.
+    rewrite /encode -N2Z.inj_lt //.
+  Qed.
+
 End ElementEncodeN.
 
-Module MSetIntervalsN <: WSetsOn N := MSetIntervals ElementEncodeN.
+Module MSetIntervalsN <: SetsOn N := MSetIntervals ElementEncodeN.
 
 
 (** *** nat *)
 Module ElementEncodeNat <: ElementEncode.
-  Module E := NPeano.Nat.
+  Module E := Nat.
 
   Definition encode (n : nat) := Z.of_nat n.
   Definition decode (z : Z) := Z.to_nat z.
@@ -2603,7 +5946,17 @@ Module ElementEncodeNat <: ElementEncode.
     intros e1 e2.
     rewrite /encode /Z.eq Nat2Z.inj_iff /E.eq //.
   Qed.
+
+  Lemma encode_lt : forall (e1 e2 : E.t),
+    (Z.lt (encode e1) (encode e2)) <-> E.lt e1 e2.
+  Proof.
+    intros e1 e2.
+    rewrite /encode -Nat2Z.inj_lt //.
+  Qed.
+
 End ElementEncodeNat.
 
-Module MSetIntervalsNat <: WSetsOn NPeano.Nat := MSetIntervals ElementEncodeNat.
+Module MSetIntervalsNat <: SetsOn Nat := MSetIntervals ElementEncodeNat.
+
+
 
diff --git a/MSetListWithDups.v b/MSetListWithDups.v
index 6ad388a..fefef58 100644
--- a/MSetListWithDups.v
+++ b/MSetListWithDups.v
@@ -34,7 +34,7 @@
 
 
 Require Export MSetInterface.
-Require Import ssreflect.
+Require Import mathcomp.ssreflect.ssreflect.
 Require Import List OrdersFacts OrdersLists.
 Require Import Sorting Permutation.
 Require Import MSetWithDups.
@@ -325,8 +325,7 @@ Module RemoveDupsFromSorted (Import X:OrderedType).
       rewrite !InA_alt.
       setoid_rewrite H_in_sort => //.
     } {
-
-      move : (LocallySorted_sort l).
+      move : (Sorted_sort l).
       rewrite /is_true /le /leb //.
     }
   Qed.
@@ -447,7 +446,7 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
   (** Logical predicates *)
   Definition In x (s : t) := SetoidList.InA E.eq x s.
 
-  Instance In_compat : Proper (E.eq==>eq==>iff) In.
+  #[local] Instance In_compat : Proper (E.eq==>eq==>iff) In.
   Proof. repeat red. intros. rewrite H H0. auto. Qed.
 
   Definition Equal s s' := forall a : elt, In a s <-> In a s'.
@@ -489,9 +488,9 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
       case (E.compare x y). {
         tauto.
       } {
-        rewrite IH; intuition; inversion H.
+        rewrite IH; intuition; inversion H1.
       } {
-        rewrite IH; intuition; inversion H.
+        rewrite IH; intuition; inversion H1.
       } 
     }
  Qed.
@@ -570,10 +569,6 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
     }
   Qed.       
 
-  Hint Resolve (@Equivalence_Reflexive _ _ E.eq_equiv).
-  Hint Immediate (@Equivalence_Symmetric _ _ E.eq_equiv).
-  Hint Resolve (@Equivalence_Transitive _ _ E.eq_equiv).
-
   Lemma rev_filter_aux_spec : forall s acc x f, compatb f ->
     (In x (rev_filter_aux acc f s) <-> (In x s /\ f x = true) \/ (In x acc)).
   Proof. 
@@ -717,7 +712,7 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
         }
         apply H.
         apply InA_cons_hd.
-        done.
+        apply (@Equivalence_Reflexive _ _ E.eq_equiv).
       }
     }
   Qed.
@@ -738,7 +733,7 @@ Module Make (E:OrderedType) <: WSetsOnWithDupsExtra E.
         exists x.
         split => //.
         apply InA_cons_hd.
-        done.
+        apply (@Equivalence_Reflexive _ _ E.eq_equiv).
       } {
         rewrite IH.
         split. {
diff --git a/MSetWithDups.v b/MSetWithDups.v
index f92023d..87d7d4b 100644
--- a/MSetWithDups.v
+++ b/MSetWithDups.v
@@ -50,7 +50,7 @@
 *)
 
 Require Import Coq.MSets.MSetInterface.
-Require Import ssreflect. 
+Require Import mathcomp.ssreflect.ssreflect.
 
 (** ** WSetsOnWithDups
 
@@ -67,7 +67,7 @@ Module Type WSetsOnWithDups (E : DecidableType).
   Include WOps E.
 
   Parameter In : elt -> t -> Prop.
-  Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.
+  #[local] Declare Instance In_compat : Proper (E.eq==>eq==>iff) In.
 
   Definition Equal s s' := forall a : elt, In a s <-> In a s'.
   Definition Subset s s' := forall a : elt, In a s -> In a s'.
diff --git a/Makefile b/Makefile
index f389935..b10449a 100644
--- a/Makefile
+++ b/Makefile
@@ -1,85 +1,953 @@
-# File: Makefile
+##########################################################################
+##         #   The Coq Proof Assistant / The Coq Development Team       ##
+##  v      #         Copyright INRIA, CNRS and contributors             ##
+## <O___,, # (see version control and CREDITS file for authors & dates) ##
+##   \VV/  ###############################################################
+##    //   #    This file is distributed under the terms of the         ##
+##         #     GNU Lesser General Public License Version 2.1          ##
+##         #     (see LICENSE file for the text of the license)         ##
+##########################################################################
+## GNUMakefile for Coq 8.16.1
+
+# For debugging purposes (must stay here, don't move below)
+INITIAL_VARS := $(.VARIABLES)
+# To implement recursion we save the name of the main Makefile
+SELF := $(lastword $(MAKEFILE_LIST))
+PARENT := $(firstword $(MAKEFILE_LIST))
+
+# This file is generated by coq_makefile and contains many variable
+# definitions, like the list of .v files or the path to Coq
+include Makefile.conf
+
+# Put in place old names
+VFILES            := $(COQMF_VFILES)
+MLIFILES          := $(COQMF_MLIFILES)
+MLFILES           := $(COQMF_MLFILES)
+MLGFILES          := $(COQMF_MLGFILES)
+MLPACKFILES       := $(COQMF_MLPACKFILES)
+MLLIBFILES        := $(COQMF_MLLIBFILES)
+METAFILE          := $(COQMF_METAFILE)
+CMDLINE_VFILES    := $(COQMF_CMDLINE_VFILES)
+INSTALLCOQDOCROOT := $(COQMF_INSTALLCOQDOCROOT)
+OTHERFLAGS        := $(COQMF_OTHERFLAGS)
+COQCORE_SRC_SUBDIRS := $(COQMF_COQ_SRC_SUBDIRS)
+OCAMLLIBS         := $(COQMF_OCAMLLIBS)
+SRC_SUBDIRS       := $(COQMF_SRC_SUBDIRS)
+COQLIBS           := $(COQMF_COQLIBS)
+COQLIBS_NOML      := $(COQMF_COQLIBS_NOML)
+CMDLINE_COQLIBS   := $(COQMF_CMDLINE_COQLIBS)
+COQLIB            := $(COQMF_COQLIB)
+COQCORELIB        := $(COQMF_COQCORELIB)
+DOCDIR            := $(COQMF_DOCDIR)
+OCAMLFIND         := $(COQMF_OCAMLFIND)
+CAMLFLAGS         := $(COQMF_CAMLFLAGS)
+HASNATDYNLINK     := $(COQMF_HASNATDYNLINK)
+OCAMLWARN         := $(COQMF_WARN)
+
+Makefile.conf: _CoqProject
+	/snap/coq-prover/31/coq-platform/bin/coq_makefile COQBIN = /snap/coq-prover/31/coq-platform/bin/ -f _CoqProject -o Makefile
+
+# This file can be created by the user to hook into double colon rules or
+# add any other Makefile code he may need
+-include Makefile.local
+
+# Parameters ##################################################################
+#
+# Parameters are make variable assignments.
+# They can be passed to (each call to) make on the command line.
+# They can also be put in Makefile.local once and for all.
+# For retro-compatibility reasons they can be put in the _CoqProject, but this
+# practice is discouraged since _CoqProject better not contain make specific
+# code (be nice to user interfaces).
+
+# Set KEEP_ERROR to have make keep files produced by failing rules.
+# By default, KEEP_ERROR is empty. So for instance if coqc creates a .vo but
+# then fails to native compile, the .vo will be deleted.
+# May confuse make so use only for debugging.
+KEEP_ERROR?=
+ifeq (,$(KEEP_ERROR))
+.DELETE_ON_ERROR:
+endif
+
+# Print shell commands (set to non empty)
+VERBOSE ?=
+
+# Time the Coq process (set to non empty), and how (see default value)
+TIMED?=
+TIMECMD?=
+# Use command time on linux, gtime on Mac OS
+TIMEFMT?="$@ (real: %e, user: %U, sys: %S, mem: %M ko)"
+ifneq (,$(TIMED))
+ifeq (0,$(shell command time -f "" true >/dev/null 2>/dev/null; echo $$?))
+STDTIME?=command time -f $(TIMEFMT)
+else
+ifeq (0,$(shell gtime -f "" true >/dev/null 2>/dev/null; echo $$?))
+STDTIME?=gtime -f $(TIMEFMT)
+else
+STDTIME?=command time
+endif
+endif
+else
+STDTIME?=command time -f $(TIMEFMT)
+endif
+
+COQBIN?=
+ifneq (,$(COQBIN))
+# add an ending /
+COQBIN:=$(COQBIN)/
+endif
+
+# Coq binaries
+COQC     ?= "$(COQBIN)coqc"
+COQTOP   ?= "$(COQBIN)coqtop"
+COQCHK   ?= "$(COQBIN)coqchk"
+COQNATIVE ?= "$(COQBIN)coqnative"
+COQDEP   ?= "$(COQBIN)coqdep"
+COQDOC   ?= "$(COQBIN)coqdoc"
+COQPP    ?= "$(COQBIN)coqpp"
+COQMKFILE ?= "$(COQBIN)coq_makefile"
+OCAMLLIBDEP ?= "$(COQBIN)ocamllibdep"
+
+# Timing scripts
+COQMAKE_ONE_TIME_FILE ?= "$(COQCORELIB)/tools/make-one-time-file.py"
+COQMAKE_BOTH_TIME_FILES ?= "$(COQCORELIB)/tools/make-both-time-files.py"
+COQMAKE_BOTH_SINGLE_TIMING_FILES ?= "$(COQCORELIB)/tools/make-both-single-timing-files.py"
+BEFORE ?=
+AFTER ?=
+
+# OCaml binaries
+CAMLC       ?= "$(OCAMLFIND)" ocamlc   -c
+CAMLOPTC    ?= "$(OCAMLFIND)" opt      -c
+CAMLLINK    ?= "$(OCAMLFIND)" ocamlc   -linkall
+CAMLOPTLINK ?= "$(OCAMLFIND)" opt      -linkall
+CAMLDOC     ?= "$(OCAMLFIND)" ocamldoc
+CAMLDEP     ?= "$(OCAMLFIND)" ocamldep -slash -ml-synonym .mlpack
+
+# DESTDIR is prepended to all installation paths
+DESTDIR ?=
+
+# Debug builds, typically -g to OCaml, -debug to Coq.
+CAMLDEBUG ?=
+COQDEBUG ?=
+
+# Extra packages to be linked in (as in findlib -package)
+CAMLPKGS ?=
+FINDLIBPKGS = -package coq-core.plugins.ltac $(CAMLPKGS)
+
+# Option for making timing files
+TIMING?=
+# Option for changing sorting of timing output file
+TIMING_SORT_BY ?= auto
+# Option for changing the fuzz parameter on the output file
+TIMING_FUZZ ?= 0
+# Option for changing whether to use real or user time for timing tables
+TIMING_REAL?=
+# Option for including the memory column(s)
+TIMING_INCLUDE_MEM?=
+# Option for sorting by the memory column
+TIMING_SORT_BY_MEM?=
+# Output file names for timed builds
+TIME_OF_BUILD_FILE               ?= time-of-build.log
+TIME_OF_BUILD_BEFORE_FILE        ?= time-of-build-before.log
+TIME_OF_BUILD_AFTER_FILE         ?= time-of-build-after.log
+TIME_OF_PRETTY_BUILD_FILE        ?= time-of-build-pretty.log
+TIME_OF_PRETTY_BOTH_BUILD_FILE   ?= time-of-build-both.log
+TIME_OF_PRETTY_BUILD_EXTRA_FILES ?= - # also output to the command line
+
+TGTS ?=
+
+# Retro compatibility (DESTDIR is standard on Unix, DSTROOT is not)
+ifdef DSTROOT
+DESTDIR := $(DSTROOT)
+endif
+
+# Substitution of the path by appending $(DESTDIR) if needed.
+# The variable $(COQMF_WINDRIVE) can be needed for Cygwin environments.
+windrive_path = $(if $(COQMF_WINDRIVE),$(subst $(COQMF_WINDRIVE),/,$(1)),$(1))
+destination_path = $(if $(DESTDIR),$(DESTDIR)/$(call windrive_path,$(1)),$(1))
+
+# Installation paths of libraries and documentation.
+COQLIBINSTALL ?= $(call destination_path,$(COQLIB)/user-contrib)
+COQDOCINSTALL ?= $(call destination_path,$(DOCDIR)/coq/user-contrib)
+COQPLUGININSTALL ?= $(call destination_path,$(COQCORELIB)/..)
+COQTOPINSTALL ?= $(call destination_path,$(COQLIB)/toploop) # FIXME: Unused variable?
+
+# findlib files installation
+FINDLIBPREINST= mkdir -p "$(COQPLUGININSTALL)/"
+FINDLIBDESTDIR= -destdir "$(COQPLUGININSTALL)/"
+
+# we need to move out of sight $(METAFILE) otherwise findlib thinks the
+# package is already installed
+findlib_install = \
+	$(HIDE)if [ "$(METAFILE)" ]; then \
+	  $(FINDLIBPREINST) && \
+	  mv "$(METAFILE)" "$(METAFILE).skip" ; \
+	  "$(OCAMLFIND)" install $(2) $(FINDLIBDESTDIR) $(FINDLIBPACKAGE) $(1); \
+	  rc=$$?; \
+	  mv "$(METAFILE).skip" "$(METAFILE)"; \
+	  exit $$rc; \
+	fi
+findlib_remove = \
+	$(HIDE)if [ ! -z "$(METAFILE)" ]; then\
+	  "$(OCAMLFIND)" remove $(FINDLIBDESTDIR) $(FINDLIBPACKAGE); \
+	fi
+
+
+########## End of parameters ##################################################
+# What follows may be relevant to you only if you need to
+# extend this Makefile.  If so, look for 'Extension point' here and
+# put in Makefile.local double colon rules accordingly.
+# E.g. to perform some work after the all target completes you can write
+#
+# post-all::
+# 	echo "All done!"
+#
+# in Makefile.local
+#
+###############################################################################
+
+
+
+
+# Flags #######################################################################
+#
+# We define a bunch of variables combining the parameters.
+# To add additional flags to coq, coqchk or coqdoc, set the
+# {COQ,COQCHK,COQDOC}EXTRAFLAGS variable to whatever you want to add.
+# To overwrite the default choice and set your own flags entirely, set the
+# {COQ,COQCHK,COQDOC}FLAGS variable.
+
+SHOW := $(if $(VERBOSE),@true "",@echo "")
+HIDE := $(if $(VERBOSE),,@)
+
+TIMER=$(if $(TIMED), $(STDTIME), $(TIMECMD))
+
+OPT?=
+
+# The DYNOBJ and DYNLIB variables are used by "coqdep -dyndep var" in .v.d
+ifeq '$(OPT)' '-byte'
+USEBYTE:=true
+DYNOBJ:=.cma
+DYNLIB:=.cma
+else
+USEBYTE:=
+DYNOBJ:=.cmxs
+DYNLIB:=.cmxs
+endif
+
+# these variables are meant to be overridden if you want to add *extra* flags
+COQEXTRAFLAGS?=
+COQCHKEXTRAFLAGS?=
+COQDOCEXTRAFLAGS?=
+
+# Find the last argument of the form "-native-compiler FLAG"
+COQUSERNATIVEFLAG:=$(strip \
+$(subst -native-compiler-,,\
+$(lastword \
+$(filter -native-compiler-%,\
+$(subst -native-compiler ,-native-compiler-,\
+$(strip $(COQEXTRAFLAGS)))))))
+
+COQFILTEREDEXTRAFLAGS:=$(strip \
+$(filter-out -native-compiler-%,\
+$(subst -native-compiler ,-native-compiler-,\
+$(strip $(COQEXTRAFLAGS)))))
+
+COQACTUALNATIVEFLAG:=$(lastword $(COQMF_COQ_NATIVE_COMPILER_DEFAULT) $(COQMF_COQPROJECTNATIVEFLAG) $(COQUSERNATIVEFLAG))
+
+ifeq '$(COQACTUALNATIVEFLAG)' 'yes'
+  COQNATIVEFLAG="-w" "-deprecated-native-compiler-option" "-native-compiler" "ondemand"
+  COQDONATIVE="yes"
+else
+ifeq '$(COQACTUALNATIVEFLAG)' 'ondemand'
+  COQNATIVEFLAG="-w" "-deprecated-native-compiler-option" "-native-compiler" "ondemand"
+  COQDONATIVE="no"
+else
+  COQNATIVEFLAG="-w" "-deprecated-native-compiler-option" "-native-compiler" "no"
+  COQDONATIVE="no"
+endif
+endif
+
+# these flags do NOT contain the libraries, to make them easier to overwrite
+COQFLAGS?=-q $(OTHERFLAGS) $(COQFILTEREDEXTRAFLAGS) $(COQNATIVEFLAG)
+COQCHKFLAGS?=-silent -o $(COQCHKEXTRAFLAGS)
+COQDOCFLAGS?=-interpolate -utf8 $(COQDOCEXTRAFLAGS)
+
+COQDOCLIBS?=$(COQLIBS_NOML)
+
+# The version of Coq being run and the version of coq_makefile that
+# generated this makefile
+COQ_VERSION:=$(shell $(COQC) --print-version | cut -d " " -f 1)
+COQMAKEFILE_VERSION:=8.16.1
+
+# COQ_SRC_SUBDIRS is for user-overriding, usually to add
+# `user-contrib/Foo` to the includes, we keep COQCORE_SRC_SUBDIRS for
+# Coq's own core libraries, which should be replaced by ocamlfind
+# options at some point.
+COQ_SRC_SUBDIRS?=
+COQSRCLIBS?= $(foreach d,$(COQ_SRC_SUBDIRS), -I "$(COQLIB)/$(d)")
+
+CAMLFLAGS+=$(OCAMLLIBS) $(COQSRCLIBS)
+# ocamldoc fails with unknown argument otherwise
+CAMLDOCFLAGS:=$(filter-out -annot, $(filter-out -bin-annot, $(CAMLFLAGS)))
+CAMLFLAGS+=$(OCAMLWARN)
+
+ifneq (,$(TIMING))
+TIMING_ARG=-time
+ifeq (after,$(TIMING))
+TIMING_EXT=after-timing
+else
+ifeq (before,$(TIMING))
+TIMING_EXT=before-timing
+else
+TIMING_EXT=timing
+endif
+endif
+else
+TIMING_ARG=
+endif
+
+# Files #######################################################################
 #
-# Author: FireEye, Inc. - Formal Methods Team <formal-methods@fireeye.com>
+# We here define a bunch of variables about the files being part of the
+# Coq project in order to ease the writing of build target and build rules
 
-COQINCLUDES=-I .
-COQC=coqtop -q $(COQINCLUDES) -batch -compile
-COQDOC=coqdoc
-COQDEP=coqdep $(COQINCLUDES)
-COQFILES=\
-  MSetWithDups.v \
-  MSetFoldWithAbort.v \
-  MSetIntervals.v \
-  MSetListWithDups.v
+VDFILE := .Makefile.d
 
+ALLSRCFILES := \
+	$(MLGFILES) \
+	$(MLFILES) \
+	$(MLPACKFILES) \
+	$(MLLIBFILES) \
+	$(MLIFILES)
 
-#######################################
-########## GLOBAL RULES ###############
+# helpers
+vo_to_obj = $(addsuffix .o,\
+  $(filter-out Warning: Error:,\
+  $(shell $(COQTOP) -q -noinit -batch -quiet -print-mod-uid $(1))))
+strip_dotslash = $(patsubst ./%,%,$(1))
 
-.DEFAULT_GOAL: all
+# without this we get undefined variables in the expansion for the
+# targets of the [deprecated,use-mllib-or-mlpack] rule
+with_undef = $(if $(filter-out undefined, $(origin $(1))),$($(1)))
 
-.PHONY: all coqhtml coqtex clean doc
+VO = vo
+VOS = vos
 
+VOFILES = $(VFILES:.v=.$(VO))
+GLOBFILES = $(VFILES:.v=.glob)
+HTMLFILES = $(VFILES:.v=.html)
+GHTMLFILES = $(VFILES:.v=.g.html)
+BEAUTYFILES = $(addsuffix .beautified,$(VFILES))
+TEXFILES = $(VFILES:.v=.tex)
+GTEXFILES = $(VFILES:.v=.g.tex)
+CMOFILES = \
+	$(MLGFILES:.mlg=.cmo) \
+	$(MLFILES:.ml=.cmo) \
+	$(MLPACKFILES:.mlpack=.cmo)
+CMXFILES = $(CMOFILES:.cmo=.cmx)
+OFILES = $(CMXFILES:.cmx=.o)
+CMAFILES = $(MLLIBFILES:.mllib=.cma) $(MLPACKFILES:.mlpack=.cma)
+CMXAFILES = $(CMAFILES:.cma=.cmxa)
+CMIFILES = \
+	$(CMOFILES:.cmo=.cmi) \
+	$(MLIFILES:.mli=.cmi)
+# the /if/ is because old _CoqProject did not list a .ml(pack|lib) but just
+# a .mlg file
+CMXSFILES = \
+	$(MLPACKFILES:.mlpack=.cmxs) \
+	$(CMXAFILES:.cmxa=.cmxs) \
+	$(if $(MLPACKFILES)$(CMXAFILES),,\
+		$(MLGFILES:.mlg=.cmxs) $(MLFILES:.ml=.cmxs))
 
-proof: $(COQFILES:.v=.vo)
-doc: coqhtml coqtex doc/main.tex
-all: proof doc
+# files that are packed into a plugin (no extension)
+PACKEDFILES = \
+	$(call strip_dotslash, \
+	  $(foreach lib, \
+	    $(call strip_dotslash, \
+	       $(MLPACKFILES:.mlpack=_MLPACK_DEPENDENCIES)),$(call with_undef,$(lib))))
+# files that are archived into a .cma (mllib)
+LIBEDFILES = \
+	$(call strip_dotslash, \
+	  $(foreach lib, \
+	    $(call strip_dotslash, \
+	       $(MLLIBFILES:.mllib=_MLLIB_DEPENDENCIES)),$(call with_undef,$(lib))))
+CMIFILESTOINSTALL = $(filter-out $(addsuffix .cmi,$(PACKEDFILES)),$(CMIFILES))
+CMOFILESTOINSTALL = $(filter-out $(addsuffix .cmo,$(PACKEDFILES)),$(CMOFILES))
+OBJFILES = $(call vo_to_obj,$(VOFILES))
+ALLNATIVEFILES = \
+	$(OBJFILES:.o=.cmi) \
+	$(OBJFILES:.o=.cmx) \
+	$(OBJFILES:.o=.cmxs)
+FINDLIBPACKAGE=$(patsubst .%,%,$(suffix $(METAFILE)))
 
+# trick: wildcard filters out non-existing files, so that `install` doesn't show
+# warnings and `clean` doesn't pass to rm a list of files that is too long for
+# the shell.
+NATIVEFILES = $(wildcard $(ALLNATIVEFILES))
+FILESTOINSTALL = \
+	$(VOFILES) \
+	$(VFILES) \
+	$(GLOBFILES) \
+	$(NATIVEFILES) \
+	$(CMXSFILES)		# to be removed when we remove legacy loading
+FINDLIBFILESTOINSTALL = \
+	$(CMIFILESTOINSTALL)
+ifeq '$(HASNATDYNLINK)' 'true'
+DO_NATDYNLINK = yes
+FINDLIBFILESTOINSTALL += $(CMXSFILES) $(CMXAFILES) $(CMOFILESTOINSTALL:.cmo=.cmx)
+else
+DO_NATDYNLINK =
+endif
 
+ALLDFILES = $(addsuffix .d,$(ALLSRCFILES)) $(VDFILE)
 
-#######################################
-########## COQ RULES ##################
+# Compilation targets #########################################################
 
-%.vo %.glob: %.v
-	$(COQC) $*
+all:
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" pre-all
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" real-all
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" post-all
+.PHONY: all
 
-%.v.d: %.v
-	$(COQDEP) "$<" > "$@" || ( RV=$$?; rm -f "$@"; exit $${RV} )
+all.timing.diff:
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" pre-all
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" real-all.timing.diff TIME_OF_PRETTY_BUILD_EXTRA_FILES=""
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" post-all
+.PHONY: all.timing.diff
 
-CLEAN_TARGETS := clean clean_doc clean_coq
+ifeq (0,$(TIMING_REAL))
+TIMING_REAL_ARG :=
+TIMING_USER_ARG := --user
+else
+ifeq (1,$(TIMING_REAL))
+TIMING_REAL_ARG := --real
+TIMING_USER_ARG :=
+else
+TIMING_REAL_ARG :=
+TIMING_USER_ARG :=
+endif
+endif
+
+ifeq (0,$(TIMING_INCLUDE_MEM))
+TIMING_INCLUDE_MEM_ARG := --no-include-mem
+else
+TIMING_INCLUDE_MEM_ARG :=
+endif
 
-ifeq (,$(filter $(CLEAN_TARGETS),$(MAKECMDGOALS)))
-  -include $(addsuffix .d,$(COQFILES))
-  .SECONDARY: $(addsuffix .d,$(COQFILES))
+ifeq (1,$(TIMING_SORT_BY_MEM))
+TIMING_SORT_BY_MEM_ARG := --sort-by-mem
+else
+TIMING_SORT_BY_MEM_ARG :=
 endif
 
-#######################################
-########## DOCUMENTATION  #############
+make-pretty-timed-before:: TIME_OF_BUILD_FILE=$(TIME_OF_BUILD_BEFORE_FILE)
+make-pretty-timed-after:: TIME_OF_BUILD_FILE=$(TIME_OF_BUILD_AFTER_FILE)
+make-pretty-timed make-pretty-timed-before make-pretty-timed-after::
+	$(HIDE)rm -f pretty-timed-success.ok
+	$(HIDE)($(MAKE) --no-print-directory -f "$(PARENT)" $(TGTS) TIMED=1 2>&1 && touch pretty-timed-success.ok) | tee -a $(TIME_OF_BUILD_FILE)
+	$(HIDE)rm pretty-timed-success.ok # must not be -f; must fail if the touch failed
+print-pretty-timed::
+	$(HIDE)$(COQMAKE_ONE_TIME_FILE) $(TIMING_INCLUDE_MEM_ARG) $(TIMING_SORT_BY_MEM_ARG) $(TIMING_REAL_ARG) $(TIME_OF_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES)
+print-pretty-timed-diff::
+	$(HIDE)$(COQMAKE_BOTH_TIME_FILES) --sort-by=$(TIMING_SORT_BY) $(TIMING_INCLUDE_MEM_ARG) $(TIMING_SORT_BY_MEM_ARG) $(TIMING_REAL_ARG) $(TIME_OF_BUILD_AFTER_FILE) $(TIME_OF_BUILD_BEFORE_FILE) $(TIME_OF_PRETTY_BOTH_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES)
+ifeq (,$(BEFORE))
+print-pretty-single-time-diff::
+	@echo 'Error: Usage: $(MAKE) print-pretty-single-time-diff AFTER=path/to/file.v.after-timing BEFORE=path/to/file.v.before-timing'
+	$(HIDE)false
+else
+ifeq (,$(AFTER))
+print-pretty-single-time-diff::
+	@echo 'Error: Usage: $(MAKE) print-pretty-single-time-diff AFTER=path/to/file.v.after-timing BEFORE=path/to/file.v.before-timing'
+	$(HIDE)false
+else
+print-pretty-single-time-diff::
+	$(HIDE)$(COQMAKE_BOTH_SINGLE_TIMING_FILES) --fuzz=$(TIMING_FUZZ) --sort-by=$(TIMING_SORT_BY) $(TIMING_USER_ARG) $(AFTER) $(BEFORE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BUILD_EXTRA_FILES)
+endif
+endif
+pretty-timed:
+	$(HIDE)$(MAKE) --no-print-directory -f "$(PARENT)" make-pretty-timed
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" print-pretty-timed
+.PHONY: pretty-timed make-pretty-timed make-pretty-timed-before make-pretty-timed-after print-pretty-timed print-pretty-timed-diff print-pretty-single-time-diff
+
+# Extension points for actions to be performed before/after the all target
+pre-all::
+	@# Extension point
+	$(HIDE)if [ "$(COQMAKEFILE_VERSION)" != "$(COQ_VERSION)" ]; then\
+	  echo "W: This Makefile was generated by Coq $(COQMAKEFILE_VERSION)";\
+	  echo "W: while the current Coq version is $(COQ_VERSION)";\
+	fi
+.PHONY: pre-all
+
+post-all::
+	@# Extension point
+.PHONY: post-all
+
+real-all: $(VOFILES) $(if $(USEBYTE),bytefiles,optfiles)
+.PHONY: real-all
+
+real-all.timing.diff: $(VOFILES:.vo=.v.timing.diff)
+.PHONY: real-all.timing.diff
+
+bytefiles: $(CMOFILES) $(CMAFILES)
+.PHONY: bytefiles
+
+optfiles: $(if $(DO_NATDYNLINK),$(CMXSFILES))
+.PHONY: optfiles
+
+# FIXME, see Ralf's bugreport
+# quick is deprecated, now renamed vio
+vio: $(VOFILES:.vo=.vio)
+.PHONY: vio
+quick: vio
+	$(warning "'make quick' is deprecated, use 'make vio' or consider using 'vos' files")
+.PHONY: quick
+
+vio2vo:
+	$(TIMER) $(COQC) $(COQDEBUG) $(COQFLAGS) $(COQLIBS) \
+		-schedule-vio2vo $(J) $(VOFILES:%.vo=%.vio)
+.PHONY: vio2vo
+
+# quick2vo is undocumented
+quick2vo:
+	$(HIDE)make -j $(J) vio
+	$(HIDE)VIOFILES=$$(for vofile in $(VOFILES); do \
+	  viofile="$$(echo "$$vofile" | sed "s/\.vo$$/.vio/")"; \
+	  if [ "$$vofile" -ot "$$viofile" -o ! -e "$$vofile" ]; then printf "$$viofile "; fi; \
+	done); \
+	echo "VIO2VO: $$VIOFILES"; \
+	if [ -n "$$VIOFILES" ]; then \
+	  $(TIMER) $(COQC) $(COQDEBUG) $(COQFLAGS) $(COQLIBS) -schedule-vio2vo $(J) $$VIOFILES; \
+	fi
+.PHONY: quick2vo
+
+checkproofs:
+	$(TIMER) $(COQC) $(COQDEBUG) $(COQFLAGS) $(COQLIBS) \
+		-schedule-vio-checking $(J) $(VOFILES:%.vo=%.vio)
+.PHONY: checkproofs
+
+vos: $(VOFILES:%.vo=%.vos)
+.PHONY: vos
+
+vok: $(VOFILES:%.vo=%.vok)
+.PHONY: vok
+
+validate: $(VOFILES)
+	$(TIMER) $(COQCHK) $(COQCHKFLAGS) $(COQLIBS_NOML) $^
+.PHONY: validate
+
+only: $(TGTS)
+.PHONY: only
+
+# Documentation targets #######################################################
+
+html: $(GLOBFILES) $(VFILES)
+	$(SHOW)'COQDOC -d html $(GAL)'
+	$(HIDE)mkdir -p html
+	$(HIDE)$(COQDOC) \
+		-toc $(COQDOCFLAGS) -html $(GAL) $(COQDOCLIBS) -d html $(VFILES)
+
+mlihtml: $(MLIFILES:.mli=.cmi)
+	$(SHOW)'CAMLDOC -d $@'
+	$(HIDE)mkdir $@ || rm -rf $@/*
+	$(HIDE)$(CAMLDOC) -html \
+		-d $@ -m A $(CAMLDEBUG) $(CAMLDOCFLAGS) $(MLIFILES) $(FINDLIBPKGS)
+
+all-mli.tex: $(MLIFILES:.mli=.cmi)
+	$(SHOW)'CAMLDOC -latex $@'
+	$(HIDE)$(CAMLDOC) -latex \
+		-o $@ -m A $(CAMLDEBUG) $(CAMLDOCFLAGS) $(MLIFILES) $(FINDLIBPKGS)
+
+all.ps: $(VFILES)
+	$(SHOW)'COQDOC -ps $(GAL)'
+	$(HIDE)$(COQDOC) \
+		-toc $(COQDOCFLAGS) -ps $(GAL) $(COQDOCLIBS) \
+		-o $@ `$(COQDEP) -sort $(VFILES)`
+
+all.pdf: $(VFILES)
+	$(SHOW)'COQDOC -pdf $(GAL)'
+	$(HIDE)$(COQDOC) \
+		-toc $(COQDOCFLAGS) -pdf $(GAL) $(COQDOCLIBS) \
+		-o $@ `$(COQDEP) -sort $(VFILES)`
+
+# FIXME: not quite right, since the output name is different
+gallinahtml: GAL=-g
+gallinahtml: html
+
+all-gal.ps: GAL=-g
+all-gal.ps: all.ps
+
+all-gal.pdf: GAL=-g
+all-gal.pdf: all.pdf
+
+# ?
+beautify: $(BEAUTYFILES)
+	for file in $^; do mv $${file%.beautified} $${file%beautified}old && mv $${file} $${file%.beautified}; done
+	@echo 'Do not do "make clean" until you are sure that everything went well!'
+	@echo 'If there were a problem, execute "for file in $$(find . -name \*.v.old -print); do mv $${file} $${file%.old}; done" in your shell/'
+.PHONY: beautify
+
+# Installation targets ########################################################
+#
+# There rules can be extended in Makefile.local
+# Extensions can't assume when they run.
+
+# findlib needs the package to not be installed, so we remove it before
+# installing it (see the call to findlib_remove)
+install: META
+	$(HIDE)code=0; for f in $(FILESTOINSTALL); do\
+	 if ! [ -f "$$f" ]; then >&2 echo $$f does not exist; code=1; fi \
+	done; exit $$code
+	$(HIDE)for f in $(FILESTOINSTALL); do\
+	 df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`";\
+	 if [ "$$?" != "0" -o -z "$$df" ]; then\
+	   echo SKIP "$$f" since it has no logical path;\
+	 else\
+	   install -d "$(COQLIBINSTALL)/$$df" &&\
+	   install -m 0644 "$$f" "$(COQLIBINSTALL)/$$df" &&\
+	   echo INSTALL "$$f" "$(COQLIBINSTALL)/$$df";\
+	 fi;\
+	done
+	$(call findlib_remove)
+	$(call findlib_install, META $(FINDLIBFILESTOINSTALL))
+	$(HIDE)$(MAKE) install-extra -f "$(SELF)"
+install-extra::
+	@# Extension point
+.PHONY: install install-extra
+
+META: $(METAFILE)
+	$(HIDE)if [ "$(METAFILE)" ]; then \
+		cat "$(METAFILE)" | grep -v 'directory.*=.*' > META; \
+	fi
+
+install-byte:
+	$(call findlib_install, $(CMAFILES) $(CMOFILESTOINSTALL), -add)
+
+install-doc:: html mlihtml
+	@# Extension point
+	$(HIDE)install -d "$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/html"
+	$(HIDE)for i in html/*; do \
+	 dest="$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/$$i";\
+	 install -m 0644 "$$i" "$$dest";\
+	 echo INSTALL "$$i" "$$dest";\
+	done
+	$(HIDE)install -d \
+		"$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/mlihtml"
+	$(HIDE)for i in mlihtml/*; do \
+	 dest="$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/$$i";\
+	 install -m 0644 "$$i" "$$dest";\
+	 echo INSTALL "$$i" "$$dest";\
+	done
+.PHONY: install-doc
+
+uninstall::
+	@# Extension point
+	$(call findlib_remove)
+	$(HIDE)for f in $(FILESTOINSTALL); do \
+	 df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`" &&\
+	 instf="$(COQLIBINSTALL)/$$df/`basename $$f`" &&\
+	 rm -f "$$instf" &&\
+	 echo RM "$$instf" ;\
+	done
+	$(HIDE)for f in $(FILESTOINSTALL); do \
+	 df="`$(COQMKFILE) -destination-of "$$f" $(COQLIBS)`" &&\
+	 echo RMDIR "$(COQLIBINSTALL)/$$df/" &&\
+	 (rmdir "$(COQLIBINSTALL)/$$df/" 2>/dev/null || true); \
+	done
+
+.PHONY: uninstall
+
+uninstall-doc::
+	@# Extension point
+	$(SHOW)'RM $(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/html'
+	$(HIDE)rm -rf "$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/html"
+	$(SHOW)'RM $(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/mlihtml'
+	$(HIDE)rm -rf "$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/mlihtml"
+	$(HIDE) rmdir "$(COQDOCINSTALL)/$(INSTALLCOQDOCROOT)/" || true
+.PHONY: uninstall-doc
+
+# Cleaning ####################################################################
+#
+# There rules can be extended in Makefile.local
+# Extensions can't assume when they run.
+
+clean::
+	@# Extension point
+	$(SHOW)'CLEAN'
+	$(HIDE)rm -f $(CMOFILES)
+	$(HIDE)rm -f $(CMIFILES)
+	$(HIDE)rm -f $(CMAFILES)
+	$(HIDE)rm -f $(CMOFILES:.cmo=.cmx)
+	$(HIDE)rm -f $(CMXAFILES)
+	$(HIDE)rm -f $(CMXSFILES)
+	$(HIDE)rm -f $(CMOFILES:.cmo=.o)
+	$(HIDE)rm -f $(CMXAFILES:.cmxa=.a)
+	$(HIDE)rm -f $(MLGFILES:.mlg=.ml)
+	$(HIDE)rm -f $(ALLDFILES)
+	$(HIDE)rm -f $(NATIVEFILES)
+	$(HIDE)find . -name .coq-native -type d -empty -delete
+	$(HIDE)rm -f $(VOFILES)
+	$(HIDE)rm -f $(VOFILES:.vo=.vio)
+	$(HIDE)rm -f $(VOFILES:.vo=.vos)
+	$(HIDE)rm -f $(VOFILES:.vo=.vok)
+	$(HIDE)rm -f $(BEAUTYFILES) $(VFILES:=.old)
+	$(HIDE)rm -f all.ps all-gal.ps all.pdf all-gal.pdf all.glob all-mli.tex
+	$(HIDE)rm -f $(VFILES:.v=.glob)
+	$(HIDE)rm -f $(VFILES:.v=.tex)
+	$(HIDE)rm -f $(VFILES:.v=.g.tex)
+	$(HIDE)rm -f pretty-timed-success.ok
+	$(HIDE)rm -f META
+	$(HIDE)rm -rf html mlihtml
+.PHONY: clean
+
+cleanall:: clean
+	@# Extension point
+	$(SHOW)'CLEAN *.aux *.timing'
+	$(HIDE)rm -f $(foreach f,$(VFILES:.v=),$(dir $(f)).$(notdir $(f)).aux)
+	$(HIDE)rm -f $(TIME_OF_BUILD_FILE) $(TIME_OF_BUILD_BEFORE_FILE) $(TIME_OF_BUILD_AFTER_FILE) $(TIME_OF_PRETTY_BUILD_FILE) $(TIME_OF_PRETTY_BOTH_BUILD_FILE)
+	$(HIDE)rm -f $(VOFILES:.vo=.v.timing)
+	$(HIDE)rm -f $(VOFILES:.vo=.v.before-timing)
+	$(HIDE)rm -f $(VOFILES:.vo=.v.after-timing)
+	$(HIDE)rm -f $(VOFILES:.vo=.v.timing.diff)
+	$(HIDE)rm -f .lia.cache .nia.cache
+.PHONY: cleanall
+
+archclean::
+	@# Extension point
+	$(SHOW)'CLEAN *.cmx *.o'
+	$(HIDE)rm -f $(NATIVEFILES)
+	$(HIDE)rm -f $(CMOFILES:%.cmo=%.cmx)
+.PHONY: archclean
+
+
+# Compilation rules ###########################################################
+
+$(MLIFILES:.mli=.cmi): %.cmi: %.mli
+	$(SHOW)'CAMLC -c $<'
+	$(HIDE)$(TIMER) $(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) $<
+
+$(MLGFILES:.mlg=.ml): %.ml: %.mlg
+	$(SHOW)'COQPP $<'
+	$(HIDE)$(COQPP) $<
+
+# Stupid hack around a deficient syntax: we cannot concatenate two expansions
+$(filter %.cmo, $(MLFILES:.ml=.cmo) $(MLGFILES:.mlg=.cmo)): %.cmo: %.ml
+	$(SHOW)'CAMLC -c $<'
+	$(HIDE)$(TIMER) $(CAMLC) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) $<
+
+# Same hack
+$(filter %.cmx, $(MLFILES:.ml=.cmx) $(MLGFILES:.mlg=.cmx)): %.cmx: %.ml
+	$(SHOW)'CAMLOPT -c $(FOR_PACK) $<'
+	$(HIDE)$(TIMER) $(CAMLOPTC) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) $(FOR_PACK) $<
+
+
+$(MLLIBFILES:.mllib=.cmxs): %.cmxs: %.cmxa
+	$(SHOW)'CAMLOPT -shared -o $@'
+	$(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) \
+		-shared -o $@ $<
+
+$(MLLIBFILES:.mllib=.cma): %.cma: | %.mllib
+	$(SHOW)'CAMLC -a -o $@'
+	$(HIDE)$(TIMER) $(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $^
+
+$(MLLIBFILES:.mllib=.cmxa): %.cmxa: | %.mllib
+	$(SHOW)'CAMLOPT -a -o $@'
+	$(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $^
+
+
+$(MLPACKFILES:.mlpack=.cmxs): %.cmxs: %.cmxa
+	$(SHOW)'CAMLOPT -shared -o $@'
+	$(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) \
+		-shared -o $@ $<
+
+$(MLPACKFILES:.mlpack=.cmxa): %.cmxa: %.cmx | %.mlpack
+	$(SHOW)'CAMLOPT -a -o $@'
+	$(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $<
+
+$(MLPACKFILES:.mlpack=.cma): %.cma: %.cmo | %.mlpack
+	$(SHOW)'CAMLC -a -o $@'
+	$(HIDE)$(TIMER) $(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -a -o $@ $^
+
+$(MLPACKFILES:.mlpack=.cmo): %.cmo: | %.mlpack
+	$(SHOW)'CAMLC -pack -o $@'
+	$(HIDE)$(TIMER) $(CAMLLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -pack -o $@ $^
+
+$(MLPACKFILES:.mlpack=.cmx): %.cmx: | %.mlpack
+	$(SHOW)'CAMLOPT -pack -o $@'
+	$(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) -pack -o $@ $^
+
+# This rule is for _CoqProject with no .mllib nor .mlpack
+$(filter-out $(MLLIBFILES:.mllib=.cmxs) $(MLPACKFILES:.mlpack=.cmxs) $(addsuffix .cmxs,$(PACKEDFILES)) $(addsuffix .cmxs,$(LIBEDFILES)),$(MLFILES:.ml=.cmxs) $(MLGFILES:.mlg=.cmxs)): %.cmxs: %.cmx
+	$(SHOW)'[deprecated,use-mllib-or-mlpack] CAMLOPT -shared -o $@'
+	$(HIDE)$(TIMER) $(CAMLOPTLINK) $(CAMLDEBUG) $(CAMLFLAGS) $(FINDLIBPKGS) \
+		-shared -o $@ $<
+
+ifneq (,$(TIMING))
+TIMING_EXTRA = > $<.$(TIMING_EXT)
+else
+TIMING_EXTRA =
+endif
+
+$(VOFILES): %.vo: %.v | $(VDFILE)
+	$(SHOW)COQC $<
+	$(HIDE)$(TIMER) $(COQC) $(COQDEBUG) $(TIMING_ARG) $(COQFLAGS) $(COQLIBS) $< $(TIMING_EXTRA)
+ifeq ($(COQDONATIVE), "yes")
+	$(SHOW)COQNATIVE $@
+	$(HIDE)$(COQNATIVE) $(COQLIBS) $@
+endif
+
+# FIXME ?merge with .vo / .vio ?
+$(GLOBFILES): %.glob: %.v
+	$(TIMER) $(COQC) $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $<
+
+$(VFILES:.v=.vio): %.vio: %.v
+	$(SHOW)COQC -vio $<
+	$(HIDE)$(TIMER) $(COQC) -vio $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $<
+
+$(VFILES:.v=.vos): %.vos: %.v
+	$(SHOW)COQC -vos $<
+	$(HIDE)$(TIMER) $(COQC) -vos $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $<
+
+$(VFILES:.v=.vok): %.vok: %.v
+	$(SHOW)COQC -vok $<
+	$(HIDE)$(TIMER) $(COQC) -vok $(COQDEBUG) $(COQFLAGS) $(COQLIBS) $<
+
+$(addsuffix .timing.diff,$(VFILES)): %.timing.diff : %.before-timing %.after-timing
+	$(SHOW)PYTHON TIMING-DIFF $*.{before,after}-timing
+	$(HIDE)$(MAKE) --no-print-directory -f "$(SELF)" print-pretty-single-time-diff BEFORE=$*.before-timing AFTER=$*.after-timing TIME_OF_PRETTY_BUILD_FILE="$@"
+
+$(BEAUTYFILES): %.v.beautified: %.v
+	$(SHOW)'BEAUTIFY $<'
+	$(HIDE)$(TIMER) $(COQC) $(COQDEBUG) $(COQFLAGS) $(COQLIBS) -beautify $<
+
+$(TEXFILES): %.tex: %.v
+	$(SHOW)'COQDOC -latex $<'
+	$(HIDE)$(COQDOC) $(COQDOCFLAGS) -latex $< -o $@
+
+$(GTEXFILES): %.g.tex: %.v
+	$(SHOW)'COQDOC -latex -g $<'
+	$(HIDE)$(COQDOC) $(COQDOCFLAGS) -latex -g $< -o $@
+
+$(HTMLFILES): %.html: %.v %.glob
+	$(SHOW)'COQDOC -html $<'
+	$(HIDE)$(COQDOC) $(COQDOCFLAGS) -html $< -o $@
+
+$(GHTMLFILES): %.g.html: %.v %.glob
+	$(SHOW)'COQDOC -html -g $<'
+	$(HIDE)$(COQDOC) $(COQDOCFLAGS)  -html -g $< -o $@
+
+# Dependency files ############################################################
+
+ifndef MAKECMDGOALS
+  -include $(ALLDFILES)
+else
+  ifneq ($(filter-out archclean clean cleanall printenv make-pretty-timed make-pretty-timed-before make-pretty-timed-after print-pretty-timed print-pretty-timed-diff print-pretty-single-time-diff,$(MAKECMDGOALS)),)
+   -include $(ALLDFILES)
+ endif
+endif
+
+.SECONDARY: $(ALLDFILES)
+
+redir_if_ok = > "$@" || ( RV=$$?; rm -f "$@"; exit $$RV )
+
+GENMLFILES:=$(MLGFILES:.mlg=.ml)
+$(addsuffix .d,$(ALLSRCFILES)): $(GENMLFILES)
+
+$(addsuffix .d,$(MLIFILES)): %.mli.d: %.mli
+	$(SHOW)'CAMLDEP $<'
+	$(HIDE)$(CAMLDEP) $(OCAMLLIBS) "$<" $(redir_if_ok)
+
+$(addsuffix .d,$(MLGFILES)): %.mlg.d: %.ml
+	$(SHOW)'CAMLDEP $<'
+	$(HIDE)$(CAMLDEP) $(OCAMLLIBS) "$<" $(redir_if_ok)
+
+$(addsuffix .d,$(MLFILES)): %.ml.d: %.ml
+	$(SHOW)'CAMLDEP $<'
+	$(HIDE)$(CAMLDEP) $(OCAMLLIBS) "$<" $(redir_if_ok)
+
+$(addsuffix .d,$(MLLIBFILES)): %.mllib.d: %.mllib
+	$(SHOW)'OCAMLLIBDEP $<'
+	$(HIDE)$(OCAMLLIBDEP) -c $(OCAMLLIBS) "$<" $(redir_if_ok)
+
+$(addsuffix .d,$(MLPACKFILES)): %.mlpack.d: %.mlpack
+	$(SHOW)'OCAMLLIBDEP $<'
+	$(HIDE)$(OCAMLLIBDEP) -c $(OCAMLLIBS) "$<" $(redir_if_ok)
+
+# If this makefile is created using a _CoqProject we have coqdep get
+# options from it. This avoids argument length limits for pathological
+# projects. Note that extra options might be on the command line.
+VDFILE_FLAGS:=$(if _CoqProject,-f _CoqProject,) $(CMDLINE_COQLIBS) $(CMDLINE_VFILES)
+
+$(VDFILE): _CoqProject $(VFILES)
+	$(SHOW)'COQDEP VFILES'
+	$(HIDE)$(COQDEP) $(if $(strip $(METAFILE)),-m "$(METAFILE)") -vos -dyndep var $(VDFILE_FLAGS) $(redir_if_ok)
 
-coqhtml:
-	mkdir -p coqdoc
-	$(COQDOC) -toc --html -g -d coqdoc $(COQFILES)
+# Misc ########################################################################
 
-coqtex:
-	mkdir -p coqdoc
-	$(COQDOC) -toc --latex -g -o coqdoc/mset_doc.tex $(COQFILES)
-	cd coqdoc; pdflatex mset_doc.tex
-	cd coqdoc; pdflatex mset_doc.tex
+byte:
+	$(HIDE)$(MAKE) all "OPT:=-byte" -f "$(SELF)"
+.PHONY: byte
 
-%.html: %.md 
-	pandoc -N $< -o $@
+opt:
+	$(HIDE)$(MAKE) all "OPT:=-opt" -f "$(SELF)"
+.PHONY:	opt
 
-%.pdf: %.md 
-	pandoc -N $< -o $@
+# This is deprecated.  To extend this makefile use
+# extension points and Makefile.local
+printenv::
+	$(warning printenv is deprecated)
+	$(warning write extensions in Makefile.local or include Makefile.conf)
+	@echo 'COQLIB = $(COQLIB)'
+	@echo 'COQCORELIB = $(COQCORELIB)'
+	@echo 'DOCDIR = $(DOCDIR)'
+	@echo 'OCAMLFIND = $(OCAMLFIND)'
+	@echo 'HASNATDYNLINK = $(HASNATDYNLINK)'
+	@echo 'SRC_SUBDIRS = $(SRC_SUBDIRS)'
+	@echo 'COQ_SRC_SUBDIRS = $(COQ_SRC_SUBDIRS)'
+	@echo 'COQCORE_SRC_SUBDIRS = $(COQCORE_SRC_SUBDIRS)'
+	@echo 'OCAMLFIND = $(OCAMLFIND)'
+	@echo 'PP = $(PP)'
+	@echo 'COQFLAGS = $(COQFLAGS)'
+	@echo 'COQLIB = $(COQLIBS)'
+	@echo 'COQLIBINSTALL = $(COQLIBINSTALL)'
+	@echo 'COQDOCINSTALL = $(COQDOCINSTALL)'
+.PHONY:	printenv
 
-doc/mset.pdf: doc/mset.tex doc/mset.bib
-	cd doc; pdflatex mset.tex
-	cd doc; bibtex mset
-	cd doc; pdflatex mset.tex
-	cd doc; pdflatex mset.tex
+# Generate a .merlin file.  If you need to append directives to this
+# file you can extend the merlin-hook target in Makefile.local
+.merlin:
+	$(SHOW)'FILL .merlin'
+	$(HIDE)echo 'FLG $(COQMF_CAMLFLAGS)' > .merlin
+	$(HIDE)echo 'B $(COQCORELIB)' >> .merlin
+	$(HIDE)echo 'S $(COQCORELIB)' >> .merlin
+	$(HIDE)$(foreach d,$(COQCORE_SRC_SUBDIRS), \
+		echo 'B $(COQCORELIB)$(d)' >> .merlin;)
+	$(HIDE)$(foreach d,$(COQ_SRC_SUBDIRS), \
+		echo 'S $(COQLIB)$(d)' >> .merlin;)
+	$(HIDE)$(foreach d,$(SRC_SUBDIRS), echo 'B $(d)' >> .merlin;)
+	$(HIDE)$(foreach d,$(SRC_SUBDIRS), echo 'S $(d)' >> .merlin;)
+	$(HIDE)$(MAKE) merlin-hook -f "$(SELF)"
+.PHONY: merlin
 
-#######################################
-########## CLEANING RULES #############
+merlin-hook::
+	@# Extension point
+.PHONY: merlin-hook
 
-clean_coq:
-	rm -f $(COQFILES:.v=.v.d) $(COQFILES:.v=.vo) $(COQFILES:.v=.glob)
+# prints all variables
+debug:
+	$(foreach v,\
+		$(sort $(filter-out $(INITIAL_VARS) INITIAL_VARS,\
+	       		$(.VARIABLES))),\
+	       	$(info $(v) = $($(v))))
+.PHONY: debug
 
-clean_doc:
-	rm -rf coqdoc
-	rm -f readme.html readme.pdf
+.DEFAULT_GOAL := all
 
-clean_tex:
-	cd doc; rm -f *.ps *~ *.dvi *.aux *.log *.idx *.toc *.nav *.out *.snm *.flc *.vrb *.bbl *.blg
+# Users can create Makefile.local-late to hook into double-colon rules
+# or add other needed Makefile code, using defined
+# variables if necessary.
+-include Makefile.local-late
 
-clean: clean_coq clean_doc clean_tex
-	rm -f *~
+# Local Variables:
+# mode: makefile-gmake
+# End:
diff --git a/README.md b/README.md
index 827d15d..a143cbe 100644
--- a/README.md
+++ b/README.md
@@ -21,17 +21,26 @@ There are the following files
 
 - `readme.md` (this readme)
 - `LICENSE` (license information, LGPL 2.1)
+- `configure.sh` 
+- `_CoqProject` 
 - `Makefile` 
 - `doc/*` (high level documentation)
-- `coqdoc/*` (generated coqdoc documentation)
 
 
-The `Makefile` provided is set up to generate documentation as well as
-process Coq files. The default target `proof` just processes the Coq
-files. For convenience coqdoc and a pdf version of top-level
-documentation is part of the repository. If you want to rebuild them
-use target `doc` or target `all`.
+The `Makefile` is generated via `coq_makefile` from `_CoqProject`. This
+is performed by `configure.sh`.
 
+## OPAM
+
+This library is available as an opam package as `coq-msets-extra`. In
+order to use it, add the coq opam repository
+
+	opam repo add coq-released https://coq.inria.fr/opam/released
+	opam update
+
+Afterwards you can install this libaray via
+
+	opam install coq-msets-extra
 
 ## Interval Sets
 
diff --git a/_CoqProject b/_CoqProject
new file mode 100644
index 0000000..2f02e9b
--- /dev/null
+++ b/_CoqProject
@@ -0,0 +1,6 @@
+-R . MSetsExtra
+
+MSetFoldWithAbort.v
+MSetIntervals.v
+MSetListWithDups.v
+MSetWithDups.v
diff --git a/all.pdf b/all.pdf
new file mode 100644
index 0000000..4442799
Binary files /dev/null and b/all.pdf differ
diff --git a/configure.sh b/configure.sh
new file mode 100755
index 0000000..9d71478
--- /dev/null
+++ b/configure.sh
@@ -0,0 +1,3 @@
+#!/bin/sh
+
+coq_makefile -f _CoqProject -o Makefile
diff --git a/coqdoc/MSetFoldWithAbort.html b/coqdoc/MSetFoldWithAbort.html
deleted file mode 100644
index 55cd5c8..0000000
--- a/coqdoc/MSetFoldWithAbort.html
+++ /dev/null
@@ -1,1361 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
-"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
-<html xmlns="http://www.w3.org/1999/xhtml">
-<head>
-<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
-<link href="coqdoc.css" rel="stylesheet" type="text/css" />
-<title>MSetFoldWithAbort</title>
-</head>
-
-<body>
-
-<div id="page">
-
-<div id="header">
-</div>
-
-<div id="main">
-
-<h1 class="libtitle">Library MSetFoldWithAbort</h1>
-
-<div class="code">
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab5"></a><h1 class="section">Fold with abort for sets</h1>
-
-
-<div class="paragraph"> </div>
-
-    This file provided an efficient fold operation for set interfaces.
-    The standard fold iterates over all elements of the set. The
-    efficient one - called <span class="inlinecode"><span class="id" title="var">foldWithAbort</span></span> - is allowed to skip
-    certain elements and thereby abort early.  
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#"><span class="id" title="library">MSetInterface</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssreflect</span>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="MSetWithDups.html#"><span class="id" title="library">MSetWithDups</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.Int.html#"><span class="id" title="library">Int</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetGenTree.html#"><span class="id" title="library">MSetGenTree</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetAVL.html#"><span class="id" title="library">MSetAVL</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetRBT.html#"><span class="id" title="library">MSetRBT</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetList.html#"><span class="id" title="library">MSetList</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetWeakList.html#"><span class="id" title="library">MSetWeakList</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab6"></a><h2 class="section">Fold With Abort Operations</h2>
-
-
-<div class="paragraph"> </div>
-
-    We want to provide an efficient folding operation. Efficieny is
-    gained by aborting the folding early, if we know that continuing
-    would not have an effect any more. Formalising this leads to the following
-    specification of <span class="inlinecode"><span class="id" title="var">foldWithAbort</span></span>. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortType"><span class="id" title="definition">foldWithAbortType</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">elt</span> </div>
-
-<div class="doc">
-element type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">t</span> </div>
-
-<div class="doc">
-type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">A</span> </div>
-
-<div class="doc">
-return type 
-</div>
-<div class="code">
- :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt;    </div>
-
-<div class="doc">
-f 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) -&gt; </div>
-
-<div class="doc">
-f_abort 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt;                  </div>
-
-<div class="doc">
-input set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt;                  </div>
-
-<div class="doc">
-base value 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortSpecPred"><span class="id" title="definition">foldWithAbortSpecPred</span></a> {<span class="id" title="var">elt</span> <span class="id" title="var">t</span> : <span class="id" title="keyword">Type</span>} <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">In</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">fold</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, (<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">foldWithAbort</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetFoldWithAbort.html#foldWithAbortType"><span class="id" title="definition">foldWithAbortType</span></a> <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-result type 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">i</span> <span class="id" title="var">i&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-base values for foldWithAbort and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-fold functions for foldWithAbort and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f_abort</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-abort function 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>) </div>
-
-<div class="doc">
-sets to fold over 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">P</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <span class="id" title="keyword">Prop</span>) </div>
-
-<div class="doc">
-equivalence relation on results 
-</div>
-<div class="code">
-,<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> is for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> compatible with the equivalence relation P 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> and <span class="inlinecode"><span class="id" title="var">f&acute;</span></span> agree for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f_abort</span></span> is OK, i.e. all other elements can be skipped without 
-        leaving the equivalence relation. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_abort"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The base values are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The results are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#foldWithAbort"><span class="id" title="variable">foldWithAbort</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_abort"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#fold"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-The specification of folding for ordered sets (as represented by
-    interface <span class="inlinecode"><span class="id" title="var">Sets</span></span>) demands that elements are visited in increasing
-    order. For ordered sets we can therefore abort folding based on
-    the weaker knowledge that greater elements have no effect on the
-    result. The following definition captures this. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortGtType"><span class="id" title="definition">foldWithAbortGtType</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">elt</span> </div>
-
-<div class="doc">
-element type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">t</span> </div>
-
-<div class="doc">
-type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">A</span> </div>
-
-<div class="doc">
-return type 
-</div>
-<div class="code">
- :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt;    </div>
-
-<div class="doc">
-f 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) -&gt; </div>
-
-<div class="doc">
-f_gt 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt;                  </div>
-
-<div class="doc">
-input set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt;                  </div>
-
-<div class="doc">
-base value 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortGtSpecPred"><span class="id" title="definition">foldWithAbortGtSpecPred</span></a> {<span class="id" title="var">elt</span> <span class="id" title="var">t</span> : <span class="id" title="keyword">Type</span>} <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">lt</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">In</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">fold</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, (<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">foldWithAbortGt</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetFoldWithAbort.html#foldWithAbortType"><span class="id" title="definition">foldWithAbortType</span></a> <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-result type 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">i</span> <span class="id" title="var">i&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-base values for foldWithAbort and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-fold functions for foldWithAbort and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-abort function 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>) </div>
-
-<div class="doc">
-sets to fold over 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">P</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <span class="id" title="keyword">Prop</span>) </div>
-
-<div class="doc">
-equivalence relation on results 
-</div>
-<div class="code">
-,<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> is for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> compatible with the equivalence relation P 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> and <span class="inlinecode"><span class="id" title="var">f&acute;</span></span> agree for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f_abort</span></span> is OK, i.e. all other elements can be skipped without 
-        leaving the equivalence relation. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt;  <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The base values are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The results are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#foldWithAbortGt"><span class="id" title="variable">foldWithAbortGt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#fold"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-For ordered sets, we can safely skip elements at the end
-    based on the knowledge that they are all greater than the current element.
-    This leads to serious performance improvements for operations like
-    filtering. It is tempting to try the symmetric operation and skip elements at
-    the beginning based on the knowledge that they are too small to be interesting.
-    So, we would like to start late as well as abort early. 
-
-<div class="paragraph"> </div>
-
-    This is indeed a very natural and efficient operation for set implementations 
-    based on binary search trees (i.e. the AVL and RBT sets). We can completely symmetrically
-    to skipping greater elements also skip smaller elements. This leads to the following
-    specification. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortGtLtType"><span class="id" title="definition">foldWithAbortGtLtType</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">elt</span> </div>
-
-<div class="doc">
-element type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">t</span> </div>
-
-<div class="doc">
-type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">A</span> </div>
-
-<div class="doc">
-return type 
-</div>
-<div class="code">
- :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) -&gt; </div>
-
-<div class="doc">
-f_lt 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt;    </div>
-
-<div class="doc">
-f 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) -&gt; </div>
-
-<div class="doc">
-f_gt 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt;                  </div>
-
-<div class="doc">
-input set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt;                  </div>
-
-<div class="doc">
-base value 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortGtLtSpecPred"><span class="id" title="definition">foldWithAbortGtLtSpecPred</span></a> {<span class="id" title="var">elt</span> <span class="id" title="var">t</span> : <span class="id" title="keyword">Type</span>} <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">lt</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">In</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">fold</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, (<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">foldWithAbortGtLt</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetFoldWithAbort.html#foldWithAbortGtLtType"><span class="id" title="definition">foldWithAbortGtLtType</span></a> <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-result type 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">i</span> <span class="id" title="var">i&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-base values for foldWithAbort and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-fold functions for foldWithAbort and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-abort functions 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>) </div>
-
-<div class="doc">
-sets to fold over 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">P</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <span class="id" title="keyword">Prop</span>) </div>
-
-<div class="doc">
-equivalence relation on results 
-</div>
-<div class="code">
-,<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> is for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> compatible with the equivalence relation P 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> and <span class="inlinecode"><span class="id" title="var">f&acute;</span></span> agree for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f_lt</span></span> is OK, i.e. smaller elements can be skipped without 
-        leaving the equivalence relation. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f_gt</span></span> is OK, i.e. greater elements can be skipped without 
-        leaving the equivalence relation. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The base values are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The results are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#foldWithAbortGtLt"><span class="id" title="variable">foldWithAbortGtLt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#fold"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-We are interested in folding with abort mainly for runtime
-    performance reasons of extracted code. The argument functions
-    <span class="inlinecode"><span class="id" title="var">f_lt</span></span>, <span class="inlinecode"><span class="id" title="var">f_gt</span></span> and <span class="inlinecode"><span class="id" title="var">f</span></span> of <span class="inlinecode"><span class="id" title="var">foldWithAbortGtLt</span></span> often share a large,
-    comparably expensive part of their computation.
-
-<div class="paragraph"> </div>
-
-    In order to further improve runtime performance, therefore another
-    version <span class="inlinecode"><span class="id" title="var">foldWithAbortPrecompute</span></span> <span class="inlinecode"><span class="id" title="var">f_precompute</span></span> <span class="inlinecode"><span class="id" title="var">f_lt</span></span> <span class="inlinecode"><span class="id" title="var">f</span></span> <span class="inlinecode"><span class="id" title="var">f_gt</span></span> that
-    uses an extra function <span class="inlinecode"><span class="id" title="var">f_precompute</span></span> to allows to compute the
-    commonly used parts of these functions only once. This leads to
-    the following definitions. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortPrecomputeType"><span class="id" title="definition">foldWithAbortPrecomputeType</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">elt</span> </div>
-
-<div class="doc">
-element type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">t</span> </div>
-
-<div class="doc">
-type of set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">A</span> </div>
-
-<div class="doc">
-return type 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">B</span> </div>
-
-<div class="doc">
-type of precomputed results 
-</div>
-<div class="code">
- := <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) -&gt;               </div>
-
-<div class="doc">
-f_precompute 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) -&gt;  </div>
-
-<div class="doc">
-f_lt 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt;     </div>
-
-<div class="doc">
-f 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) -&gt;  </div>
-
-<div class="doc">
-f_gt 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt;                        </div>
-
-<div class="doc">
-input set 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt;                        </div>
-
-<div class="doc">
-base value 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-The specification is similar to the one without precompute,
-    but uses <span class="inlinecode"><span class="id" title="var">f_precompute</span></span> so avoid doing computations multiple times 
-</div>
-<div class="code">
-<span class="id" title="keyword">Definition</span> <a name="foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> {<span class="id" title="var">elt</span> <span class="id" title="var">t</span> : <span class="id" title="keyword">Type</span>} <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">lt</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">In</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">fold</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}, (<a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) -&gt; <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">foldWithAbortPrecompute</span> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetFoldWithAbort.html#foldWithAbortPrecomputeType"><span class="id" title="definition">foldWithAbortPrecomputeType</span></a> <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-result type 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">i</span> <span class="id" title="var">i&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-base values for foldWithAbortPrecompute and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f_precompute</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-precompute function 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-fold functions for foldWithAbortPrecompute and fold 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#elt"><span class="id" title="variable">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-abort functions 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>) </div>
-
-<div class="doc">
-sets to fold over 
-</div>
-<div class="code">
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">P</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <span class="id" title="keyword">Prop</span>) </div>
-
-<div class="doc">
-equivalence relation on results 
-</div>
-<div class="code">
-,<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> is for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> compatible with the equivalence relation P 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f</span></span> and <span class="inlinecode"><span class="id" title="var">f&acute;</span></span> agree for the elements of <span class="inlinecode"><span class="id" title="var">s</span></span> 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f_lt</span></span> is OK, i.e. smaller elements can be skipped without 
-        leaving the equivalence relation. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">f_gt</span></span> is OK, i.e. greater elements can be skipped without 
-        leaving the equivalence relation. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#In"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#lt"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The base values are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a> -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-The results are in equivalence relation 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#foldWithAbortPrecompute"><span class="id" title="variable">foldWithAbortPrecompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_precompute"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#fold"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab7"></a><h3 class="section">Module Types</h3>
-
-<div class="paragraph"> </div>
-
- We now define a module type for <span class="inlinecode"><span class="id" title="var">foldWithAbort</span></span>. This module
-    type demands only the existence of the precompute version, since
-    the other ones can be easily defined via this most efficient one. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) (<span class="id" title="keyword">Import</span> <span class="id" title="var">C</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="HasFoldWithAbort.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> : <span class="id" title="keyword">forall</span> {<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>}, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecomputeType"><span class="id" title="definition">foldWithAbortPrecomputeType</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="HasFoldWithAbort.foldWithAbortPrecomputeSpec"><span class="id" title="axiom">foldWithAbortPrecomputeSpec</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a>).<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab8"></a><h2 class="section">Derived operations</h2>
-
-
-<div class="paragraph"> </div>
-
-    Using these efficient fold operations, many operations can 
-    be implemented efficiently. We provide lemmata and efficient implementations
-    of useful algorithms via module <span class="inlinecode"><span class="id" title="var">HasFoldWithAbortOps</span></span>. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">FT</span> : <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetFoldWithAbort.html#C"><span class="id" title="module">C</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <span class="id" title="var">FT</span>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <span class="id" title="var">C</span>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab9"></a><h3 class="section">First lets define the other folding with abort variants</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a> {<span class="id" title="var">A</span>} <span class="id" title="var">f_lt</span> (<span class="id" title="var">f</span> : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)) <span class="id" title="var">f_gt</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortGtLtSpec"><span class="id" title="lemma">foldWithAbortGtLtSpec</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpecPred"><span class="id" title="definition">foldWithAbortGtLtSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a> {<span class="id" title="var">A</span>} (<span class="id" title="var">f</span> : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)) <span class="id" title="var">f_gt</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortGtSpec"><span class="id" title="lemma">foldWithAbortGtSpec</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpecPred"><span class="id" title="definition">foldWithAbortGtSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a> {<span class="id" title="var">A</span>} (<span class="id" title="var">f</span> : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>)) <span class="id" title="var">f_abort</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f_abort"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">_</span> <span class="id" title="var">st</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#f_abort"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortSpec"><span class="id" title="lemma">foldWithAbortSpec</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecPred"><span class="id" title="definition">foldWithAbortSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab10"></a><h3 class="section">Specialisations for equality</h3>
-
-<div class="paragraph"> </div>
-
- Let's provide simplified specifications, which use equality instead
-      of an arbitrary equivalence relation on results. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal"><span class="id" title="lemma">foldWithAbortPrecomputeSpec_Equal</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_pre</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal"><span class="id" title="lemma">foldWithAbortGtLtSpec_Equal</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortGtSpec_Equal"><span class="id" title="lemma">foldWithAbortGtSpec_Equal</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.foldWithAbortSpec_Equal"><span class="id" title="lemma">foldWithAbortSpec_Equal</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_abort</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_abort"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_abort"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab11"></a><h3 class="section">FoldWithAbortSpecArgs</h3>
-
-<div class="paragraph"> </div>
-
- While folding, we are often interested in skipping elements that do not
-      satisfy a certain property <span class="inlinecode"><span class="id" title="var">P</span></span>. This needs expressing in terms of
-      skips of smaller of larger elements in order to be done efficiently by
-      our folding functions. Formally, this leads to the definition
-      of <span class="inlinecode"><span class="id" title="var">foldWithAbortSpecForPred</span></span>. 
-
-<div class="paragraph"> </div>
-
-      Given a <span class="inlinecode"><span class="id" title="var">FoldWithAbortSpecArg</span></span> for a predicate <span class="inlinecode"><span class="id" title="var">P</span></span> and a
-      set <span class="inlinecode"><span class="id" title="var">s</span></span>, many operations can be implemented efficiently. Below we will provide
-      efficient versions of <span class="inlinecode"><span class="id" title="var">filter</span></span>, <span class="inlinecode"><span class="id" title="var">choose</span></span>, <span class="inlinecode"><span class="id" title="tactic">exists</span></span>, <span class="inlinecode"><span class="id" title="keyword">forall</span></span> and more.
-    
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Record</span> <a name="HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> {<span class="id" title="var">B</span>} := {<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a name="HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>);         </div>
-
-<div class="doc">
-The precompute function 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a name="HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a>  : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>); </div>
-
-<div class="doc">
-f_lt without state argument 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a name="HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a>  : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>); </div>
-
-<div class="doc">
-f_gt without state argument 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a name="HasFoldWithAbortOps.fwasa_P&acute;"><span class="id" title="projection">fwasa_P&acute;</span></a>    : (<a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)  </div>
-
-<div class="doc">
-the predicate P 
-</div>
-<div class="code">
-&nbsp;&nbsp;}.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">foldWithAbortSpecForPred</span></span> <span class="inlinecode"><span class="id" title="var">s</span></span> <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">fwasa</span></span> holds, if
-      the argument <span class="inlinecode"><span class="id" title="var">fwasa</span></span> fits the predicate <span class="inlinecode"><span class="id" title="var">P</span></span> for set <span class="inlinecode"><span class="id" title="var">s</span></span>. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>} <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">fwasa</span> : @<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) :=<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-the predicate <span class="inlinecode"><span class="id" title="var">P&acute;</span></span> coincides for <span class="inlinecode"><span class="id" title="var">s</span></span> and the given precomputation with <span class="inlinecode"><span class="id" title="var">P</span></span> 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P&acute;"><span class="id" title="projection">fwasa_P&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-If <span class="inlinecode"><span class="id" title="var">fwasa_f_lt</span></span> holds, all elements smaller than the current one
-         don't satisfy predicate P. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
-
-<div class="doc">
-If <span class="inlinecode"><span class="id" title="var">fwasa_f_gt</span></span> holds, all elements greater than the current one
-         don't satisfy predicate P. 
-</div>
-<div class="code">
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab12"></a><h3 class="section">Filter with abort</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.filter_with_abort"><span class="id" title="definition">filter_with_abort</span></a> {<span class="id" title="var">B</span>} (<span class="id" title="var">fwasa</span> : @<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">p</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#p"><span class="id" title="variable">p</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">e_pre</span> <span class="id" title="var">s</span> =&gt; <span class="id" title="keyword">if</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P&acute;"><span class="id" title="projection">fwasa_P&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#e_pre"><span class="id" title="variable">e_pre</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="MSetFoldWithAbort.html#C.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">p</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#C.empty"><span class="id" title="axiom">empty</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.filter_with_abort_spec"><span class="id" title="lemma">filter_with_abort_spec</span></a> {<span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> <span class="id" title="var">fwasa</span> <span class="id" title="var">P</span> <span class="id" title="var">s</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort"><span class="id" title="definition">filter_with_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#C.filter"><span class="id" title="axiom">filter</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab13"></a><h3 class="section">Choose with abort</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> {<span class="id" title="var">B</span>} (<span class="id" title="var">fwasa</span> : @<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">p</span> <span class="id" title="var">st</span> =&gt; <span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> =&gt; (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#p"><span class="id" title="variable">p</span></a>) | <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">end</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">e_pre</span> <span class="id" title="var">st</span> =&gt; <span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> =&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P&acute;"><span class="id" title="projection">fwasa_P&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#e_pre"><span class="id" title="variable">e_pre</span></a><span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> | <span class="id" title="var">_</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">end</span>) <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> <span class="id" title="var">p</span> <span class="id" title="var">st</span> =&gt; <span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> =&gt; (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#p"><span class="id" title="variable">p</span></a>) | <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">end</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.choose_with_abort_spec"><span class="id" title="lemma">choose_with_abort_spec</span></a> {<span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> <span class="id" title="var">fwasa</span> <span class="id" title="var">P</span> <span class="id" title="var">s</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">match</span> (<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> =&gt; (<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">e</span> =&gt; <a class="idref" href="MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <span class="id" title="var">e</span> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab14"></a><h3 class="section">Exists and Forall with abort</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a> {<span class="id" title="var">B</span>} (<span class="id" title="var">fwasa</span> : @<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.exists_with_abort_spec"><span class="id" title="lemma">exists_with_abort_spec</span></a> {<span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> <span class="id" title="var">fwasa</span> <span class="id" title="var">P</span> <span class="id" title="var">s</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.exists_"><span class="id" title="axiom">exists_</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>).<br/>
-</div>
-
-<div class="doc">
-Negation leads to forall. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="HasFoldWithAbortOps.forall_with_abort"><span class="id" title="definition">forall_with_abort</span></a> {<span class="id" title="var">B</span>} <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="HasFoldWithAbortOps.forall_with_abort_spec"><span class="id" title="lemma">forall_with_abort_spec</span></a> {<span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort"><span class="id" title="definition">forall_with_abort</span></a> <a class="idref" href="MSetFoldWithAbort.html#fwasa"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#C.for_all"><span class="id" title="axiom">for_all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">e</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>)) <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>).<br/>
- <span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab15"></a><h2 class="section">Modules Types For Sets with Fold with Abort</h2>
-
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="WSetsWithDupsFoldA"><span class="id" title="module">WSetsWithDupsFoldA</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a name="WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithDupsFoldA"><span class="id" title="module">WSetsWithDupsFoldA</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="WSetsWithFoldA"><span class="id" title="module">WSetsWithFoldA</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSets"><span class="id" title="module">WSets</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a name="WSetsWithFoldA.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#WSetsWithFoldA"><span class="id" title="module">WSetsWithFoldA</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#Sets"><span class="id" title="module">Sets</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a name="SetsWithFoldA.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#SetsOn"><span class="id" title="module">SetsOn</span></a> <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab16"></a><h2 class="section">Implementations</h2>
-
-<div class="paragraph"> </div>
-
-<a name="lab17"></a><h3 class="section">GenTree implementation</h3>
-
-    Finally, provide such a fold with abort operation for generic trees. 
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a> (<span class="id" title="keyword">Import</span> <span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) (<span class="id" title="keyword">Import</span> <span class="id" title="var">I</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetGenTree.html#InfoTyp"><span class="id" title="module">InfoTyp</span></a>) <br/>
-&nbsp;&nbsp;(<span class="id" title="keyword">Import</span> <span class="id" title="var">Raw</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetGenTree.html#Ops"><span class="id" title="module">Ops</span></a> <a class="idref" href="MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetFoldWithAbort.html#I"><span class="id" title="module">I</span></a>)<br/>
-&nbsp;&nbsp;(<span class="id" title="var">M</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetGenTree.html#Props"><span class="id" title="module">MSetGenTree.Props</span></a> <a class="idref" href="MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetFoldWithAbort.html#I"><span class="id" title="module">I</span></a> <a class="idref" href="MSetFoldWithAbort.html#Raw"><span class="id" title="module">Raw</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f_pre</span> : <a class="idref" href="MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">f_lt</span> (<span class="id" title="var">f</span>: <a class="idref" href="MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">f_gt</span> <span class="id" title="var">t</span> (<span class="id" title="var">base</span>: <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetFoldWithAbort.html#Raw.Leaf"><span class="id" title="constructor">Raw.Leaf</span></a> =&gt; <a class="idref" href="MSetFoldWithAbort.html#base"><span class="id" title="variable">base</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetFoldWithAbort.html#Raw.Node"><span class="id" title="constructor">Raw.Node</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> <span class="id" title="var">x</span> <span class="id" title="var">r</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">x_pre</span> := <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">in</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">st0</span> := <span class="id" title="keyword">if</span> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetFoldWithAbort.html#x_pre"><span class="id" title="variable">x_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#base"><span class="id" title="variable">base</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="MSetFoldWithAbort.html#base"><span class="id" title="variable">base</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="MSetFoldWithAbort.html#foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">l</span> <a class="idref" href="MSetFoldWithAbort.html#base"><span class="id" title="variable">base</span></a> <span class="id" title="keyword">in</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">st1</span> := <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetFoldWithAbort.html#x_pre"><span class="id" title="variable">x_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#st0"><span class="id" title="variable">st0</span></a> <span class="id" title="keyword">in</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">st2</span> := <span class="id" title="keyword">if</span> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetFoldWithAbort.html#x_pre"><span class="id" title="variable">x_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#st1"><span class="id" title="variable">st1</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="MSetFoldWithAbort.html#st1"><span class="id" title="variable">st1</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="MSetFoldWithAbort.html#foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">r</span> <a class="idref" href="MSetFoldWithAbort.html#st1"><span class="id" title="variable">st1</span></a> <span class="id" title="keyword">in</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#st2"><span class="id" title="variable">st2</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MakeGenTreeFoldA.foldWithAbort_RawSpec"><span class="id" title="lemma">foldWithAbort_RawSpec</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> <span class="id" title="var">i&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_pre</span> : <a class="idref" href="MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span> : <a class="idref" href="MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f&acute;</span> : <a class="idref" href="MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetFoldWithAbort.html#Raw.tree"><span class="id" title="inductive">Raw.tree</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">P</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <span class="id" title="keyword">Prop</span>),<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetFoldWithAbort.html#M.bst"><span class="id" title="inductive">M.bst</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a>) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e</span>, <a class="idref" href="MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>)) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span>, <a class="idref" href="MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a>)) -&gt;       <br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">st&acute;</span> <span class="id" title="var">e2</span>, <a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#st"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#e2"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetFoldWithAbort.html#st&acute;"><span class="id" title="variable">st&acute;</span></a>))) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetFoldWithAbort.html#f&acute;"><span class="id" title="variable">f&acute;</span></a> <a class="idref" href="MSetFoldWithAbort.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetFoldWithAbort.html#i&acute;"><span class="id" title="variable">i&acute;</span></a>).<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab18"></a><h3 class="section">AVL implementation</h3>
-
-    The generic tree implementation naturally leads to an AVL one. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="MakeAVLSetsWithFoldA"><span class="id" title="module">MakeAVLSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetAVL.html#Make"><span class="id" title="module">MSetAVL.Make</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.Int.html#Z_as_Int"><span class="id" title="module">Z_as_Int</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="MakeAVLSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> (<span class="id" title="var">f</span>: <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">f_gt</span> <span class="id" title="var">t</span> (<span class="id" title="var">base</span>: <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> (<a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.this"><span class="id" title="projection">this</span></a>)) <a class="idref" href="MSetFoldWithAbort.html#base"><span class="id" title="variable">base</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA"><span class="id" title="module">MakeAVLSetsWithFoldA</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab19"></a><h3 class="section">RBT implementation</h3>
-
-    The generic tree implementation naturally leads to an RBT one. 
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="MakeRBTSetsWithFoldA"><span class="id" title="module">MakeRBTSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetRBT.html#Make"><span class="id" title="module">MSetRBT.Make</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetRBT.html#Color"><span class="id" title="module">Color</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="MakeRBTSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> (<span class="id" title="var">f</span>: <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) <span class="id" title="var">f_gt</span> <span class="id" title="var">t</span> (<span class="id" title="var">base</span>: <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> (<a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.this"><span class="id" title="projection">this</span></a>)) <a class="idref" href="MSetFoldWithAbort.html#base"><span class="id" title="variable">base</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA"><span class="id" title="module">MakeRBTSetsWithFoldA</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab20"></a><h2 class="section">Sorted Lists Implementation</h2>
-
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="MakeListSetsWithFoldA"><span class="id" title="module">MakeListSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetList.html#Make"><span class="id" title="module">MSetList.Make</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="MakeListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f_pre</span> : <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">f_lt</span> : <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span>: <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">t</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a>) (<span class="id" title="var">acc</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> :=<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> =&gt; (<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">pre_x</span> := <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">in</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">acc</span> := <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> (<a class="idref" href="MSetFoldWithAbort.html#pre_x"><span class="id" title="variable">pre_x</span></a>) <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a> <span class="id" title="keyword">in</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetFoldWithAbort.html#pre_x"><span class="id" title="variable">pre_x</span></a> <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><span class="id" title="notation">)</span> <span class="id" title="keyword">then</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">xs</span> <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;)<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="MakeListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">t</span> <span class="id" title="var">acc</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.this"><span class="id" title="projection">this</span></a>) <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#MakeListSetsWithFoldA"><span class="id" title="module">MakeListSetsWithFoldA</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab21"></a><h3 class="section">Unsorted Lists without Dups Implementation</h3>
-
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="MakeWeakListSetsWithFoldA"><span class="id" title="module">MakeWeakListSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetFoldWithAbort.html#WSetsWithFoldA"><span class="id" title="module">WSetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="MakeWeakListSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetWeakList.html#MakeRaw"><span class="id" title="module">MSetWeakList.MakeRaw</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="MakeWeakListSetsWithFoldA.E"><span class="id" title="module">E</span></a> := <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WRaw2SetsOn"><span class="id" title="module">WRaw2SetsOn</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="MakeWeakListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f_pre</span> : <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">f_lt</span> : <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">f</span>: <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f_gt</span> : <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">t</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a>) (<span class="id" title="var">acc</span> : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> :=<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> =&gt; (<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">pre_x</span> := <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">in</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">acc</span> := <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> (<a class="idref" href="MSetFoldWithAbort.html#pre_x"><span class="id" title="variable">pre_x</span></a>) <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a> <span class="id" title="keyword">in</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetFoldWithAbort.html#pre_x"><span class="id" title="variable">pre_x</span></a> <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetFoldWithAbort.html#pre_x"><span class="id" title="variable">pre_x</span></a> <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">then</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetFoldWithAbort.html#foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">xs</span> <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;)<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="MakeWeakListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>: <span class="id" title="keyword">Type</span>} <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">t</span> <span class="id" title="var">acc</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetFoldWithAbort.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetFoldWithAbort.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_pre"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_lt"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetFoldWithAbort.html#f_gt"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetFoldWithAbort.html#t"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.this"><span class="id" title="projection">this</span></a>) <a class="idref" href="MSetFoldWithAbort.html#acc"><span class="id" title="variable">acc</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA"><span class="id" title="module">MakeWeakListSetsWithFoldA</span></a>.<br/>
-
-<br/>
-</div>
-</div>
-
-<div id="footer">
-<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
-</div>
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-</div>
-
-</body>
-</html>
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-"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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-<head>
-<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
-<link href="coqdoc.css" rel="stylesheet" type="text/css" />
-<title>MSetIntervals</title>
-</head>
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-<body>
-
-<div id="page">
-
-<div id="header">
-</div>
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-<div id="main">
-
-<h1 class="libtitle">Library MSetIntervals</h1>
-
-<div class="code">
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab22"></a><h1 class="section">Weak sets implemented by interval lists</h1>
-
-
-<div class="paragraph"> </div>
-
-    This file contains an implementation of the weak set interface
-    <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span> which uses internally intervals of Z. This allows some
-    large sets, which naturally map to intervals of integers to be
-    represented very efficiently.
-
-<div class="paragraph"> </div>
-
-    Internally intervals of Z are used. However, via an encoding
-    and decoding layer, other types of elements can be handled as
-    well. There are instantiations for Z, N and nat currently.
-    More can be easily added.
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#"><span class="id" title="library">MSetInterface</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.OrdersFacts.html#"><span class="id" title="library">OrdersFacts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.OrdersLists.html#"><span class="id" title="library">OrdersLists</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#"><span class="id" title="library">BinNat</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssreflect</span>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.Natural.Abstract.NOrder.html#"><span class="id" title="library">NOrder</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.DecidableTypeEx.html#"><span class="id" title="library">DecidableTypeEx</span></a>.<br/>
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="NOP"><span class="id" title="module">NOP</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.Natural.Abstract.NOrder.html#NOrderProp"><span class="id" title="module">NOrderProp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N"><span class="id" title="module">N</span></a>.<br/>
-<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">Z_scope</span>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab23"></a><h2 class="section">Auxiliary stuff</h2>
-
-<div class="paragraph"> </div>
-
- Simple auxiliary lemmata 
-</div>
-<div class="code">
-<span class="id" title="keyword">Lemma</span> <a name="Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Lemma</span> <a name="add_add_sub_eq"><span class="id" title="lemma">add_add_sub_eq</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a>).<br/>
-
-<br/>
-<span class="id" title="keyword">Lemma</span> <a name="NoDupA_map"><span class="id" title="lemma">NoDupA_map</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> (<span class="id" title="var">eqA</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">eqB</span> : <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a> -&gt; <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">l</span>,<br/>
-&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetIntervals.html#eqA"><span class="id" title="variable">eqA</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> -&gt;<br/>
-&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>,  <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x2"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#eqB"><span class="id" title="variable">eqB</span></a> (<a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x1"><span class="id" title="variable">x1</span></a>) (<a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x2"><span class="id" title="variable">x2</span></a>) -&gt; <a class="idref" href="MSetIntervals.html#eqA"><span class="id" title="variable">eqA</span></a> <a class="idref" href="MSetIntervals.html#x1"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetIntervals.html#x2"><span class="id" title="variable">x2</span></a>) -&gt;<br/>
-&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetIntervals.html#eqB"><span class="id" title="variable">eqB</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab24"></a><h3 class="section">rev_map</h3>
-
-
-<div class="paragraph"> </div>
-
-    rev_map is used for efficiency. 
-</div>
-<div class="code">
-<span class="id" title="keyword">Fixpoint</span> <a name="rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">acc</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>) :=<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> =&gt; <a class="idref" href="MSetIntervals.html#rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">)::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-<span class="id" title="keyword">Definition</span> <a name="rev_map"><span class="id" title="definition">rev_map</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a> := <a class="idref" href="MSetIntervals.html#rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-Lemmata about rev_map 
-</div>
-<div class="code">
-<span class="id" title="keyword">Lemma</span> <a name="rev_map_aux_alt_def"><span class="id" title="lemma">rev_map_aux_alt_def</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">l</span> <span class="id" title="var">acc</span>,<br/>
-&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>.<br/>
-<span class="id" title="keyword">Lemma</span> <a name="rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> {<span class="id" title="var">A</span> <span class="id" title="var">B</span>} : <span class="id" title="keyword">forall</span> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) <span class="id" title="var">l</span>,<br/>
-&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#rev_map"><span class="id" title="definition">rev_map</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab25"></a><h2 class="section">Encoding Elements</h2>
-
-<div class="paragraph"> </div>
-
- We want to encode not only elements of type Z, but other types
-    as well. In order to do so, an encoding / decoding layer is used.
-    This layer is represented by module type <span class="inlinecode"><span class="id" title="var">ElementEncode</span></span>. It
-    provides encode and decode function. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a name="ElementEncode.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="ElementEncode.encode"><span class="id" title="axiom">encode</span></a> : <a class="idref" href="MSetIntervals.html#ElementEncode.E.t"><span class="id" title="axiom">E.t</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="ElementEncode.decode"><span class="id" title="axiom">decode</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> -&gt; <a class="idref" href="MSetIntervals.html#ElementEncode.E.t"><span class="id" title="axiom">E.t</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-Decoding is the inverse of encoding. Notice that
-      the reverse is not demanded. This means that
-      we do need to provide for all integers <span class="inlinecode"><span class="id" title="var">z</span></span> an
-      element <span class="inlinecode"><span class="id" title="var">e</span></span> with <span class="inlinecode"><span class="id" title="var">encode</span></span> <span class="inlinecode"><span class="id" title="var">v</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">z</span></span>. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a name="ElementEncode.decode_encode_ok"><span class="id" title="axiom">decode_encode_ok</span></a>: <span class="id" title="keyword">forall</span> (<span class="id" title="var">e</span> : <a class="idref" href="MSetIntervals.html#ElementEncode.E.t"><span class="id" title="axiom">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#ElementEncode.decode"><span class="id" title="axiom">decode</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-Encoding is compatible with the equality of elements. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a name="ElementEncode.encode_eq"><span class="id" title="axiom">encode_eq</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">e1</span> <span class="id" title="var">e2</span> : <a class="idref" href="MSetIntervals.html#ElementEncode.E.t"><span class="id" title="axiom">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#ElementEncode.E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab26"></a><h2 class="section">Set Operations</h2>
-
-
-<div class="paragraph"> </div>
-
-    We represent sets of Z via lists of intervals. The intervals are all
-    in increasing order and non-overlapping. Moreover, we require the most compact
-    representation, i.e. no two intervals can be merged. For example
-
-<div class="paragraph"> </div>
-
-    <span class="inlinecode">0-2,</span> <span class="inlinecode">4-4,</span> <span class="inlinecode">6-8</span> is a valid interval list for the set {0;1;2;4;6;7;8}
-
-<div class="paragraph"> </div>
-
-    In contrast
-
-<div class="paragraph"> </div>
-
-    <span class="inlinecode">4-4,</span> <span class="inlinecode">0-2,</span> <span class="inlinecode">6-8</span> is a invalid because the intervals are not ordered andb
-    <span class="inlinecode">0-2,</span> <span class="inlinecode">4-5,</span> <span class="inlinecode">6-8</span> is a invalid because it is not compact (<span class="inlinecode">0-2,</span> <span class="inlinecode">4--8</span> is valid).
-
-<div class="paragraph"> </div>
-
-    Intervals we represent by tuples <span class="inlinecode">(<span class="id" title="var">Z</span>,</span> <span class="inlinecode"><span class="id" title="var">N</span>)</span>. The tuple <span class="inlinecode">(<span class="id" title="var">z</span>,</span> <span class="inlinecode"><span class="id" title="var">c</span>)</span>  represents the
-    interval <span class="inlinecode"><span class="id" title="var">z</span>-(<span class="id" title="var">z</span>+<span class="id" title="var">c</span>)</span>.
-
-<div class="paragraph"> </div>
-
-    We apply the encode function before adding an element to such interval sets and
-    the decode function when checking it. This allows for sets with other element types
-    than Z.
- 
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="Ops"><span class="id" title="module">Ops</span></a> (<span class="id" title="var">Enc</span> : <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>) &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WOps"><span class="id" title="module">WOps</span></a> <a class="idref" href="MSetIntervals.html#Enc.E"><span class="id" title="module">Enc.E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetIntervals.html#Enc.E.t"><span class="id" title="axiom">Enc.E.t</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-The empty list is trivial to define and check for. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.is_empty"><span class="id" title="definition">is_empty</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> | <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-Defining the list of elements, is much more tricky, especially, 
-      if it needs to be executable. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Ops.acc_pred"><span class="id" title="lemma">acc_pred</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span>, <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <a class="idref" href="MSetIntervals.html#p"><span class="id" title="variable">p</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.lt"><span class="id" title="definition">N.lt</span></a> <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.lt"><span class="id" title="definition">N.lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.pred"><span class="id" title="definition">N.pred</span></a> <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.elementsZ_aux&acute;&acute;"><span class="id" title="definition">elementsZ_aux&acute;&acute;</span></a> (<span class="id" title="var">acc</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.lt"><span class="id" title="definition">N.lt</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) { <span class="id" title="keyword">struct</span> <span class="id" title="var">H</span> } : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">c0</span> <span class="id" title="keyword">return</span> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#c0"><span class="id" title="variable">c0</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N0"><span class="id" title="constructor">N0</span></a> =&gt; <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">c</span> =&gt; <span class="id" title="keyword">fun</span> <span class="id" title="var">Heq</span> =&gt; <a class="idref" href="MSetIntervals.html#elementsZ_aux&acute;&acute;"><span class="id" title="definition">elementsZ_aux&acute;&acute;</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.pred"><span class="id" title="definition">N.pred</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) (<a class="idref" href="MSetIntervals.html#Ops.acc_pred"><span class="id" title="lemma">acc_pred</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetIntervals.html#Heq"><span class="id" title="variable">Heq</span></a> <a class="idref" href="MSetIntervals.html#H"><span class="id" title="variable">H</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Extraction</span> <span class="id" title="keyword">Inline</span> <span class="id" title="var">elementsZ_aux&acute;&acute;</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> <span class="id" title="var">acc</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> := <a class="idref" href="MSetIntervals.html#Ops.elementsZ_aux&acute;&acute;"><span class="id" title="definition">elementsZ_aux&acute;&acute;</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> (<a class="idref" href="MSetIntervals.html#NOP.lt_wf_0"><span class="id" title="lemma">lt_wf_0</span></a> <span class="id" title="var">_</span>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.elementsZ_aux"><span class="id" title="definition">elementsZ_aux</span></a> <span class="id" title="var">acc</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">s&acute;</span> =&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#elementsZ_aux"><span class="id" title="definition">elementsZ_aux</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <span class="id" title="var">x</span> <span class="id" title="var">c</span>) <span class="id" title="var">s&acute;</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Ops.elementsZ_aux"><span class="id" title="definition">elementsZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elements"><span class="id" title="definition">elements</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Ops.rev_map"><span class="id" title="definition">rev_map</span></a> <a class="idref" href="MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-membership is easily defined 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.memZ"><span class="id" title="definition">memZ</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span><span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)<span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#memZ"><span class="id" title="definition">memZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.mem"><span class="id" title="definition">mem</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#Ops.memZ"><span class="id" title="definition">memZ</span></a> (<a class="idref" href="MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-adding an element needs to be defined carefully again in order to
-      generate efficient code 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> (<span class="id" title="var">acc</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <span class="id" title="var">y</span>) <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> =&gt; <span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)) <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> =&gt; <a class="idref" href="MSetIntervals.html#addZ_aux"><span class="id" title="definition">addZ_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c&acute;</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l&acute;</span> =&gt; <span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eqb"><span class="id" title="definition">Z.eqb</span></a> <span class="id" title="var">z</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>)<span class="id" title="notation">)</span> <span class="id" title="keyword">then</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> (<span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a><span class="id" title="var">c&acute;</span>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l&acute;</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>                 <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.addZ"><span class="id" title="definition">addZ</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := <a class="idref" href="MSetIntervals.html#Ops.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.add"><span class="id" title="definition">add</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := <a class="idref" href="MSetIntervals.html#Ops.addZ"><span class="id" title="definition">addZ</span></a> (<a class="idref" href="MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">add_list</span></span> simple extension to add many elements, which then allows to
-      define from_elements. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.add_list"><span class="id" title="definition">add_list</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#Ops.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.from_elements"><span class="id" title="definition">from_elements</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#Ops.add_list"><span class="id" title="definition">add_list</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#Ops.empty"><span class="id" title="definition">empty</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">singleton</span></span> is trivial to define 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.singleton"><span class="id" title="definition">singleton</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Ops.singleton_alt_def"><span class="id" title="lemma">singleton_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetIntervals.html#Ops.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#Ops.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#Ops.empty"><span class="id" title="definition">empty</span></a>.<br/>
- 
-<br/>
-</div>
-
-<div class="doc">
-removing needs to be done with code extraction in mind again. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> 0<span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> (<span class="id" title="var">acc</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span><span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)<span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <span class="id" title="notation">(</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="MSetIntervals.html#Ops.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">)-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">)</span></a>)) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Ops.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a><span class="id" title="var">y</span>)) <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>)) <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">)</span> <span class="id" title="keyword">else</span> <a class="idref" href="MSetIntervals.html#removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.removeZ"><span class="id" title="definition">removeZ</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#Ops.removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.remove"><span class="id" title="definition">remove</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#Ops.removeZ"><span class="id" title="definition">removeZ</span></a> (<a class="idref" href="MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.remove_list"><span class="id" title="definition">remove_list</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#Ops.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-all other operations are defined trivially (if not always efficiently)
-      in terms of already defined ones. In the future it might be worth implementing
-      some of them more efficiently. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.union"><span class="id" title="definition">union</span></a> (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Ops.add_list"><span class="id" title="definition">add_list</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s1"><span class="id" title="variable">s1</span></a>) <a class="idref" href="MSetIntervals.html#s2"><span class="id" title="variable">s2</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Ops.from_elements"><span class="id" title="definition">from_elements</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#filter"><span class="id" title="definition">List.filter</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.inter"><span class="id" title="definition">inter</span></a> (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Ops.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#Ops.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="MSetIntervals.html#s1"><span class="id" title="variable">s1</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.diff"><span class="id" title="definition">diff</span></a> (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Ops.remove_list"><span class="id" title="definition">remove_list</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="MSetIntervals.html#s1"><span class="id" title="variable">s1</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.subset"><span class="id" title="definition">subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#forallb"><span class="id" title="definition">List.forallb</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#Ops.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.equal"><span class="id" title="definition">equal</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">cx</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">xs</span>), (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">cy</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">ys</span>) =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eqb"><span class="id" title="definition">Z.eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <span class="id" title="var">cx</span> <span class="id" title="var">cy</span>) (<a class="idref" href="MSetIntervals.html#equal"><span class="id" title="definition">equal</span></a> <span class="id" title="var">xs</span> <span class="id" title="var">ys</span>))<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.fold"><span class="id" title="definition">fold</span></a> {<span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a>) : <a class="idref" href="MSetIntervals.html#B"><span class="id" title="variable">B</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetIntervals.html#i"><span class="id" title="variable">i</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.for_all"><span class="id" title="definition">for_all</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#forallb"><span class="id" title="definition">List.forallb</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.exists_"><span class="id" title="definition">exists_</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#existsb"><span class="id" title="definition">List.existsb</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.partition"><span class="id" title="definition">partition</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Ops.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#Ops.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.cardinalN"><span class="id" title="definition">cardinalN</span></a> <span class="id" title="var">c</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">cx</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">xs</span> =&gt; <a class="idref" href="MSetIntervals.html#cardinalN"><span class="id" title="definition">cardinalN</span></a> (<a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a> <span class="id" title="var">cx</span>)%<span class="id" title="var">N</span> <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.cardinal"><span class="id" title="definition">cardinal</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> (<a class="idref" href="MSetIntervals.html#Ops.cardinalN"><span class="id" title="definition">cardinalN</span></a> (0%<span class="id" title="var">N</span>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.chooseZ"><span class="id" title="definition">chooseZ</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> (<a class="idref" href="MSetIntervals.html#Ops.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.choose"><span class="id" title="definition">choose</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">e</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#Ops"><span class="id" title="module">Ops</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab27"></a><h2 class="section">Raw Module</h2>
-
-
-<div class="paragraph"> </div>
-
-    Following the idea of <span class="inlinecode"><span class="id" title="var">MSetInterface.RawSets</span></span>, we first
-    define a module <span class="inlinecode"><span class="id" title="var">Raw</span></span> proves all the required properties with
-    respect to an explicitly provided invariant. In a next step, this
-    invariant is then moved into the set type. This allows to instantiate
-    the <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span> interface. 
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="Raw"><span class="id" title="module">Raw</span></a> (<span class="id" title="var">Enc</span> : <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> (<a class="idref" href="MSetIntervals.html#Ops"><span class="id" title="module">Ops</span></a> <a class="idref" href="MSetIntervals.html#Enc"><span class="id" title="module">Enc</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab28"></a><h3 class="section">Defining invariant <span class="inlinecode"><span class="id" title="var">IsOk</span></span></h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.inf"><span class="id" title="definition">inf</span></a> (<span class="id" title="var">x</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">l</span>: <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) :=<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span><br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Raw.isok"><span class="id" title="definition">isok</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) :=<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span> =&gt; <a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> (<span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a>) <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <span class="id" title="var">c</span> 0) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="MSetIntervals.html#isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a><br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">exists</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <span class="id" title="var">s</span> := (<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab29"></a><h3 class="section">Defining notations</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a name="Raw.ForNotations"><span class="id" title="section">ForNotations</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Class</span> <a name="Raw.Ok"><span class="id" title="record">Ok</span></a> (<span class="id" title="var">s</span>:<a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) : <span class="id" title="keyword">Prop</span> := <a name="Raw.ok"><span class="id" title="projection">ok</span></a> : <a class="idref" href="MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hint Resolve</span> @<a class="idref" href="MSetIntervals.html#Raw.ok"><span class="id" title="definition">ok</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hint Unfold</span> <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a name="Raw.IsOk_Ok"><span class="id" title="instance">IsOk_Ok</span></a> <span class="id" title="var">s</span> `(<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> := { <a class="idref" href="MSetIntervals.html#Raw.ok"><span class="id" title="definition">ok</span></a> := <span class="id" title="var">Hs</span> }.<br/>
-
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.In"><span class="id" title="definition">In</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">SetoidList.InA</span></a> <a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.Empty"><span class="id" title="definition">Empty</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <span class="id" title="var">s</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">exists</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#Raw.ForNotations"><span class="id" title="section">ForNotations</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab30"></a><h3 class="section">elements list properties</h3>
-
-
-<div class="paragraph"> </div>
-
-      The functions <span class="inlinecode"><span class="id" title="var">elementsZ</span></span>, <span class="inlinecode"><span class="id" title="var">elementsZ_single</span></span>, 
-      <span class="inlinecode"><span class="id" title="var">elements</span></span> and <span class="inlinecode"><span class="id" title="var">elements_single</span></span> are crucial and used
-      everywhere. Therefore, we first establish a few properties of
-      these important functions. 
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a> : (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="notation">:</span> <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a> : (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="notation">:</span> <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> (<span class="id" title="var">x</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.peano_rec"><span class="id" title="definition">N.peano_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">n</span> <span class="id" title="var">ls</span> =&gt; (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ls"><span class="id" title="variable">ls</span></a>) <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.elements_single"><span class="id" title="definition">elements_single</span></a> <span class="id" title="var">x</span> <span class="id" title="var">c</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (0%<span class="id" title="var">N</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_single_add"><span class="id" title="lemma">elementsZ_single_add</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c2</span> <span class="id" title="var">c1</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#c1"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a> <a class="idref" href="MSetIntervals.html#c2"><span class="id" title="variable">c2</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c1"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c1"><span class="id" title="variable">c1</span></a>) <a class="idref" href="MSetIntervals.html#c2"><span class="id" title="variable">c2</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">c</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (1%<span class="id" title="var">N</span>)) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.length_elementsZ_single"><span class="id" title="lemma">length_elementsZ_single</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">cx</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_aux&acute;&acute;_irrel"><span class="id" title="lemma">elementsZ_aux&acute;&acute;_irrel</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">c</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;&acute;"><span class="id" title="definition">elementsZ_aux&acute;&acute;</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetIntervals.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;&acute;"><span class="id" title="definition">elementsZ_aux&acute;&acute;</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetIntervals.html#H2"><span class="id" title="variable">H2</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_aux&acute;_pos"><span class="id" title="lemma">elementsZ_aux&acute;_pos</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">p</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="MSetIntervals.html#p"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.PArith.BinPos.html#Pos.pred_N"><span class="id" title="definition">Pos.pred_N</span></a> <a class="idref" href="MSetIntervals.html#p"><span class="id" title="variable">p</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_aux&acute;_zero"><span class="id" title="lemma">elementsZ_aux&acute;_zero</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (0%<span class="id" title="var">N</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_aux&acute;_succ"><span class="id" title="lemma">elementsZ_aux&acute;_succ</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_single_intro"><span class="id" title="lemma">elementsZ_single_intro</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux&acute;"><span class="id" title="definition">elementsZ_aux&acute;</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_aux_alt_def"><span class="id" title="lemma">elementsZ_aux_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">acc</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ_aux"><span class="id" title="definition">elementsZ_aux</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">:</span> <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elements_cons"><span class="id" title="lemma">elements_cons</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> (<span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><span class="id" title="notation">)</span> <span class="id" title="notation">:</span> <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Raw.elements_single"><span class="id" title="definition">elements_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) <span class="id" title="var">x</span>, (<a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> 0)%<span class="id" title="var">N</span> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab31"></a><h3 class="section">Alternative definition of addZ</h3>
-
-<div class="paragraph"> </div>
-
- <span class="inlinecode"><span class="id" title="var">addZ</span></span> is defined with efficient execution in mind.
-      We derive first an alternative definition that demonstrates
-      the intention better and is better suited for proofs. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.addZ_ind"><span class="id" title="lemma">addZ_ind</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) -&gt; <span class="id" title="keyword">Prop</span>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)  (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> <span class="id" title="var">c&acute;</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c&acute;"><span class="id" title="variable">c&acute;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">)</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c&acute;</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#b4b39b4b0d7e7e7359e0985cb2b2cef5"><span class="id" title="notation">=?</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c&acute;"><span class="id" title="variable">c&acute;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>))) -&gt;<br/>
-<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>)) -&gt;<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)(<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#f12180ef2a874e1992519421c4388969"><span class="id" title="notation">?=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>)) -&gt;<br/>
-<br/>
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.addZ_aux_alt_def"><span class="id" title="lemma">addZ_aux_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">acc</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.addZ_alt_def"><span class="id" title="lemma">addZ_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a>1) <span class="id" title="var">y</span>) <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> (<span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a>1)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> =&gt; <span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)) <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> =&gt; <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <span class="id" title="keyword">match</span> <span class="id" title="var">l</span> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> (<span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a>1)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c&acute;</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l&acute;</span> =&gt; <span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eqb"><span class="id" title="definition">Z.eqb</span></a> <span class="id" title="var">z</span> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)<span class="id" title="notation">)</span> <span class="id" title="keyword">then</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> (<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a> <span class="id" title="var">c&acute;</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l&acute;</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a>(<span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a>1)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c&acute;</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l&acute;</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>                 <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>  <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab32"></a><h3 class="section">Alternative definition of removeZ</h3>
-
-<div class="paragraph"> </div>
-
- <span class="inlinecode"><span class="id" title="var">removeZ</span></span> is defined with efficient execution in mind.
-      We derive first an alternative definition that demonstrates
-      the intention better and is better suited for proofs. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.removeZ_aux_alt_def"><span class="id" title="lemma">removeZ_aux_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetIntervals.html#acc"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.removeZ_alt_def"><span class="id" title="lemma">removeZ_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span><span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">else</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)<span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <span class="id" title="notation">(</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">(</span><a class="idref" href="MSetIntervals.html#Raw.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a><span class="id" title="var">y</span>)) <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">)-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">)</span></a>)) <span class="id" title="var">l</span>)<span class="id" title="notation">)</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="notation">)</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab33"></a><h3 class="section">Auxiliary Lemmata about Invariant</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.inf_impl"><span class="id" title="lemma">inf_impl</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) -&gt; <a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.Ok_nil"><span class="id" title="lemma">Ok_nil</span></a> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.is_encoded_elems_list_app"><span class="id" title="lemma">is_encoded_elems_list_app</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetIntervals.html#l1"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetIntervals.html#l2"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> <a class="idref" href="MSetIntervals.html#l1"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> <a class="idref" href="MSetIntervals.html#l2"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.is_encoded_elems_list_rev"><span class="id" title="lemma">is_encoded_elems_list_rev</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">l</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.isok_cons"><span class="id" title="lemma">isok_cons</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="var">s&acute;</span>, <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> (<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a>(<a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> 0)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.Ok_cons"><span class="id" title="lemma">Ok_cons</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="var">s&acute;</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> (<a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a>(<a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> 0)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">y</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~(</span></a><a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.isok_inf_nin"><span class="id" title="lemma">isok_inf_nin</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">)</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab34"></a><h3 class="section">Properties of In and InZ</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.In_alt_def"><span class="id" title="lemma">In_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> (<a class="idref" href="MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab35"></a><h3 class="section">Membership specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.memZ_spec"><span class="id" title="lemma">memZ_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.memZ"><span class="id" title="definition">memZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.mem_spec"><span class="id" title="lemma">mem_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab36"></a><h3 class="section">add specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.addZ_spec"><span class="id" title="lemma">addZ_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.addZ_isok"><span class="id" title="lemma">addZ_isok</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.add_ok"><span class="id" title="instance">add_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.add_spec"><span class="id" title="lemma">add_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab37"></a><h3 class="section">remove specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.isok_removeZ_aux_insert_guarded"><span class="id" title="lemma">isok_removeZ_aux_insert_guarded</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> (<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.inf_removeZ_aux_insert_guarded"><span class="id" title="lemma">inf_removeZ_aux_insert_guarded</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><span class="id" title="keyword">if</span> (<a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#815de9cef818373773d49eaaae63867a"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span> <span class="id" title="keyword">then</span> <span class="id" title="notation">(</span><a class="idref" href="MSetIntervals.html#Raw.inf"><span class="id" title="definition">inf</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><span class="id" title="notation">)</span> <span class="id" title="keyword">else</span> <span class="id" title="notation">(</span><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><span class="id" title="notation">)</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.removeZ_counter_pos_aux"><span class="id" title="lemma">removeZ_counter_pos_aux</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0 <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.removeZ_isok"><span class="id" title="lemma">removeZ_isok</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_removeZ_aux_insert_guarded"><span class="id" title="lemma">elementsZ_removeZ_aux_insert_guarded</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="MSetIntervals.html#Raw.removeZ_aux_insert_guarded"><span class="id" title="definition">removeZ_aux_insert_guarded</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.removeZ_spec"><span class="id" title="lemma">removeZ_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.remove_ok"><span class="id" title="instance">remove_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.remove_spec"><span class="id" title="lemma">remove_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab38"></a><h3 class="section">empty specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.empty_ok"><span class="id" title="instance">empty_ok</span></a> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.empty_spec&acute;"><span class="id" title="lemma">empty_spec&acute;</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, (<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.empty_spec"><span class="id" title="lemma">empty_spec</span></a> : <a class="idref" href="MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab39"></a><h3 class="section">is_empty specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.is_empty_spec"><span class="id" title="lemma">is_empty_spec</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.is_empty"><span class="id" title="definition">is_empty</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab40"></a><h3 class="section">singleton specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.singleton_ok"><span class="id" title="instance">singleton_ok</span></a> <span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.singleton_spec"><span class="id" title="lemma">singleton_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#Raw.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab41"></a><h3 class="section">add_list specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.add_list_ok"><span class="id" title="lemma">add_list_ok</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">l</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.add_list"><span class="id" title="definition">add_list</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.add_list_spec"><span class="id" title="lemma">add_list_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">l</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.add_list"><span class="id" title="definition">add_list</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">SetoidList.InA</span></a> <a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab42"></a><h3 class="section">remove_list specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.remove_list_ok"><span class="id" title="lemma">remove_list_ok</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">l</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.remove_list"><span class="id" title="definition">remove_list</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.remove_list_spec"><span class="id" title="lemma">remove_list_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">l</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.remove_list"><span class="id" title="definition">remove_list</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab43"></a><h3 class="section">union specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.union_ok"><span class="id" title="instance">union_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.union_spec"><span class="id" title="lemma">union_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) (<span class="id" title="var">Hs&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab44"></a><h3 class="section">filter specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.filter_ok"><span class="id" title="instance">filter_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">f</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.filter_spec"><span class="id" title="lemma">filter_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab45"></a><h3 class="section">inter specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.inter_ok"><span class="id" title="instance">inter_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.inter_spec"><span class="id" title="lemma">inter_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) (<span class="id" title="var">Hs&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab46"></a><h3 class="section">diff specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.diff_ok"><span class="id" title="instance">diff_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.diff_spec"><span class="id" title="lemma">diff_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) (<span class="id" title="var">Hs&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab47"></a><h3 class="section">subset specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.subset_spec"><span class="id" title="lemma">subset_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) (<span class="id" title="var">Hs&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.Subset"><span class="id" title="definition">Subset</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab48"></a><h3 class="section">elements and elementsZ specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.NoDupA_elementsZ_single"><span class="id" title="lemma">NoDupA_elementsZ_single</span></a>: <span class="id" title="keyword">forall</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_spec2w"><span class="id" title="lemma">elementsZ_spec2w</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elements_spec2w"><span class="id" title="lemma">elements_spec2w</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-</div>
-
-<div class="doc">
-<a name="lab49"></a><h3 class="section">equal specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.equal_alt_def"><span class="id" title="lemma">equal_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetIntervals.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetIntervals.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#s1"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#s2"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_cons_le_start"><span class="id" title="lemma">elementsZ_cons_le_start</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">cx</span> <span class="id" title="var">xs</span> <span class="id" title="var">y</span> <span class="id" title="var">cy</span> <span class="id" title="var">ys</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#xs"><span class="id" title="variable">xs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ys"><span class="id" title="variable">ys</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">z</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ys"><span class="id" title="variable">ys</span></a>)) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#xs"><span class="id" title="variable">xs</span></a>))) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_cons_le_end"><span class="id" title="lemma">elementsZ_cons_le_end</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">cx</span> <span class="id" title="var">xs</span> <span class="id" title="var">y</span> <span class="id" title="var">cy</span> <span class="id" title="var">ys</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#xs"><span class="id" title="variable">xs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ys"><span class="id" title="variable">ys</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a>) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">z</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#xs"><span class="id" title="variable">xs</span></a>)) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ys"><span class="id" title="variable">ys</span></a>))) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">&lt;=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_cons_equiv_hd"><span class="id" title="lemma">elementsZ_cons_equiv_hd</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">cx</span> <span class="id" title="var">xs</span> <span class="id" title="var">y</span> <span class="id" title="var">cy</span> <span class="id" title="var">ys</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#xs"><span class="id" title="variable">xs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.isok"><span class="id" title="definition">isok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ys"><span class="id" title="variable">ys</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">z</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#xs"><span class="id" title="variable">xs</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetIntervals.html#ys"><span class="id" title="variable">ys</span></a>))) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.elementsZ_single_equiv"><span class="id" title="lemma">elementsZ_single_equiv</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">cx</span> <span class="id" title="var">y</span> <span class="id" title="var">cy</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> 0)%<span class="id" title="var">N</span> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">&lt;&gt;</span></a> 0)%<span class="id" title="var">N</span> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">z</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a>)) -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#cx"><span class="id" title="variable">cx</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#cy"><span class="id" title="variable">cy</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.equal_elementsZ"><span class="id" title="lemma">equal_elementsZ</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) {<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>} {<span class="id" title="var">Hs&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>},<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, (<a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>)) -&gt; (<a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.equal_spec"><span class="id" title="lemma">equal_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) {<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>} {<span class="id" title="var">Hs&acute;</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>},<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-</div>
-
-<div class="doc">
-<a name="lab50"></a><h3 class="section">choose specification</h3>
-
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.choose_alt_def"><span class="id" title="lemma">choose_alt_def</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="MSetIntervals.html#Raw.chooseZ"><span class="id" title="definition">chooseZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">e</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">e</span>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.choose_spec1"><span class="id" title="definition">choose_spec1</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>), <a class="idref" href="MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Raw.choose_spec2"><span class="id" title="definition">choose_spec2</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.chooseZ_min"><span class="id" title="lemma">chooseZ_min</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.chooseZ"><span class="id" title="definition">chooseZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> <a class="idref" href="MSetIntervals.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.chooseZ_InZ"><span class="id" title="lemma">chooseZ_InZ</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.chooseZ"><span class="id" title="definition">chooseZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.chooseZ_spec3"><span class="id" title="lemma">chooseZ_spec3</span></a>: <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> <span class="id" title="var">x</span> <span class="id" title="var">x&acute;</span>, <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.chooseZ"><span class="id" title="definition">chooseZ</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.chooseZ"><span class="id" title="definition">chooseZ</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetIntervals.html#x&acute;"><span class="id" title="variable">x&acute;</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> -&gt; <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#x&acute;"><span class="id" title="variable">x&acute;</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab51"></a><h3 class="section">fold specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.fold_spec"><span class="id" title="lemma">fold_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetIntervals.html#i"><span class="id" title="variable">i</span></a>.<br/>
-
-<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab52"></a><h3 class="section">cardinal specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.cardinalN_spec"><span class="id" title="lemma">cardinalN_spec</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">c</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>), <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.cardinalN"><span class="id" title="definition">cardinalN</span></a> <a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="MSetIntervals.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#3eae0a7077c32c35751dcbc966d4030e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N.of_nat"><span class="id" title="definition">N.of_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)))%<span class="id" title="var">N</span>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.cardinal_spec"><span class="id" title="lemma">cardinal_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab53"></a><h3 class="section">for_all specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.for_all_spec"><span class="id" title="lemma">for_all_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab54"></a><h3 class="section">exists specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.exists_spec"><span class="id" title="lemma">exists_spec</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">Hs</span> : <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#Raw.exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#Raw.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab55"></a><h3 class="section">partition specification</h3>
-
-</div>
-<div class="code">
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.partition_ok1"><span class="id" title="instance">partition_ok1</span></a> <span class="id" title="var">s</span> <span class="id" title="var">f</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a name="Raw.partition_ok2"><span class="id" title="instance">partition_ok2</span></a> <span class="id" title="var">s</span> <span class="id" title="var">f</span> : <span class="id" title="keyword">forall</span> `(<a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.partition_spec1"><span class="id" title="lemma">partition_spec1</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> -&gt; <a class="idref" href="MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Raw.partition_spec2"><span class="id" title="lemma">partition_spec2</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#Raw"><span class="id" title="module">Raw</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab56"></a><h2 class="section">Main Module</h2>
-
-
-<div class="paragraph"> </div>
-
-    We can now build the invariant into the set type to obtain an instantiation
-    of module type <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> (<span class="id" title="var">Enc</span> : <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>) &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="MSetIntervals.html#Enc.E"><span class="id" title="module">Enc.E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="MSetIntervals.E"><span class="id" title="module">E</span></a> := <a class="idref" href="MSetIntervals.html#Enc.E"><span class="id" title="module">Enc.E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="MSetIntervals.Raw"><span class="id" title="module">Raw</span></a> := <a class="idref" href="MSetIntervals.html#Raw"><span class="id" title="module">Raw</span></a> <a class="idref" href="MSetIntervals.html#Enc"><span class="id" title="module">Enc</span></a>.<br/>
-
-<br/>
-&nbsp;<span class="id" title="keyword">Local</span>&nbsp;<span class="id" title="keyword">Local</span>
-<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.elt"><span class="id" title="definition">Raw.elt</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Record</span> <a name="MSetIntervals.t_"><span class="id" title="record">t_</span></a> := <a name="MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> {<a name="MSetIntervals.this"><span class="id" title="projection">this</span></a> :&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.t"><span class="id" title="definition">Raw.t</span></a>; <a name="MSetIntervals.is_ok"><span class="id" title="projection">is_ok</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.Ok"><span class="id" title="class">Raw.Ok</span></a> <a class="idref" href="MSetIntervals.html#this"><span class="id" title="method">this</span></a>}.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.t_"><span class="id" title="record">t_</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Hint Resolve</span> <a class="idref" href="MSetIntervals.html#MSetIntervals.is_ok"><span class="id" title="projection">is_ok</span></a> : <span class="id" title="var">typeclass_instances</span>.<br/>
-
-<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.In"><span class="id" title="definition">In</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.Subset"><span class="id" title="definition">Subset</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">exists</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.mem"><span class="id" title="definition">mem</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.mem"><span class="id" title="definition">Raw.mem</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.add"><span class="id" title="definition">add</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.add"><span class="id" title="definition">Raw.add</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>)).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.remove"><span class="id" title="definition">remove</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.remove"><span class="id" title="definition">Raw.remove</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>)).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.singleton"><span class="id" title="definition">singleton</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.singleton"><span class="id" title="definition">Raw.singleton</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.union"><span class="id" title="definition">union</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.union"><span class="id" title="definition">Raw.union</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.inter"><span class="id" title="definition">inter</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.inter"><span class="id" title="definition">Raw.inter</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.diff"><span class="id" title="definition">diff</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.diff"><span class="id" title="definition">Raw.diff</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.equal"><span class="id" title="definition">equal</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.equal"><span class="id" title="definition">Raw.equal</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.subset"><span class="id" title="definition">subset</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.subset"><span class="id" title="definition">Raw.subset</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.(<a class="idref" href="MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.empty"><span class="id" title="definition">Raw.empty</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.is_empty"><span class="id" title="definition">is_empty</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.is_empty"><span class="id" title="definition">Raw.is_empty</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.elements"><span class="id" title="definition">elements</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.elements"><span class="id" title="definition">Raw.elements</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.choose"><span class="id" title="definition">choose</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.choose"><span class="id" title="definition">Raw.choose</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.fold"><span class="id" title="definition">fold</span></a> {<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>}(<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.fold"><span class="id" title="definition">Raw.fold</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.cardinal"><span class="id" title="definition">cardinal</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.cardinal"><span class="id" title="definition">Raw.cardinal</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.filter"><span class="id" title="definition">Raw.filter</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.for_all"><span class="id" title="definition">for_all</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.for_all"><span class="id" title="definition">Raw.for_all</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.exists_"><span class="id" title="definition">exists_</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.exists_"><span class="id" title="definition">Raw.exists_</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.partition"><span class="id" title="definition">partition</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<span class="id" title="var">s</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">p</span> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Raw.partition"><span class="id" title="definition">Raw.partition</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">in</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="MSetIntervals.html#p"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="MSetIntervals.html#p"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;<span class="id" title="keyword">Instance</span> <a name="MSetIntervals.In_compat"><span class="id" title="instance">In_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a>.<br/>
- 
-<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.eq"><span class="id" title="definition">eq</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> -&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> -&gt; <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
-
-<br/>
-&nbsp;<span class="id" title="keyword">Instance</span> <a name="MSetIntervals.eq_equiv"><span class="id" title="instance">eq_equiv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a>.<br/>
- 
-<br/>
-&nbsp;<span class="id" title="keyword">Definition</span> <a name="MSetIntervals.eq_dec"><span class="id" title="definition">eq_dec</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span>:<a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}+{</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetIntervals.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a>.<br/>
-
-<br/>
-&nbsp;<span class="id" title="keyword">Section</span> <a name="MSetIntervals.Spec"><span class="id" title="section">Spec</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a name="MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a name="MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a name="MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a name="MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a name="MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.mem_spec"><span class="id" title="lemma">mem_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.equal_spec"><span class="id" title="lemma">equal_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.subset_spec"><span class="id" title="lemma">subset_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Subset"><span class="id" title="definition">Subset</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.empty_spec"><span class="id" title="lemma">empty_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.empty"><span class="id" title="definition">empty</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.is_empty_spec"><span class="id" title="lemma">is_empty_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.is_empty"><span class="id" title="definition">is_empty</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.add_spec"><span class="id" title="lemma">add_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.remove_spec"><span class="id" title="lemma">remove_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.singleton_spec"><span class="id" title="lemma">singleton_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.union_spec"><span class="id" title="lemma">union_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.inter_spec"><span class="id" title="lemma">inter_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.diff_spec"><span class="id" title="lemma">diff_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.fold_spec"><span class="id" title="lemma">fold_spec</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetIntervals.html#A"><span class="id" title="variable">A</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#MSetIntervals.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetIntervals.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">a</span> <span class="id" title="var">e</span> =&gt; <a class="idref" href="MSetIntervals.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetIntervals.html#a"><span class="id" title="variable">a</span></a>) (<a class="idref" href="MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetIntervals.html#i"><span class="id" title="variable">i</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.cardinal_spec"><span class="id" title="lemma">cardinal_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.filter_spec"><span class="id" title="lemma">filter_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.for_all_spec"><span class="id" title="lemma">for_all_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#MSetIntervals.for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.exists_spec"><span class="id" title="lemma">exists_spec</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetIntervals.html#MSetIntervals.exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.partition_spec1"><span class="id" title="lemma">partition_spec1</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> -&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.partition_spec2"><span class="id" title="lemma">partition_spec2</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetIntervals.html#x"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.elements_spec2w"><span class="id" title="lemma">elements_spec2w</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.choose_spec1"><span class="id" title="lemma">choose_spec1</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="MSetIntervals.choose_spec2"><span class="id" title="lemma">choose_spec2</span></a> : <a class="idref" href="MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> -&gt; <a class="idref" href="MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#MSetIntervals.Spec"><span class="id" title="section">Spec</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab57"></a><h2 class="section">Instantiations</h2>
-
-
-<div class="paragraph"> </div>
-
-    It remains to provide instantiations for commonly used datatypes. 
-<div class="paragraph"> </div>
-
-<a name="lab58"></a><h3 class="section">Z</h3>
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="ElementEncodeZ"><span class="id" title="module">ElementEncodeZ</span></a> &lt;: <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="ElementEncodeZ.E"><span class="id" title="module">E</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z"><span class="id" title="module">Z</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> (<span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="ElementEncodeZ.decode"><span class="id" title="definition">decode</span></a> (<span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="ElementEncodeZ.decode_encode_ok"><span class="id" title="lemma">decode_encode_ok</span></a>: <span class="id" title="keyword">forall</span> (<span class="id" title="var">e</span> : <a class="idref" href="MSetIntervals.html#ElementEncodeZ.E.t"><span class="id" title="definition">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#ElementEncodeZ.decode"><span class="id" title="definition">decode</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="ElementEncodeZ.encode_eq"><span class="id" title="lemma">encode_eq</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">e1</span> <span class="id" title="var">e2</span> : <a class="idref" href="MSetIntervals.html#ElementEncodeZ.E.t"><span class="id" title="definition">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#ElementEncodeZ.E.eq"><span class="id" title="definition">E.eq</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>.<br/>
- 
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#ElementEncodeZ"><span class="id" title="module">ElementEncodeZ</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="MSetIntervalsZ"><span class="id" title="module">MSetIntervalsZ</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z"><span class="id" title="module">Z</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> <a class="idref" href="MSetIntervals.html#ElementEncodeZ"><span class="id" title="module">ElementEncodeZ</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab59"></a><h3 class="section">N</h3>
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="ElementEncodeN"><span class="id" title="module">ElementEncodeN</span></a> &lt;: <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="ElementEncodeN.E"><span class="id" title="module">E</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N"><span class="id" title="module">N</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> (<span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="ElementEncodeN.decode"><span class="id" title="definition">decode</span></a> (<span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="ElementEncodeN.decode_encode_ok"><span class="id" title="lemma">decode_encode_ok</span></a>: <span class="id" title="keyword">forall</span> (<span class="id" title="var">e</span> : <a class="idref" href="MSetIntervals.html#ElementEncodeN.E.t"><span class="id" title="definition">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#ElementEncodeN.decode"><span class="id" title="definition">decode</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="ElementEncodeN.encode_eq"><span class="id" title="lemma">encode_eq</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">e1</span> <span class="id" title="var">e2</span> : <a class="idref" href="MSetIntervals.html#ElementEncodeN.E.t"><span class="id" title="definition">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="axiom">Z.eq</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#ElementEncodeN.E.eq"><span class="id" title="definition">E.eq</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>.<br/>
- <span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#ElementEncodeN"><span class="id" title="module">ElementEncodeN</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="MSetIntervalsN"><span class="id" title="module">MSetIntervalsN</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.NArith.BinNat.html#N"><span class="id" title="module">N</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> <a class="idref" href="MSetIntervals.html#ElementEncodeN"><span class="id" title="module">ElementEncodeN</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab60"></a><h3 class="section">nat</h3>
-
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="ElementEncodeNat"><span class="id" title="module">ElementEncodeNat</span></a> &lt;: <a class="idref" href="MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="ElementEncodeNat.E"><span class="id" title="module">E</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.Natural.Peano.NPeano.html#Nat"><span class="id" title="module">NPeano.Nat</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> (<span class="id" title="var">n</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.of_nat"><span class="id" title="definition">Z.of_nat</span></a> <a class="idref" href="MSetIntervals.html#n"><span class="id" title="variable">n</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="ElementEncodeNat.decode"><span class="id" title="definition">decode</span></a> (<span class="id" title="var">z</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.to_nat"><span class="id" title="definition">Z.to_nat</span></a> <a class="idref" href="MSetIntervals.html#z"><span class="id" title="variable">z</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="ElementEncodeNat.decode_encode_ok"><span class="id" title="lemma">decode_encode_ok</span></a>: <span class="id" title="keyword">forall</span> (<span class="id" title="var">e</span> : <a class="idref" href="MSetIntervals.html#ElementEncodeNat.E.t"><span class="id" title="definition">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetIntervals.html#ElementEncodeNat.decode"><span class="id" title="definition">decode</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetIntervals.html#e"><span class="id" title="variable">e</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="ElementEncodeNat.encode_eq"><span class="id" title="lemma">encode_eq</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">e1</span> <span class="id" title="var">e2</span> : <a class="idref" href="MSetIntervals.html#ElementEncodeNat.E.t"><span class="id" title="definition">E.t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="axiom">Z.eq</span></a> (<a class="idref" href="MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetIntervals.html#ElementEncodeNat.E.eq"><span class="id" title="definition">E.eq</span></a> <a class="idref" href="MSetIntervals.html#e1"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetIntervals.html#e2"><span class="id" title="variable">e2</span></a>.<br/>
- <span class="id" title="keyword">End</span> <a class="idref" href="MSetIntervals.html#ElementEncodeNat"><span class="id" title="module">ElementEncodeNat</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="MSetIntervalsNat"><span class="id" title="module">MSetIntervalsNat</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Numbers.Natural.Peano.NPeano.html#Nat"><span class="id" title="module">NPeano.Nat</span></a> := <a class="idref" href="MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> <a class="idref" href="MSetIntervals.html#ElementEncodeNat"><span class="id" title="module">ElementEncodeNat</span></a>.<br/>
-
-<br/>
-</div>
-</div>
-
-<div id="footer">
-<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
-</div>
-
-</div>
-
-</body>
-</html>
\ No newline at end of file
diff --git a/coqdoc/MSetListWithDups.html b/coqdoc/MSetListWithDups.html
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--- a/coqdoc/MSetListWithDups.html
+++ /dev/null
@@ -1,469 +0,0 @@
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
-"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
-<html xmlns="http://www.w3.org/1999/xhtml">
-<head>
-<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
-<link href="coqdoc.css" rel="stylesheet" type="text/css" />
-<title>MSetListWithDups</title>
-</head>
-
-<body>
-
-<div id="page">
-
-<div id="header">
-</div>
-
-<div id="main">
-
-<h1 class="libtitle">Library MSetListWithDups</h1>
-
-<div class="code">
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab61"></a><h1 class="section">Weak sets implemented as lists with duplicates</h1>
-
-
-<div class="paragraph"> </div>
-
-    This file contains an implementation of the weak set interface
-    <span class="inlinecode"><span class="id" title="var">WSetsOnWithDupsExtra</span></span>. As a datatype unsorted lists are used
-    that might contain duplicates. 
-
-<div class="paragraph"> </div>
-
-    This implementation is useful, if one needs very efficient
-    insert and union operation, and can guarantee that one does not
-    add too many duplicates. The operation <span class="inlinecode"><span class="id" title="var">elements_dist</span></span> is implemented
-    by sorting the list first. Therefore this instantiation can only
-    be used if the element type is ordered. 
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#"><span class="id" title="library">MSetInterface</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssreflect</span>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.OrdersFacts.html#"><span class="id" title="library">OrdersFacts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.OrdersLists.html#"><span class="id" title="library">OrdersLists</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Sorting.Sorting.html#"><span class="id" title="library">Sorting</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Sorting.Permutation.html#"><span class="id" title="library">Permutation</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="MSetWithDups.html#"><span class="id" title="library">MSetWithDups</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab62"></a><h2 class="section">Removing duplicates from sorted lists</h2>
-
-
-<div class="paragraph"> </div>
-
-    The following module <span class="inlinecode"><span class="id" title="var">RemoveDupsFromSorted</span></span> defines an operation
-    <span class="inlinecode"><span class="id" title="var">remove_dups_from_sortedA</span></span> that removes duplicates from a sorted
-    list. In order to talk about sorted lists, the element type needs
-    to be ordered.
-
-<div class="paragraph"> </div>
-
-    This function is combined with a sort function to get a function
-    <span class="inlinecode"><span class="id" title="var">remove_dups_by_sortingA</span></span> to sort unsorted lists and then remove
-    duplicates.  
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a> (<span class="id" title="keyword">Import</span> <span class="id" title="var">X</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-First, we need some infrastructure for our ordered type 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="RemoveDupsFromSorted.MX"><span class="id" title="module">MX</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.OrdersFacts.html#OrderedTypeFacts"><span class="id" title="module">OrderedTypeFacts</span></a> <a class="idref" href="MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="RemoveDupsFromSorted.XTotalLeBool"><span class="id" title="module">XTotalLeBool</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#TotalLeBool"><span class="id" title="module">TotalLeBool</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>.<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a name="30402bd5327406053496dee97ab5979c"><span class="id" title="notation">&quot;</span></a>&lt;=?" := <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35).<br/>
-
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a name="RemoveDupsFromSorted.XTotalLeBool.leb_total"><span class="id" title="lemma">leb_total</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="MSetListWithDups.html#30402bd5327406053496dee97ab5979c"><span class="id" title="notation">&lt;=?</span></a> <a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="MSetListWithDups.html#30402bd5327406053496dee97ab5979c"><span class="id" title="notation">&lt;=?</span></a> <a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="RemoveDupsFromSorted.XTotalLeBool.le"><span class="id" title="definition">le</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> := (<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool"><span class="id" title="module">XTotalLeBool</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="RemoveDupsFromSorted.eqb_eq_alt"><span class="id" title="lemma">eqb_eq_alt</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.MX.eqb"><span class="id" title="definition">eqb</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-Now we can define our main function 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> (<span class="id" title="var">acc</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> =&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">with</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev&acute;"><span class="id" title="definition">List.rev&acute;</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a>)<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys</span> =&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.MX.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="keyword">then</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a> <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> (<span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> := <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="notation">:</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-We can prove some technical lemmata 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt"><span class="id" title="lemma">remove_dups_from_sortedA_aux_alt</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>) <span class="id" title="var">acc</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#5d35a99a6abff1d64bf70538feb70e36"><span class="id" title="notation">)</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="RemoveDupsFromSorted.remove_dups_from_sortedA_alt"><span class="id" title="lemma">remove_dups_from_sortedA_alt</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> =&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">with</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys</span> =&gt; <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.MX.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="keyword">then</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">else</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="RemoveDupsFromSorted.remove_dups_from_sortedA_hd"><span class="id" title="lemma">remove_dups_from_sortedA_hd</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">xs</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">exists</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">(</span></a><span class="id" title="var">x&acute;</span>:<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>) <span class="id" title="var">xs&acute;</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> (<a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#xs"><span class="id" title="variable">xs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#x&acute;"><span class="id" title="variable">x&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#xs&acute;"><span class="id" title="variable">xs&acute;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.MX.eqb"><span class="id" title="definition">eqb</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#x&acute;"><span class="id" title="variable">x&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
-</div>
-
-<div class="doc">
-Finally we get our main result for removing duplicates from sorted lists 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="RemoveDupsFromSorted.remove_dups_from_sortedA_spec"><span class="id" title="lemma">remove_dups_from_sortedA_spec</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.le"><span class="id" title="definition">le</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">l&acute;</span> := <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">in</span> (<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetListWithDups.html#X.lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="MSetListWithDups.html#l&acute;"><span class="id" title="variable">l&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#l&acute;"><span class="id" title="variable">l&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#l&acute;"><span class="id" title="variable">l&acute;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>).<br/>
-</div>
-
-<div class="doc">
-Next, we combine it with sorting 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="RemoveDupsFromSorted.XSort"><span class="id" title="module">XSort</span></a> := <span class="id" title="keyword">Sort</span> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool"><span class="id" title="module">XTotalLeBool</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="RemoveDupsFromSorted.remove_dups_by_sortingA"><span class="id" title="definition">remove_dups_by_sortingA</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> (<a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XSort.sort"><span class="id" title="definition">XSort.sort</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="RemoveDupsFromSorted.remove_dups_by_sortingA_spec"><span class="id" title="lemma">remove_dups_by_sortingA_spec</span></a> : <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> (<span class="id" title="var">l</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">l&acute;</span> := <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA"><span class="id" title="definition">remove_dups_by_sortingA</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">in</span> (<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetListWithDups.html#X.lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="MSetListWithDups.html#l&acute;"><span class="id" title="variable">l&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#l&acute;"><span class="id" title="variable">l&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#l&acute;"><span class="id" title="variable">l&acute;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>).<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab63"></a><h2 class="section">Operations Module</h2>
-
-
-<div class="paragraph"> </div>
-
-With removing duplicates defined, we can implement
-the operations for our set implementation easily. 
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="Ops"><span class="id" title="module">Ops</span></a> (<span class="id" title="var">X</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WOps"><span class="id" title="module">WOps</span></a> <a class="idref" href="MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a name="Ops.RDFS"><span class="id" title="module">RDFS</span></a> := <a class="idref" href="MSetListWithDups.html#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a> <a class="idref" href="MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="Ops.MX"><span class="id" title="module">MX</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.OrdersFacts.html#OrderedTypeFacts"><span class="id" title="module">OrderedTypeFacts</span></a> <a class="idref" href="MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.is_empty"><span class="id" title="definition">is_empty</span></a> (<span class="id" title="var">l</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> | <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">end</span>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.mem"><span class="id" title="definition">mem</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> =&gt; <a class="idref" href="MSetListWithDups.html#mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.add"><span class="id" title="definition">add</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.singleton"><span class="id" title="definition">singleton</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a>) := <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <span class="id" title="var">acc</span> (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span>) =&gt; <a class="idref" href="MSetListWithDups.html#rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> (<span class="id" title="keyword">if</span> <span class="id" title="notation">(</span><a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span><span class="id" title="notation">)</span> <span class="id" title="keyword">then</span> <span class="id" title="notation">(</span><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a><span class="id" title="notation">)</span> <span class="id" title="keyword">else</span> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">xs</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> := <a class="idref" href="MSetListWithDups.html#Ops.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetListWithDups.html#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.remove"><span class="id" title="definition">remove</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s</span> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">y</span> =&gt; <span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> | <span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">end</span>) <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.union"><span class="id" title="definition">union</span></a> (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetListWithDups.html#s2"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetListWithDups.html#s1"><span class="id" title="variable">s1</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.inter"><span class="id" title="definition">inter</span></a> (<span class="id" title="var">s1</span> <span class="id" title="var">s2</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">y</span> =&gt; <a class="idref" href="MSetListWithDups.html#Ops.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetListWithDups.html#s2"><span class="id" title="variable">s2</span></a>) <a class="idref" href="MSetListWithDups.html#s1"><span class="id" title="variable">s1</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elements"><span class="id" title="definition">elements</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.elements_dist"><span class="id" title="definition">elements_dist</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#Ops.RDFS.remove_dups_by_sortingA"><span class="id" title="definition">RDFS.remove_dups_by_sortingA</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.fold"><span class="id" title="definition">fold</span></a> {<span class="id" title="var">B</span> : <span class="id" title="keyword">Type</span>} (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#B"><span class="id" title="variable">B</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#B"><span class="id" title="variable">B</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetListWithDups.html#B"><span class="id" title="variable">B</span></a>) : <a class="idref" href="MSetListWithDups.html#B"><span class="id" title="variable">B</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetListWithDups.html#i"><span class="id" title="variable">i</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.diff"><span class="id" title="definition">diff</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetListWithDups.html#Ops.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetListWithDups.html#Ops.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.subset"><span class="id" title="definition">subset</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#forallb"><span class="id" title="definition">List.forallb</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetListWithDups.html#Ops.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.equal"><span class="id" title="definition">equal</span></a> (<span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="MSetListWithDups.html#Ops.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) (<a class="idref" href="MSetListWithDups.html#Ops.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.for_all"><span class="id" title="definition">for_all</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt; <span class="id" title="keyword">if</span> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">then</span> <a class="idref" href="MSetListWithDups.html#for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.exists_"><span class="id" title="definition">exists_</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt; <span class="id" title="keyword">if</span> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="MSetListWithDups.html#exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a name="Ops.partition_aux"><span class="id" title="definition">partition_aux</span></a> (<span class="id" title="var">a1</span> <span class="id" title="var">a2</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">*</span></a> <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> =&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <span class="id" title="keyword">then</span> <a class="idref" href="MSetListWithDups.html#partition_aux"><span class="id" title="definition">partition_aux</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a>) <a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span> <span class="id" title="keyword">else</span> <br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#partition_aux"><span class="id" title="definition">partition_aux</span></a> <a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a>) <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.partition"><span class="id" title="definition">partition</span></a> := <a class="idref" href="MSetListWithDups.html#Ops.partition_aux"><span class="id" title="definition">partition_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.cardinal"><span class="id" title="definition">cardinal</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetListWithDups.html#Ops.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Ops.choose"><span class="id" title="definition">choose</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> :=<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#0a8150c3c4df34d205c5ed9a427e2ed5"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetListWithDups.html#Ops"><span class="id" title="module">Ops</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab64"></a><h2 class="section">Main Module</h2>
-
-
-<div class="paragraph"> </div>
-
-    Using these operations, we can define the main functor. For this,
-    we need to prove that the provided operations do indeed satisfy
-    the weak set interface. This is mostly straightforward and
-    unsurprising. The only interesting part is that removing
-    duplicates from a sorted list behaves as expected. This has
-    however already been proved in module <span class="inlinecode"><span class="id" title="var">RemoveDupsFromSorted</span></span>.
-  
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <a name="Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">E</span>:<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a> <a class="idref" href="MSetListWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetListWithDups.html#Ops"><span class="id" title="module">Ops</span></a> <a class="idref" href="MSetListWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <span class="id" title="var">MX</span>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab65"></a><h2 class="section">Proofs of set operation specifications.</h2>
- Logical predicates 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.In"><span class="id" title="definition">In</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">s</span> : <a class="idref" href="MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">SetoidList.InA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a name="Make.In_compat"><span class="id" title="instance">In_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.Empty"><span class="id" title="definition">Empty</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <span class="id" title="var">s</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <span class="id" title="var">s</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">exists</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetListWithDups.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="dce525a765dab06eb193bd383e9264ac"><span class="id" title="notation">&quot;</span></a>s  [=]  t" := (<a class="idref" href="MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="25d383999de34b2e68fd5cd20a7b85ae"><span class="id" title="notation">&quot;</span></a>s  [&lt;=]  t" := (<a class="idref" href="MSetListWithDups.html#Make.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="Make.eq"><span class="id" title="definition">eq</span></a> : <a class="idref" href="MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a> -&gt; <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.eq_equiv"><span class="id" title="lemma">eq_equiv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetListWithDups.html#Make.eq"><span class="id" title="definition">eq</span></a>.<br/>
- </div>
-
-<div class="doc">
-Specifications of set operators 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="Make.compatb"><span class="id" title="abbreviation">compatb</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.mem_spec"><span class="id" title="lemma">mem_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.subset_spec"><span class="id" title="lemma">subset_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span>, <a class="idref" href="MSetListWithDups.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="MSetListWithDups.html#25d383999de34b2e68fd5cd20a7b85ae"><span class="id" title="notation">[&lt;=]</span></a><a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.equal_spec"><span class="id" title="lemma">equal_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span>, <a class="idref" href="MSetListWithDups.html#Make.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="MSetListWithDups.html#dce525a765dab06eb193bd383e9264ac"><span class="id" title="notation">[=]</span></a><a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.eq_dec"><span class="id" title="lemma">eq_dec</span></a> :  <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="MSetListWithDups.html#Make.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="MSetListWithDups.html#Make.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Specif.html#5bf2050e90b21ebc82dc5463d1ba338e"><span class="id" title="notation">}</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.empty_spec"><span class="id" title="lemma">empty_spec</span></a> : <a class="idref" href="MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetListWithDups.html#Make.empty"><span class="id" title="definition">empty</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.is_empty_spec"><span class="id" title="lemma">is_empty_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetListWithDups.html#Make.is_empty"><span class="id" title="definition">is_empty</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.add_spec"><span class="id" title="lemma">add_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetListWithDups.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.singleton_spec"><span class="id" title="lemma">singleton_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetListWithDups.html#Make.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Hint Resolve</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetListWithDups.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Hint Immediate</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence_Symmetric"><span class="id" title="method">Equivalence_Symmetric</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetListWithDups.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Hint Resolve</span> (@<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.RelationClasses.html#Equivalence_Transitive"><span class="id" title="method">Equivalence_Transitive</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetListWithDups.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.rev_filter_aux_spec"><span class="id" title="lemma">rev_filter_aux_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span> <span class="id" title="var">f</span>, <a class="idref" href="MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#acc"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.filter_spec"><span class="id" title="lemma">filter_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">f</span>, <a class="idref" href="MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.remove_spec"><span class="id" title="lemma">remove_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetListWithDups.html#Make.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.union_spec"><span class="id" title="lemma">union_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.inter_spec"><span class="id" title="lemma">inter_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.fold_spec"><span class="id" title="lemma">fold_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetListWithDups.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#A"><span class="id" title="variable">A</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetListWithDups.html#Make.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetListWithDups.html#i"><span class="id" title="variable">i</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.diff_spec"><span class="id" title="lemma">diff_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.cardinal_spec"><span class="id" title="lemma">cardinal_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetListWithDups.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.for_all_spec"><span class="id" title="lemma">for_all_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>, <a class="idref" href="MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetListWithDups.html#Make.for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.exists_spec"><span class="id" title="lemma">exists_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>, <a class="idref" href="MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetListWithDups.html#Make.exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetListWithDups.html#Make.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.partition_aux_spec"><span class="id" title="lemma">partition_aux_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetListWithDups.html#Make.partition_aux"><span class="id" title="definition">partition_aux</span></a> <a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="MSetListWithDups.html#a1"><span class="id" title="variable">a1</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="MSetListWithDups.html#a2"><span class="id" title="variable">a2</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>).<br/>
- &nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.partition_spec1"><span class="id" title="lemma">partition_spec1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>, <a class="idref" href="MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetListWithDups.html#Make.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetListWithDups.html#dce525a765dab06eb193bd383e9264ac"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.partition_spec2"><span class="id" title="lemma">partition_spec2</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>, <a class="idref" href="MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetListWithDups.html#Make.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetListWithDups.html#dce525a765dab06eb193bd383e9264ac"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetListWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.choose_spec1"><span class="id" title="lemma">choose_spec1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetListWithDups.html#Make.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.choose_spec2"><span class="id" title="lemma">choose_spec2</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetListWithDups.html#Make.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> -&gt; <a class="idref" href="MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.elements_dist_spec_full"><span class="id" title="lemma">elements_dist_spec_full</span></a> :<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>,<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetListWithDups.html#E.lt"><span class="id" title="axiom">E.lt</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a><span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.elements_dist_spec1"><span class="id" title="lemma">elements_dist_spec1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetListWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="Make.elements_dist_spec2w"><span class="id" title="lemma">elements_dist_spec2w</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetListWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
- 
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetListWithDups.html#Make"><span class="id" title="module">Make</span></a>.<br/>
-</div>
-</div>
-
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-"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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-<head>
-<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
-<link href="coqdoc.css" rel="stylesheet" type="text/css" />
-<title>MSetWithDups</title>
-</head>
-
-<body>
-
-<div id="page">
-
-<div id="header">
-</div>
-
-<div id="main">
-
-<h1 class="libtitle">Library MSetWithDups</h1>
-
-<div class="code">
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab1"></a><h1 class="section">Signature for weak sets which may contain duplicates</h1>
-
-
-<div class="paragraph"> </div>
-
-    The interface <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span> demands that <span class="inlinecode"><span class="id" title="var">elements</span></span> returns a list
-    without duplicates and that the fold function iterates over this
-    result. Another potential problem is that the function <span class="inlinecode"><span class="id" title="var">cardinal</span></span>
-    is supposed to return the length of the elements list.
-
-<div class="paragraph"> </div>
-
-    Therefore, implementations that store duplicates internally and for
-    which the fold function would visit elements multiple times are
-    ruled out.  Such implementations might be desirable for performance
-    reasons, though. One such (sometimes useful) example are unsorted
-    lists with duplicates. They have a very efficient insert and union
-    operation. If they are used in such a way that not too many
-    membership tests happen and that not too many duplicates
-    accumulate, it might be a very efficient datastructure.
-
-<div class="paragraph"> </div>
-
-    In order to allow efficient weak set implementations that use
-    duplicates internally, we provide new module types in this
-    file. There is <span class="inlinecode"><span class="id" title="var">WSetsOnWithDups</span></span>, which is a proper subset of
-    <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>. It just removes the problematic properties of <span class="inlinecode"><span class="id" title="var">elements</span></span>
-    and <span class="inlinecode"><span class="id" title="var">cardinal</span></span>.
-
-<div class="paragraph"> </div>
-
-    Since one is of course interested in specifying the cardinality
-    and in computing a list of elements without duplicates, there is
-    also an extension <span class="inlinecode"><span class="id" title="var">WSetsOnWithDupsExtra</span></span> of <span class="inlinecode"><span class="id" title="var">WSetsOnWithDups</span></span>. This
-    extension introduces a new operation <span class="inlinecode"><span class="id" title="var">elements_dist</span></span>, which is a
-    version of <span class="inlinecode"><span class="id" title="var">elements</span></span> without duplicates. This allows to
-    specify <span class="inlinecode"><span class="id" title="var">cardinality</span></span> with respect to <span class="inlinecode"><span class="id" title="var">elements_dist</span></span>.
-
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#"><span class="id" title="library">Coq.MSets.MSetInterface</span></a>.<br/>
-<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">ssreflect</span>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab2"></a><h2 class="section">WSetsOnWithDups</h2>
-
-
-<div class="paragraph"> </div>
-
-    The module type <span class="inlinecode"><span class="id" title="var">WSetOnWithDups</span></span> is a proper subset of <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>;
-    the problematic parameters <span class="inlinecode"><span class="id" title="var">cardinal_spec</span></span> and <span class="inlinecode"><span class="id" title="var">elements_spec2w</span></span>
-    are missing. 
-
-<div class="paragraph"> </div>
-
-    We use this approach to be as noninvasive as possible. If we had the
-    liberty to modify the existing MSet library, it might be better to
-    define WSetsOnWithDups as below and define WSetOn by adding the two
-    extra parameters.
- 
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WOps"><span class="id" title="module">WOps</span></a> <a class="idref" href="MSetWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a> -&gt; <span class="id" title="keyword">Prop</span>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Declare Instance</span> <a name="WSetsOnWithDups.In_compat"><span class="id" title="instance">In_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOnWithDups.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOnWithDups.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s&acute;</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetWithDups.html#s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <span class="id" title="var">s</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">a</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOnWithDups.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <span class="id" title="var">s</span> := <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span>, <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a> -&gt; <a class="idref" href="MSetWithDups.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOnWithDups.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="var">P</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> -&gt; <span class="id" title="keyword">Prop</span>) <span class="id" title="var">s</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">exists</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetWithDups.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="217aae9075a91af245df2f2a31191fab"><span class="id" title="notation">&quot;</span></a>s  [=]  t" := (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="93b79b206f9e047c003978d66496371a"><span class="id" title="notation">&quot;</span></a>s  [&lt;=]  t" := (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOnWithDups.eq"><span class="id" title="definition">eq</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a> -&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a> -&gt; <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Equalities.html#IsEq"><span class="id" title="module">IsEq</span></a>. </div>
-
-<div class="doc">
-<span class="inlinecode"><span class="id" title="var">eq</span></span> is obviously an equivalence, for subtyping only 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Equalities.html#HasEqDec"><span class="id" title="module">HasEqDec</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a name="WSetsOnWithDups.Spec"><span class="id" title="section">Spec</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a name="WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a name="WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>: <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a name="WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a name="WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a name="WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a name="WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Classes.Morphisms.html#5bcff5fca4c3a87e683dd8ca07bab63f"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.mem_spec"><span class="id" title="axiom">mem_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.mem"><span class="id" title="axiom">mem</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.equal_spec"><span class="id" title="axiom">equal_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.equal"><span class="id" title="axiom">equal</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="MSetWithDups.html#217aae9075a91af245df2f2a31191fab"><span class="id" title="notation">[=]</span></a><a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.subset_spec"><span class="id" title="axiom">subset_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.subset"><span class="id" title="axiom">subset</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="MSetWithDups.html#93b79b206f9e047c003978d66496371a"><span class="id" title="notation">[&lt;=]</span></a><a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.empty_spec"><span class="id" title="axiom">empty_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.empty"><span class="id" title="axiom">empty</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.is_empty_spec"><span class="id" title="axiom">is_empty_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.is_empty"><span class="id" title="axiom">is_empty</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.add_spec"><span class="id" title="axiom">add_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.remove_spec"><span class="id" title="axiom">remove_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.remove"><span class="id" title="axiom">remove</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.singleton_spec"><span class="id" title="axiom">singleton_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.singleton"><span class="id" title="axiom">singleton</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.union_spec"><span class="id" title="axiom">union_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.union"><span class="id" title="axiom">union</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">\/</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.inter_spec"><span class="id" title="axiom">inter_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.inter"><span class="id" title="axiom">inter</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.diff_spec"><span class="id" title="axiom">diff_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.diff"><span class="id" title="axiom">diff</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">~</span></a><a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;"><span class="id" title="variable">s&acute;</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.fold_spec"><span class="id" title="axiom">fold_spec</span></a> : <span class="id" title="keyword">forall</span> (<span class="id" title="var">A</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">i</span> : <a class="idref" href="MSetWithDups.html#A"><span class="id" title="variable">A</span></a>) (<span class="id" title="var">f</span> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> -&gt; <a class="idref" href="MSetWithDups.html#A"><span class="id" title="variable">A</span></a> -&gt; <a class="idref" href="MSetWithDups.html#A"><span class="id" title="variable">A</span></a>),<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetWithDups.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetWithDups.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetWithDups.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetWithDups.html#i"><span class="id" title="variable">i</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.filter_spec"><span class="id" title="axiom">filter_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.filter"><span class="id" title="axiom">filter</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">/\</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.for_all_spec"><span class="id" title="axiom">for_all_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.for_all"><span class="id" title="axiom">for_all</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.exists_spec"><span class="id" title="axiom">exists_spec</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.exists_"><span class="id" title="axiom">exists_</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.partition_spec1"><span class="id" title="axiom">partition_spec1</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.partition"><span class="id" title="axiom">partition</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetWithDups.html#217aae9075a91af245df2f2a31191fab"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.filter"><span class="id" title="axiom">filter</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.partition_spec2"><span class="id" title="axiom">partition_spec2</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> -&gt;<br/>
-&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.partition"><span class="id" title="axiom">partition</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetWithDups.html#217aae9075a91af245df2f2a31191fab"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.filter"><span class="id" title="axiom">filter</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> =&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.elements_spec1"><span class="id" title="axiom">elements_spec1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDups.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.choose_spec1"><span class="id" title="axiom">choose_spec1</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.choose"><span class="id" title="axiom">choose</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> -&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDups.choose_spec2"><span class="id" title="axiom">choose_spec2</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.choose"><span class="id" title="axiom">choose</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> -&gt; <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups.Spec"><span class="id" title="section">Spec</span></a>.<br/>
-
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab3"></a><h2 class="section">WSetsOnWithDupsExtra</h2>
-
-
-<div class="paragraph"> </div>
-
-    <span class="inlinecode"><span class="id" title="var">WSetsOnWithDupsExtra</span></span> introduces <span class="inlinecode"><span class="id" title="var">elements_dist</span></span> in order to
-    specify cardinality and in order to get an operation similar to
-    the original behavior of <span class="inlinecode"><span class="id" title="var">elements</span></span>. 
-</div>
-<div class="code">
-<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a name="WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>).<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-An operation for getting an elements list without duplicates 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> : <a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.t"><span class="id" title="axiom">t</span></a> -&gt; <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.elt"><span class="id" title="definition">elt</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDupsExtra.elements_dist_spec1"><span class="id" title="axiom">elements_dist_spec1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDupsExtra.elements_dist_spec2w"><span class="id" title="axiom">elements_dist_spec2w</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-Cardinality can then be specified with respect to <span class="inlinecode"><span class="id" title="var">elements_dist</span></span>. 
-</div>
-<div class="code">
-&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a name="WSetsOnWithDupsExtra.cardinal_spec"><span class="id" title="axiom">cardinal_spec</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.cardinal"><span class="id" title="axiom">cardinal</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a>.<br/>
-
-<br/>
-</div>
-
-<div class="doc">
-<a name="lab4"></a><h2 class="section">WSetOn to WSetsOnWithDupsExtra</h2>
-
-
-<div class="paragraph"> </div>
-
-    Since <span class="inlinecode"><span class="id" title="var">WSetsOnWithDupsExtra</span></span> is morally a weaker version of <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>
-    that allows the fold operation to visit elements multiple time, we can write then
-    following conversion. 
-</div>
-<div class="code">
-
-<br/>
-<span class="id" title="keyword">Module</span> <a name="WSetsOn_TO_WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOn_TO_WSetsOnWithDupsExtra</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>) (<span class="id" title="var">W</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="MSetWithDups.html#E"><span class="id" title="module">E</span></a>) &lt;: <br/>
-&nbsp;&nbsp;<a class="idref" href="MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a> <a class="idref" href="MSetWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetWithDups.html#W"><span class="id" title="module">W</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a name="WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a> := <a class="idref" href="MSetWithDups.html#W.elements"><span class="id" title="axiom">W.elements</span></a>.<br/>
-
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1"><span class="id" title="lemma">elements_dist_spec1</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Init.Logic.html#df1ced36fc33ce188051218bca314374"><span class="id" title="notation">&lt;-&gt;</span></a><br/>
-&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetWithDups.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
- 
-<br/>
-&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a name="WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w"><span class="id" title="lemma">elements_dist_spec2w</span></a> : <span class="id" title="keyword">forall</span> <span class="id" title="var">s</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.4pl6/stdlib/Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetWithDups.html#s"><span class="id" title="variable">s</span></a>).<br/>
- 
-<br/>
-<span class="id" title="keyword">End</span> <a class="idref" href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOn_TO_WSetsOnWithDupsExtra</span></a>.<br/>
-
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-</div>
-</div>
-
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-<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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diff --git a/coqdoc/coqdoc.sty b/coqdoc/coqdoc.sty
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diff --git a/coqdoc/index.html b/coqdoc/index.html
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--- a/coqdoc/index.html
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-<head>
-<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
-<link href="coqdoc.css" rel="stylesheet" type="text/css" />
-<title>Index</title>
-</head>
-
-<body>
-
-<div id="page">
-
-<div id="header">
-</div>
-
-<div id="main">
-
-<table>
-<tr>
-<td>Global Index</td>
-<td><a href="index.html#global_A">A</a></td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#global_E">E</a></td>
-<td><a href="index.html#global_F">F</a></td>
-<td>G</td>
-<td><a href="index.html#global_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#global_M">M</a></td>
-<td><a href="index.html#global_N">N</a></td>
-<td><a href="index.html#global_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#global_R">R</a></td>
-<td><a href="index.html#global_S">S</a></td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#global_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td><a href="index.html#global_Z">Z</a></td>
-<td>_</td>
-<td>other</td>
-<td>(451 entries)</td>
-</tr>
-<tr>
-<td>Notation Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#notation_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#notation_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#notation_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(5 entries)</td>
-</tr>
-<tr>
-<td>Module Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#module_E">E</a></td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#module_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#module_M">M</a></td>
-<td><a href="index.html#module_N">N</a></td>
-<td><a href="index.html#module_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#module_R">R</a></td>
-<td><a href="index.html#module_S">S</a></td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#module_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(43 entries)</td>
-</tr>
-<tr>
-<td>Variable Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#variable_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
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-<td>V</td>
-<td><a href="index.html#variable_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(10 entries)</td>
-</tr>
-<tr>
-<td>Library Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
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-<td>K</td>
-<td>L</td>
-<td><a href="index.html#library_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(4 entries)</td>
-</tr>
-<tr>
-<td>Lemma Index</td>
-<td><a href="index.html#lemma_A">A</a></td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#lemma_E">E</a></td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#lemma_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#lemma_M">M</a></td>
-<td><a href="index.html#lemma_N">N</a></td>
-<td><a href="index.html#lemma_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#lemma_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#lemma_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td><a href="index.html#lemma_Z">Z</a></td>
-<td>_</td>
-<td>other</td>
-<td>(168 entries)</td>
-</tr>
-<tr>
-<td>Axiom Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#axiom_E">E</a></td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#axiom_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td>M</td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#axiom_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(31 entries)</td>
-</tr>
-<tr>
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-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#constructor_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
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-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(2 entries)</td>
-</tr>
-<tr>
-<td>Projection Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#projection_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#projection_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#projection_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(7 entries)</td>
-</tr>
-<tr>
-<td>Inductive Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td>M</td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#inductive_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(1 entry)</td>
-</tr>
-<tr>
-<td>Section Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#section_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#section_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#section_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(3 entries)</td>
-</tr>
-<tr>
-<td>Instance Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#instance_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#instance_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#instance_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(15 entries)</td>
-</tr>
-<tr>
-<td>Abbreviation Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#abbreviation_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#abbreviation_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(3 entries)</td>
-</tr>
-<tr>
-<td>Definition Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#definition_E">E</a></td>
-<td><a href="index.html#definition_F">F</a></td>
-<td>G</td>
-<td><a href="index.html#definition_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#definition_M">M</a></td>
-<td>N</td>
-<td><a href="index.html#definition_O">O</a></td>
-<td>P</td>
-<td>Q</td>
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-<td>S</td>
-<td>T</td>
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-<td>V</td>
-<td><a href="index.html#definition_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(156 entries)</td>
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-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#record_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#record_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#record_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(3 entries)</td>
-</tr>
-</table>
-<hr/>
-<h1>Global Index</h1>
-<a name="global_A"></a><h2>A </h2>
-<a href="MSetIntervals.html#add_add_sub_eq">add_add_sub_eq</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="global_E"></a><h2>E </h2>
-<a href="MSetIntervals.html#ElementEncode">ElementEncode</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN">ElementEncodeN</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat">ElementEncodeNat</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.decode">ElementEncodeNat.decode</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.decode_encode_ok">ElementEncodeNat.decode_encode_ok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.E">ElementEncodeNat.E</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.encode">ElementEncodeNat.encode</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.encode_eq">ElementEncodeNat.encode_eq</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.decode">ElementEncodeN.decode</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.decode_encode_ok">ElementEncodeN.decode_encode_ok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.E">ElementEncodeN.E</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.encode">ElementEncodeN.encode</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.encode_eq">ElementEncodeN.encode_eq</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ">ElementEncodeZ</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.decode">ElementEncodeZ.decode</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.decode_encode_ok">ElementEncodeZ.decode_encode_ok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.E">ElementEncodeZ.E</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.encode">ElementEncodeZ.encode</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.encode_eq">ElementEncodeZ.encode_eq</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.decode">ElementEncode.decode</a> [axiom, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.decode_encode_ok">ElementEncode.decode_encode_ok</a> [axiom, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.E">ElementEncode.E</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.encode">ElementEncode.encode</a> [axiom, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.encode_eq">ElementEncode.encode_eq</a> [axiom, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="global_F"></a><h2>F </h2>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtLtSpecPred">foldWithAbortGtLtSpecPred</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtLtType">foldWithAbortGtLtType</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtSpecPred">foldWithAbortGtSpecPred</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtType">foldWithAbortGtType</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred">foldWithAbortPrecomputeSpecPred</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortPrecomputeType">foldWithAbortPrecomputeType</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortSpecPred">foldWithAbortSpecPred</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortType">foldWithAbortType</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="global_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbort">HasFoldWithAbort</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps">HasFoldWithAbortOps</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort_spec">HasFoldWithAbortOps.choose_with_abort_spec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort">HasFoldWithAbortOps.choose_with_abort</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort_spec">HasFoldWithAbortOps.exists_with_abort_spec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort">HasFoldWithAbortOps.exists_with_abort</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort_spec">HasFoldWithAbortOps.filter_with_abort_spec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort">HasFoldWithAbortOps.filter_with_abort</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort">HasFoldWithAbortOps.foldWithAbort</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt">HasFoldWithAbortOps.foldWithAbortGt</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt">HasFoldWithAbortOps.foldWithAbortGtLt</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec">HasFoldWithAbortOps.foldWithAbortGtLtSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec">HasFoldWithAbortOps.foldWithAbortGtSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtSpec_Equal</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal">HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec">HasFoldWithAbortOps.foldWithAbortSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg">HasFoldWithAbortOps.FoldWithAbortSpecArg</a> [record, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred">HasFoldWithAbortOps.foldWithAbortSpecArgsForPred</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec_Equal">HasFoldWithAbortOps.foldWithAbortSpec_Equal</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort_spec">HasFoldWithAbortOps.forall_with_abort_spec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort">HasFoldWithAbortOps.forall_with_abort</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P&acute;">HasFoldWithAbortOps.fwasa_P'</a> [projection, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt">HasFoldWithAbortOps.fwasa_f_gt</a> [projection, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt">HasFoldWithAbortOps.fwasa_f_lt</a> [projection, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre">HasFoldWithAbortOps.fwasa_f_pre</a> [projection, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecompute">HasFoldWithAbort.foldWithAbortPrecompute</a> [axiom, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecomputeSpec">HasFoldWithAbort.foldWithAbortPrecomputeSpec</a> [axiom, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="global_M"></a><h2>M </h2>
-<a href="MSetListWithDups.html#Make">Make</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA">MakeAVLSetsWithFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute">MakeAVLSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeGenTreeFoldA">MakeGenTreeFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_RawSpec">MakeGenTreeFoldA.foldWithAbort_RawSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw">MakeGenTreeFoldA.foldWithAbort_Raw</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA">MakeListSetsWithFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute">MakeListSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortRaw">MakeListSetsWithFoldA.foldWithAbortRaw</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA">MakeRBTSetsWithFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute">MakeRBTSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA">MakeWeakListSetsWithFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.E">MakeWeakListSetsWithFoldA.E</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute">MakeWeakListSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortRaw">MakeWeakListSetsWithFoldA.foldWithAbortRaw</a> [definition, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.Raw">MakeWeakListSetsWithFoldA.Raw</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetListWithDups.html#Make.add_spec">Make.add_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.cardinal_spec">Make.cardinal_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.choose_spec2">Make.choose_spec2</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.choose_spec1">Make.choose_spec1</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.compatb">Make.compatb</a> [abbreviation, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.diff_spec">Make.diff_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_dist_spec2w">Make.elements_dist_spec2w</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_dist_spec1">Make.elements_dist_spec1</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_dist_spec_full">Make.elements_dist_spec_full</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_spec1">Make.elements_spec1</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Empty">Make.Empty</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.empty_spec">Make.empty_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.eq">Make.eq</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Equal">Make.Equal</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.equal_spec">Make.equal_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.eq_dec">Make.eq_dec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.eq_equiv">Make.eq_equiv</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Exists">Make.Exists</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.exists_spec">Make.exists_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.filter_spec">Make.filter_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.fold_spec">Make.fold_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.for_all_spec">Make.for_all_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.For_all">Make.For_all</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.In">Make.In</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.inter_spec">Make.inter_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.In_compat">Make.In_compat</a> [instance, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.is_empty_spec">Make.is_empty_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.mem_spec">Make.mem_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.partition_spec2">Make.partition_spec2</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.partition_spec1">Make.partition_spec1</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.partition_aux_spec">Make.partition_aux_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.remove_spec">Make.remove_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.rev_filter_aux_spec">Make.rev_filter_aux_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.singleton_spec">Make.singleton_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Subset">Make.Subset</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.subset_spec">Make.subset_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.union_spec">Make.union_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#25d383999de34b2e68fd5cd20a7b85ae">_ [<=] _</a> [notation, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#dce525a765dab06eb193bd383e9264ac">_ [=] _</a> [notation, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a> [library]<br/>
-<a href="MSetIntervals.html#MSetIntervals">MSetIntervals</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html">MSetIntervals</a> [library]<br/>
-<a href="MSetIntervals.html#MSetIntervalsN">MSetIntervalsN</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervalsNat">MSetIntervalsNat</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervalsZ">MSetIntervalsZ</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.add">MSetIntervals.add</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.add_spec">MSetIntervals.add_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.cardinal">MSetIntervals.cardinal</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.cardinal_spec">MSetIntervals.cardinal_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.choose">MSetIntervals.choose</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.choose_spec2">MSetIntervals.choose_spec2</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.choose_spec1">MSetIntervals.choose_spec1</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.compatb">MSetIntervals.compatb</a> [abbreviation, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.diff">MSetIntervals.diff</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.diff_spec">MSetIntervals.diff_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.E">MSetIntervals.E</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elements">MSetIntervals.elements</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elements_spec2w">MSetIntervals.elements_spec2w</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elements_spec1">MSetIntervals.elements_spec1</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elt">MSetIntervals.elt</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.empty">MSetIntervals.empty</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Empty">MSetIntervals.Empty</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.empty_spec">MSetIntervals.empty_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.eq">MSetIntervals.eq</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.equal">MSetIntervals.equal</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Equal">MSetIntervals.Equal</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.equal_spec">MSetIntervals.equal_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.eq_dec">MSetIntervals.eq_dec</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.eq_equiv">MSetIntervals.eq_equiv</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Exists">MSetIntervals.Exists</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.exists_spec">MSetIntervals.exists_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.exists_">MSetIntervals.exists_</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.filter">MSetIntervals.filter</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.filter_spec">MSetIntervals.filter_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.fold">MSetIntervals.fold</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.fold_spec">MSetIntervals.fold_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.for_all_spec">MSetIntervals.for_all_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.for_all">MSetIntervals.for_all</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.For_all">MSetIntervals.For_all</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.In">MSetIntervals.In</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.inter">MSetIntervals.inter</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.inter_spec">MSetIntervals.inter_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.In_compat">MSetIntervals.In_compat</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.is_empty_spec">MSetIntervals.is_empty_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.is_empty">MSetIntervals.is_empty</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.is_ok">MSetIntervals.is_ok</a> [projection, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.mem">MSetIntervals.mem</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.mem_spec">MSetIntervals.mem_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Mkt">MSetIntervals.Mkt</a> [constructor, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.partition">MSetIntervals.partition</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.partition_spec2">MSetIntervals.partition_spec2</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.partition_spec1">MSetIntervals.partition_spec1</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Raw">MSetIntervals.Raw</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.remove">MSetIntervals.remove</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.remove_spec">MSetIntervals.remove_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.singleton">MSetIntervals.singleton</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.singleton_spec">MSetIntervals.singleton_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec">MSetIntervals.Spec</a> [section, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.f">MSetIntervals.Spec.f</a> [variable, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.s">MSetIntervals.Spec.s</a> [variable, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.s&acute;">MSetIntervals.Spec.s'</a> [variable, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.x">MSetIntervals.Spec.x</a> [variable, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.y">MSetIntervals.Spec.y</a> [variable, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.subset">MSetIntervals.subset</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Subset">MSetIntervals.Subset</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.subset_spec">MSetIntervals.subset_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.t">MSetIntervals.t</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.this">MSetIntervals.this</a> [projection, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.t_">MSetIntervals.t_</a> [record, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.union">MSetIntervals.union</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.union_spec">MSetIntervals.union_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html">MSetListWithDups</a> [library]<br/>
-<a href="MSetWithDups.html">MSetWithDups</a> [library]<br/>
-<br/><br/><a name="global_N"></a><h2>N </h2>
-<a href="MSetIntervals.html#NoDupA_map">NoDupA_map</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#NOP">NOP</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="global_O"></a><h2>O </h2>
-<a href="MSetIntervals.html#Ops">Ops</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops">Ops</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.acc_pred">Ops.acc_pred</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.add">Ops.add</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.add">Ops.add</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.addZ">Ops.addZ</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.addZ_aux">Ops.addZ_aux</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.add_list">Ops.add_list</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.cardinal">Ops.cardinal</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.cardinal">Ops.cardinal</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.cardinalN">Ops.cardinalN</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.choose">Ops.choose</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.choose">Ops.choose</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.chooseZ">Ops.chooseZ</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.diff">Ops.diff</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.diff">Ops.diff</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.elements">Ops.elements</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.elements">Ops.elements</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ">Ops.elementsZ</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ_aux">Ops.elementsZ_aux</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ_aux&acute;">Ops.elementsZ_aux'</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ_aux&acute;&acute;">Ops.elementsZ_aux''</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.elements_dist">Ops.elements_dist</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.elt">Ops.elt</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.elt">Ops.elt</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.empty">Ops.empty</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.empty">Ops.empty</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.equal">Ops.equal</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.equal">Ops.equal</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.exists_">Ops.exists_</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.exists_">Ops.exists_</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.filter">Ops.filter</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.filter">Ops.filter</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.fold">Ops.fold</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.fold">Ops.fold</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.for_all">Ops.for_all</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.for_all">Ops.for_all</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.from_elements">Ops.from_elements</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.inter">Ops.inter</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.inter">Ops.inter</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.is_empty">Ops.is_empty</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.is_empty">Ops.is_empty</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.mem">Ops.mem</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.mem">Ops.mem</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.memZ">Ops.memZ</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.MX">Ops.MX</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.partition">Ops.partition</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.partition">Ops.partition</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.partition_aux">Ops.partition_aux</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.RDFS">Ops.RDFS</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.remove">Ops.remove</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.remove">Ops.remove</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.removeZ">Ops.removeZ</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.removeZ_aux">Ops.removeZ_aux</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.removeZ_aux_insert_guarded">Ops.removeZ_aux_insert_guarded</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.remove_list">Ops.remove_list</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.rev_filter">Ops.rev_filter</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.rev_filter_aux">Ops.rev_filter_aux</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.singleton">Ops.singleton</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.singleton">Ops.singleton</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.singleton_alt_def">Ops.singleton_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.subset">Ops.subset</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.subset">Ops.subset</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.t">Ops.t</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.t">Ops.t</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.union">Ops.union</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.union">Ops.union</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<br/><br/><a name="global_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw">Raw</a> [module, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_isok">Raw.addZ_isok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_spec">Raw.addZ_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_alt_def">Raw.addZ_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_aux_alt_def">Raw.addZ_aux_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_ind">Raw.addZ_ind</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_list_spec">Raw.add_list_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_list_ok">Raw.add_list_ok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_spec">Raw.add_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_ok">Raw.add_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.cardinalN_spec">Raw.cardinalN_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.cardinal_spec">Raw.cardinal_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.chooseZ_spec3">Raw.chooseZ_spec3</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.chooseZ_InZ">Raw.chooseZ_InZ</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.chooseZ_min">Raw.chooseZ_min</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.choose_spec2">Raw.choose_spec2</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.choose_spec1">Raw.choose_spec1</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.choose_alt_def">Raw.choose_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.diff_spec">Raw.diff_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.diff_ok">Raw.diff_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_equiv">Raw.elementsZ_single_equiv</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons_equiv_hd">Raw.elementsZ_cons_equiv_hd</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons_le_end">Raw.elementsZ_cons_le_end</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons_le_start">Raw.elementsZ_cons_le_start</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_spec2w">Raw.elementsZ_spec2w</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_removeZ_aux_insert_guarded">Raw.elementsZ_removeZ_aux_insert_guarded</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons">Raw.elementsZ_cons</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux_alt_def">Raw.elementsZ_aux_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_intro">Raw.elementsZ_single_intro</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;_succ">Raw.elementsZ_aux'_succ</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;_zero">Raw.elementsZ_aux'_zero</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;_pos">Raw.elementsZ_aux'_pos</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;&acute;_irrel">Raw.elementsZ_aux''_irrel</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_succ_front">Raw.elementsZ_single_succ_front</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_add">Raw.elementsZ_single_add</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_succ">Raw.elementsZ_single_succ</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_base">Raw.elementsZ_single_base</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single">Raw.elementsZ_single</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_nil">Raw.elementsZ_nil</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_spec2w">Raw.elements_spec2w</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_spec1">Raw.elements_spec1</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_cons">Raw.elements_cons</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_single">Raw.elements_single</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_nil">Raw.elements_nil</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Empty">Raw.Empty</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.empty_spec">Raw.empty_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.empty_spec&acute;">Raw.empty_spec'</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.empty_ok">Raw.empty_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Equal">Raw.Equal</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.equal_spec">Raw.equal_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.equal_elementsZ">Raw.equal_elementsZ</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.equal_alt_def">Raw.equal_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Exists">Raw.Exists</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.exists_spec">Raw.exists_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.filter_spec">Raw.filter_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.filter_ok">Raw.filter_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.fold_spec">Raw.fold_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.ForNotations">Raw.ForNotations</a> [section, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.for_all_spec">Raw.for_all_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.For_all">Raw.For_all</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In">Raw.In</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inf">Raw.inf</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inf_removeZ_aux_insert_guarded">Raw.inf_removeZ_aux_insert_guarded</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inf_impl">Raw.inf_impl</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inter_spec">Raw.inter_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inter_ok">Raw.inter_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.InZ">Raw.InZ</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_InZ">Raw.In_InZ</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_alt_def">Raw.In_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_elementsZ_single_hd">Raw.In_elementsZ_single_hd</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_elementsZ_single1">Raw.In_elementsZ_single1</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_elementsZ_single">Raw.In_elementsZ_single</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.IsOk">Raw.IsOk</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok">Raw.isok</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok_removeZ_aux_insert_guarded">Raw.isok_removeZ_aux_insert_guarded</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok_inf_nin">Raw.isok_inf_nin</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok_cons">Raw.isok_cons</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.IsOk_Ok">Raw.IsOk_Ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_empty_spec">Raw.is_empty_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_encoded_elems_list_rev">Raw.is_encoded_elems_list_rev</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_encoded_elems_list_app">Raw.is_encoded_elems_list_app</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_encoded_elems_list">Raw.is_encoded_elems_list</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.length_elementsZ_single">Raw.length_elementsZ_single</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.memZ_spec">Raw.memZ_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.mem_spec">Raw.mem_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Nin_elements_greater">Raw.Nin_elements_greater</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.NoDupA_elementsZ_single">Raw.NoDupA_elementsZ_single</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.ok">Raw.ok</a> [projection, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Ok">Raw.Ok</a> [record, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.ok">Raw.ok</a> [constructor, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Ok">Raw.Ok</a> [inductive, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Ok_cons">Raw.Ok_cons</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Ok_nil">Raw.Ok_nil</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_spec2">Raw.partition_spec2</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_spec1">Raw.partition_spec1</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_ok2">Raw.partition_ok2</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_ok1">Raw.partition_ok1</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_spec">Raw.removeZ_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_isok">Raw.removeZ_isok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_counter_pos_aux">Raw.removeZ_counter_pos_aux</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_alt_def">Raw.removeZ_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_aux_alt_def">Raw.removeZ_aux_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_list_spec">Raw.remove_list_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_list_ok">Raw.remove_list_ok</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_spec">Raw.remove_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_ok">Raw.remove_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.singleton_spec">Raw.singleton_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.singleton_ok">Raw.singleton_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Subset">Raw.Subset</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.subset_spec">Raw.subset_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.union_spec">Raw.union_spec</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.union_ok">Raw.union_ok</a> [instance, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted">RemoveDupsFromSorted</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt">RemoveDupsFromSorted.eqb_eq_alt</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.MX">RemoveDupsFromSorted.MX</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA_spec">RemoveDupsFromSorted.remove_dups_by_sortingA_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA">RemoveDupsFromSorted.remove_dups_by_sortingA</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_spec">RemoveDupsFromSorted.remove_dups_from_sortedA_spec</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_hd">RemoveDupsFromSorted.remove_dups_from_sortedA_hd</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_alt</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA">RemoveDupsFromSorted.remove_dups_from_sortedA</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux">RemoveDupsFromSorted.remove_dups_from_sortedA_aux</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XSort">RemoveDupsFromSorted.XSort</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool">RemoveDupsFromSorted.XTotalLeBool</a> [module, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.le">RemoveDupsFromSorted.XTotalLeBool.le</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb">RemoveDupsFromSorted.XTotalLeBool.leb</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb_total">RemoveDupsFromSorted.XTotalLeBool.leb_total</a> [lemma, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t">RemoveDupsFromSorted.XTotalLeBool.t</a> [definition, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#30402bd5327406053496dee97ab5979c">_ <=? _</a> [notation, in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#rev_map_alt_def">rev_map_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#rev_map_aux_alt_def">rev_map_aux_alt_def</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#rev_map">rev_map</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#rev_map_aux">rev_map_aux</a> [definition, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="global_S"></a><h2>S </h2>
-<a href="MSetFoldWithAbort.html#SetsWithFoldA">SetsWithFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#SetsWithFoldA.E">SetsWithFoldA.E</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="global_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups">WSetsOnWithDups</a> [module, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra">WSetsOnWithDupsExtra</a> [module, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.cardinal_spec">WSetsOnWithDupsExtra.cardinal_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist_spec2w">WSetsOnWithDupsExtra.elements_dist_spec2w</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist_spec1">WSetsOnWithDupsExtra.elements_dist_spec1</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist">WSetsOnWithDupsExtra.elements_dist</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.add_spec">WSetsOnWithDups.add_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.choose_spec2">WSetsOnWithDups.choose_spec2</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.choose_spec1">WSetsOnWithDups.choose_spec1</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.compatb">WSetsOnWithDups.compatb</a> [abbreviation, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.diff_spec">WSetsOnWithDups.diff_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.elements_spec1">WSetsOnWithDups.elements_spec1</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Empty">WSetsOnWithDups.Empty</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.empty_spec">WSetsOnWithDups.empty_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.eq">WSetsOnWithDups.eq</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Equal">WSetsOnWithDups.Equal</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.equal_spec">WSetsOnWithDups.equal_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Exists">WSetsOnWithDups.Exists</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.exists_spec">WSetsOnWithDups.exists_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.filter_spec">WSetsOnWithDups.filter_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.fold_spec">WSetsOnWithDups.fold_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.for_all_spec">WSetsOnWithDups.for_all_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.For_all">WSetsOnWithDups.For_all</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.In">WSetsOnWithDups.In</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.inter_spec">WSetsOnWithDups.inter_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.In_compat">WSetsOnWithDups.In_compat</a> [instance, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.is_empty_spec">WSetsOnWithDups.is_empty_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.mem_spec">WSetsOnWithDups.mem_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.partition_spec2">WSetsOnWithDups.partition_spec2</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.partition_spec1">WSetsOnWithDups.partition_spec1</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.remove_spec">WSetsOnWithDups.remove_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.singleton_spec">WSetsOnWithDups.singleton_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec">WSetsOnWithDups.Spec</a> [section, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.f">WSetsOnWithDups.Spec.f</a> [variable, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.s">WSetsOnWithDups.Spec.s</a> [variable, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;">WSetsOnWithDups.Spec.s'</a> [variable, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.x">WSetsOnWithDups.Spec.x</a> [variable, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.y">WSetsOnWithDups.Spec.y</a> [variable, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Subset">WSetsOnWithDups.Subset</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.subset_spec">WSetsOnWithDups.subset_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.union_spec">WSetsOnWithDups.union_spec</a> [axiom, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#93b79b206f9e047c003978d66496371a">_ [<=] _</a> [notation, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#217aae9075a91af245df2f2a31191fab">_ [=] _</a> [notation, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w</a> [lemma, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1</a> [lemma, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist</a> [definition, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra">WSetsOn_TO_WSetsOnWithDupsExtra</a> [module, in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithDupsFoldA">WSetsWithDupsFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithDupsFoldA.E">WSetsWithDupsFoldA.E</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithFoldA">WSetsWithFoldA</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithFoldA.E">WSetsWithFoldA.E</a> [module, in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="global_Z"></a><h2>Z </h2>
-<a href="MSetIntervals.html#Z_le_add_r">Z_le_add_r</a> [lemma, in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><hr/>
-<h1>Notation Index</h1>
-<a name="notation_M"></a><h2>M </h2>
-<a href="MSetListWithDups.html#25d383999de34b2e68fd5cd20a7b85ae">_ [<=] _</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#dce525a765dab06eb193bd383e9264ac">_ [=] _</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<br/><br/><a name="notation_R"></a><h2>R </h2>
-<a href="MSetListWithDups.html#30402bd5327406053496dee97ab5979c">_ <=? _</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<br/><br/><a name="notation_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#93b79b206f9e047c003978d66496371a">_ [<=] _</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#217aae9075a91af245df2f2a31191fab">_ [=] _</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Module Index</h1>
-<a name="module_E"></a><h2>E </h2>
-<a href="MSetIntervals.html#ElementEncode">ElementEncode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN">ElementEncodeN</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat">ElementEncodeNat</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.E">ElementEncodeNat.E</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.E">ElementEncodeN.E</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ">ElementEncodeZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.E">ElementEncodeZ.E</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.E">ElementEncode.E</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="module_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbort">HasFoldWithAbort</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps">HasFoldWithAbortOps</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="module_M"></a><h2>M </h2>
-<a href="MSetListWithDups.html#Make">Make</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA">MakeAVLSetsWithFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeGenTreeFoldA">MakeGenTreeFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA">MakeListSetsWithFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA">MakeRBTSetsWithFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA">MakeWeakListSetsWithFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.E">MakeWeakListSetsWithFoldA.E</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.Raw">MakeWeakListSetsWithFoldA.Raw</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals">MSetIntervals</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervalsN">MSetIntervalsN</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervalsNat">MSetIntervalsNat</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervalsZ">MSetIntervalsZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.E">MSetIntervals.E</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Raw">MSetIntervals.Raw</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="module_N"></a><h2>N </h2>
-<a href="MSetIntervals.html#NOP">NOP</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="module_O"></a><h2>O </h2>
-<a href="MSetIntervals.html#Ops">Ops</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops">Ops</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.MX">Ops.MX</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.RDFS">Ops.RDFS</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<br/><br/><a name="module_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw">Raw</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted">RemoveDupsFromSorted</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.MX">RemoveDupsFromSorted.MX</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XSort">RemoveDupsFromSorted.XSort</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool">RemoveDupsFromSorted.XTotalLeBool</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<br/><br/><a name="module_S"></a><h2>S </h2>
-<a href="MSetFoldWithAbort.html#SetsWithFoldA">SetsWithFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#SetsWithFoldA.E">SetsWithFoldA.E</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="module_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups">WSetsOnWithDups</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra">WSetsOnWithDupsExtra</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra">WSetsOn_TO_WSetsOnWithDupsExtra</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithDupsFoldA">WSetsWithDupsFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithDupsFoldA.E">WSetsWithDupsFoldA.E</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithFoldA">WSetsWithFoldA</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#WSetsWithFoldA.E">WSetsWithFoldA.E</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><hr/>
-<h1>Variable Index</h1>
-<a name="variable_M"></a><h2>M </h2>
-<a href="MSetIntervals.html#MSetIntervals.Spec.f">MSetIntervals.Spec.f</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.s">MSetIntervals.Spec.s</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.s&acute;">MSetIntervals.Spec.s'</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.x">MSetIntervals.Spec.x</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Spec.y">MSetIntervals.Spec.y</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="variable_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.f">WSetsOnWithDups.Spec.f</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.s">WSetsOnWithDups.Spec.s</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.s&acute;">WSetsOnWithDups.Spec.s'</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.x">WSetsOnWithDups.Spec.x</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec.y">WSetsOnWithDups.Spec.y</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Library Index</h1>
-<a name="library_M"></a><h2>M </h2>
-<a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a> <br/>
-<a href="MSetIntervals.html">MSetIntervals</a> <br/>
-<a href="MSetListWithDups.html">MSetListWithDups</a> <br/>
-<a href="MSetWithDups.html">MSetWithDups</a> <br/>
-<br/><br/><hr/>
-<h1>Lemma Index</h1>
-<a name="lemma_A"></a><h2>A </h2>
-<a href="MSetIntervals.html#add_add_sub_eq">add_add_sub_eq</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="lemma_E"></a><h2>E </h2>
-<a href="MSetIntervals.html#ElementEncodeNat.decode_encode_ok">ElementEncodeNat.decode_encode_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.encode_eq">ElementEncodeNat.encode_eq</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.decode_encode_ok">ElementEncodeN.decode_encode_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.encode_eq">ElementEncodeN.encode_eq</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.decode_encode_ok">ElementEncodeZ.decode_encode_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.encode_eq">ElementEncodeZ.encode_eq</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="lemma_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort_spec">HasFoldWithAbortOps.choose_with_abort_spec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort_spec">HasFoldWithAbortOps.exists_with_abort_spec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort_spec">HasFoldWithAbortOps.filter_with_abort_spec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec">HasFoldWithAbortOps.foldWithAbortGtLtSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec">HasFoldWithAbortOps.foldWithAbortGtSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtSpec_Equal</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal">HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec">HasFoldWithAbortOps.foldWithAbortSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec_Equal">HasFoldWithAbortOps.foldWithAbortSpec_Equal</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort_spec">HasFoldWithAbortOps.forall_with_abort_spec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="lemma_M"></a><h2>M </h2>
-<a href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_RawSpec">MakeGenTreeFoldA.foldWithAbort_RawSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetListWithDups.html#Make.add_spec">Make.add_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.cardinal_spec">Make.cardinal_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.choose_spec2">Make.choose_spec2</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.choose_spec1">Make.choose_spec1</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.diff_spec">Make.diff_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_dist_spec2w">Make.elements_dist_spec2w</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_dist_spec1">Make.elements_dist_spec1</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_dist_spec_full">Make.elements_dist_spec_full</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.elements_spec1">Make.elements_spec1</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.empty_spec">Make.empty_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.equal_spec">Make.equal_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.eq_dec">Make.eq_dec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.eq_equiv">Make.eq_equiv</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.exists_spec">Make.exists_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.filter_spec">Make.filter_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.fold_spec">Make.fold_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.for_all_spec">Make.for_all_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.inter_spec">Make.inter_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.is_empty_spec">Make.is_empty_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.mem_spec">Make.mem_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.partition_spec2">Make.partition_spec2</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.partition_spec1">Make.partition_spec1</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.partition_aux_spec">Make.partition_aux_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.remove_spec">Make.remove_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.rev_filter_aux_spec">Make.rev_filter_aux_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.singleton_spec">Make.singleton_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.subset_spec">Make.subset_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.union_spec">Make.union_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.add_spec">MSetIntervals.add_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.cardinal_spec">MSetIntervals.cardinal_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.choose_spec2">MSetIntervals.choose_spec2</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.choose_spec1">MSetIntervals.choose_spec1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.diff_spec">MSetIntervals.diff_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elements_spec2w">MSetIntervals.elements_spec2w</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elements_spec1">MSetIntervals.elements_spec1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.empty_spec">MSetIntervals.empty_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.equal_spec">MSetIntervals.equal_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.exists_spec">MSetIntervals.exists_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.filter_spec">MSetIntervals.filter_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.fold_spec">MSetIntervals.fold_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.for_all_spec">MSetIntervals.for_all_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.inter_spec">MSetIntervals.inter_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.is_empty_spec">MSetIntervals.is_empty_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.mem_spec">MSetIntervals.mem_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.partition_spec2">MSetIntervals.partition_spec2</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.partition_spec1">MSetIntervals.partition_spec1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.remove_spec">MSetIntervals.remove_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.singleton_spec">MSetIntervals.singleton_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.subset_spec">MSetIntervals.subset_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.union_spec">MSetIntervals.union_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="lemma_N"></a><h2>N </h2>
-<a href="MSetIntervals.html#NoDupA_map">NoDupA_map</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="lemma_O"></a><h2>O </h2>
-<a href="MSetIntervals.html#Ops.acc_pred">Ops.acc_pred</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.singleton_alt_def">Ops.singleton_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="lemma_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.addZ_isok">Raw.addZ_isok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_spec">Raw.addZ_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_alt_def">Raw.addZ_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_aux_alt_def">Raw.addZ_aux_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.addZ_ind">Raw.addZ_ind</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_list_spec">Raw.add_list_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_list_ok">Raw.add_list_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.add_spec">Raw.add_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.cardinalN_spec">Raw.cardinalN_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.cardinal_spec">Raw.cardinal_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.chooseZ_spec3">Raw.chooseZ_spec3</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.chooseZ_InZ">Raw.chooseZ_InZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.chooseZ_min">Raw.chooseZ_min</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.choose_alt_def">Raw.choose_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.diff_spec">Raw.diff_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_equiv">Raw.elementsZ_single_equiv</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons_equiv_hd">Raw.elementsZ_cons_equiv_hd</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons_le_end">Raw.elementsZ_cons_le_end</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons_le_start">Raw.elementsZ_cons_le_start</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_spec2w">Raw.elementsZ_spec2w</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_removeZ_aux_insert_guarded">Raw.elementsZ_removeZ_aux_insert_guarded</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_cons">Raw.elementsZ_cons</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux_alt_def">Raw.elementsZ_aux_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_intro">Raw.elementsZ_single_intro</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;_succ">Raw.elementsZ_aux'_succ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;_zero">Raw.elementsZ_aux'_zero</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;_pos">Raw.elementsZ_aux'_pos</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_aux&acute;&acute;_irrel">Raw.elementsZ_aux''_irrel</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_succ_front">Raw.elementsZ_single_succ_front</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_add">Raw.elementsZ_single_add</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_succ">Raw.elementsZ_single_succ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single_base">Raw.elementsZ_single_base</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_nil">Raw.elementsZ_nil</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_spec2w">Raw.elements_spec2w</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_spec1">Raw.elements_spec1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_cons">Raw.elements_cons</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_nil">Raw.elements_nil</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.empty_spec">Raw.empty_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.empty_spec&acute;">Raw.empty_spec'</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.equal_spec">Raw.equal_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.equal_elementsZ">Raw.equal_elementsZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.equal_alt_def">Raw.equal_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.exists_spec">Raw.exists_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.filter_spec">Raw.filter_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.fold_spec">Raw.fold_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.for_all_spec">Raw.for_all_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inf_removeZ_aux_insert_guarded">Raw.inf_removeZ_aux_insert_guarded</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inf_impl">Raw.inf_impl</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inter_spec">Raw.inter_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_InZ">Raw.In_InZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_alt_def">Raw.In_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_elementsZ_single_hd">Raw.In_elementsZ_single_hd</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_elementsZ_single1">Raw.In_elementsZ_single1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In_elementsZ_single">Raw.In_elementsZ_single</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok_removeZ_aux_insert_guarded">Raw.isok_removeZ_aux_insert_guarded</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok_inf_nin">Raw.isok_inf_nin</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok_cons">Raw.isok_cons</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_empty_spec">Raw.is_empty_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_encoded_elems_list_rev">Raw.is_encoded_elems_list_rev</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_encoded_elems_list_app">Raw.is_encoded_elems_list_app</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.length_elementsZ_single">Raw.length_elementsZ_single</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.memZ_spec">Raw.memZ_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.mem_spec">Raw.mem_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Nin_elements_greater">Raw.Nin_elements_greater</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.NoDupA_elementsZ_single">Raw.NoDupA_elementsZ_single</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Ok_cons">Raw.Ok_cons</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Ok_nil">Raw.Ok_nil</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_spec2">Raw.partition_spec2</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_spec1">Raw.partition_spec1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_spec">Raw.removeZ_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_isok">Raw.removeZ_isok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_counter_pos_aux">Raw.removeZ_counter_pos_aux</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_alt_def">Raw.removeZ_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.removeZ_aux_alt_def">Raw.removeZ_aux_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_list_spec">Raw.remove_list_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_list_ok">Raw.remove_list_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_spec">Raw.remove_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.singleton_spec">Raw.singleton_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.subset_spec">Raw.subset_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.union_spec">Raw.union_spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt">RemoveDupsFromSorted.eqb_eq_alt</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA_spec">RemoveDupsFromSorted.remove_dups_by_sortingA_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_spec">RemoveDupsFromSorted.remove_dups_from_sortedA_spec</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_hd">RemoveDupsFromSorted.remove_dups_from_sortedA_hd</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_alt</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb_total">RemoveDupsFromSorted.XTotalLeBool.leb_total</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#rev_map_alt_def">rev_map_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#rev_map_aux_alt_def">rev_map_aux_alt_def</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="lemma_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><a name="lemma_Z"></a><h2>Z </h2>
-<a href="MSetIntervals.html#Z_le_add_r">Z_le_add_r</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><hr/>
-<h1>Axiom Index</h1>
-<a name="axiom_E"></a><h2>E </h2>
-<a href="MSetIntervals.html#ElementEncode.decode">ElementEncode.decode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.decode_encode_ok">ElementEncode.decode_encode_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.encode">ElementEncode.encode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncode.encode_eq">ElementEncode.encode_eq</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="axiom_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecompute">HasFoldWithAbort.foldWithAbortPrecompute</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecomputeSpec">HasFoldWithAbort.foldWithAbortPrecomputeSpec</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="axiom_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.cardinal_spec">WSetsOnWithDupsExtra.cardinal_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist_spec2w">WSetsOnWithDupsExtra.elements_dist_spec2w</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist_spec1">WSetsOnWithDupsExtra.elements_dist_spec1</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist">WSetsOnWithDupsExtra.elements_dist</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.add_spec">WSetsOnWithDups.add_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.choose_spec2">WSetsOnWithDups.choose_spec2</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.choose_spec1">WSetsOnWithDups.choose_spec1</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.diff_spec">WSetsOnWithDups.diff_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.elements_spec1">WSetsOnWithDups.elements_spec1</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.empty_spec">WSetsOnWithDups.empty_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.equal_spec">WSetsOnWithDups.equal_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.exists_spec">WSetsOnWithDups.exists_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.filter_spec">WSetsOnWithDups.filter_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.fold_spec">WSetsOnWithDups.fold_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.for_all_spec">WSetsOnWithDups.for_all_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.In">WSetsOnWithDups.In</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.inter_spec">WSetsOnWithDups.inter_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.is_empty_spec">WSetsOnWithDups.is_empty_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.mem_spec">WSetsOnWithDups.mem_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.partition_spec2">WSetsOnWithDups.partition_spec2</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.partition_spec1">WSetsOnWithDups.partition_spec1</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.remove_spec">WSetsOnWithDups.remove_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.singleton_spec">WSetsOnWithDups.singleton_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.subset_spec">WSetsOnWithDups.subset_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.union_spec">WSetsOnWithDups.union_spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Constructor Index</h1>
-<a name="constructor_M"></a><h2>M </h2>
-<a href="MSetIntervals.html#MSetIntervals.Mkt">MSetIntervals.Mkt</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="constructor_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.ok">Raw.ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><hr/>
-<h1>Projection Index</h1>
-<a name="projection_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P&acute;">HasFoldWithAbortOps.fwasa_P'</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt">HasFoldWithAbortOps.fwasa_f_gt</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt">HasFoldWithAbortOps.fwasa_f_lt</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre">HasFoldWithAbortOps.fwasa_f_pre</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="projection_M"></a><h2>M </h2>
-<a href="MSetIntervals.html#MSetIntervals.is_ok">MSetIntervals.is_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.this">MSetIntervals.this</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="projection_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.ok">Raw.ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><hr/>
-<h1>Inductive Index</h1>
-<a name="inductive_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.Ok">Raw.Ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><hr/>
-<h1>Section Index</h1>
-<a name="section_M"></a><h2>M </h2>
-<a href="MSetIntervals.html#MSetIntervals.Spec">MSetIntervals.Spec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="section_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.ForNotations">Raw.ForNotations</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="section_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups.Spec">WSetsOnWithDups.Spec</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Instance Index</h1>
-<a name="instance_M"></a><h2>M </h2>
-<a href="MSetListWithDups.html#Make.In_compat">Make.In_compat</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.eq_equiv">MSetIntervals.eq_equiv</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.In_compat">MSetIntervals.In_compat</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="instance_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.add_ok">Raw.add_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.diff_ok">Raw.diff_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.empty_ok">Raw.empty_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.filter_ok">Raw.filter_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inter_ok">Raw.inter_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.IsOk_Ok">Raw.IsOk_Ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_ok2">Raw.partition_ok2</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.partition_ok1">Raw.partition_ok1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.remove_ok">Raw.remove_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.singleton_ok">Raw.singleton_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.union_ok">Raw.union_ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="instance_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups.In_compat">WSetsOnWithDups.In_compat</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Abbreviation Index</h1>
-<a name="abbreviation_M"></a><h2>M </h2>
-<a href="MSetListWithDups.html#Make.compatb">Make.compatb</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.compatb">MSetIntervals.compatb</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="abbreviation_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups.compatb">WSetsOnWithDups.compatb</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Definition Index</h1>
-<a name="definition_E"></a><h2>E </h2>
-<a href="MSetIntervals.html#ElementEncodeNat.decode">ElementEncodeNat.decode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeNat.encode">ElementEncodeNat.encode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.decode">ElementEncodeN.decode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeN.encode">ElementEncodeN.encode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.decode">ElementEncodeZ.decode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#ElementEncodeZ.encode">ElementEncodeZ.encode</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="definition_F"></a><h2>F </h2>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtLtSpecPred">foldWithAbortGtLtSpecPred</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtLtType">foldWithAbortGtLtType</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtSpecPred">foldWithAbortGtSpecPred</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortGtType">foldWithAbortGtType</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred">foldWithAbortPrecomputeSpecPred</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortPrecomputeType">foldWithAbortPrecomputeType</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortSpecPred">foldWithAbortSpecPred</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#foldWithAbortType">foldWithAbortType</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="definition_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort">HasFoldWithAbortOps.choose_with_abort</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort">HasFoldWithAbortOps.exists_with_abort</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort">HasFoldWithAbortOps.filter_with_abort</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort">HasFoldWithAbortOps.foldWithAbort</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt">HasFoldWithAbortOps.foldWithAbortGt</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt">HasFoldWithAbortOps.foldWithAbortGtLt</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred">HasFoldWithAbortOps.foldWithAbortSpecArgsForPred</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort">HasFoldWithAbortOps.forall_with_abort</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="definition_M"></a><h2>M </h2>
-<a href="MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute">MakeAVLSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw">MakeGenTreeFoldA.foldWithAbort_Raw</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute">MakeListSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortRaw">MakeListSetsWithFoldA.foldWithAbortRaw</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute">MakeRBTSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute">MakeWeakListSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortRaw">MakeWeakListSetsWithFoldA.foldWithAbortRaw</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<a href="MSetListWithDups.html#Make.Empty">Make.Empty</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.eq">Make.eq</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Equal">Make.Equal</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Exists">Make.Exists</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.For_all">Make.For_all</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.In">Make.In</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Make.Subset">Make.Subset</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.add">MSetIntervals.add</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.cardinal">MSetIntervals.cardinal</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.choose">MSetIntervals.choose</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.diff">MSetIntervals.diff</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elements">MSetIntervals.elements</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.elt">MSetIntervals.elt</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.empty">MSetIntervals.empty</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Empty">MSetIntervals.Empty</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.eq">MSetIntervals.eq</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.equal">MSetIntervals.equal</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Equal">MSetIntervals.Equal</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.eq_dec">MSetIntervals.eq_dec</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Exists">MSetIntervals.Exists</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.exists_">MSetIntervals.exists_</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.filter">MSetIntervals.filter</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.fold">MSetIntervals.fold</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.for_all">MSetIntervals.for_all</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.For_all">MSetIntervals.For_all</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.In">MSetIntervals.In</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.inter">MSetIntervals.inter</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.is_empty">MSetIntervals.is_empty</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.mem">MSetIntervals.mem</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.partition">MSetIntervals.partition</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.remove">MSetIntervals.remove</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.singleton">MSetIntervals.singleton</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.subset">MSetIntervals.subset</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.Subset">MSetIntervals.Subset</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.t">MSetIntervals.t</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#MSetIntervals.union">MSetIntervals.union</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="definition_O"></a><h2>O </h2>
-<a href="MSetIntervals.html#Ops.add">Ops.add</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.add">Ops.add</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.addZ">Ops.addZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.addZ_aux">Ops.addZ_aux</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.add_list">Ops.add_list</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.cardinal">Ops.cardinal</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.cardinal">Ops.cardinal</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.cardinalN">Ops.cardinalN</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.choose">Ops.choose</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.choose">Ops.choose</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.chooseZ">Ops.chooseZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.diff">Ops.diff</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.diff">Ops.diff</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.elements">Ops.elements</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.elements">Ops.elements</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ">Ops.elementsZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ_aux">Ops.elementsZ_aux</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ_aux&acute;">Ops.elementsZ_aux'</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.elementsZ_aux&acute;&acute;">Ops.elementsZ_aux''</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.elements_dist">Ops.elements_dist</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.elt">Ops.elt</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.elt">Ops.elt</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.empty">Ops.empty</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.empty">Ops.empty</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.equal">Ops.equal</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.equal">Ops.equal</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.exists_">Ops.exists_</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.exists_">Ops.exists_</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.filter">Ops.filter</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.filter">Ops.filter</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.fold">Ops.fold</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.fold">Ops.fold</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.for_all">Ops.for_all</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.for_all">Ops.for_all</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.from_elements">Ops.from_elements</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.inter">Ops.inter</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.inter">Ops.inter</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.is_empty">Ops.is_empty</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.is_empty">Ops.is_empty</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.mem">Ops.mem</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.mem">Ops.mem</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.memZ">Ops.memZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.partition">Ops.partition</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.partition">Ops.partition</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.partition_aux">Ops.partition_aux</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.remove">Ops.remove</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.remove">Ops.remove</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.removeZ">Ops.removeZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.removeZ_aux">Ops.removeZ_aux</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.removeZ_aux_insert_guarded">Ops.removeZ_aux_insert_guarded</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Ops.remove_list">Ops.remove_list</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.rev_filter">Ops.rev_filter</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#Ops.rev_filter_aux">Ops.rev_filter_aux</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.singleton">Ops.singleton</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.singleton">Ops.singleton</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.subset">Ops.subset</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.subset">Ops.subset</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.t">Ops.t</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.t">Ops.t</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#Ops.union">Ops.union</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#Ops.union">Ops.union</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<br/><br/><a name="definition_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.choose_spec2">Raw.choose_spec2</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.choose_spec1">Raw.choose_spec1</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elementsZ_single">Raw.elementsZ_single</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.elements_single">Raw.elements_single</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Empty">Raw.Empty</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Equal">Raw.Equal</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Exists">Raw.Exists</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.For_all">Raw.For_all</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.In">Raw.In</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.inf">Raw.inf</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.InZ">Raw.InZ</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.IsOk">Raw.IsOk</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.isok">Raw.isok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.is_encoded_elems_list">Raw.is_encoded_elems_list</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#Raw.Subset">Raw.Subset</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA">RemoveDupsFromSorted.remove_dups_by_sortingA</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA">RemoveDupsFromSorted.remove_dups_from_sortedA</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux">RemoveDupsFromSorted.remove_dups_from_sortedA_aux</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.le">RemoveDupsFromSorted.XTotalLeBool.le</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb">RemoveDupsFromSorted.XTotalLeBool.leb</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t">RemoveDupsFromSorted.XTotalLeBool.t</a> [in <a href="MSetListWithDups.html">MSetListWithDups</a>]<br/>
-<a href="MSetIntervals.html#rev_map">rev_map</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<a href="MSetIntervals.html#rev_map_aux">rev_map_aux</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="definition_W"></a><h2>W </h2>
-<a href="MSetWithDups.html#WSetsOnWithDups.Empty">WSetsOnWithDups.Empty</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.eq">WSetsOnWithDups.eq</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Equal">WSetsOnWithDups.Equal</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Exists">WSetsOnWithDups.Exists</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.For_all">WSetsOnWithDups.For_all</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOnWithDups.Subset">WSetsOnWithDups.Subset</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<a href="MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist</a> [in <a href="MSetWithDups.html">MSetWithDups</a>]<br/>
-<br/><br/><hr/>
-<h1>Record Index</h1>
-<a name="record_H"></a><h2>H </h2>
-<a href="MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg">HasFoldWithAbortOps.FoldWithAbortSpecArg</a> [in <a href="MSetFoldWithAbort.html">MSetFoldWithAbort</a>]<br/>
-<br/><br/><a name="record_M"></a><h2>M </h2>
-<a href="MSetIntervals.html#MSetIntervals.t_">MSetIntervals.t_</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><a name="record_R"></a><h2>R </h2>
-<a href="MSetIntervals.html#Raw.Ok">Raw.Ok</a> [in <a href="MSetIntervals.html">MSetIntervals</a>]<br/>
-<br/><br/><hr/><table>
-<tr>
-<td>Global Index</td>
-<td><a href="index.html#global_A">A</a></td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#global_E">E</a></td>
-<td><a href="index.html#global_F">F</a></td>
-<td>G</td>
-<td><a href="index.html#global_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#global_M">M</a></td>
-<td><a href="index.html#global_N">N</a></td>
-<td><a href="index.html#global_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#global_R">R</a></td>
-<td><a href="index.html#global_S">S</a></td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#global_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td><a href="index.html#global_Z">Z</a></td>
-<td>_</td>
-<td>other</td>
-<td>(451 entries)</td>
-</tr>
-<tr>
-<td>Notation Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#notation_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#notation_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#notation_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(5 entries)</td>
-</tr>
-<tr>
-<td>Module Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#module_E">E</a></td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#module_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#module_M">M</a></td>
-<td><a href="index.html#module_N">N</a></td>
-<td><a href="index.html#module_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#module_R">R</a></td>
-<td><a href="index.html#module_S">S</a></td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#module_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(43 entries)</td>
-</tr>
-<tr>
-<td>Variable Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#variable_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#variable_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(10 entries)</td>
-</tr>
-<tr>
-<td>Library Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#library_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(4 entries)</td>
-</tr>
-<tr>
-<td>Lemma Index</td>
-<td><a href="index.html#lemma_A">A</a></td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#lemma_E">E</a></td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#lemma_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#lemma_M">M</a></td>
-<td><a href="index.html#lemma_N">N</a></td>
-<td><a href="index.html#lemma_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#lemma_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#lemma_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td><a href="index.html#lemma_Z">Z</a></td>
-<td>_</td>
-<td>other</td>
-<td>(168 entries)</td>
-</tr>
-<tr>
-<td>Axiom Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#axiom_E">E</a></td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#axiom_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td>M</td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#axiom_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(31 entries)</td>
-</tr>
-<tr>
-<td>Constructor Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#constructor_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#constructor_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(2 entries)</td>
-</tr>
-<tr>
-<td>Projection Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#projection_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#projection_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#projection_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(7 entries)</td>
-</tr>
-<tr>
-<td>Inductive Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td>M</td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#inductive_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(1 entry)</td>
-</tr>
-<tr>
-<td>Section Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#section_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#section_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#section_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(3 entries)</td>
-</tr>
-<tr>
-<td>Instance Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#instance_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#instance_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#instance_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(15 entries)</td>
-</tr>
-<tr>
-<td>Abbreviation Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td>H</td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#abbreviation_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td>R</td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#abbreviation_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(3 entries)</td>
-</tr>
-<tr>
-<td>Definition Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td><a href="index.html#definition_E">E</a></td>
-<td><a href="index.html#definition_F">F</a></td>
-<td>G</td>
-<td><a href="index.html#definition_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#definition_M">M</a></td>
-<td>N</td>
-<td><a href="index.html#definition_O">O</a></td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#definition_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td><a href="index.html#definition_W">W</a></td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(156 entries)</td>
-</tr>
-<tr>
-<td>Record Index</td>
-<td>A</td>
-<td>B</td>
-<td>C</td>
-<td>D</td>
-<td>E</td>
-<td>F</td>
-<td>G</td>
-<td><a href="index.html#record_H">H</a></td>
-<td>I</td>
-<td>J</td>
-<td>K</td>
-<td>L</td>
-<td><a href="index.html#record_M">M</a></td>
-<td>N</td>
-<td>O</td>
-<td>P</td>
-<td>Q</td>
-<td><a href="index.html#record_R">R</a></td>
-<td>S</td>
-<td>T</td>
-<td>U</td>
-<td>V</td>
-<td>W</td>
-<td>X</td>
-<td>Y</td>
-<td>Z</td>
-<td>_</td>
-<td>other</td>
-<td>(3 entries)</td>
-</tr>
-</table>
-<hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
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-<div id="toc">
-<a href="MSetWithDups.html"><h2>Library MSetWithDups</h2></a>
-<ul class="doclist">
-<li><a href="MSetWithDups.html#lab1">Signature for weak sets which may contain duplicates</a>
-<ul class="doclist">
-<li><a href="MSetWithDups.html#lab2">WSetsOnWithDups</a>
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-</li>
-<li><a href="MSetWithDups.html#lab3">WSetsOnWithDupsExtra</a>
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-</li>
-<li><a href="MSetWithDups.html#lab4">WSetOn to WSetsOnWithDupsExtra</a>
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-</li>
-</ul>
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-</li>
-</ul>
-<a href="MSetFoldWithAbort.html"><h2>Library MSetFoldWithAbort</h2></a>
-<ul class="doclist">
-<li><a href="MSetFoldWithAbort.html#lab5">Fold with abort for sets</a>
-<ul class="doclist">
-<li><a href="MSetFoldWithAbort.html#lab6">Fold With Abort Operations</a>
-<ul class="doclist">
-<li><a href="MSetFoldWithAbort.html#lab7">Module Types</a>
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-</li>
-</ul>
-
-</li>
-<li><a href="MSetFoldWithAbort.html#lab8">Derived operations</a>
-<ul class="doclist">
-<li><a href="MSetFoldWithAbort.html#lab9">First lets define the other folding with abort variants</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab10">Specialisations for equality</a>
-
-</li>
-<li><a href="MSetFoldWithAbort.html#lab11">FoldWithAbortSpecArgs</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab12">Filter with abort</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab13">Choose with abort</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab14">Exists and Forall with abort</a>
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-</li>
-</ul>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab15">Modules Types For Sets with Fold with Abort</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab16">Implementations</a>
-<ul class="doclist">
-<li><a href="MSetFoldWithAbort.html#lab17">GenTree implementation</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab18">AVL implementation</a>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab19">RBT implementation</a>
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-</li>
-</ul>
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-</li>
-<li><a href="MSetFoldWithAbort.html#lab20">Sorted Lists Implementation</a>
-<ul class="doclist">
-<li><a href="MSetFoldWithAbort.html#lab21">Unsorted Lists without Dups Implementation</a>
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-</li>
-</ul>
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-</li>
-</ul>
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-</li>
-</ul>
-<a href="MSetIntervals.html"><h2>Library MSetIntervals</h2></a>
-<ul class="doclist">
-<li><a href="MSetIntervals.html#lab22">Weak sets implemented by interval lists</a>
-<ul class="doclist">
-<li><a href="MSetIntervals.html#lab23">Auxiliary stuff</a>
-<ul class="doclist">
-<li><a href="MSetIntervals.html#lab24">rev_map</a>
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-</li>
-</ul>
-
-</li>
-<li><a href="MSetIntervals.html#lab25">Encoding Elements</a>
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-</li>
-<li><a href="MSetIntervals.html#lab26">Set Operations</a>
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-</li>
-<li><a href="MSetIntervals.html#lab27">Raw Module</a>
-<ul class="doclist">
-<li><a href="MSetIntervals.html#lab28">Defining invariant <span class="inlinecode"><span class="id" title="var">IsOk</span></span></a>
-
-</li>
-<li><a href="MSetIntervals.html#lab29">Defining notations</a>
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-</li>
-<li><a href="MSetIntervals.html#lab30">elements list properties</a>
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-</li>
-<li><a href="MSetIntervals.html#lab31">Alternative definition of addZ</a>
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-</li>
-<li><a href="MSetIntervals.html#lab32">Alternative definition of removeZ</a>
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-</li>
-<li><a href="MSetIntervals.html#lab33">Auxiliary Lemmata about Invariant</a>
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-</li>
-<li><a href="MSetIntervals.html#lab34">Properties of In and InZ</a>
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-</li>
-<li><a href="MSetIntervals.html#lab35">Membership specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab36">add specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab37">remove specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab38">empty specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab39">is_empty specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab40">singleton specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab41">add_list specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab42">remove_list specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab43">union specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab44">filter specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab45">inter specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab46">diff specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab47">subset specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab48">elements and elementsZ specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab49">equal specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab50">choose specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab51">fold specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab52">cardinal specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab53">for_all specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab54">exists specification</a>
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-</li>
-<li><a href="MSetIntervals.html#lab55">partition specification</a>
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-</li>
-</ul>
-
-</li>
-<li><a href="MSetIntervals.html#lab56">Main Module</a>
-
-</li>
-<li><a href="MSetIntervals.html#lab57">Instantiations</a>
-<ul class="doclist">
-<li><a href="MSetIntervals.html#lab58">Z</a>
-
-</li>
-<li><a href="MSetIntervals.html#lab59">N</a>
-
-</li>
-<li><a href="MSetIntervals.html#lab60">nat</a>
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-</li>
-</ul>
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-</li>
-</ul>
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-</li>
-</ul>
-<a href="MSetListWithDups.html"><h2>Library MSetListWithDups</h2></a>
-<ul class="doclist">
-<li><a href="MSetListWithDups.html#lab61">Weak sets implemented as lists with duplicates</a>
-<ul class="doclist">
-<li><a href="MSetListWithDups.html#lab62">Removing duplicates from sorted lists</a>
-
-</li>
-<li><a href="MSetListWithDups.html#lab63">Operations Module</a>
-
-</li>
-<li><a href="MSetListWithDups.html#lab64">Main Module</a>
-
-</li>
-<li><a href="MSetListWithDups.html#lab65">Proofs of set operation specifications.</a>
-
-</li>
-</ul>
-
-</li>
-</ul>
-</div>
-<hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
-</div>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>MSetsExtra.MSetFoldWithAbort</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library MSetsExtra.MSetFoldWithAbort</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab1"></a><h1 class="section">Fold with abort for sets</h1>
+
+
+<div class="paragraph"> </div>
+
+    This file provided an efficient fold operation for set interfaces.
+    The standard fold iterates over all elements of the set. The
+    efficient one - called <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a></span> - is allowed to skip
+    certain elements and thereby abort early.  
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#"><span class="id" title="library">MSetInterface</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">mathcomp.ssreflect.ssreflect</span>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#"><span class="id" title="library">MSetWithDups</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Int.html#"><span class="id" title="library">Int</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetGenTree.html#"><span class="id" title="library">MSetGenTree</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetAVL.html#"><span class="id" title="library">MSetAVL</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetRBT.html#"><span class="id" title="library">MSetRBT</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetList.html#"><span class="id" title="library">MSetList</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetWeakList.html#"><span class="id" title="library">MSetWeakList</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab2"></a><h2 class="section">Fold With Abort Operations</h2>
+
+
+<div class="paragraph"> </div>
+
+    We want to provide an efficient folding operation. Efficieny is
+    gained by aborting the folding early, if we know that continuing
+    would not have an effect any more. Formalising this leads to the following
+    specification of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a></span>. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortType" class="idref" href="#foldWithAbortType"><span class="id" title="definition">foldWithAbortType</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="elt:1" class="idref" href="#elt:1"><span class="id" title="binder">elt</span></a> </div>
+
+<div class="doc">
+element type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="t:2" class="idref" href="#t:2"><span class="id" title="binder">t</span></a> </div>
+
+<div class="doc">
+type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="A:3" class="idref" href="#A:3"><span class="id" title="binder">A</span></a> </div>
+
+<div class="doc">
+return type </div>
+<div class="code">
+ :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:1"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:3"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:3"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>    </div>
+
+<div class="doc">
+f 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:1"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:3"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> </div>
+
+<div class="doc">
+f_abort 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:2"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                  </div>
+
+<div class="doc">
+input set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:3"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                  </div>
+
+<div class="doc">
+base value 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:3"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortSpecPred" class="idref" href="#foldWithAbortSpecPred"><span class="id" title="definition">foldWithAbortSpecPred</span></a> {<a id="elt:4" class="idref" href="#elt:4"><span class="id" title="binder">elt</span></a> <a id="t:5" class="idref" href="#t:5"><span class="id" title="binder">t</span></a> : <span class="id" title="keyword">Type</span>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="In:6" class="idref" href="#In:6"><span class="id" title="binder">In</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:4"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fold:8" class="idref" href="#fold:8"><span class="id" title="binder">fold</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:7" class="idref" href="#A:7"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:4"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:7"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:7"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:7"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:7"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="foldWithAbort:10" class="idref" href="#foldWithAbort:10"><span class="id" title="binder">foldWithAbort</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:9" class="idref" href="#A:9"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortType"><span class="id" title="definition">foldWithAbortType</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:4"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:5"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:9"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:11" class="idref" href="#A:11"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+result type 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="i:12" class="idref" href="#i:12"><span class="id" title="binder">i</span></a> <a id="i':13" class="idref" href="#i':13"><span class="id" title="binder">i'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+base values for foldWithAbort and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:14" class="idref" href="#f:14"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:4"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a>) (<a id="f':15" class="idref" href="#f':15"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:4"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+fold functions for foldWithAbort and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f_abort:16" class="idref" href="#f_abort:16"><span class="id" title="binder">f_abort</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:4"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+abort function 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:17" class="idref" href="#s:17"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:5"><span class="id" title="variable">t</span></a>) </div>
+
+<div class="doc">
+sets to fold over 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="P:18" class="idref" href="#P:18"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:11"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) </div>
+
+<div class="doc">
+equivalence relation on results </div>
+<div class="code">
+,<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> is for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> compatible with the equivalence relation P 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:19" class="idref" href="#st:19"><span class="id" title="binder">st</span></a> <a id="st':20" class="idref" href="#st':20"><span class="id" title="binder">st'</span></a> <a id="e:21" class="idref" href="#e:21"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:6"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:21"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:19"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':20"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:14"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:21"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:19"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:14"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:21"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':20"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> and <span class="inlinecode"><span class="id" title="var">f'</span></span> agree for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:22" class="idref" href="#e:22"><span class="id" title="binder">e</span></a> <a id="st:23" class="idref" href="#st:23"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:6"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:22"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:14"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:22"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:23"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':15"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:22"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:23"><span class="id" title="variable">st</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">f_abort</span></span> is OK, i.e. all other elements can be skipped without 
+        leaving the equivalence relation. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:24" class="idref" href="#e1:24"><span class="id" title="binder">e1</span></a> <a id="st:25" class="idref" href="#st:25"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:6"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:24"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_abort:16"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:24"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:25"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':26" class="idref" href="#st':26"><span class="id" title="binder">st'</span></a> <a id="e2:27" class="idref" href="#e2:27"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:25"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':26"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:6"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:27"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:27"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:24"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:25"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:14"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:27"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':26"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The base values are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:12"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':13"><span class="id" title="variable">i'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The results are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:18"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbort:10"><span class="id" title="variable">foldWithAbort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:14"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_abort:16"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:12"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fold:8"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':15"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:17"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':13"><span class="id" title="variable">i'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The specification of folding for ordered sets (as represented by
+    interface <span class="inlinecode"><span class="id" title="var">Sets</span></span>) demands that elements are visited in increasing
+    order. For ordered sets we can therefore abort folding based on
+    the weaker knowledge that greater elements have no effect on the
+    result. The following definition captures this. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortGtType" class="idref" href="#foldWithAbortGtType"><span class="id" title="definition">foldWithAbortGtType</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="elt:28" class="idref" href="#elt:28"><span class="id" title="binder">elt</span></a> </div>
+
+<div class="doc">
+element type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="t:29" class="idref" href="#t:29"><span class="id" title="binder">t</span></a> </div>
+
+<div class="doc">
+type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="A:30" class="idref" href="#A:30"><span class="id" title="binder">A</span></a> </div>
+
+<div class="doc">
+return type </div>
+<div class="code">
+ :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:28"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:30"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:30"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>    </div>
+
+<div class="doc">
+f 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:28"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:30"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> </div>
+
+<div class="doc">
+f_gt 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:29"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                  </div>
+
+<div class="doc">
+input set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:30"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                  </div>
+
+<div class="doc">
+base value 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:30"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortGtSpecPred" class="idref" href="#foldWithAbortGtSpecPred"><span class="id" title="definition">foldWithAbortGtSpecPred</span></a> {<a id="elt:31" class="idref" href="#elt:31"><span class="id" title="binder">elt</span></a> <a id="t:32" class="idref" href="#t:32"><span class="id" title="binder">t</span></a> : <span class="id" title="keyword">Type</span>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:33" class="idref" href="#lt:33"><span class="id" title="binder">lt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="In:34" class="idref" href="#In:34"><span class="id" title="binder">In</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:32"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fold:36" class="idref" href="#fold:36"><span class="id" title="binder">fold</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:35" class="idref" href="#A:35"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:35"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:35"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:32"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:35"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:35"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="foldWithAbortGt:38" class="idref" href="#foldWithAbortGt:38"><span class="id" title="binder">foldWithAbortGt</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:37" class="idref" href="#A:37"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortType"><span class="id" title="definition">foldWithAbortType</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:32"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:37"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:39" class="idref" href="#A:39"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+result type 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="i:40" class="idref" href="#i:40"><span class="id" title="binder">i</span></a> <a id="i':41" class="idref" href="#i':41"><span class="id" title="binder">i'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+base values for foldWithAbort and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:42" class="idref" href="#f:42"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a>) (<a id="f':43" class="idref" href="#f':43"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+fold functions for foldWithAbort and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f_gt:44" class="idref" href="#f_gt:44"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:31"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+abort function 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:45" class="idref" href="#s:45"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:32"><span class="id" title="variable">t</span></a>) </div>
+
+<div class="doc">
+sets to fold over 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="P:46" class="idref" href="#P:46"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:39"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) </div>
+
+<div class="doc">
+equivalence relation on results </div>
+<div class="code">
+,<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> is for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> compatible with the equivalence relation P 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:47" class="idref" href="#st:47"><span class="id" title="binder">st</span></a> <a id="st':48" class="idref" href="#st':48"><span class="id" title="binder">st'</span></a> <a id="e:49" class="idref" href="#e:49"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:34"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:49"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:47"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':48"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:49"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:47"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:49"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':48"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> and <span class="inlinecode"><span class="id" title="var">f'</span></span> agree for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:50" class="idref" href="#e:50"><span class="id" title="binder">e</span></a> <a id="st:51" class="idref" href="#st:51"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:34"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:50"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:50"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:51"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':43"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:50"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:51"><span class="id" title="variable">st</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">f_abort</span></span> is OK, i.e. all other elements can be skipped without 
+        leaving the equivalence relation. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:52" class="idref" href="#e1:52"><span class="id" title="binder">e1</span></a> <a id="st:53" class="idref" href="#st:53"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:34"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:52"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:44"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:52"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:53"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':54" class="idref" href="#st':54"><span class="id" title="binder">st'</span></a> <a id="e2:55" class="idref" href="#e2:55"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:53"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':54"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:34"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:55"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#lt:33"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:52"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:55"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:53"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:55"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':54"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The base values are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:40"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':41"><span class="id" title="variable">i'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The results are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:46"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGt:38"><span class="id" title="variable">foldWithAbortGt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:42"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:44"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:40"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fold:36"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':43"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:45"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':41"><span class="id" title="variable">i'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+For ordered sets, we can safely skip elements at the end
+    based on the knowledge that they are all greater than the current element.
+    This leads to serious performance improvements for operations like
+    filtering. It is tempting to try the symmetric operation and skip elements at
+    the beginning based on the knowledge that they are too small to be interesting.
+    So, we would like to start late as well as abort early. 
+
+<div class="paragraph"> </div>
+
+    This is indeed a very natural and efficient operation for set implementations 
+    based on binary search trees (i.e. the AVL and RBT sets). We can completely symmetrically
+    to skipping greater elements also skip smaller elements. This leads to the following
+    specification. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortGtLtType" class="idref" href="#foldWithAbortGtLtType"><span class="id" title="definition">foldWithAbortGtLtType</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="elt:56" class="idref" href="#elt:56"><span class="id" title="binder">elt</span></a> </div>
+
+<div class="doc">
+element type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="t:57" class="idref" href="#t:57"><span class="id" title="binder">t</span></a> </div>
+
+<div class="doc">
+type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="A:58" class="idref" href="#A:58"><span class="id" title="binder">A</span></a> </div>
+
+<div class="doc">
+return type </div>
+<div class="code">
+ :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:56"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> </div>
+
+<div class="doc">
+f_lt 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:56"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:58"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>    </div>
+
+<div class="doc">
+f 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:56"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> </div>
+
+<div class="doc">
+f_gt 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:57"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                  </div>
+
+<div class="doc">
+input set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:58"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                  </div>
+
+<div class="doc">
+base value 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:58"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortGtLtSpecPred" class="idref" href="#foldWithAbortGtLtSpecPred"><span class="id" title="definition">foldWithAbortGtLtSpecPred</span></a> {<a id="elt:59" class="idref" href="#elt:59"><span class="id" title="binder">elt</span></a> <a id="t:60" class="idref" href="#t:60"><span class="id" title="binder">t</span></a> : <span class="id" title="keyword">Type</span>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:61" class="idref" href="#lt:61"><span class="id" title="binder">lt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="In:62" class="idref" href="#In:62"><span class="id" title="binder">In</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:60"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fold:64" class="idref" href="#fold:64"><span class="id" title="binder">fold</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:63" class="idref" href="#A:63"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:63"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:63"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:60"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:63"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:63"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="foldWithAbortGtLt:66" class="idref" href="#foldWithAbortGtLt:66"><span class="id" title="binder">foldWithAbortGtLt</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:65" class="idref" href="#A:65"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtLtType"><span class="id" title="definition">foldWithAbortGtLtType</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:60"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:65"><span class="id" title="variable">A</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:67" class="idref" href="#A:67"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+result type 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="i:68" class="idref" href="#i:68"><span class="id" title="binder">i</span></a> <a id="i':69" class="idref" href="#i':69"><span class="id" title="binder">i'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+base values for foldWithAbort and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:70" class="idref" href="#f:70"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a>) (<a id="f':71" class="idref" href="#f':71"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+fold functions for foldWithAbort and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f_lt:72" class="idref" href="#f_lt:72"><span class="id" title="binder">f_lt</span></a> <a id="f_gt:73" class="idref" href="#f_gt:73"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:59"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+abort functions 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:74" class="idref" href="#s:74"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:60"><span class="id" title="variable">t</span></a>) </div>
+
+<div class="doc">
+sets to fold over 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="P:75" class="idref" href="#P:75"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:67"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) </div>
+
+<div class="doc">
+equivalence relation on results </div>
+<div class="code">
+,<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> is for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> compatible with the equivalence relation P 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:76" class="idref" href="#st:76"><span class="id" title="binder">st</span></a> <a id="st':77" class="idref" href="#st':77"><span class="id" title="binder">st'</span></a> <a id="e:78" class="idref" href="#e:78"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:62"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:78"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:76"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':77"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:78"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:76"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:78"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':77"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> and <span class="inlinecode"><span class="id" title="var">f'</span></span> agree for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:79" class="idref" href="#e:79"><span class="id" title="binder">e</span></a> <a id="st:80" class="idref" href="#st:80"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:62"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:79"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:79"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:80"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':71"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:79"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:80"><span class="id" title="variable">st</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">f_lt</span></span> is OK, i.e. smaller elements can be skipped without 
+        leaving the equivalence relation. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:81" class="idref" href="#e1:81"><span class="id" title="binder">e1</span></a> <a id="st:82" class="idref" href="#st:82"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:62"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:81"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:72"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:81"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:82"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':83" class="idref" href="#st':83"><span class="id" title="binder">st'</span></a> <a id="e2:84" class="idref" href="#e2:84"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:82"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':83"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:62"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:84"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#lt:61"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:84"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:81"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:82"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:84"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':83"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">f_gt</span></span> is OK, i.e. greater elements can be skipped without 
+        leaving the equivalence relation. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:85" class="idref" href="#e1:85"><span class="id" title="binder">e1</span></a> <a id="st:86" class="idref" href="#st:86"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:62"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:85"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:73"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:85"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:86"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':87" class="idref" href="#st':87"><span class="id" title="binder">st'</span></a> <a id="e2:88" class="idref" href="#e2:88"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:86"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':87"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:62"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:88"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#lt:61"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:85"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:88"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:86"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:88"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':87"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The base values are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:68"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':69"><span class="id" title="variable">i'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The results are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:75"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtLt:66"><span class="id" title="variable">foldWithAbortGtLt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:72"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:70"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:73"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:68"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fold:64"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':71"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:74"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':69"><span class="id" title="variable">i'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We are interested in folding with abort mainly for runtime
+    performance reasons of extracted code. The argument functions
+    <span class="inlinecode"><span class="id" title="var">f_lt</span></span>, <span class="inlinecode"><span class="id" title="var">f_gt</span></span> and <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a></span> often share a large,
+    comparably expensive part of their computation.
+
+<div class="paragraph"> </div>
+
+    In order to further improve runtime performance, therefore another
+    version <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a></span> <span class="inlinecode"><span class="id" title="var">f_precompute</span></span> <span class="inlinecode"><span class="id" title="var">f_lt</span></span> <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> <span class="inlinecode"><span class="id" title="var">f_gt</span></span> that
+    uses an extra function <span class="inlinecode"><span class="id" title="var">f_precompute</span></span> to allows to compute the
+    commonly used parts of these functions only once. This leads to
+    the following definitions. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortPrecomputeType" class="idref" href="#foldWithAbortPrecomputeType"><span class="id" title="definition">foldWithAbortPrecomputeType</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="elt:89" class="idref" href="#elt:89"><span class="id" title="binder">elt</span></a> </div>
+
+<div class="doc">
+element type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="t:90" class="idref" href="#t:90"><span class="id" title="binder">t</span></a> </div>
+
+<div class="doc">
+type of set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="A:91" class="idref" href="#A:91"><span class="id" title="binder">A</span></a> </div>
+
+<div class="doc">
+return type 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a id="B:92" class="idref" href="#B:92"><span class="id" title="binder">B</span></a> </div>
+
+<div class="doc">
+type of precomputed results </div>
+<div class="code">
+ := <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:89"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:92"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>               </div>
+
+<div class="doc">
+f_precompute 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:89"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:92"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:91"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  </div>
+
+<div class="doc">
+f_lt 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:89"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:92"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:91"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:91"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>     </div>
+
+<div class="doc">
+f 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:89"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:92"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:91"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  </div>
+
+<div class="doc">
+f_gt 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:90"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                        </div>
+
+<div class="doc">
+input set 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:91"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>                        </div>
+
+<div class="doc">
+base value 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:91"><span class="id" title="variable">A</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The specification is similar to the one without precompute,
+    but uses <span class="inlinecode"><span class="id" title="var">f_precompute</span></span> so avoid doing computations multiple times 
+</div>
+<div class="code">
+<span class="id" title="keyword">Definition</span> <a id="foldWithAbortPrecomputeSpecPred" class="idref" href="#foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> {<a id="elt:93" class="idref" href="#elt:93"><span class="id" title="binder">elt</span></a> <a id="t:94" class="idref" href="#t:94"><span class="id" title="binder">t</span></a> : <span class="id" title="keyword">Type</span>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="lt:95" class="idref" href="#lt:95"><span class="id" title="binder">lt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="In:96" class="idref" href="#In:96"><span class="id" title="binder">In</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:94"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="fold:98" class="idref" href="#fold:98"><span class="id" title="binder">fold</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:97" class="idref" href="#A:97"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:97"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:97"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:94"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:97"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:97"><span class="id" title="variable">A</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="foldWithAbortPrecompute:101" class="idref" href="#foldWithAbortPrecompute:101"><span class="id" title="binder">foldWithAbortPrecompute</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:99" class="idref" href="#A:99"><span class="id" title="binder">A</span></a> <a id="B:100" class="idref" href="#B:100"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>}, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeType"><span class="id" title="definition">foldWithAbortPrecomputeType</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:94"><span class="id" title="variable">t</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:99"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:100"><span class="id" title="variable">B</span></a>) : <span class="id" title="keyword">Prop</span> := <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="A:102" class="idref" href="#A:102"><span class="id" title="binder">A</span></a> <a id="B:103" class="idref" href="#B:103"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+result type 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="i:104" class="idref" href="#i:104"><span class="id" title="binder">i</span></a> <a id="i':105" class="idref" href="#i':105"><span class="id" title="binder">i'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+base values for foldWithAbortPrecompute and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f_precompute:106" class="idref" href="#f_precompute:106"><span class="id" title="binder">f_precompute</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:103"><span class="id" title="variable">B</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+precompute function 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:107" class="idref" href="#f:107"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:103"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a>) (<a id="f':108" class="idref" href="#f':108"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+fold functions for foldWithAbortPrecompute and fold 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f_lt:109" class="idref" href="#f_lt:109"><span class="id" title="binder">f_lt</span></a> <a id="f_gt:110" class="idref" href="#f_gt:110"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#elt:93"><span class="id" title="variable">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:103"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+abort functions 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:111" class="idref" href="#s:111"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:94"><span class="id" title="variable">t</span></a>) </div>
+
+<div class="doc">
+sets to fold over 
+</div>
+<div class="code">
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="P:112" class="idref" href="#P:112"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:102"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) </div>
+
+<div class="doc">
+equivalence relation on results </div>
+<div class="code">
+,<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">P</span></span> is an equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> is for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> compatible with the equivalence relation P 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:113" class="idref" href="#st:113"><span class="id" title="binder">st</span></a> <a id="st':114" class="idref" href="#st':114"><span class="id" title="binder">st'</span></a> <a id="e:115" class="idref" href="#e:115"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:96"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:115"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:113"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':114"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:115"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:115"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:113"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:115"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:115"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':114"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a></span> and <span class="inlinecode"><span class="id" title="var">f'</span></span> agree for the elements of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:116" class="idref" href="#e:116"><span class="id" title="binder">e</span></a> <a id="st:117" class="idref" href="#st:117"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:96"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:116"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:116"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:116"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:117"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':108"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:116"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:117"><span class="id" title="variable">st</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">f_lt</span></span> is OK, i.e. smaller elements can be skipped without 
+        leaving the equivalence relation. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:118" class="idref" href="#e1:118"><span class="id" title="binder">e1</span></a> <a id="st:119" class="idref" href="#st:119"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:96"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:118"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:109"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:118"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:118"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:119"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':120" class="idref" href="#st':120"><span class="id" title="binder">st'</span></a> <a id="e2:121" class="idref" href="#e2:121"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:119"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':120"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:96"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:121"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#lt:95"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:121"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:118"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:119"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:121"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:121"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':120"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">f_gt</span></span> is OK, i.e. greater elements can be skipped without 
+        leaving the equivalence relation. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:122" class="idref" href="#e1:122"><span class="id" title="binder">e1</span></a> <a id="st:123" class="idref" href="#st:123"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:96"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:122"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:110"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:122"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:122"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:123"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':124" class="idref" href="#st':124"><span class="id" title="binder">st'</span></a> <a id="e2:125" class="idref" href="#e2:125"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:123"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':124"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#In:96"><span class="id" title="variable">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:125"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#lt:95"><span class="id" title="variable">lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:122"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:125"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:123"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:125"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:125"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':124"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The base values are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:104"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':105"><span class="id" title="variable">i'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+The results are in equivalence relation 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:112"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecompute:101"><span class="id" title="variable">foldWithAbortPrecompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_precompute:106"><span class="id" title="variable">f_precompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:109"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:107"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:110"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:104"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fold:98"><span class="id" title="variable">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':108"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:111"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':105"><span class="id" title="variable">i'</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab3"></a><h3 class="section">Module Types</h3>
+
+<div class="paragraph"> </div>
+
+ We now define a module type for <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a></span>. This module
+    type demands only the existence of the precompute version, since
+    the other ones can be easily defined via this most efficient one. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="HasFoldWithAbort" class="idref" href="#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) (<span class="id" title="keyword">Import</span> <span class="id" title="var">C</span> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="HasFoldWithAbort.foldWithAbortPrecompute" class="idref" href="#HasFoldWithAbort.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:126" class="idref" href="#A:126"><span class="id" title="binder">A</span></a> <a id="B:127" class="idref" href="#B:127"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>}, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeType"><span class="id" title="definition">foldWithAbortPrecomputeType</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:126"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:127"><span class="id" title="variable">B</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="HasFoldWithAbort.foldWithAbortPrecomputeSpec" class="idref" href="#HasFoldWithAbort.foldWithAbortPrecomputeSpec"><span class="id" title="axiom">foldWithAbortPrecomputeSpec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a>).<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab4"></a><h2 class="section">Derived operations</h2>
+
+
+<div class="paragraph"> </div>
+
+    Using these efficient fold operations, many operations can 
+    be implemented efficiently. We provide lemmata and efficient implementations
+    of useful algorithms via module <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a></span>. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="HasFoldWithAbortOps" class="idref" href="#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) (<span class="id" title="var">C</span> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="var">FT</span> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C"><span class="id" title="module">C</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT"><span class="id" title="module">FT</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C"><span class="id" title="module">C</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab5"></a><h3 class="section">First lets define the other folding with abort variants</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.foldWithAbortGtLt" class="idref" href="#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a> {<a id="A:130" class="idref" href="#A:130"><span class="id" title="binder">A</span></a>} <a id="f_lt:131" class="idref" href="#f_lt:131"><span class="id" title="binder">f_lt</span></a> (<a id="f:132" class="idref" href="#f:132"><span class="id" title="binder">f</span></a> : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:130"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:130"><span class="id" title="variable">A</span></a>)) <a id="f_gt:133" class="idref" href="#f_gt:133"><span class="id" title="binder">f_gt</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:138" class="idref" href="#e:138"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:139" class="idref" href="#st:139"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:131"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:138"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:139"><span class="id" title="variable">st</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:136" class="idref" href="#e:136"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:137" class="idref" href="#st:137"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:132"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:136"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:137"><span class="id" title="variable">st</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:134" class="idref" href="#e:134"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:135" class="idref" href="#st:135"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:133"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:134"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:135"><span class="id" title="variable">st</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortGtLtSpec" class="idref" href="#HasFoldWithAbortOps.foldWithAbortGtLtSpec"><span class="id" title="lemma">foldWithAbortGtLtSpec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtLtSpecPred"><span class="id" title="definition">foldWithAbortGtLtSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtLtSpecPred"><span class="id" title="definition">foldWithAbortGtLtSpecPred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_f_compat</span> <span class="id" title="var">H_ff'</span> <span class="id" title="var">H_lt</span> <span class="id" title="var">H_gt</span> <span class="id" title="var">H_ii'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecomputeSpec"><span class="id" title="axiom">foldWithAbortPrecomputeSpec</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.foldWithAbortGt" class="idref" href="#HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a> {<a id="A:140" class="idref" href="#A:140"><span class="id" title="binder">A</span></a>} (<a id="f:141" class="idref" href="#f:141"><span class="id" title="binder">f</span></a> : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:140"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:140"><span class="id" title="variable">A</span></a>)) <a id="f_gt:142" class="idref" href="#f_gt:142"><span class="id" title="binder">f_gt</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:145" class="idref" href="#e:145"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:146" class="idref" href="#st:146"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:141"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:145"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:146"><span class="id" title="variable">st</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:143" class="idref" href="#e:143"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:144" class="idref" href="#st:144"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:142"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:143"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:144"><span class="id" title="variable">st</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortGtSpec" class="idref" href="#HasFoldWithAbortOps.foldWithAbortGtSpec"><span class="id" title="lemma">foldWithAbortGtSpec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtSpecPred"><span class="id" title="definition">foldWithAbortGtSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtSpecPred"><span class="id" title="definition">foldWithAbortGtSpecPred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_f_compat</span> <span class="id" title="var">H_ff'</span> <span class="id" title="var">H_gt</span> <span class="id" title="var">H_ii'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecomputeSpec"><span class="id" title="axiom">foldWithAbortPrecomputeSpec</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.foldWithAbort" class="idref" href="#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a> {<a id="A:147" class="idref" href="#A:147"><span class="id" title="binder">A</span></a>} (<a id="f:148" class="idref" href="#f:148"><span class="id" title="binder">f</span></a> : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:147"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:147"><span class="id" title="variable">A</span></a>)) <a id="f_abort:149" class="idref" href="#f_abort:149"><span class="id" title="binder">f_abort</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:154" class="idref" href="#e:154"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:155" class="idref" href="#st:155"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_abort:149"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:154"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:155"><span class="id" title="variable">st</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:152" class="idref" href="#e:152"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:153" class="idref" href="#st:153"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:148"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:152"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:153"><span class="id" title="variable">st</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:150" class="idref" href="#e:150"><span class="id" title="binder">e</span></a> <span class="id" title="var">_</span> <a id="st:151" class="idref" href="#st:151"><span class="id" title="binder">st</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_abort:149"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:150"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:151"><span class="id" title="variable">st</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortSpec" class="idref" href="#HasFoldWithAbortOps.foldWithAbortSpec"><span class="id" title="lemma">foldWithAbortSpec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortSpecPred"><span class="id" title="definition">foldWithAbortSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a>) (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtSpecPred"><span class="id" title="definition">foldWithAbortGtSpecPred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_abort</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_equiv_P</span> <span class="id" title="var">H_f_compat</span> <span class="id" title="var">H_ff'</span> <span class="id" title="var">H_abort</span> <span class="id" title="var">H_ii'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_lt_neq</span>: (<span class="id" title="keyword">∀</span> <a id="e1:156" class="idref" href="#e1:156"><span class="id" title="binder">e1</span></a> <a id="e2:157" class="idref" href="#e2:157"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:156"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:157"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:156"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:157"><span class="id" title="variable">e2</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_lt</span> <span class="id" title="var">H_e12_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_e12_eq</span> <span class="id" title="tactic">in</span> <span class="id" title="var">H_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : ( <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Irreflexive"><span class="id" title="class">Irreflexive</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a>) <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#StrictOrder_Irreflexive"><span class="id" title="method">StrictOrder_Irreflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Irreflexive"><span class="id" title="class">Irreflexive</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Reflexive"><span class="id" title="class">Reflexive</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#complement"><span class="id" title="definition">complement</span></a> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H</span>, <span class="id" title="var">H_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecomputeSpec"><span class="id" title="axiom">foldWithAbortPrecomputeSpec</span></a> ⇒ //; (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_in_e1</span> <span class="id" title="var">H_abort_e1</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_P</span> <span class="id" title="var">H_in_e2</span> /<span class="id" title="var">H_lt_neq</span> <span class="id" title="var">H_lt</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<span class="id" title="var">H_abort</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_in_e1</span> <span class="id" title="var">H_abort_e1</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_P</span> <span class="id" title="var">H_in_e2</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">auto</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab6"></a><h3 class="section">Specialisations for equality</h3>
+
+<div class="paragraph"> </div>
+
+ Let's provide simplified specifications, which use equality instead
+      of an arbitrary equivalence relation on results. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal" class="idref" href="#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal"><span class="id" title="lemma">foldWithAbortPrecomputeSpec_Equal</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:158" class="idref" href="#A:158"><span class="id" title="binder">A</span></a> <a id="B:159" class="idref" href="#B:159"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:160" class="idref" href="#i:160"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:158"><span class="id" title="variable">A</span></a>) (<a id="f_pre:161" class="idref" href="#f_pre:161"><span class="id" title="binder">f_pre</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:159"><span class="id" title="variable">B</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:162" class="idref" href="#f:162"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:159"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:158"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:158"><span class="id" title="variable">A</span></a>) (<a id="f':163" class="idref" href="#f':163"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:158"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:158"><span class="id" title="variable">A</span></a>) (<a id="f_lt:164" class="idref" href="#f_lt:164"><span class="id" title="binder">f_lt</span></a> <a id="f_gt:165" class="idref" href="#f_gt:165"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:159"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:158"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:166" class="idref" href="#s:166"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:167" class="idref" href="#e:167"><span class="id" title="binder">e</span></a> <a id="st:168" class="idref" href="#st:168"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:167"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:162"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:167"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:161"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:167"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:168"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':163"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:167"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:168"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:169" class="idref" href="#e1:169"><span class="id" title="binder">e1</span></a> <a id="st:170" class="idref" href="#st:170"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:169"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:164"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:169"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:161"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:169"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:170"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:171" class="idref" href="#e2:171"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:171"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:171"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:169"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:162"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:171"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:161"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:171"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:170"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:170"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:172" class="idref" href="#e1:172"><span class="id" title="binder">e1</span></a> <a id="st:173" class="idref" href="#st:173"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:172"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:165"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:172"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:161"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:172"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:173"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:174" class="idref" href="#e2:174"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:174"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:172"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:174"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:162"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:174"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:161"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:174"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:173"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:173"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:161"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:164"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:162"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:165"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:160"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':163"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:166"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:160"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_lt</span> <span class="id" title="var">H_gt</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecomputeSpec"><span class="id" title="axiom">foldWithAbortPrecomputeSpec</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">i</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span>); <span class="id" title="tactic">eauto</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#eq_equivalence"><span class="id" title="instance">eq_equivalence</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">st1</span> <span class="id" title="var">st2</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span> → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_smaller</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_lt</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_smaller</span> <span class="id" title="var">e2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_greater</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_gt</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_greater</span> <span class="id" title="var">e2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal" class="idref" href="#HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal"><span class="id" title="lemma">foldWithAbortGtLtSpec_Equal</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:175" class="idref" href="#A:175"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:176" class="idref" href="#i:176"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:175"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:177" class="idref" href="#f:177"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:175"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:175"><span class="id" title="variable">A</span></a>) (<a id="f':178" class="idref" href="#f':178"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:175"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:175"><span class="id" title="variable">A</span></a>) (<a id="f_lt:179" class="idref" href="#f_lt:179"><span class="id" title="binder">f_lt</span></a> <a id="f_gt:180" class="idref" href="#f_gt:180"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:175"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:181" class="idref" href="#s:181"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:182" class="idref" href="#e:182"><span class="id" title="binder">e</span></a> <a id="st:183" class="idref" href="#st:183"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:182"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:177"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:182"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:183"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':178"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:182"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:183"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:184" class="idref" href="#e1:184"><span class="id" title="binder">e1</span></a> <a id="st:185" class="idref" href="#st:185"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:184"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:179"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:184"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:185"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:186" class="idref" href="#e2:186"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:186"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:186"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:184"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:177"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:186"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:185"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:185"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:187" class="idref" href="#e1:187"><span class="id" title="binder">e1</span></a> <a id="st:188" class="idref" href="#st:188"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:187"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:180"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:187"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:188"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:189" class="idref" href="#e2:189"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:189"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:187"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:189"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:177"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:189"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:188"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:188"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt"><span class="id" title="definition">foldWithAbortGtLt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:179"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:177"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:180"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:176"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':178"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:181"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:176"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_lt</span> <span class="id" title="var">H_gt</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec"><span class="id" title="lemma">foldWithAbortGtLtSpec</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span>); <span class="id" title="tactic">eauto</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#eq_equivalence"><span class="id" title="instance">eq_equivalence</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">st1</span> <span class="id" title="var">st2</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span> → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_smaller</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_lt</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_smaller</span> <span class="id" title="var">e2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_greater</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_gt</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_greater</span> <span class="id" title="var">e2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortGtSpec_Equal" class="idref" href="#HasFoldWithAbortOps.foldWithAbortGtSpec_Equal"><span class="id" title="lemma">foldWithAbortGtSpec_Equal</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:190" class="idref" href="#A:190"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:191" class="idref" href="#i:191"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:190"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:192" class="idref" href="#f:192"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:190"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:190"><span class="id" title="variable">A</span></a>) (<a id="f':193" class="idref" href="#f':193"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:190"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:190"><span class="id" title="variable">A</span></a>) (<a id="f_gt:194" class="idref" href="#f_gt:194"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:190"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:195" class="idref" href="#s:195"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:196" class="idref" href="#e:196"><span class="id" title="binder">e</span></a> <a id="st:197" class="idref" href="#st:197"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:196"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:192"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:196"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:197"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':193"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:196"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:197"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:198" class="idref" href="#e1:198"><span class="id" title="binder">e1</span></a> <a id="st:199" class="idref" href="#st:199"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:198"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:194"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:198"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:199"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:200" class="idref" href="#e2:200"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:200"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:198"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:200"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:192"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:200"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:199"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:199"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt"><span class="id" title="definition">foldWithAbortGt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:192"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:194"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:191"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':193"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:195"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:191"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_gt</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec"><span class="id" title="lemma">foldWithAbortGtSpec</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span>); <span class="id" title="tactic">eauto</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#eq_equivalence"><span class="id" title="instance">eq_equivalence</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">st1</span> <span class="id" title="var">st2</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span> → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_greater</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_gt</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_greater</span> <span class="id" title="var">e2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.foldWithAbortSpec_Equal" class="idref" href="#HasFoldWithAbortOps.foldWithAbortSpec_Equal"><span class="id" title="lemma">foldWithAbortSpec_Equal</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:201" class="idref" href="#A:201"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:202" class="idref" href="#i:202"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:201"><span class="id" title="variable">A</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:203" class="idref" href="#f:203"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:201"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:201"><span class="id" title="variable">A</span></a>) (<a id="f':204" class="idref" href="#f':204"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:201"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:201"><span class="id" title="variable">A</span></a>) (<a id="f_abort:205" class="idref" href="#f_abort:205"><span class="id" title="binder">f_abort</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:201"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:206" class="idref" href="#s:206"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>),<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:207" class="idref" href="#e:207"><span class="id" title="binder">e</span></a> <a id="st:208" class="idref" href="#st:208"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:207"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:206"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:203"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:207"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:208"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':204"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:207"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:208"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:209" class="idref" href="#e1:209"><span class="id" title="binder">e1</span></a> <a id="st:210" class="idref" href="#st:210"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:209"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:206"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_abort:205"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:209"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:210"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:211" class="idref" href="#e2:211"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:211"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:206"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:209"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:211"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:203"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:211"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:210"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:210"><span class="id" title="variable">st</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort"><span class="id" title="definition">foldWithAbort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:203"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_abort:205"><span class="id" title="variable">f_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:206"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:202"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':204"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:206"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:202"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_abort</span> <span class="id" title="var">s</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_abort</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec"><span class="id" title="lemma">foldWithAbortSpec</span></a> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span>); <span class="id" title="tactic">eauto</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#eq_equivalence"><span class="id" title="instance">eq_equivalence</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">st1</span> <span class="id" title="var">st2</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span> → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_abort</span> <span class="id" title="var">st'</span> <span class="id" title="var">e2</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_abort</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_do_abort</span> <span class="id" title="var">e2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab7"></a><h3 class="section">FoldWithAbortSpecArgs</h3>
+
+<div class="paragraph"> </div>
+
+ While folding, we are often interested in skipping elements that do not
+      satisfy a certain property <span class="inlinecode"><span class="id" title="var">P</span></span>. This needs expressing in terms of
+      skips of smaller of larger elements in order to be done efficiently by
+      our folding functions. Formally, this leads to the definition
+      of <span class="inlinecode"><span class="id" title="var">foldWithAbortSpecForPred</span></span>. 
+
+<div class="paragraph"> </div>
+
+      Given a <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a></span> for a predicate <span class="inlinecode"><span class="id" title="var">P</span></span> and a
+      set <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span>, many operations can be implemented efficiently. Below we will provide
+      efficient versions of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.filter"><span class="id" title="definition">filter</span></a></span>, <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.choose"><span class="id" title="definition">choose</span></a></span>, <span class="inlinecode"><span class="id" title="tactic">∃</span></span>, <span class="inlinecode"><span class="id" title="keyword">∀</span></span> and more.
+    
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Record</span> <a id="HasFoldWithAbortOps.FoldWithAbortSpecArg" class="idref" href="#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> {<a id="B:212" class="idref" href="#B:212"><span class="id" title="binder">B</span></a>} := {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="HasFoldWithAbortOps.fwasa_f_pre" class="idref" href="#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:212"><span class="id" title="variable">B</span></a>);         </div>
+
+<div class="doc">
+The precompute function 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="HasFoldWithAbortOps.fwasa_f_lt" class="idref" href="#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a>  : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:212"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>); </div>
+
+<div class="doc">
+f_lt without state argument 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="HasFoldWithAbortOps.fwasa_f_gt" class="idref" href="#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a>  : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:212"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>); </div>
+
+<div class="doc">
+f_gt without state argument 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="HasFoldWithAbortOps.fwasa_P'" class="idref" href="#HasFoldWithAbortOps.fwasa_P'"><span class="id" title="projection">fwasa_P'</span></a>    : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:212"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)  </div>
+
+<div class="doc">
+the predicate P 
+</div>
+<div class="code">
+&nbsp;&nbsp;}.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><span class="id" title="var">foldWithAbortSpecForPred</span></span> <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">fwasa</span></span> holds, if
+      the argument <span class="inlinecode"><span class="id" title="var">fwasa</span></span> fits the predicate <span class="inlinecode"><span class="id" title="var">P</span></span> for set <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span>. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.foldWithAbortSpecArgsForPred" class="idref" href="#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> {<a id="A:218" class="idref" href="#A:218"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="s:219" class="idref" href="#s:219"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>) (<a id="P:220" class="idref" href="#P:220"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="fwasa:221" class="idref" href="#fwasa:221"><span class="id" title="binder">fwasa</span></a> : @<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:218"><span class="id" title="variable">A</span></a>) :=<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+the predicate <span class="inlinecode"><span class="id" title="var">P'</span></span> coincides for <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a></span> and the given precomputation with <span class="inlinecode"><span class="id" title="var">P</span></span> 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:222" class="idref" href="#e:222"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:222"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P'"><span class="id" title="projection">fwasa_P'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:221"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:222"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:221"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:222"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:220"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:222"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+If <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a></span> holds, all elements smaller than the current one
+         don't satisfy predicate P. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:223" class="idref" href="#e1:223"><span class="id" title="binder">e1</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:223"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:221"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:223"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:221"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:223"><span class="id" title="variable">e1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:224" class="idref" href="#e2:224"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:224"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:224"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:223"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:220"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:224"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</div>
+
+<div class="doc">
+If <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a></span> holds, all elements greater than the current one
+         don't satisfy predicate P. 
+</div>
+<div class="code">
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:225" class="idref" href="#e1:225"><span class="id" title="binder">e1</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:225"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:221"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:225"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:221"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:225"><span class="id" title="variable">e1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e2:226" class="idref" href="#e2:226"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:226"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:219"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:225"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:226"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:220"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:226"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab8"></a><h3 class="section">Filter with abort</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.filter_with_abort" class="idref" href="#HasFoldWithAbortOps.filter_with_abort"><span class="id" title="definition">filter_with_abort</span></a> {<a id="B:227" class="idref" href="#B:227"><span class="id" title="binder">B</span></a>} (<a id="fwasa:228" class="idref" href="#fwasa:228"><span class="id" title="binder">fwasa</span></a> : @<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:227"><span class="id" title="variable">B</span></a>) <a id="s:229" class="idref" href="#s:229"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:227"><span class="id" title="variable">B</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:228"><span class="id" title="variable">fwasa</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:230" class="idref" href="#e:230"><span class="id" title="binder">e</span></a> <a id="p:231" class="idref" href="#p:231"><span class="id" title="binder">p</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:228"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:230"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#p:231"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:232" class="idref" href="#e:232"><span class="id" title="binder">e</span></a> <a id="e_pre:233" class="idref" href="#e_pre:233"><span class="id" title="binder">e_pre</span></a> <a id="s:234" class="idref" href="#s:234"><span class="id" title="binder">s</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P'"><span class="id" title="projection">fwasa_P'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:228"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:232"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e_pre:233"><span class="id" title="variable">e_pre</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:232"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:234"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:234"><span class="id" title="variable">s</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:235" class="idref" href="#e:235"><span class="id" title="binder">e</span></a> <a id="p:236" class="idref" href="#p:236"><span class="id" title="binder">p</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:228"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:235"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#p:236"><span class="id" title="variable">p</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:229"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.empty"><span class="id" title="axiom">empty</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.filter_with_abort_spec" class="idref" href="#HasFoldWithAbortOps.filter_with_abort_spec"><span class="id" title="lemma">filter_with_abort_spec</span></a> {<a id="B:237" class="idref" href="#B:237"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> <a id="fwasa:238" class="idref" href="#fwasa:238"><span class="id" title="binder">fwasa</span></a> <a id="P:239" class="idref" href="#P:239"><span class="id" title="binder">P</span></a> <a id="s:240" class="idref" href="#s:240"><span class="id" title="binder">s</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:237"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:240"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:239"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:238"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:239"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort"><span class="id" title="definition">filter_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:238"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:240"><span class="id" title="variable">s</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.filter"><span class="id" title="axiom">filter</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:239"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:240"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">P'</span> <span class="id" title="var">P</span> <span class="id" title="var">s</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_f'</span>] [<span class="id" title="var">H_lt</span>] <span class="id" title="var">H_gt</span> <span class="id" title="var">H_proper</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort"><span class="id" title="definition">filter_with_abort</span></a> /=.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> <span class="id" title="var">f_pre</span> (<span class="id" title="keyword">fun</span> <a id="e:246" class="idref" href="#e:246"><span class="id" title="binder">e</span></a> <a id="p:247" class="idref" href="#p:247"><span class="id" title="binder">p</span></a> <span class="id" title="var">_</span> ⇒ <span class="id" title="var">f_lt</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:246"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#p:247"><span class="id" title="variable">p</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="e:243" class="idref" href="#e:243"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a>) (<a id="e_pre:244" class="idref" href="#e_pre:244"><span class="id" title="binder">e_pre</span></a> : <span class="id" title="var">B</span>) (<a id="s0:245" class="idref" href="#s0:245"><span class="id" title="binder">s0</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">P'</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:243"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e_pre:244"><span class="id" title="variable">e_pre</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:243"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s0:245"><span class="id" title="variable">s0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s0:245"><span class="id" title="variable">s0</span></a>) (<span class="id" title="keyword">fun</span> <a id="e:241" class="idref" href="#e:241"><span class="id" title="binder">e</span></a> <a id="p:242" class="idref" href="#p:242"><span class="id" title="binder">p</span></a> <span class="id" title="var">_</span> ⇒ <span class="id" title="var">f_gt</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:241"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#p:242"><span class="id" title="variable">p</span></a>) <span class="id" title="var">s</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.empty"><span class="id" title="axiom">empty</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> (<span class="id" title="keyword">fun</span> <a id="e:248" class="idref" href="#e:248"><span class="id" title="binder">e</span></a> <a id="s0:249" class="idref" href="#s0:249"><span class="id" title="binder">s0</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">P</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:248"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:248"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s0:249"><span class="id" title="variable">s0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s0:249"><span class="id" title="variable">s0</span></a>) <span class="id" title="var">s</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.empty"><span class="id" title="axiom">empty</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal"><span class="id" title="lemma">foldWithAbortPrecomputeSpec_Equal</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_f_lt_eq</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_e2_in</span> <span class="id" title="var">H_lt_e2_e1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">H_f'</span> <span class="id" title="var">_</span> <span class="id" title="var">H_e2_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : (<span class="id" title="var">P</span> <span class="id" title="var">e2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<span class="id" title="var">H_lt</span> <span class="id" title="var">e1</span>); <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e1</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_f_gt_eq</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_e2_in</span> <span class="id" title="var">H_gt_e2_e1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">H_f'</span> <span class="id" title="var">_</span> <span class="id" title="var">H_e2_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : (<span class="id" title="var">P</span> <span class="id" title="var">e2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<span class="id" title="var">H_gt</span> <span class="id" title="var">e1</span>); <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.Equal"><span class="id" title="definition">Equal</span></a> ⇒ <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold_spec"><span class="id" title="axiom">fold_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.filter_spec"><span class="id" title="axiom">filter_spec</span></a> ⇒ //.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <span class="id" title="keyword">∀</span> <a id="acc:250" class="idref" href="#acc:250"><span class="id" title="binder">acc</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <span class="id" title="var">e</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> (<span class="id" title="keyword">fun</span> (<a id="e0:251" class="idref" href="#e0:251"><span class="id" title="binder">e0</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a>) (<a id="s0:252" class="idref" href="#s0:252"><span class="id" title="binder">s0</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.t"><span class="id" title="axiom">t</span></a>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">P</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:251"><span class="id" title="variable">e0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:251"><span class="id" title="variable">e0</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s0:252"><span class="id" title="variable">s0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s0:252"><span class="id" title="variable">s0</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements"><span class="id" title="axiom">elements</span></a> <span class="id" title="var">s</span>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:250"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <span class="id" title="var">e</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements"><span class="id" title="axiom">elements</span></a> <span class="id" title="var">s</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P</span> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <span class="id" title="var">e</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:250"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements_spec1"><span class="id" title="axiom">elements_spec1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <span class="id" title="var">e</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.empty"><span class="id" title="axiom">empty</span></a>) <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.empty_spec"><span class="id" title="axiom">empty_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements"><span class="id" title="axiom">elements</span></a> <span class="id" title="var">s</span>) <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">P</span> <span class="id" title="var">x</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.add_spec"><span class="id" title="axiom">add_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab9"></a><h3 class="section">Choose with abort</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.choose_with_abort" class="idref" href="#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> {<a id="B:253" class="idref" href="#B:253"><span class="id" title="binder">B</span></a>} (<a id="fwasa:254" class="idref" href="#fwasa:254"><span class="id" title="binder">fwasa</span></a> : @<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:253"><span class="id" title="variable">B</span></a>) <a id="s:255" class="idref" href="#s:255"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#FT.foldWithAbortPrecompute"><span class="id" title="axiom">foldWithAbortPrecompute</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:254"><span class="id" title="variable">fwasa</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:264" class="idref" href="#e:264"><span class="id" title="binder">e</span></a> <a id="p:265" class="idref" href="#p:265"><span class="id" title="binder">p</span></a> <a id="st:266" class="idref" href="#st:266"><span class="id" title="binder">st</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:266"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:254"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:264"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#p:265"><span class="id" title="variable">p</span></a>) | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:260" class="idref" href="#e:260"><span class="id" title="binder">e</span></a> <a id="e_pre:261" class="idref" href="#e_pre:261"><span class="id" title="binder">e_pre</span></a> <a id="st:262" class="idref" href="#st:262"><span class="id" title="binder">st</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:262"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P'"><span class="id" title="projection">fwasa_P'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:254"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:260"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e_pre:261"><span class="id" title="variable">e_pre</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:260"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:262"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">end</span>) <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> <a id="e:256" class="idref" href="#e:256"><span class="id" title="binder">e</span></a> <a id="p:257" class="idref" href="#p:257"><span class="id" title="binder">p</span></a> <a id="st:258" class="idref" href="#st:258"><span class="id" title="binder">st</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:258"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:254"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:256"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#p:257"><span class="id" title="variable">p</span></a>) | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">end</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:255"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.choose_with_abort_spec" class="idref" href="#HasFoldWithAbortOps.choose_with_abort_spec"><span class="id" title="lemma">choose_with_abort_spec</span></a> {<a id="B:268" class="idref" href="#B:268"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> <a id="fwasa:269" class="idref" href="#fwasa:269"><span class="id" title="binder">fwasa</span></a> <a id="P:270" class="idref" href="#P:270"><span class="id" title="binder">P</span></a> <a id="s:271" class="idref" href="#s:271"><span class="id" title="binder">s</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:268"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:271"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:270"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:269"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:270"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:269"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:271"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ (<span class="id" title="keyword">∀</span> <a id="e:272" class="idref" href="#e:272"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:272"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:271"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:270"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:272"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">e</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.In"><span class="id" title="axiom">In</span></a> <span class="id" title="var">e</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:271"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:270"><span class="id" title="variable">P</span></a> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">P'</span> <span class="id" title="var">P</span> <span class="id" title="var">s</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_f'</span>] [<span class="id" title="var">H_lt</span>] <span class="id" title="var">H_gt</span> <span class="id" title="var">H_proper</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">fwasa</span> := {|<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre"><span class="id" title="projection">fwasa_f_pre</span></a> := <span class="id" title="var">f_pre</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt"><span class="id" title="projection">fwasa_f_lt</span></a> := <span class="id" title="var">f_lt</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt"><span class="id" title="projection">fwasa_f_gt</span></a> := <span class="id" title="var">f_gt</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P'"><span class="id" title="projection">fwasa_P'</span></a> := <span class="id" title="var">P'</span> |}.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ (<span class="id" title="keyword">∀</span> <a id="e:273" class="idref" href="#e:273"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:273"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements"><span class="id" title="axiom">elements</span></a> <span class="id" title="var">s</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">P</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:273"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">e</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <span class="id" title="var">e</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements"><span class="id" title="axiom">elements</span></a> <span class="id" title="var">s</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="var">P</span> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements_spec1"><span class="id" title="axiom">elements_spec1</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements_spec1"><span class="id" title="axiom">elements_spec1</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold"><span class="id" title="axiom">fold</span></a> (<span class="id" title="keyword">fun</span> <a id="e:274" class="idref" href="#e:274"><span class="id" title="binder">e</span></a> <a id="st:275" class="idref" href="#st:275"><span class="id" title="binder">st</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:275"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">P</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:274"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:274"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:275"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">end</span>) <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal"><span class="id" title="lemma">foldWithAbortPrecomputeSpec_Equal</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">st</span> <span class="id" title="var">H_e_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">st</span> ⇒ //=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> [] //= <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_f_lt_eq</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_e2_in</span> <span class="id" title="var">H_lt_e2_e1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">H_f'</span> <span class="id" title="var">_</span> <span class="id" title="var">H_e2_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">P</span> <span class="id" title="var">e2</span>) ⇒ // <span class="id" title="var">H_P_e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_P_e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#not_true_iff_false"><span class="id" title="lemma">not_true_iff_false</span></a>, (<span class="id" title="var">H_lt</span> <span class="id" title="var">e1</span>); <span class="id" title="tactic">auto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e1</span> [] //= <span class="id" title="var">H_e1_in</span> <span class="id" title="var">H_f_gt_eq</span> <span class="id" title="var">e2</span> <span class="id" title="var">H_e2_in</span> <span class="id" title="var">H_gt_e2_e1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">H_f'</span> <span class="id" title="var">_</span> <span class="id" title="var">H_e2_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">P</span> <span class="id" title="var">e2</span>) ⇒ // <span class="id" title="var">H_P_e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_P_e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#not_true_iff_false"><span class="id" title="lemma">not_true_iff_false</span></a>, (<span class="id" title="var">H_gt</span> <span class="id" title="var">e1</span>); <span class="id" title="tactic">auto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.fold_spec"><span class="id" title="axiom">fold_spec</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elements"><span class="id" title="axiom">elements</span></a> <span class="id" title="var">s</span>) <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">P</span> <span class="id" title="var">x</span>) ⇒ <span class="id" title="var">H_Px</span>; <span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">H_Px</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : <span class="id" title="keyword">∀</span> <a id="xs:277" class="idref" href="#xs:277"><span class="id" title="binder">xs</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<span class="id" title="keyword">fun</span> (<a id="x0:278" class="idref" href="#x0:278"><span class="id" title="binder">x0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a>) (<a id="y:279" class="idref" href="#y:279"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x0:278"><span class="id" title="variable">x0</span></a> <span class="id" title="keyword">with</span> | <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x0:278"><span class="id" title="variable">x0</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">P</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#y:279"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#y:279"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#xs:277"><span class="id" title="variable">xs</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">ys</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">ys</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="x0:284" class="idref" href="#x0:284"><span class="id" title="binder">x0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a>) (<a id="y:285" class="idref" href="#y:285"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.elt"><span class="id" title="definition">elt</span></a>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x0:281"><span class="id" title="variable">x0</span></a> <span class="id" title="keyword">with</span> | <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x0:281"><span class="id" title="variable">x0</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">P</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#y:282"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#y:282"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>). {<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e</span> [<span class="id" title="var">H_e_in</span>] <span class="id" title="var">H_Pe</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_e_nin</span> <span class="id" title="var">e</span> <span class="id" title="var">H_e_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <span class="id" title="var">e</span> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <span class="id" title="var">e</span> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">inversion</span> <span class="id" title="var">H_e_in</span>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_e_nin</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab10"></a><h3 class="section">Exists and Forall with abort</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.exists_with_abort" class="idref" href="#HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a> {<a id="B:287" class="idref" href="#B:287"><span class="id" title="binder">B</span></a>} (<a id="fwasa:288" class="idref" href="#fwasa:288"><span class="id" title="binder">fwasa</span></a> : @<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg"><span class="id" title="record">FoldWithAbortSpecArg</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:287"><span class="id" title="variable">B</span></a>) <a id="s:289" class="idref" href="#s:289"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:288"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:289"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.exists_with_abort_spec" class="idref" href="#HasFoldWithAbortOps.exists_with_abort_spec"><span class="id" title="lemma">exists_with_abort_spec</span></a> {<a id="B:290" class="idref" href="#B:290"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> <a id="fwasa:291" class="idref" href="#fwasa:291"><span class="id" title="binder">fwasa</span></a> <a id="P:292" class="idref" href="#P:292"><span class="id" title="binder">P</span></a> <a id="s:293" class="idref" href="#s:293"><span class="id" title="binder">s</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:290"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:293"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:292"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:291"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:292"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:291"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:293"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.exists_"><span class="id" title="axiom">exists_</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:292"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:293"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">fwasa</span> <span class="id" title="var">P</span> <span class="id" title="var">s</span> <span class="id" title="var">H_fwasa</span> <span class="id" title="var">H_proper</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq_sym"><span class="id" title="lemma">Logic.eq_sym</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort_spec"><span class="id" title="lemma">choose_with_abort_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_fwasa</span> <span class="id" title="var">H_proper</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort"><span class="id" title="definition">choose_with_abort</span></a> <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">e</span> [<span class="id" title="var">H_e_in</span>] <span class="id" title="var">H_Pe</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.exists_spec"><span class="id" title="axiom">exists_spec</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.Exists"><span class="id" title="definition">Exists</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_not_ex</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#not_true_iff_false"><span class="id" title="lemma">not_true_iff_false</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.exists_spec"><span class="id" title="axiom">exists_spec</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.Exists"><span class="id" title="definition">Exists</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">e</span>] [<span class="id" title="var">H_in</span>] <span class="id" title="var">H_pe</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_not_ex</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_pe</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Negation leads to forall. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="HasFoldWithAbortOps.forall_with_abort" class="idref" href="#HasFoldWithAbortOps.forall_with_abort"><span class="id" title="definition">forall_with_abort</span></a> {<a id="B:294" class="idref" href="#B:294"><span class="id" title="binder">B</span></a>} <a id="fwasa:295" class="idref" href="#fwasa:295"><span class="id" title="binder">fwasa</span></a> <a id="s:296" class="idref" href="#s:296"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort"><span class="id" title="definition">exists_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:294"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:295"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:296"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="HasFoldWithAbortOps.forall_with_abort_spec" class="idref" href="#HasFoldWithAbortOps.forall_with_abort_spec"><span class="id" title="lemma">forall_with_abort_spec</span></a> {<a id="B:297" class="idref" href="#B:297"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> <a id="fwasa:298" class="idref" href="#fwasa:298"><span class="id" title="binder">fwasa</span></a> <a id="s:299" class="idref" href="#s:299"><span class="id" title="binder">s</span></a> <a id="P:300" class="idref" href="#P:300"><span class="id" title="binder">P</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred"><span class="id" title="definition">foldWithAbortSpecArgsForPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:297"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:299"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:300"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:298"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:300"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort"><span class="id" title="definition">forall_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#fwasa:298"><span class="id" title="variable">fwasa</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:299"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.for_all"><span class="id" title="axiom">for_all</span></a> (<span class="id" title="keyword">fun</span> <a id="e:301" class="idref" href="#e:301"><span class="id" title="binder">e</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:300"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:301"><span class="id" title="variable">e</span></a>)) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:299"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">fwasa</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span> <span class="id" title="var">H_ok</span> <span class="id" title="var">H_proper</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort"><span class="id" title="definition">forall_with_abort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort_spec"><span class="id" title="lemma">exists_with_abort_spec</span></a>; <span class="id" title="tactic">auto</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#eq_iff_eq_true"><span class="id" title="lemma">eq_iff_eq_true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#negb_true_iff"><span class="id" title="lemma">negb_true_iff</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#not_true_iff_false"><span class="id" title="lemma">not_true_iff_false</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.exists_spec"><span class="id" title="axiom">exists_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.for_all_spec"><span class="id" title="axiom">for_all_spec</span></a>; <span class="id" title="var">last</span> <span class="id" title="var">solve_proper</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.Exists"><span class="id" title="definition">Exists</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#C.For_all"><span class="id" title="definition">For_all</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">x</span> <span class="id" title="var">H_x_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#negb_true_iff"><span class="id" title="lemma">negb_true_iff</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#not_true_iff_false"><span class="id" title="lemma">not_true_iff_false</span></a> ⇒ <span class="id" title="var">H_Px</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_pre</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> [<span class="id" title="var">x</span>] [<span class="id" title="var">H_x_in</span>] <span class="id" title="var">H_P_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_pre</span> <span class="id" title="var">x</span> <span class="id" title="var">H_x_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_P_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab11"></a><h2 class="section">Modules Types For Sets with Fold with Abort</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="WSetsWithDupsFoldA" class="idref" href="#WSetsWithDupsFoldA"><span class="id" title="module">WSetsWithDupsFoldA</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="WSetsWithDupsFoldA.E" class="idref" href="#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithDupsFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithDupsFoldA"><span class="id" title="module">WSetsWithDupsFoldA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="WSetsWithFoldA" class="idref" href="#WSetsWithFoldA"><span class="id" title="module">WSetsWithFoldA</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#WSets"><span class="id" title="module">WSets</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="WSetsWithFoldA.E" class="idref" href="#WSetsWithFoldA.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA"><span class="id" title="module">WSetsWithFoldA</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="SetsWithFoldA" class="idref" href="#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#Sets"><span class="id" title="module">Sets</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="SetsWithFoldA.E" class="idref" href="#SetsWithFoldA.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#SetsOn"><span class="id" title="module">SetsOn</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort"><span class="id" title="module">HasFoldWithAbort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA.E"><span class="id" title="module">E</span></a>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab12"></a><h2 class="section">Implementations</h2>
+
+<div class="paragraph"> </div>
+
+<a id="lab13"></a><h3 class="section">GenTree implementation</h3>
+
+    Finally, provide such a fold with abort operation for generic trees. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="MakeGenTreeFoldA" class="idref" href="#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a> (<span class="id" title="keyword">Import</span> <span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) (<span class="id" title="keyword">Import</span> <span class="id" title="var">I</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetGenTree.html#InfoTyp"><span class="id" title="module">InfoTyp</span></a>) <br/>
+&nbsp;&nbsp;(<span class="id" title="keyword">Import</span> <span class="id" title="var">Raw</span>: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetGenTree.html#Ops"><span class="id" title="module">MSetGenTree.Ops</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#I"><span class="id" title="module">I</span></a>)<br/>
+&nbsp;&nbsp;(<span class="id" title="var">M</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetGenTree.html#Props"><span class="id" title="module">MSetGenTree.Props</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#I"><span class="id" title="module">I</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw"><span class="id" title="module">Raw</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="MakeGenTreeFoldA.foldWithAbort_Raw" class="idref" href="#MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> {<a id="A:302" class="idref" href="#A:302"><span class="id" title="binder">A</span></a> <a id="B:303" class="idref" href="#B:303"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="f_pre:304" class="idref" href="#f_pre:304"><span class="id" title="binder">f_pre</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:303"><span class="id" title="variable">B</span></a>) <a id="f_lt:305" class="idref" href="#f_lt:305"><span class="id" title="binder">f_lt</span></a> (<a id="f:306" class="idref" href="#f:306"><span class="id" title="binder">f</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:303"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:302"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:302"><span class="id" title="variable">A</span></a>) <a id="f_gt:307" class="idref" href="#f_gt:307"><span class="id" title="binder">f_gt</span></a> <a id="t:308" class="idref" href="#t:308"><span class="id" title="binder">t</span></a> (<a id="base:309" class="idref" href="#base:309"><span class="id" title="binder">base</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:302"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:302"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:308"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.Leaf"><span class="id" title="constructor">Raw.Leaf</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#base:309"><span class="id" title="variable">base</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.Node"><span class="id" title="constructor">Raw.Node</span></a> <span class="id" title="var">_</span> <span class="id" title="var">l</span> <span class="id" title="var">x</span> <span class="id" title="var">r</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="x_pre:312" class="idref" href="#x_pre:312"><span class="id" title="binder">x_pre</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:304"><span class="id" title="variable">f_pre</span></a> <span class="id" title="var">x</span> <span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="st0:313" class="idref" href="#st0:313"><span class="id" title="binder">st0</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:305"><span class="id" title="variable">f_lt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x_pre:312"><span class="id" title="variable">x_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#base:309"><span class="id" title="variable">base</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#base:309"><span class="id" title="variable">base</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbort_Raw:310"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:304"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:305"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:306"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:307"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">l</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#base:309"><span class="id" title="variable">base</span></a> <span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="st1:314" class="idref" href="#st1:314"><span class="id" title="binder">st1</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:306"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x_pre:312"><span class="id" title="variable">x_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st0:313"><span class="id" title="variable">st0</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="st2:315" class="idref" href="#st2:315"><span class="id" title="binder">st2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:307"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#x_pre:312"><span class="id" title="variable">x_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st1:314"><span class="id" title="variable">st1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st1:314"><span class="id" title="variable">st1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbort_Raw:310"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:304"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:305"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:306"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:307"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">r</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st1:314"><span class="id" title="variable">st1</span></a> <span class="id" title="tactic">in</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st2:315"><span class="id" title="variable">st2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MakeGenTreeFoldA.foldWithAbort_RawSpec" class="idref" href="#MakeGenTreeFoldA.foldWithAbort_RawSpec"><span class="id" title="lemma">foldWithAbort_RawSpec</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:316" class="idref" href="#A:316"><span class="id" title="binder">A</span></a> <a id="B:317" class="idref" href="#B:317"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:318" class="idref" href="#i:318"><span class="id" title="binder">i</span></a> <a id="i':319" class="idref" href="#i':319"><span class="id" title="binder">i'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a>) (<a id="f_pre:320" class="idref" href="#f_pre:320"><span class="id" title="binder">f_pre</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:317"><span class="id" title="variable">B</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:321" class="idref" href="#f:321"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:317"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a>) (<a id="f':322" class="idref" href="#f':322"><span class="id" title="binder">f'</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a>) (<a id="f_lt:323" class="idref" href="#f_lt:323"><span class="id" title="binder">f_lt</span></a> <a id="f_gt:324" class="idref" href="#f_gt:324"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:317"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:325" class="idref" href="#s:325"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.tree"><span class="id" title="inductive">Raw.tree</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="P:326" class="idref" href="#P:326"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:316"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.bst"><span class="id" title="inductive">M.bst</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st:327" class="idref" href="#st:327"><span class="id" title="binder">st</span></a> <a id="st':328" class="idref" href="#st':328"><span class="id" title="binder">st'</span></a> <a id="e:329" class="idref" href="#e:329"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:329"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:327"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':328"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:329"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:329"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:327"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:329"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:329"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':328"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e:330" class="idref" href="#e:330"><span class="id" title="binder">e</span></a> <a id="st:331" class="idref" href="#st:331"><span class="id" title="binder">st</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:330"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:330"><span class="id" title="variable">e</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:330"><span class="id" title="variable">e</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:331"><span class="id" title="variable">st</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':322"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:330"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:331"><span class="id" title="variable">st</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>       <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:332" class="idref" href="#e1:332"><span class="id" title="binder">e1</span></a> <a id="st:333" class="idref" href="#st:333"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:332"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:323"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:332"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:332"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:333"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':334" class="idref" href="#st':334"><span class="id" title="binder">st'</span></a> <a id="e2:335" class="idref" href="#e2:335"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:333"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':334"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:335"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:335"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:332"><span class="id" title="variable">e1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:333"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:335"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:335"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':334"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="e1:336" class="idref" href="#e1:336"><span class="id" title="binder">e1</span></a> <a id="st:337" class="idref" href="#st:337"><span class="id" title="binder">st</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:336"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:324"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:336"><span class="id" title="variable">e1</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:336"><span class="id" title="variable">e1</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:337"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="st':338" class="idref" href="#st':338"><span class="id" title="binder">st'</span></a> <a id="e2:339" class="idref" href="#e2:339"><span class="id" title="binder">e2</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:337"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':338"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:339"><span class="id" title="variable">e2</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e1:336"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:339"><span class="id" title="variable">e2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:337"><span class="id" title="variable">st</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:339"><span class="id" title="variable">e2</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e2:339"><span class="id" title="variable">e2</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st':338"><span class="id" title="variable">st'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:318"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':319"><span class="id" title="variable">i'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#P:326"><span class="id" title="variable">P</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:320"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:323"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:324"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i:318"><span class="id" title="variable">i</span></a>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f':322"><span class="id" title="variable">f'</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#s:325"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#i':319"><span class="id" title="variable">i'</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_bst</span> <span class="id" title="var">H_equiv_P</span> <span class="id" title="var">H_P_f</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_RL</span> <span class="id" title="var">H_RG</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">base</span> := <span class="id" title="var">s</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">i</span> <span class="id" title="var">i'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : (<span class="id" title="keyword">∀</span> <a id="e:340" class="idref" href="#e:340"><span class="id" title="binder">e</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:340"><span class="id" title="variable">e</span></a> <span class="id" title="var">base</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:340"><span class="id" title="variable">e</span></a> <span class="id" title="var">s</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">In</span></a> /<span class="id" title="var">base</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.bst"><span class="id" title="inductive">M.bst</span></a> <span class="id" title="var">base</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_bst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">base</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H_bst</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">base</span> <span class="id" title="keyword">as</span> [|<span class="id" title="var">c</span> <span class="id" title="var">l</span> <span class="id" title="var">IHl</span> <span class="id" title="var">e</span> <span class="id" title="var">r</span> <span class="id" title="var">IHr</span>] <span class="id" title="keyword">using</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.tree_ind"><span class="id" title="definition">M.tree_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">Raw.fold</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_bst</span> <span class="id" title="var">H_sub</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">H_P_ii'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_bst_l</span> [<span class="id" title="var">H_bst_r</span> [<span class="id" title="var">H_lt_tree_l</span> <span class="id" title="var">H_gt_tree_r</span>]]]: <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.bst"><span class="id" title="inductive">M.bst</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.bst"><span class="id" title="inductive">M.bst</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.lt_tree"><span class="id" title="definition">M.lt_tree</span></a> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.gt_tree"><span class="id" title="definition">M.gt_tree</span></a> <span class="id" title="var">e</span> <span class="id" title="var">r</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">inversion</span> <span class="id" title="var">H_bst</span>. <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_sub_l</span> : (<span class="id" title="keyword">∀</span> <a id="e0:341" class="idref" href="#e0:341"><span class="id" title="binder">e0</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:341"><span class="id" title="variable">e0</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:341"><span class="id" title="variable">e0</span></a> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:341"><span class="id" title="variable">e0</span></a> <span class="id" title="var">e</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e0</span> <span class="id" title="var">H_in_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_lt_tree_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_sub</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IHl</span> <span class="id" title="var">H_bst_l</span>) ⇒ {} <span class="id" title="var">IHl</span> {<span class="id" title="var">H_bst_l</span>} {<span class="id" title="var">H_lt_tree_l</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_sub_r</span> : (<span class="id" title="keyword">∀</span> <a id="e0:342" class="idref" href="#e0:342"><span class="id" title="binder">e0</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.t"><span class="id" title="axiom">E.t</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:342"><span class="id" title="variable">e0</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:342"><span class="id" title="variable">e0</span></a> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <span class="id" title="var">e</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e0:342"><span class="id" title="variable">e0</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e0</span> <span class="id" title="var">H_in_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gt_tree_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_sub</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IHr</span> <span class="id" title="var">H_bst_r</span>) ⇒ {} <span class="id" title="var">IHr</span> {<span class="id" title="var">H_bst_r</span>} {<span class="id" title="var">H_gt_tree_r</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_e</span> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <span class="id" title="var">e</span> <span class="id" title="var">s</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_sub</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {<span class="id" title="var">H_sub</span>}.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">st0</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">f_lt</span> <span class="id" title="var">e</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e</span>) <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">l</span> <span class="id" title="var">i</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">st0'</span> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">Raw.fold</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">l</span> <span class="id" title="var">i'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">st1</span> := <span class="id" title="var">f</span> <span class="id" title="var">e</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e</span>) <span class="id" title="var">st0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">st1'</span> := <span class="id" title="var">f'</span> <span class="id" title="var">e</span> <span class="id" title="var">st0'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">st2</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <span class="id" title="var">f_gt</span> <span class="id" title="var">e</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e</span>) <span class="id" title="var">st1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <span class="id" title="var">st1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <span class="id" title="var">f_pre</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">r</span> <span class="id" title="var">st1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">st2'</span> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">Raw.fold</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">r</span> <span class="id" title="var">st1'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_P_st0</span> : <span class="id" title="var">P</span> <span class="id" title="var">st0</span> <span class="id" title="var">st0'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">st0</span> /<span class="id" title="var">st0'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f_lt</span> <span class="id" title="var">e</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e</span>) <span class="id" title="var">i</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_fl_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> :  <span class="id" title="var">H_P_ii'</span> <span class="id" title="var">H_sub_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> :  <span class="id" title="var">H_equiv_P</span> <span class="id" title="var">H_f'</span> (<span class="id" title="var">H_RL</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in_e</span> <span class="id" title="var">H_fl_eq</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.lt_tree"><span class="id" title="definition">M.lt_tree</span></a>. <span class="id" title="tactic">clear</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_equiv_P</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_RL</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">i'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [|<span class="id" title="var">c</span> <span class="id" title="var">l</span> <span class="id" title="var">IHl</span> <span class="id" title="var">e'</span> <span class="id" title="var">r</span> <span class="id" title="var">IHr</span>] <span class="id" title="keyword">using</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.tree_ind"><span class="id" title="definition">M.tree_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">i'</span> <span class="id" title="var">H_P_ii'</span> <span class="id" title="var">H_sub_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHr</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_sub_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [] : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <span class="id" title="var">e'</span> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <span class="id" title="var">e'</span> <span class="id" title="var">e</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_sub_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_e'_in</span> <span class="id" title="var">H_lt_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_P_i</span> : (<span class="id" title="var">P</span> <span class="id" title="var">i</span> (<span class="id" title="var">f</span> <span class="id" title="var">e'</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e'</span>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">fold</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">l</span> <span class="id" title="var">i'</span>))). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Transitive"><span class="id" title="method">Equivalence_Transitive</span></a>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_P_i</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_f'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_RL</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHl</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_sub_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHl</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_sub_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_P_st1</span> : <span class="id" title="var">P</span> <span class="id" title="var">st1</span> <span class="id" title="var">st1'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">st1</span> /<span class="id" title="var">st1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_P_f</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_P_st2</span> : <span class="id" title="var">P</span> <span class="id" title="var">st2</span> <span class="id" title="var">st2'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">st2</span> /<span class="id" title="var">st2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">clearbody</span> <span class="id" title="var">st1</span> <span class="id" title="var">st1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f_gt</span> <span class="id" title="var">e</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e</span>) <span class="id" title="var">st1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_gt_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> :  <span class="id" title="var">H_P_st1</span> <span class="id" title="var">H_sub_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_equiv_P</span> (<span class="id" title="var">H_RG</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in_e</span> <span class="id" title="var">H_gt_eq</span>) <span class="id" title="var">H_f'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.gt_tree"><span class="id" title="definition">M.gt_tree</span></a>. <span class="id" title="tactic">clear</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_equiv_P</span> <span class="id" title="var">H_RG</span> <span class="id" title="var">H_f'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">st1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">r</span> <span class="id" title="keyword">as</span> [|<span class="id" title="var">c</span> <span class="id" title="var">l</span> <span class="id" title="var">IHl</span> <span class="id" title="var">e'</span> <span class="id" title="var">r</span> <span class="id" title="var">IHr</span>] <span class="id" title="keyword">using</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.tree_ind"><span class="id" title="definition">M.tree_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">st1'</span> <span class="id" title="var">H_P_st1</span> <span class="id" title="var">H_sub_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHr</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_sub_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [] : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <span class="id" title="var">e'</span> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#E.lt"><span class="id" title="axiom">E.lt</span></a> <span class="id" title="var">e</span> <span class="id" title="var">e'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_sub_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_e'_in</span> <span class="id" title="var">H_lt_ee'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_P_st1_aux</span> : (<span class="id" title="var">P</span> <span class="id" title="var">st1</span> (<span class="id" title="var">f</span> <span class="id" title="var">e'</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">e'</span>) (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#Raw.fold"><span class="id" title="definition">fold</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">l</span> <span class="id" title="var">st1'</span>))). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Transitive"><span class="id" title="method">Equivalence_Transitive</span></a>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_P_st1_aux</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_f'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_RG</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHl</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_sub_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In"><span class="id" title="definition">M.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#M.In_node_iff"><span class="id" title="lemma">M.In_node_iff</span></a>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHr</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">H_sub_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab14"></a><h3 class="section">AVL implementation</h3>
+
+    The generic tree implementation naturally leads to an AVL one. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="MakeAVLSetsWithFoldA" class="idref" href="#MakeAVLSetsWithFoldA"><span class="id" title="module">MakeAVLSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetAVL.html#Make"><span class="id" title="module">MSetAVL.Make</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Int.html#Z_as_Int"><span class="id" title="module">Z_as_Int</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="MakeAVLSetsWithFoldA.foldWithAbortPrecompute" class="idref" href="#MakeAVLSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<a id="A:343" class="idref" href="#A:343"><span class="id" title="binder">A</span></a> <a id="B:344" class="idref" href="#B:344"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} <a id="f_pre:345" class="idref" href="#f_pre:345"><span class="id" title="binder">f_pre</span></a> <a id="f_lt:346" class="idref" href="#f_lt:346"><span class="id" title="binder">f_lt</span></a> (<a id="f:347" class="idref" href="#f:347"><span class="id" title="binder">f</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:344"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:343"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:343"><span class="id" title="variable">A</span></a>) <a id="f_gt:348" class="idref" href="#f_gt:348"><span class="id" title="binder">f_gt</span></a> <a id="t:349" class="idref" href="#t:349"><span class="id" title="binder">t</span></a> (<a id="base:350" class="idref" href="#base:350"><span class="id" title="binder">base</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:343"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:343"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:345"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:346"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:347"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:348"><span class="id" title="variable">f_gt</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:349"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.this"><span class="id" title="projection">this</span></a>)) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#base:350"><span class="id" title="variable">base</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec" class="idref" href="#MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_P_f</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_RL</span> <span class="id" title="var">H_RG</span> <span class="id" title="var">H_P_ii'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbort_RawSpec"><span class="id" title="lemma">foldWithAbort_RawSpec</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span>. <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.this"><span class="id" title="projection">this</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.Ok"><span class="id" title="class">Raw.Ok</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA"><span class="id" title="module">MakeAVLSetsWithFoldA</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab15"></a><h3 class="section">RBT implementation</h3>
+
+    The generic tree implementation naturally leads to an RBT one. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="MakeRBTSetsWithFoldA" class="idref" href="#MakeRBTSetsWithFoldA"><span class="id" title="module">MakeRBTSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetRBT.html#Make"><span class="id" title="module">MSetRBT.Make</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA"><span class="id" title="module">MakeGenTreeFoldA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetRBT.html#Color"><span class="id" title="module">Color</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="MakeRBTSetsWithFoldA.foldWithAbortPrecompute" class="idref" href="#MakeRBTSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<a id="A:351" class="idref" href="#A:351"><span class="id" title="binder">A</span></a> <a id="B:352" class="idref" href="#B:352"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} <a id="f_pre:353" class="idref" href="#f_pre:353"><span class="id" title="binder">f_pre</span></a> <a id="f_lt:354" class="idref" href="#f_lt:354"><span class="id" title="binder">f_lt</span></a> (<a id="f:355" class="idref" href="#f:355"><span class="id" title="binder">f</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:352"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:351"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:351"><span class="id" title="variable">A</span></a>) <a id="f_gt:356" class="idref" href="#f_gt:356"><span class="id" title="binder">f_gt</span></a> <a id="t:357" class="idref" href="#t:357"><span class="id" title="binder">t</span></a> (<a id="base:358" class="idref" href="#base:358"><span class="id" title="binder">base</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:351"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:351"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbort_Raw"><span class="id" title="definition">foldWithAbort_Raw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:353"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:354"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:355"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:356"><span class="id" title="variable">f_gt</span></a> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:357"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.this"><span class="id" title="projection">this</span></a>)) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#base:358"><span class="id" title="variable">base</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec" class="idref" href="#MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span> <span class="id" title="var">s</span> <span class="id" title="var">P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_P_f</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_RL</span> <span class="id" title="var">H_RG</span> <span class="id" title="var">H_P_ii'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbort_RawSpec"><span class="id" title="lemma">foldWithAbort_RawSpec</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span>. <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.this"><span class="id" title="projection">this</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.Ok"><span class="id" title="class">Raw.Ok</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA"><span class="id" title="module">MakeRBTSetsWithFoldA</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab16"></a><h2 class="section">Sorted Lists Implementation</h2>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="MakeListSetsWithFoldA" class="idref" href="#MakeListSetsWithFoldA"><span class="id" title="module">MakeListSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA"><span class="id" title="module">SetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetList.html#Make"><span class="id" title="module">MSetList.Make</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="MakeListSetsWithFoldA.foldWithAbortRaw" class="idref" href="#MakeListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> {<a id="A:359" class="idref" href="#A:359"><span class="id" title="binder">A</span></a> <a id="B:360" class="idref" href="#B:360"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="f_pre:361" class="idref" href="#f_pre:361"><span class="id" title="binder">f_pre</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:360"><span class="id" title="variable">B</span></a>) (<a id="f_lt:362" class="idref" href="#f_lt:362"><span class="id" title="binder">f_lt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:360"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:359"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:363" class="idref" href="#f:363"><span class="id" title="binder">f</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:360"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:359"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:359"><span class="id" title="variable">A</span></a>) (<a id="f_gt:364" class="idref" href="#f_gt:364"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:360"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:359"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="t:365" class="idref" href="#t:365"><span class="id" title="binder">t</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a>) (<a id="acc:366" class="idref" href="#acc:366"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:359"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:359"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:365"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:366"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> ⇒ (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="pre_x:369" class="idref" href="#pre_x:369"><span class="id" title="binder">pre_x</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:361"><span class="id" title="variable">f_pre</span></a> <span class="id" title="var">x</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="acc:370" class="idref" href="#acc:370"><span class="id" title="binder">acc</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:363"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#pre_x:369"><span class="id" title="variable">pre_x</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:366"><span class="id" title="variable">acc</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:364"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#pre_x:369"><span class="id" title="variable">pre_x</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:370"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:370"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortRaw:367"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:361"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:362"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:363"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:364"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">xs</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:370"><span class="id" title="variable">acc</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="MakeListSetsWithFoldA.foldWithAbortPrecompute" class="idref" href="#MakeListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<a id="A:371" class="idref" href="#A:371"><span class="id" title="binder">A</span></a> <a id="B:372" class="idref" href="#B:372"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} <a id="f_pre:373" class="idref" href="#f_pre:373"><span class="id" title="binder">f_pre</span></a> <a id="f_lt:374" class="idref" href="#f_lt:374"><span class="id" title="binder">f_lt</span></a> <a id="f:375" class="idref" href="#f:375"><span class="id" title="binder">f</span></a> <a id="f_gt:376" class="idref" href="#f_gt:376"><span class="id" title="binder">f_gt</span></a> <a id="t:377" class="idref" href="#t:377"><span class="id" title="binder">t</span></a> <a id="acc:378" class="idref" href="#acc:378"><span class="id" title="binder">acc</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:371"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:372"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:373"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:374"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:375"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:376"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:377"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.this"><span class="id" title="projection">this</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:378"><span class="id" title="variable">acc</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec" class="idref" href="#MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">l</span> <span class="id" title="var">H_is_ok_l</span> <span class="id" title="var">P</span> <span class="id" title="var">H_equiv_P</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.In"><span class="id" title="definition">In</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.this"><span class="id" title="projection">this</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.In"><span class="id" title="definition">Raw.In</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.fold"><span class="id" title="definition">Raw.fold</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_P_f</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_RL</span> <span class="id" title="var">H_RG</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">base</span> := <span class="id" title="var">l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">i</span> <span class="id" title="var">i'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : (<span class="id" title="keyword">∀</span> <a id="e:379" class="idref" href="#e:379"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:379"><span class="id" title="variable">e</span></a> <span class="id" title="var">base</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:379"><span class="id" title="variable">e</span></a> <span class="id" title="var">l</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">base</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#sort"><span class="id" title="abbreviation">sort</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <span class="id" title="var">base</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.isok_iff"><span class="id" title="lemma">Raw.isok_iff</span></a> /<span class="id" title="var">base</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H_is_ok_l</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">base</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_sort</span> <span class="id" title="var">H_in_xxs</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">Pii'</span> /=.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_sort_xs</span> <span class="id" title="var">H_hd_rel</span> {<span class="id" title="var">H_sort</span>}] : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#HdRel"><span class="id" title="inductive">HdRel</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">xs</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">inversion</span> <span class="id" title="var">H_sort</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_hd_rel</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.Inf_alt"><span class="id" title="lemma">Raw.ML.Inf_alt</span></a> <span class="id" title="var">x</span> <span class="id" title="var">H_sort_xs</span>) ⇒ <span class="id" title="var">H_lt_xs</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x_in_l</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">l</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xxs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq_equiv"><span class="id" title="axiom">X.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_xs</span> : (<span class="id" title="keyword">∀</span> <a id="e:380" class="idref" href="#e:380"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:380"><span class="id" title="variable">e</span></a> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:380"><span class="id" title="variable">e</span></a> <span class="id" title="var">l</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xxs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_P_next</span> : <span class="id" title="var">P</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">i'</span> <span class="id" title="var">x</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> -<span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_P_f</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f_gt</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>)); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_gt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_suff</span> : (<span class="id" title="keyword">∀</span> <a id="st:381" class="idref" href="#st:381"><span class="id" title="binder">st</span></a>, <span class="id" title="var">P</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:381"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">P</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <span class="id" title="var">f'</span>) <span class="id" title="var">xs</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:381"><span class="id" title="variable">st</span></a>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_suff</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in_xs</span> <span class="id" title="var">H_lt_xs</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">H_in_xxs</span> <span class="id" title="var">H_sort_xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_RG</span> <span class="id" title="var">x</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x_in_l</span> <span class="id" title="var">H_gt</span>) ⇒ <span class="id" title="var">H_RG_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x'</span> <span class="id" title="var">xs'</span> <span class="id" title="var">IH'</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">H_in_xs</span> <span class="id" title="var">H_lt_xs</span> <span class="id" title="var">st</span> <span class="id" title="var">H_P_st</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x'_in_l</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">x'</span> <span class="id" title="var">l</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq_equiv"><span class="id" title="axiom">X.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_lt_xs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> -<span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_RG_x</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_lt_xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq_equiv"><span class="id" title="axiom">X.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA"><span class="id" title="module">MakeListSetsWithFoldA</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab17"></a><h3 class="section">Unsorted Lists without Dups Implementation</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="MakeWeakListSetsWithFoldA" class="idref" href="#MakeWeakListSetsWithFoldA"><span class="id" title="module">MakeWeakListSetsWithFoldA</span></a> (<span class="id" title="var">X</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA"><span class="id" title="module">WSetsWithFoldA</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">Module</span> <span class="id" title="var">E</span> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="MakeWeakListSetsWithFoldA.Raw" class="idref" href="#MakeWeakListSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetWeakList.html#MakeRaw"><span class="id" title="module">MSetWeakList.MakeRaw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="MakeWeakListSetsWithFoldA.E" class="idref" href="#MakeWeakListSetsWithFoldA.E"><span class="id" title="module">E</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#WRaw2SetsOn"><span class="id" title="module">WRaw2SetsOn</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.E"><span class="id" title="module">E</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.Raw"><span class="id" title="module">Raw</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="MakeWeakListSetsWithFoldA.foldWithAbortRaw" class="idref" href="#MakeWeakListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> {<a id="A:382" class="idref" href="#A:382"><span class="id" title="binder">A</span></a> <a id="B:383" class="idref" href="#B:383"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} (<a id="f_pre:384" class="idref" href="#f_pre:384"><span class="id" title="binder">f_pre</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:383"><span class="id" title="variable">B</span></a>) (<a id="f_lt:385" class="idref" href="#f_lt:385"><span class="id" title="binder">f_lt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:383"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:382"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a id="f:386" class="idref" href="#f:386"><span class="id" title="binder">f</span></a>: <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:383"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:382"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:382"><span class="id" title="variable">A</span></a>) (<a id="f_gt:387" class="idref" href="#f_gt:387"><span class="id" title="binder">f_gt</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:383"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:382"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="t:388" class="idref" href="#t:388"><span class="id" title="binder">t</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a>) (<a id="acc:389" class="idref" href="#acc:389"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:382"><span class="id" title="variable">A</span></a>) : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:382"><span class="id" title="variable">A</span></a> :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:388"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:389"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> ⇒ (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="pre_x:392" class="idref" href="#pre_x:392"><span class="id" title="binder">pre_x</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:384"><span class="id" title="variable">f_pre</span></a> <span class="id" title="var">x</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="acc:393" class="idref" href="#acc:393"><span class="id" title="binder">acc</span></a> := <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:386"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#pre_x:392"><span class="id" title="variable">pre_x</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:389"><span class="id" title="variable">acc</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:387"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#pre_x:392"><span class="id" title="variable">pre_x</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:393"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:385"><span class="id" title="variable">f_lt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#pre_x:392"><span class="id" title="variable">pre_x</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:393"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:393"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortRaw:390"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:384"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:385"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:386"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:387"><span class="id" title="variable">f_gt</span></a> <span class="id" title="var">xs</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:393"><span class="id" title="variable">acc</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;)<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="MakeWeakListSetsWithFoldA.foldWithAbortPrecompute" class="idref" href="#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> {<a id="A:394" class="idref" href="#A:394"><span class="id" title="binder">A</span></a> <a id="B:395" class="idref" href="#B:395"><span class="id" title="binder">B</span></a>: <span class="id" title="keyword">Type</span>} <a id="f_pre:396" class="idref" href="#f_pre:396"><span class="id" title="binder">f_pre</span></a> <a id="f_lt:397" class="idref" href="#f_lt:397"><span class="id" title="binder">f_lt</span></a> <a id="f:398" class="idref" href="#f:398"><span class="id" title="binder">f</span></a> <a id="f_gt:399" class="idref" href="#f_gt:399"><span class="id" title="binder">f_gt</span></a> <a id="t:400" class="idref" href="#t:400"><span class="id" title="binder">t</span></a> <a id="acc:401" class="idref" href="#acc:401"><span class="id" title="binder">acc</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortRaw"><span class="id" title="definition">foldWithAbortRaw</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#A:394"><span class="id" title="variable">A</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#B:395"><span class="id" title="variable">B</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_pre:396"><span class="id" title="variable">f_pre</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_lt:397"><span class="id" title="variable">f_lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f:398"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#f_gt:399"><span class="id" title="variable">f_gt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#t:400"><span class="id" title="variable">t</span></a>.(<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.this"><span class="id" title="projection">this</span></a>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#acc:401"><span class="id" title="variable">acc</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec" class="idref" href="#MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec"><span class="id" title="lemma">foldWithAbortPrecomputeSpec</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred"><span class="id" title="definition">foldWithAbortPrecomputeSpecPred</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> (@<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">f_pre</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> <span class="id" title="var">f_lt</span> <span class="id" title="var">f_gt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">l</span> <span class="id" title="var">H_is_ok_l</span> <span class="id" title="var">P</span> <span class="id" title="var">H_P_equiv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.fold"><span class="id" title="definition">fold</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute"><span class="id" title="definition">foldWithAbortPrecompute</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.In"><span class="id" title="definition">In</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.this"><span class="id" title="projection">this</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.In"><span class="id" title="definition">Raw.In</span></a> /<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.fold"><span class="id" title="definition">Raw.fold</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_P_f</span> <span class="id" title="var">H_f'</span> <span class="id" title="var">H_RL</span> <span class="id" title="var">H_RG</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">base</span> := <span class="id" title="var">l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">i</span> <span class="id" title="var">i'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : (<span class="id" title="keyword">∀</span> <a id="e:402" class="idref" href="#e:402"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:402"><span class="id" title="variable">e</span></a> <span class="id" title="var">base</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:402"><span class="id" title="variable">e</span></a> <span class="id" title="var">l</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">base</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">base</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_is_ok_l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H_is_ok_l</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">base</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_nodup_xxs</span> <span class="id" title="var">H_in_xxs</span> <span class="id" title="var">i</span> <span class="id" title="var">i'</span> <span class="id" title="var">Pii'</span> /=.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_nin_x_xs</span> <span class="id" title="var">H_nodup_xs</span> {<span class="id" title="var">H_nodup_xxs</span>}] : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">xs</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">inversion</span> <span class="id" title="var">H_nodup_xxs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x_in_l</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">l</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xxs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq_equiv"><span class="id" title="axiom">X.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_xs</span> : (<span class="id" title="keyword">∀</span> <a id="e:403" class="idref" href="#e:403"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.t"><span class="id" title="axiom">X.t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:403"><span class="id" title="variable">e</span></a> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:403"><span class="id" title="variable">e</span></a> <span class="id" title="var">l</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xxs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_P_next</span> : <span class="id" title="var">P</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">i'</span> <span class="id" title="var">x</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> -<span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_P_f</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f_gt</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">f_lt</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>)); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">andb_true_iff</span></a> [<span class="id" title="var">H_gt</span> <span class="id" title="var">H_lt</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_suff</span> : (<span class="id" title="keyword">∀</span> <a id="st:404" class="idref" href="#st:404"><span class="id" title="binder">st</span></a>, <span class="id" title="var">P</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:404"><span class="id" title="variable">st</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">P</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span> (<span class="id" title="var">f_pre</span> <span class="id" title="var">x</span>) <span class="id" title="var">i</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <span class="id" title="var">f'</span>) <span class="id" title="var">xs</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#st:404"><span class="id" title="variable">st</span></a>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_suff</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_neq_xs</span> : <span class="id" title="keyword">∀</span> <a id="e:405" class="idref" href="#e:405"><span class="id" title="binder">e</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:405"><span class="id" title="variable">e</span></a> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:405"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.lt"><span class="id" title="axiom">X.lt</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#e:405"><span class="id" title="variable">e</span></a> <span class="id" title="var">x</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H_in</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.compare_spec"><span class="id" title="axiom">X.compare_spec</span></a> <span class="id" title="var">x</span> <span class="id" title="var">e</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">e</span>) ⇒ <span class="id" title="var">H_cmp</span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">H_cmp</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_nin_x_xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in_xs</span> <span class="id" title="var">H_neq_xs</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">H_in_xxs</span> <span class="id" title="var">H_nodup_xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_RG</span> <span class="id" title="var">x</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x_in_l</span> <span class="id" title="var">H_gt</span>) ⇒ <span class="id" title="var">H_RG_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_RL</span> <span class="id" title="var">x</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x_in_l</span> <span class="id" title="var">H_lt</span>) ⇒ <span class="id" title="var">H_RL_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x'</span> <span class="id" title="var">xs'</span> <span class="id" title="var">IH'</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">H_in_xs</span> <span class="id" title="var">H_neq_xs</span> <span class="id" title="var">st</span> <span class="id" title="var">H_P_st</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x'_in_xxs'</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">x'</span> (<span class="id" title="var">x'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x'_in_l</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq"><span class="id" title="axiom">X.eq</span></a> <span class="id" title="var">x'</span> <span class="id" title="var">l</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>, <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X.eq_equiv"><span class="id" title="axiom">X.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_nin_x_xs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_xs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_tl"><span class="id" title="constructor">InA_cons_tl</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_neq_xs</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_cons"><span class="id" title="lemma">in_cons</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> -<span class="id" title="var">H_f'</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_neq_xs</span> <span class="id" title="var">x'</span> <span class="id" title="var">H_x'_in_xxs'</span>) ⇒ [] <span class="id" title="var">H_cmp</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_RG_x</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_RL_x</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps"><span class="id" title="module">HasFoldWithAbortOps</span></a> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA"><span class="id" title="module">MakeWeakListSetsWithFoldA</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
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+
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+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>MSetsExtra.MSetIntervals</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library MSetsExtra.MSetIntervals</h1>
+
+<div class="code">
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab18"></a><h1 class="section">Weak sets implemented by interval lists</h1>
+
+
+<div class="paragraph"> </div>
+
+    This file contains an implementation of the set interface
+    <span class="inlinecode"><span class="id" title="var">SetsOn</span></span> which uses internally intervals of Z. This allows some
+    large sets, which naturally map to intervals of integers to be
+    represented very efficiently.
+
+<div class="paragraph"> </div>
+
+    Internally intervals of Z are used. However, via an encoding
+    and decoding layer, other types of elements can be handled as
+    well. There are instantiations for Z, N and nat currently.
+    More can be easily added.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#"><span class="id" title="library">MSetInterface</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.OrdersFacts.html#"><span class="id" title="library">OrdersFacts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.OrdersLists.html#"><span class="id" title="library">OrdersLists</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#"><span class="id" title="library">BinNat</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">mathcomp.ssreflect.ssreflect</span>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.NArith.html#"><span class="id" title="library">NArith</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.ZArith.html#"><span class="id" title="library">ZArith</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.Natural.Abstract.NOrder.html#"><span class="id" title="library">NOrder</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.DecidableTypeEx.html#"><span class="id" title="library">DecidableTypeEx</span></a>.<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a id="NOP" class="idref" href="#NOP"><span class="id" title="module">NOP</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.Natural.Abstract.NOrder.html#NOrderProp"><span class="id" title="module">NOrderProp</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N"><span class="id" title="module">N</span></a>.<br/>
+<span class="id" title="keyword">Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">Z_scope</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab19"></a><h2 class="section">Auxiliary stuff</h2>
+
+<div class="paragraph"> </div>
+
+ Simple auxiliary lemmata 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="Z_le_add_r" class="idref" href="#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a> : <span class="id" title="keyword">∀</span> (<a id="z:1" class="idref" href="#z:1"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#z:1"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:1"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:2"><span class="id" title="variable">n</span></a>.<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">z</span> <span class="id" title="var">n</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">n</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_0_r"><span class="id" title="lemma">Z.add_0_r</span></a> //.<br/>
+&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zplus_le_compat_l"><span class="id" title="lemma">Zplus_le_compat_l</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.is_nonneg"><span class="id" title="lemma">N2Z.is_nonneg</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z_lt_add_r" class="idref" href="#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a> : <span class="id" title="keyword">∀</span> (<a id="z:3" class="idref" href="#z:3"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
+&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#z:3"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:3"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:4"><span class="id" title="variable">n</span></a>.<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">z</span> <span class="id" title="var">n</span> <span class="id" title="var">H_neq_0</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">n</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_0_r"><span class="id" title="lemma">Z.add_0_r</span></a> //.<br/>
+&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_lt_mono_l"><span class="id" title="lemma">Z.add_lt_mono_l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_lt"><span class="id" title="lemma">N2Z.inj_lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_0_lt_0"><span class="id" title="lemma">N.neq_0_lt_0</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z_lt_le_add_r" class="idref" href="#Z_lt_le_add_r"><span class="id" title="lemma">Z_lt_le_add_r</span></a> : <span class="id" title="keyword">∀</span> <a id="y1:5" class="idref" href="#y1:5"><span class="id" title="binder">y1</span></a> <a id="y2:6" class="idref" href="#y2:6"><span class="id" title="binder">y2</span></a> <a id="c:7" class="idref" href="#c:7"><span class="id" title="binder">c</span></a>, <br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y1:5"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:6"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y1:5"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:6"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:7"><span class="id" title="variable">c</span></a>.<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;}<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Z_to_N_minus_neq_0" class="idref" href="#Z_to_N_minus_neq_0"><span class="id" title="lemma">Z_to_N_minus_neq_0</span></a> : <span class="id" title="keyword">∀</span> (<a id="x:8" class="idref" href="#x:8"><span class="id" title="binder">x</span></a> <a id="y:9" class="idref" href="#y:9"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y:9"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:8"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:9"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span>.<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_0_lt_0"><span class="id" title="lemma">N.neq_0_lt_0</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_lt"><span class="id" title="lemma">N2Z.inj_lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H</span> :  0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_0_sub"><span class="id" title="lemma">Z.lt_0_sub</span></a>.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="add_add_sub_eq" class="idref" href="#add_add_sub_eq"><span class="id" title="lemma">add_add_sub_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="x:10" class="idref" href="#x:10"><span class="id" title="binder">x</span></a> <a id="y:11" class="idref" href="#y:11"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:10"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:11"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:10"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:11"><span class="id" title="variable">y</span></a>).<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_sub_assoc"><span class="id" title="lemma">Z.add_sub_assoc</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span>  <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_sub_swap"><span class="id" title="lemma">Z.add_sub_swap</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.sub_diag"><span class="id" title="lemma">Z.sub_diag</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_0_l"><span class="id" title="lemma">Z.add_0_l</span></a> //.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="NoDupA_map" class="idref" href="#NoDupA_map"><span class="id" title="lemma">NoDupA_map</span></a> {<a id="A:12" class="idref" href="#A:12"><span class="id" title="binder">A</span></a> <a id="B:13" class="idref" href="#B:13"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> (<a id="eqA:14" class="idref" href="#eqA:14"><span class="id" title="binder">eqA</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:12"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:12"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="eqB:15" class="idref" href="#eqB:15"><span class="id" title="binder">eqB</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#B:13"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:13"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="f:16" class="idref" href="#f:16"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:12"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:13"><span class="id" title="variable">B</span></a>) <a id="l:17" class="idref" href="#l:17"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#eqA:14"><span class="id" title="variable">eqA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:17"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:18" class="idref" href="#x1:18"><span class="id" title="binder">x1</span></a> <a id="x2:19" class="idref" href="#x2:19"><span class="id" title="binder">x2</span></a>,  <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:18"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:17"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:19"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:17"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#eqB:15"><span class="id" title="variable">eqB</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:16"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:18"><span class="id" title="variable">x1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:16"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:19"><span class="id" title="variable">x2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#eqA:14"><span class="id" title="variable">eqA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:18"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:19"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#eqB:15"><span class="id" title="variable">eqB</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:16"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:17"><span class="id" title="variable">l</span></a>).<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">eqA</span> <span class="id" title="var">eqB</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">_</span>; <span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_nil"><span class="id" title="constructor">NoDupA_nil</span></a>.<br/>
+&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">H_eqA_impl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_nin_x</span> <span class="id" title="var">H_no_dup_xs</span>] : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <span class="id" title="var">eqA</span> <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <span class="id" title="var">eqA</span> <span class="id" title="var">xs</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">H_pre</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_cons"><span class="id" title="constructor">NoDupA_cons</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_in_x1</span> <span class="id" title="var">H_in_x2</span> <span class="id" title="var">H_eqB</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_eqA_impl</span> ⇒ //=; <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_map</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H_nin_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in_map</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> ⇒ [[<span class="id" title="var">y</span>] [<span class="id" title="var">H_eqB_y</span>]].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_map_iff"><span class="id" title="lemma">in_map_iff</span></a> ⇒ [[<span class="id" title="var">y'</span>] [<span class="id" title="var">H_y_eq</span>] <span class="id" title="var">H_y'_in</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_eqA_impl</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;}<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab20"></a><h3 class="section">rev_map</h3>
+
+
+<div class="paragraph"> </div>
+
+    rev_map is used for efficiency. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Fixpoint</span> <a id="rev_map_aux" class="idref" href="#rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> {<a id="A:20" class="idref" href="#A:20"><span class="id" title="binder">A</span></a> <a id="B:21" class="idref" href="#B:21"><span class="id" title="binder">B</span></a>} (<a id="f:22" class="idref" href="#f:22"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:20"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:21"><span class="id" title="variable">B</span></a>) (<a id="acc:23" class="idref" href="#acc:23"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:21"><span class="id" title="variable">B</span></a>) (<a id="l:24" class="idref" href="#l:24"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:20"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:24"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:23"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_aux:25"><span class="id" title="definition">rev_map_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:22"><span class="id" title="variable">f</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:22"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:23"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Definition</span> <a id="rev_map" class="idref" href="#rev_map"><span class="id" title="definition">rev_map</span></a> {<a id="A:27" class="idref" href="#A:27"><span class="id" title="binder">A</span></a> <a id="B:28" class="idref" href="#B:28"><span class="id" title="binder">B</span></a>} (<a id="f:29" class="idref" href="#f:29"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:27"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:28"><span class="id" title="variable">B</span></a>) (<a id="l:30" class="idref" href="#l:30"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:27"><span class="id" title="variable">A</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:28"><span class="id" title="variable">B</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:29"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:30"><span class="id" title="variable">l</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Lemmata about rev_map 
+</div>
+<div class="code">
+<span class="id" title="keyword">Lemma</span> <a id="rev_map_aux_alt_def" class="idref" href="#rev_map_aux_alt_def"><span class="id" title="lemma">rev_map_aux_alt_def</span></a> {<a id="A:31" class="idref" href="#A:31"><span class="id" title="binder">A</span></a> <a id="B:32" class="idref" href="#B:32"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> (<a id="f:33" class="idref" href="#f:33"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:31"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:32"><span class="id" title="variable">B</span></a>) <a id="l:34" class="idref" href="#l:34"><span class="id" title="binder">l</span></a> <a id="acc:35" class="idref" href="#acc:35"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_aux"><span class="id" title="definition">rev_map_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:33"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:35"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:34"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:33"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:34"><span class="id" title="variable">l</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:35"><span class="id" title="variable">acc</span></a>.<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span> //.<br/>
+&nbsp;&nbsp;}<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="rev_map_alt_def" class="idref" href="#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> {<a id="A:36" class="idref" href="#A:36"><span class="id" title="binder">A</span></a> <a id="B:37" class="idref" href="#B:37"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> (<a id="f:38" class="idref" href="#f:38"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:36"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:37"><span class="id" title="variable">B</span></a>) <a id="l:39" class="idref" href="#l:39"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map"><span class="id" title="definition">rev_map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:38"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:39"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:38"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:39"><span class="id" title="variable">l</span></a>).<br/>
+<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">l</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map"><span class="id" title="definition">rev_map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_aux_alt_def"><span class="id" title="lemma">rev_map_aux_alt_def</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_alt"><span class="id" title="lemma">rev_alt</span></a> //.<br/>
+<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab21"></a><h2 class="section">Encoding Elements</h2>
+
+<div class="paragraph"> </div>
+
+ We want to encode not only elements of type Z, but other types
+    as well. In order to do so, an encoding / decoding layer is used.
+    This layer is represented by module type <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a></span>. It
+    provides encode and decode function. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="ElementEncode" class="idref" href="#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Declare Module</span> <a id="ElementEncode.E" class="idref" href="#ElementEncode.E"><span class="id" title="module">E</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="ElementEncode.encode" class="idref" href="#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.t"><span class="id" title="axiom">E.t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="ElementEncode.decode" class="idref" href="#ElementEncode.decode"><span class="id" title="axiom">decode</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.t"><span class="id" title="axiom">E.t</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Decoding is the inverse of encoding. Notice that
+      the reverse is not demanded. This means that
+      we do need to provide for all integers <span class="inlinecode"><span class="id" title="var">z</span></span> an
+      element <span class="inlinecode"><span class="id" title="var">e</span></span> with <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a></span> <span class="inlinecode"><span class="id" title="var">v</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">z</span></span>. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="ElementEncode.decode_encode_ok" class="idref" href="#ElementEncode.decode_encode_ok"><span class="id" title="axiom">decode_encode_ok</span></a>: <span class="id" title="keyword">∀</span> (<a id="e:42" class="idref" href="#e:42"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.t"><span class="id" title="axiom">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.decode"><span class="id" title="axiom">decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:42"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:42"><span class="id" title="variable">e</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Encoding is compatible with the equality of elements. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="ElementEncode.encode_eq" class="idref" href="#ElementEncode.encode_eq"><span class="id" title="axiom">encode_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:44" class="idref" href="#e1:44"><span class="id" title="binder">e1</span></a> <a id="e2:45" class="idref" href="#e2:45"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.t"><span class="id" title="axiom">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:44"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:45"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:44"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:45"><span class="id" title="variable">e2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Encoding is compatible with the order of elements. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Axiom</span> <a id="ElementEncode.encode_lt" class="idref" href="#ElementEncode.encode_lt"><span class="id" title="axiom">encode_lt</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:47" class="idref" href="#e1:47"><span class="id" title="binder">e1</span></a> <a id="e2:48" class="idref" href="#e2:48"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.t"><span class="id" title="axiom">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:47"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.encode"><span class="id" title="axiom">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:48"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:47"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:48"><span class="id" title="variable">e2</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab22"></a><h2 class="section">Set Operations</h2>
+
+
+<div class="paragraph"> </div>
+
+    We represent sets of Z via lists of intervals. The intervals are all
+    in increasing order and non-overlapping. Moreover, we require the most compact
+    representation, i.e. no two intervals can be merged. For example
+
+<div class="paragraph"> </div>
+
+    <span class="inlinecode">0-2,</span> <span class="inlinecode">4-4,</span> <span class="inlinecode">6-8</span> is a valid interval list for the set {0;1;2;4;6;7;8}
+
+<div class="paragraph"> </div>
+
+    In contrast
+
+<div class="paragraph"> </div>
+
+    <span class="inlinecode">4-4,</span> <span class="inlinecode">0-2,</span> <span class="inlinecode">6-8</span> is a invalid because the intervals are not ordered andb
+    <span class="inlinecode">0-2,</span> <span class="inlinecode">4-5,</span> <span class="inlinecode">6-8</span> is a invalid because it is not compact (<span class="inlinecode">0-2,</span> <span class="inlinecode">4--8</span> is valid).
+
+<div class="paragraph"> </div>
+
+    Intervals we represent by tuples <span class="inlinecode">(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,</span> <span class="inlinecode"><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)</span>. The tuple <span class="inlinecode">(<span class="id" title="var">z</span>,</span> <span class="inlinecode"><span class="id" title="var">c</span>)</span>  represents the
+    interval <span class="inlinecode"><span class="id" title="var">z</span>-(<span class="id" title="var">z</span>+<span class="id" title="var">c</span>)</span>.
+
+<div class="paragraph"> </div>
+
+    We apply the encode function before adding an element to such interval sets and
+    the decode function when checking it. This allows for sets with other element types
+    than Z.
+ 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Ops" class="idref" href="#Ops"><span class="id" title="module">Ops</span></a> (<span class="id" title="var">Enc</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>) &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#Ops"><span class="id" title="module">Ops</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E"><span class="id" title="module">Enc.E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.elt" class="idref" href="#Ops.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.t"><span class="id" title="axiom">Enc.E.t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.t" class="idref" href="#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+The empty list is trivial to define and check for. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.empty" class="idref" href="#Ops.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.is_empty" class="idref" href="#Ops.is_empty"><span class="id" title="definition">is_empty</span></a> (<a id="l:50" class="idref" href="#l:50"><span class="id" title="binder">l</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:50"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Defining the list of elements, is much more tricky, especially, 
+      if it needs to be executable. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Ops.acc_pred" class="idref" href="#Ops.acc_pred"><span class="id" title="lemma">acc_pred</span></a> : <span class="id" title="keyword">∀</span> <a id="n:52" class="idref" href="#n:52"><span class="id" title="binder">n</span></a> <a id="p:53" class="idref" href="#p:53"><span class="id" title="binder">p</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Npos"><span class="id" title="constructor">Npos</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#p:53"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt"><span class="id" title="definition">N.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:52"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt"><span class="id" title="definition">N.lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.pred"><span class="id" title="definition">N.pred</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:52"><span class="id" title="variable">n</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span> <span class="id" title="var">p</span> <span class="id" title="var">H0</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt_pred_l"><span class="id" title="lemma">N.lt_pred_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">discriminate</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.fold_elementsZ_aux" class="idref" href="#Ops.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a> {<a id="A:54" class="idref" href="#A:54"><span class="id" title="binder">A</span></a>} (<a id="f:55" class="idref" href="#f:55"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:54"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:54"><span class="id" title="variable">A</span></a>) (<a id="acc:56" class="idref" href="#acc:56"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:54"><span class="id" title="variable">A</span></a>) (<a id="x:57" class="idref" href="#x:57"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c:58" class="idref" href="#c:58"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="H:59" class="idref" href="#H:59"><span class="id" title="binder">H</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Wf.html#Acc"><span class="id" title="inductive">Acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt"><span class="id" title="definition">N.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:58"><span class="id" title="variable">c</span></a>) { <span class="id" title="keyword">struct</span> <span class="id" title="var">H</span> } : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:54"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:58"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">as</span> <span class="id" title="var">c0</span> <span class="id" title="keyword">return</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:58"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c0:61"><span class="id" title="variable">c0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:54"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N0"><span class="id" title="constructor">N0</span></a> ⇒ <span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:56"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">c</span> ⇒ <span class="id" title="keyword">fun</span> <a id="Heq:62" class="idref" href="#Heq:62"><span class="id" title="binder">Heq</span></a> ⇒ <span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:55"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:56"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:57"><span class="id" title="variable">x</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:56"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">acc'</span> ⇒             <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#fold_elementsZ_aux:60"><span class="id" title="definition">fold_elementsZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:55"><span class="id" title="variable">f</span></a> <span class="id" title="var">acc'</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:57"><span class="id" title="variable">x</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.pred"><span class="id" title="definition">N.pred</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:58"><span class="id" title="variable">c</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.acc_pred"><span class="id" title="lemma">acc_pred</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Heq:62"><span class="id" title="variable">Heq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#H:59"><span class="id" title="variable">H</span></a>) <span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.fold_elementsZ_single" class="idref" href="#Ops.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> {<a id="A:63" class="idref" href="#A:63"><span class="id" title="binder">A</span></a>} <a id="f:64" class="idref" href="#f:64"><span class="id" title="binder">f</span></a> (<a id="acc:65" class="idref" href="#acc:65"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:63"><span class="id" title="variable">A</span></a>) <a id="x:66" class="idref" href="#x:66"><span class="id" title="binder">x</span></a> <a id="c:67" class="idref" href="#c:67"><span class="id" title="binder">c</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:64"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:65"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:66"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:67"><span class="id" title="variable">c</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#NOP.lt_wf_0"><span class="id" title="lemma">lt_wf_0</span></a> <span class="id" title="var">_</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.fold_elementsZ" class="idref" href="#Ops.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> {<a id="A:68" class="idref" href="#A:68"><span class="id" title="binder">A</span></a>} <a id="f:69" class="idref" href="#f:69"><span class="id" title="binder">f</span></a> (<a id="acc:70" class="idref" href="#acc:70"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:68"><span class="id" title="variable">A</span></a>) (<a id="s:71" class="idref" href="#s:71"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:68"><span class="id" title="variable">A</span></a>) := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:71"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:70"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:69"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:70"><span class="id" title="variable">acc</span></a> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#fold_elementsZ:72"><span class="id" title="definition">fold_elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:69"><span class="id" title="variable">f</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.elementsZ" class="idref" href="#Ops.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a id="s:74" class="idref" href="#s:74"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> (<span class="id" title="keyword">fun</span> <a id="l:75" class="idref" href="#l:75"><span class="id" title="binder">l</span></a> <a id="x:76" class="idref" href="#x:76"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:76"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:75"><span class="id" title="variable">l</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:74"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.elements" class="idref" href="#Ops.elements"><span class="id" title="definition">elements</span></a> (<a id="s:77" class="idref" href="#s:77"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map"><span class="id" title="definition">rev_map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:77"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+membership is easily defined 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.memZ" class="idref" href="#Ops.memZ"><span class="id" title="definition">memZ</span></a> (<a id="x:78" class="idref" href="#x:78"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="s:79" class="idref" href="#s:79"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:79"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:78"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:78"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#memZ:80"><span class="id" title="definition">memZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:78"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.mem" class="idref" href="#Ops.mem"><span class="id" title="definition">mem</span></a> (<a id="x:82" class="idref" href="#x:82"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<a id="s:83" class="idref" href="#s:83"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.memZ"><span class="id" title="definition">memZ</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:82"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:83"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Comparing intervals 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Inductive</span> <a id="Ops.interval_compare_result" class="idref" href="#Ops.interval_compare_result"><span class="id" title="inductive">interval_compare_result</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a id="Ops.ICR_before" class="idref" href="#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_before_touch" class="idref" href="#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_overlap_before" class="idref" href="#Ops.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_overlap_after" class="idref" href="#Ops.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_equal" class="idref" href="#Ops.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_subsume_1" class="idref" href="#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_subsume_2" class="idref" href="#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_after" class="idref" href="#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a id="Ops.ICR_after_touch" class="idref" href="#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.interval_compare" class="idref" href="#Ops.interval_compare"><span class="id" title="definition">interval_compare</span></a> (<a id="i1:86" class="idref" href="#i1:86"><span class="id" title="binder">i1</span></a> <a id="i2:87" class="idref" href="#i2:87"><span class="id" title="binder">i2</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_compare_result"><span class="id" title="inductive">interval_compare_result</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#i1:86"><span class="id" title="variable">i1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#i2:87"><span class="id" title="variable">i2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="yc2:88" class="idref" href="#yc2:88"><span class="id" title="binder">yc2</span></a> := (<span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#yc2:88"><span class="id" title="variable">yc2</span></a> <span class="id" title="var">y1</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <span class="id" title="keyword">let</span> <a id="yc1:89" class="idref" href="#yc1:89"><span class="id" title="binder">yc1</span></a> := (<span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#yc1:89"><span class="id" title="variable">yc1</span></a> <span class="id" title="var">y2</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#yc1:89"><span class="id" title="variable">yc1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#yc2:88"><span class="id" title="variable">yc2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.interval_1_compare" class="idref" href="#Ops.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> (<a id="y1:90" class="idref" href="#y1:90"><span class="id" title="binder">y1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="i:91" class="idref" href="#i:91"><span class="id" title="binder">i</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_compare_result"><span class="id" title="inductive">interval_compare_result</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#i:91"><span class="id" title="variable">i</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="yc2:93" class="idref" href="#yc2:93"><span class="id" title="binder">yc2</span></a> := (<span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#yc2:93"><span class="id" title="variable">yc2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:90"><span class="id" title="variable">y1</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:90"><span class="id" title="variable">y1</span></a>) <span class="id" title="var">y2</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.compare" class="idref" href="#Ops.compare"><span class="id" title="definition">compare</span></a> (<a id="s1:94" class="idref" href="#s1:94"><span class="id" title="binder">s1</span></a> <a id="s2:95" class="idref" href="#s2:95"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:94"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:95"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s1'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s2'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare"><span class="id" title="definition">N.compare</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#compare:96"><span class="id" title="definition">compare</span></a> <span class="id" title="var">s1'</span> <span class="id" title="var">s2'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Auxiliary functions for inserting at front and merging intervals 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.merge_interval_size" class="idref" href="#Ops.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> (<a id="x1:97" class="idref" href="#x1:97"><span class="id" title="binder">x1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c1:98" class="idref" href="#c1:98"><span class="id" title="binder">c1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="x2:99" class="idref" href="#x2:99"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:100" class="idref" href="#c2:100"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.max"><span class="id" title="definition">N.max</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:98"><span class="id" title="variable">c1</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x2:99"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:100"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:97"><span class="id" title="variable">x1</span></a>))).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.insert_interval_begin" class="idref" href="#Ops.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> (<a id="x:101" class="idref" href="#x:101"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c:102" class="idref" href="#c:102"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="l:103" class="idref" href="#l:103"><span class="id" title="binder">l</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:103"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:101"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c:102"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:101"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:102"><span class="id" title="variable">c</span></a>) <span class="id" title="var">y</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:101"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:102"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:103"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:101"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c:102"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c'</span>)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#insert_interval_begin:104"><span class="id" title="definition">insert_interval_begin</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:101"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:101"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:102"><span class="id" title="variable">c</span></a> <span class="id" title="var">y</span> <span class="id" title="var">c'</span>) <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+adding an element needs to be defined carefully again in order to
+      generate efficient code 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.addZ_aux" class="idref" href="#Ops.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> (<a id="acc:106" class="idref" href="#acc:106"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="x:107" class="idref" href="#x:107"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="s:108" class="idref" href="#s:108"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:108"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:107"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:106"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:107"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:107"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:106"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:108"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:107"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:106"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#addZ_aux:109"><span class="id" title="definition">addZ_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:106"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#x:107"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:106"><span class="id" title="variable">acc</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span>) <span class="id" title="var">l</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:106"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.addZ" class="idref" href="#Ops.addZ"><span class="id" title="definition">addZ</span></a> <a id="x:111" class="idref" href="#x:111"><span class="id" title="binder">x</span></a> <a id="s:112" class="idref" href="#s:112"><span class="id" title="binder">s</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:111"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:112"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.add" class="idref" href="#Ops.add"><span class="id" title="definition">add</span></a> <a id="x:113" class="idref" href="#x:113"><span class="id" title="binder">x</span></a> <a id="s:114" class="idref" href="#s:114"><span class="id" title="binder">s</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.addZ"><span class="id" title="definition">addZ</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:113"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:114"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.add_list"><span class="id" title="definition">add_list</span></a></span> is a simple extension to add many elements.
+      This is used to define the function <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.from_elements"><span class="id" title="definition">from_elements</span></a></span>. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.add_list" class="idref" href="#Ops.add_list"><span class="id" title="definition">add_list</span></a> (<a id="l:115" class="idref" href="#l:115"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<a id="s:116" class="idref" href="#s:116"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<span class="id" title="keyword">fun</span> <a id="s:117" class="idref" href="#s:117"><span class="id" title="binder">s</span></a> <a id="x:118" class="idref" href="#x:118"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:118"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:117"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#l:115"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:116"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.from_elements" class="idref" href="#Ops.from_elements"><span class="id" title="definition">from_elements</span></a> (<a id="l:119" class="idref" href="#l:119"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.add_list"><span class="id" title="definition">add_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:119"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.empty"><span class="id" title="definition">empty</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.singleton"><span class="id" title="definition">singleton</span></a></span> is trivial to define 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.singleton" class="idref" href="#Ops.singleton"><span class="id" title="definition">singleton</span></a> (<a id="x:120" class="idref" href="#x:120"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:120"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Ops.singleton_alt_def" class="idref" href="#Ops.singleton_alt_def"><span class="id" title="lemma">singleton_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="x:121" class="idref" href="#x:121"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:121"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:121"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.empty"><span class="id" title="definition">empty</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">by</span> []. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+removing needs to be done with code extraction in mind again. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.insert_intervalZ_guarded" class="idref" href="#Ops.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> (<a id="x:122" class="idref" href="#x:122"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c:123" class="idref" href="#c:123"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) <a id="s:124" class="idref" href="#s:124"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:123"><span class="id" title="variable">c</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:124"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:122"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:123"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:124"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.removeZ_aux" class="idref" href="#Ops.removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> (<a id="acc:125" class="idref" href="#acc:125"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="x:126" class="idref" href="#x:126"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="s:127" class="idref" href="#s:127"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:127"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:125"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:126"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:125"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:127"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:126"><span class="id" title="variable">x</span></a> (<span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:126"><span class="id" title="variable">x</span></a>) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:126"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)</span></a>)) <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:126"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a><span class="id" title="var">y</span>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:125"><span class="id" title="variable">acc</span></a>)) <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#removeZ_aux:128"><span class="id" title="definition">removeZ_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:125"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#x:126"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.removeZ" class="idref" href="#Ops.removeZ"><span class="id" title="definition">removeZ</span></a> (<a id="x:130" class="idref" href="#x:130"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="s:131" class="idref" href="#s:131"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:130"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:131"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.remove" class="idref" href="#Ops.remove"><span class="id" title="definition">remove</span></a> (<a id="x:132" class="idref" href="#x:132"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<a id="s:133" class="idref" href="#s:133"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.removeZ"><span class="id" title="definition">removeZ</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:132"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:133"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.remove_list" class="idref" href="#Ops.remove_list"><span class="id" title="definition">remove_list</span></a> (<a id="l:134" class="idref" href="#l:134"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<a id="s:135" class="idref" href="#s:135"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">List.fold_left</span></a> (<span class="id" title="keyword">fun</span> <a id="s:136" class="idref" href="#s:136"><span class="id" title="binder">s</span></a> <a id="x:137" class="idref" href="#x:137"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:137"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:136"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#l:134"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:135"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+union 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.union_aux" class="idref" href="#Ops.union_aux"><span class="id" title="definition">union_aux</span></a> (<a id="s1:138" class="idref" href="#s1:138"><span class="id" title="binder">s1</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">aux</span> (<a id="s2:140" class="idref" href="#s2:140"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) (<a id="acc:141" class="idref" href="#acc:141"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:138"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:140"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:140"><span class="id" title="variable">s2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:138"><span class="id" title="variable">s1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:140"><span class="id" title="variable">s2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> ((<span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c2</span>)%<span class="id" title="var">N</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#aux:142"><span class="id" title="variable">aux</span></a> <span class="id" title="var">l2</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> ((<span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c2</span>)%<span class="id" title="var">N</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:140"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#union_aux:139"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:140"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#aux:142"><span class="id" title="variable">aux</span></a> <span class="id" title="var">l2</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:141"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.union" class="idref" href="#Ops.union"><span class="id" title="definition">union</span></a> <a id="s1:143" class="idref" href="#s1:143"><span class="id" title="binder">s1</span></a> <a id="s2:144" class="idref" href="#s2:144"><span class="id" title="binder">s2</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.union_aux"><span class="id" title="definition">union_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:143"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:144"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+diff 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.diff_aux" class="idref" href="#Ops.diff_aux"><span class="id" title="definition">diff_aux</span></a> (<a id="y2:145" class="idref" href="#y2:145"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:146" class="idref" href="#c2:146"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="acc:147" class="idref" href="#acc:147"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="s:148" class="idref" href="#s:148"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:148"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:147"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
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+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:145"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c2:146"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
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+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <span class="id" title="var">y1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#y2:145"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:147"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#y2:145"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:146"><span class="id" title="variable">c2</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:145"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:146"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)</span></a>)) <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.diff_aux2" class="idref" href="#Ops.diff_aux2"><span class="id" title="definition">diff_aux2</span></a> (<a id="acc:151" class="idref" href="#acc:151"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="s1:152" class="idref" href="#s1:152"><span class="id" title="binder">s1</span></a> <a id="s2:153" class="idref" href="#s2:153"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:152"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:153"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:151"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:152"><span class="id" title="variable">s1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:151"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:152"><span class="id" title="variable">s1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:151"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:152"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s1'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#diff_aux2:154"><span class="id" title="definition">diff_aux2</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">s1'</span> <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.diff" class="idref" href="#Ops.diff"><span class="id" title="definition">diff</span></a> <a id="s1:155" class="idref" href="#s1:155"><span class="id" title="binder">s1</span></a> <a id="s2:156" class="idref" href="#s2:156"><span class="id" title="binder">s2</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.diff_aux2"><span class="id" title="definition">diff_aux2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:155"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:156"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+subset 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.subset" class="idref" href="#Ops.subset"><span class="id" title="definition">subset</span></a> (<a id="s1:157" class="idref" href="#s1:157"><span class="id" title="binder">s1</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">aux</span> (<a id="s2:159" class="idref" href="#s2:159"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:157"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:159"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#aux:160"><span class="id" title="variable">aux</span></a> <span class="id" title="var">l2</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#subset:158"><span class="id" title="definition">subset</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#subset:158"><span class="id" title="definition">subset</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:159"><span class="id" title="variable">s2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+equal 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.equal" class="idref" href="#Ops.equal"><span class="id" title="definition">equal</span></a> (<a id="s:161" class="idref" href="#s:161"><span class="id" title="binder">s</span></a> <a id="s':162" class="idref" href="#s':162"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:161"><span class="id" title="variable">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#s':162"><span class="id" title="variable">s'</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">cx</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">xs</span>), (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">cy</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">ys</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eqb"><span class="id" title="definition">Z.eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <span class="id" title="var">cx</span> <span class="id" title="var">cy</span>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#equal:163"><span class="id" title="definition">equal</span></a> <span class="id" title="var">xs</span> <span class="id" title="var">ys</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+inter 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.inter_aux" class="idref" href="#Ops.inter_aux"><span class="id" title="definition">inter_aux</span></a> (<a id="y2:166" class="idref" href="#y2:166"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:167" class="idref" href="#c2:167"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="acc:168" class="idref" href="#acc:168"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="s:169" class="idref" href="#s:169"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:169"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l1</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#inter_aux:170"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a> <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#inter_aux:170"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a> <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:169"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:169"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#inter_aux:170"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:169"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#inter_aux:170"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">l1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:166"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:167"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:168"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:169"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.inter_aux2" class="idref" href="#Ops.inter_aux2"><span class="id" title="definition">inter_aux2</span></a> (<a id="acc:172" class="idref" href="#acc:172"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="s1:173" class="idref" href="#s1:173"><span class="id" title="binder">s1</span></a> <a id="s2:174" class="idref" href="#s2:174"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:173"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:174"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:172"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:172"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.inter_aux"><span class="id" title="definition">inter_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:172"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:173"><span class="id" title="variable">s1</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s1'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#inter_aux2:175"><span class="id" title="definition">inter_aux2</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">s1'</span> <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.inter" class="idref" href="#Ops.inter"><span class="id" title="definition">inter</span></a> <a id="s1:176" class="idref" href="#s1:176"><span class="id" title="binder">s1</span></a> <a id="s2:177" class="idref" href="#s2:177"><span class="id" title="binder">s2</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.inter_aux2"><span class="id" title="definition">inter_aux2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:176"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:177"><span class="id" title="variable">s2</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Partition and filter 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partitionZ_fold_insert" class="idref" href="#Ops.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cur:178" class="idref" href="#cur:178"><span class="id" title="binder">cur</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) (<a id="x:179" class="idref" href="#x:179"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#cur:178"><span class="id" title="variable">cur</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:179"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partitionZ_fold_skip" class="idref" href="#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> (<a id="acc:181" class="idref" href="#acc:181"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a id="cur:182" class="idref" href="#cur:182"><span class="id" title="binder">cur</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#cur:182"><span class="id" title="variable">cur</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:181"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">yc</span> ⇒ <span class="id" title="var">yc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:181"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partitionZ_fold_fun" class="idref" href="#Ops.partitionZ_fold_fun"><span class="id" title="definition">partitionZ_fold_fun</span></a> <a id="f:184" class="idref" href="#f:184"><span class="id" title="binder">f</span></a> <a id="st:185" class="idref" href="#st:185"><span class="id" title="binder">st</span></a> (<a id="x:186" class="idref" href="#x:186"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#st:185"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:184"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:186"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> <span class="id" title="var">c_t</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:186"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <span class="id" title="var">acc_f</span> <span class="id" title="var">c_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <span class="id" title="var">acc_t</span> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> <span class="id" title="var">c_f</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:186"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partitionZ_single_aux" class="idref" href="#Ops.partitionZ_single_aux"><span class="id" title="definition">partitionZ_single_aux</span></a> <a id="f:188" class="idref" href="#f:188"><span class="id" title="binder">f</span></a> <a id="st:189" class="idref" href="#st:189"><span class="id" title="binder">st</span></a> (<a id="x:190" class="idref" href="#x:190"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c:191" class="idref" href="#c:191"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> (<span class="id" title="keyword">fun</span> <a id="st:192" class="idref" href="#st:192"><span class="id" title="binder">st</span></a> <a id="x:193" class="idref" href="#x:193"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_fun"><span class="id" title="definition">partitionZ_fold_fun</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:188"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#st:192"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:193"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#st:189"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:190"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:191"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partitionZ_single" class="idref" href="#Ops.partitionZ_single"><span class="id" title="definition">partitionZ_single</span></a> <a id="f:194" class="idref" href="#f:194"><span class="id" title="binder">f</span></a> <a id="acc_t:195" class="idref" href="#acc_t:195"><span class="id" title="binder">acc_t</span></a> <a id="acc_f:196" class="idref" href="#acc_f:196"><span class="id" title="binder">acc_f</span></a> <a id="x:197" class="idref" href="#x:197"><span class="id" title="binder">x</span></a> <a id="c:198" class="idref" href="#c:198"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_single_aux"><span class="id" title="definition">partitionZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:194"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:195"><span class="id" title="variable">acc_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:196"><span class="id" title="variable">acc_f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:197"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:198"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:195"><span class="id" title="variable">acc_t</span></a> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:196"><span class="id" title="variable">acc_f</span></a> <span class="id" title="var">c_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.partitionZ_aux" class="idref" href="#Ops.partitionZ_aux"><span class="id" title="definition">partitionZ_aux</span></a> <a id="acc_t:199" class="idref" href="#acc_t:199"><span class="id" title="binder">acc_t</span></a> <a id="acc_f:200" class="idref" href="#acc_f:200"><span class="id" title="binder">acc_f</span></a> <a id="f:201" class="idref" href="#f:201"><span class="id" title="binder">f</span></a> <a id="s:202" class="idref" href="#s:202"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:202"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:199"><span class="id" title="variable">acc_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:200"><span class="id" title="variable">acc_f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_single"><span class="id" title="definition">partitionZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:201"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:199"><span class="id" title="variable">acc_t</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:200"><span class="id" title="variable">acc_f</span></a> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">acc_f'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#partitionZ_aux:203"><span class="id" title="definition">partitionZ_aux</span></a> <span class="id" title="var">acc_t'</span> <span class="id" title="var">acc_f'</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:201"><span class="id" title="variable">f</span></a> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partitionZ" class="idref" href="#Ops.partitionZ"><span class="id" title="definition">partitionZ</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_aux"><span class="id" title="definition">partitionZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partition" class="idref" href="#Ops.partition"><span class="id" title="definition">partition</span></a> (<a id="f:205" class="idref" href="#f:205"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ"><span class="id" title="definition">partitionZ</span></a> (<span class="id" title="keyword">fun</span> <a id="z:206" class="idref" href="#z:206"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#f:205"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:206"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.filterZ_fold_fun" class="idref" href="#Ops.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a> <a id="f:207" class="idref" href="#f:207"><span class="id" title="binder">f</span></a> <a id="st:208" class="idref" href="#st:208"><span class="id" title="binder">st</span></a> (<a id="x:209" class="idref" href="#x:209"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#st:208"><span class="id" title="variable">st</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:207"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:209"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> <span class="id" title="var">c_t</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:209"><span class="id" title="variable">x</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <span class="id" title="var">acc_t</span> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.filterZ_single_aux" class="idref" href="#Ops.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <a id="f:211" class="idref" href="#f:211"><span class="id" title="binder">f</span></a> <a id="st:212" class="idref" href="#st:212"><span class="id" title="binder">st</span></a> (<a id="x:213" class="idref" href="#x:213"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c:214" class="idref" href="#c:214"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> (<span class="id" title="keyword">fun</span> <a id="st:215" class="idref" href="#st:215"><span class="id" title="binder">st</span></a> <a id="x:216" class="idref" href="#x:216"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:211"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#st:215"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:216"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#st:212"><span class="id" title="variable">st</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:213"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:214"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.filterZ_single" class="idref" href="#Ops.filterZ_single"><span class="id" title="definition">filterZ_single</span></a> <a id="f:217" class="idref" href="#f:217"><span class="id" title="binder">f</span></a> <a id="acc:218" class="idref" href="#acc:218"><span class="id" title="binder">acc</span></a> <a id="x:219" class="idref" href="#x:219"><span class="id" title="binder">x</span></a> <a id="c:220" class="idref" href="#c:220"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:217"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:218"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:219"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:220"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:218"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:220"><span class="id" title="variable">c</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.filterZ_aux" class="idref" href="#Ops.filterZ_aux"><span class="id" title="definition">filterZ_aux</span></a> <a id="acc:221" class="idref" href="#acc:221"><span class="id" title="binder">acc</span></a> <a id="f:222" class="idref" href="#f:222"><span class="id" title="binder">f</span></a> <a id="s:223" class="idref" href="#s:223"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:223"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:221"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#filterZ_aux:224"><span class="id" title="definition">filterZ_aux</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.filterZ_single"><span class="id" title="definition">filterZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:222"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:221"><span class="id" title="variable">acc</span></a> <span class="id" title="var">y</span> <span class="id" title="var">c</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#f:222"><span class="id" title="variable">f</span></a> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.filterZ" class="idref" href="#Ops.filterZ"><span class="id" title="definition">filterZ</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.filterZ_aux"><span class="id" title="definition">filterZ_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.filter" class="idref" href="#Ops.filter"><span class="id" title="definition">filter</span></a> (<a id="f:226" class="idref" href="#f:226"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.filterZ"><span class="id" title="definition">filterZ</span></a> (<span class="id" title="keyword">fun</span> <a id="z:227" class="idref" href="#z:227"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#f:226"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:227"><span class="id" title="variable">z</span></a>)).<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Simple wrappers 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.fold" class="idref" href="#Ops.fold"><span class="id" title="definition">fold</span></a> {<a id="B:228" class="idref" href="#B:228"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>} (<a id="f:229" class="idref" href="#f:229"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:228"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:228"><span class="id" title="variable">B</span></a>) (<a id="s:230" class="idref" href="#s:230"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) (<a id="i:231" class="idref" href="#i:231"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#B:228"><span class="id" title="variable">B</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#B:228"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> (<span class="id" title="keyword">fun</span> <a id="b:232" class="idref" href="#b:232"><span class="id" title="binder">b</span></a> <a id="z:233" class="idref" href="#z:233"><span class="id" title="binder">z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:229"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:233"><span class="id" title="variable">z</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#b:232"><span class="id" title="variable">b</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#i:231"><span class="id" title="variable">i</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:230"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.for_all" class="idref" href="#Ops.for_all"><span class="id" title="definition">for_all</span></a> (<a id="f:234" class="idref" href="#f:234"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:235" class="idref" href="#s:235"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> (<span class="id" title="keyword">fun</span> <a id="b:236" class="idref" href="#b:236"><span class="id" title="binder">b</span></a> <a id="z:237" class="idref" href="#z:237"><span class="id" title="binder">z</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#b:236"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:234"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:237"><span class="id" title="variable">z</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:235"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.exists_" class="idref" href="#Ops.exists_"><span class="id" title="definition">exists_</span></a> (<a id="f:238" class="idref" href="#f:238"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:239" class="idref" href="#s:239"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> (<span class="id" title="keyword">fun</span> <a id="b:240" class="idref" href="#b:240"><span class="id" title="binder">b</span></a> <a id="z:241" class="idref" href="#z:241"><span class="id" title="binder">z</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#b:240"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:238"><span class="id" title="variable">f</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:241"><span class="id" title="variable">z</span></a>))) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:239"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.cardinalN" class="idref" href="#Ops.cardinalN"><span class="id" title="definition">cardinalN</span></a> <a id="c:242" class="idref" href="#c:242"><span class="id" title="binder">c</span></a> (<a id="s:243" class="idref" href="#s:243"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:243"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#c:242"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">cx</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">xs</span> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#cardinalN:244"><span class="id" title="definition">cardinalN</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c:242"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <span class="id" title="var">cx</span>)%<span class="id" title="var">N</span> <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.cardinal" class="idref" href="#Ops.cardinal"><span class="id" title="definition">cardinal</span></a> (<a id="s:246" class="idref" href="#s:246"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.cardinalN"><span class="id" title="definition">cardinalN</span></a> (0%<span class="id" title="var">N</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:246"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.min_eltZ" class="idref" href="#Ops.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> (<a id="s:247" class="idref" href="#s:247"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:247"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.min_elt" class="idref" href="#Ops.min_elt"><span class="id" title="definition">min_elt</span></a> (<a id="s:249" class="idref" href="#s:249"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:249"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.choose" class="idref" href="#Ops.choose"><span class="id" title="definition">choose</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.min_elt"><span class="id" title="definition">min_elt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.max_eltZ" class="idref" href="#Ops.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> (<a id="s:250" class="idref" href="#s:250"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:250"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.pred"><span class="id" title="definition">Z.pred</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s'</span> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#max_eltZ:251"><span class="id" title="definition">max_eltZ</span></a> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.max_elt" class="idref" href="#Ops.max_elt"><span class="id" title="definition">max_elt</span></a> (<a id="s:253" class="idref" href="#s:253"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:253"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Ops"><span class="id" title="module">Ops</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab23"></a><h2 class="section">Raw Module</h2>
+
+
+<div class="paragraph"> </div>
+
+    Following the idea of <span class="inlinecode"><span class="id" title="var">MSetInterface.RawSets</span></span>, we first
+    define a module <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw"><span class="id" title="module">Raw</span></a></span> proves all the required properties with
+    respect to an explicitly provided invariant. In a next step, this
+    invariant is then moved into the set type. This allows to instantiate
+    the <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span> interface. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Raw" class="idref" href="#Raw"><span class="id" title="module">Raw</span></a> (<span class="id" title="var">Enc</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Ops"><span class="id" title="module">Ops</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc"><span class="id" title="module">Enc</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab24"></a><h3 class="section">Defining invariant <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a></span></h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.is_encoded_elems_list" class="idref" href="#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a id="l:254" class="idref" href="#l:254"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="x:255" class="idref" href="#x:255"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:255"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:254"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="e:256" class="idref" href="#e:256"><span class="id" title="binder">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:256"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:255"><span class="id" title="variable">x</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.interval_list_elements_greater" class="idref" href="#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> (<a id="x:257" class="idref" href="#x:257"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="l:258" class="idref" href="#l:258"><span class="id" title="binder">l</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:258"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb"><span class="id" title="definition">Z.ltb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:257"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Raw.interval_list_invariant" class="idref" href="#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a id="l:260" class="idref" href="#l:260"><span class="id" title="binder">l</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:260"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l'</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">)</span></a>) <span class="id" title="var">l'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <span class="id" title="var">c</span> 0) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::bool_scope:x_'&amp;&amp;'_x"><span class="id" title="notation">&amp;&amp;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#interval_list_invariant:261"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">l'</span><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.IsOk" class="idref" href="#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <a id="s:263" class="idref" href="#s:263"><span class="id" title="binder">s</span></a> := (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:263"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:263"><span class="id" title="variable">s</span></a>)).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab25"></a><h3 class="section">Defining notations</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="Raw.ForNotations" class="idref" href="#Raw.ForNotations"><span class="id" title="section">ForNotations</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Class</span> <a id="Raw.Ok" class="idref" href="#Raw.Ok"><span class="id" title="record">Ok</span></a> (<a id="s:264" class="idref" href="#s:264"><span class="id" title="binder">s</span></a>:<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) : <span class="id" title="keyword">Prop</span> := <a id="Raw.ok" class="idref" href="#Raw.ok"><span class="id" title="projection">ok</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:264"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Instance</span> <a id="Raw.IsOk_Ok" class="idref" href="#Raw.IsOk_Ok"><span class="id" title="instance">IsOk_Ok</span></a> <a id="s:267" class="idref" href="#s:267"><span class="id" title="binder">s</span></a> `(<a id="Hs:268" class="idref" href="#Hs:268"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:267"><span class="id" title="variable">s</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:267"><span class="id" title="variable">s</span></a> := { <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ok"><span class="id" title="definition">ok</span></a> := <span class="id" title="var">Hs</span> }.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.In" class="idref" href="#Raw.In"><span class="id" title="definition">In</span></a> <a id="x:269" class="idref" href="#x:269"><span class="id" title="binder">x</span></a> <a id="s:270" class="idref" href="#s:270"><span class="id" title="binder">s</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">SetoidList.InA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:269"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:270"><span class="id" title="variable">s</span></a>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.InZ" class="idref" href="#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a id="x:271" class="idref" href="#x:271"><span class="id" title="binder">x</span></a> <a id="s:272" class="idref" href="#s:272"><span class="id" title="binder">s</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:271"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:272"><span class="id" title="variable">s</span></a>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.Equal" class="idref" href="#Raw.Equal"><span class="id" title="definition">Equal</span></a> <a id="s:273" class="idref" href="#s:273"><span class="id" title="binder">s</span></a> <a id="s':274" class="idref" href="#s':274"><span class="id" title="binder">s'</span></a> := <span class="id" title="keyword">∀</span> <a id="a:275" class="idref" href="#a:275"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:275"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:273"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:275"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':274"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.Subset" class="idref" href="#Raw.Subset"><span class="id" title="definition">Subset</span></a> <a id="s:276" class="idref" href="#s:276"><span class="id" title="binder">s</span></a> <a id="s':277" class="idref" href="#s':277"><span class="id" title="binder">s'</span></a> := <span class="id" title="keyword">∀</span> <a id="a:278" class="idref" href="#a:278"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:278"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:276"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:278"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':277"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.Empty" class="idref" href="#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a id="s:279" class="idref" href="#s:279"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">∀</span> <a id="a:280" class="idref" href="#a:280"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:280"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:279"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.For_all" class="idref" href="#Raw.For_all"><span class="id" title="definition">For_all</span></a> (<a id="P:281" class="idref" href="#P:281"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="s:282" class="idref" href="#s:282"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">∀</span> <a id="x:283" class="idref" href="#x:283"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:283"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:282"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#P:281"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:283"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.Exists" class="idref" href="#Raw.Exists"><span class="id" title="definition">Exists</span></a> (<a id="P:284" class="idref" href="#P:284"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="s:285" class="idref" href="#s:285"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:286" class="idref" href="#x:286"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:286"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:285"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#P:284"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:286"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ForNotations"><span class="id" title="section">ForNotations</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab26"></a><h3 class="section">elements list properties</h3>
+
+
+<div class="paragraph"> </div>
+
+      The functions <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Ops.elementsZ"><span class="id" title="definition">elementsZ</span></a></span>, <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a></span>, 
+      <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a></span> and <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_single"><span class="id" title="definition">elements_single</span></a></span> are crucial and used
+      everywhere. Therefore, we first establish a few properties of
+      these important functions. 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_nil" class="idref" href="#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elements_nil" class="idref" href="#Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.elementsZ_single" class="idref" href="#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> (<a id="x:287" class="idref" href="#x:287"><span class="id" title="binder">x</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c:288" class="idref" href="#c:288"><span class="id" title="binder">c</span></a>:<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_rec"><span class="id" title="definition">N.peano_rec</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<span class="id" title="keyword">fun</span> <a id="n:289" class="idref" href="#n:289"><span class="id" title="binder">n</span></a> <a id="ls:290" class="idref" href="#ls:290"><span class="id" title="binder">ls</span></a> ⇒ (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:287"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:289"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ls:290"><span class="id" title="variable">ls</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#c:288"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.elements_single" class="idref" href="#Raw.elements_single"><span class="id" title="definition">elements_single</span></a> <a id="x:291" class="idref" href="#x:291"><span class="id" title="binder">x</span></a> <a id="c:292" class="idref" href="#c:292"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">List.map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:291"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:292"><span class="id" title="variable">c</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_single_base" class="idref" href="#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> : <span class="id" title="keyword">∀</span> <a id="x:293" class="idref" href="#x:293"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:293"><span class="id" title="variable">x</span></a> (0%<span class="id" title="var">N</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_single_succ" class="idref" href="#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> : <span class="id" title="keyword">∀</span> <a id="x:294" class="idref" href="#x:294"><span class="id" title="binder">x</span></a> <a id="c:295" class="idref" href="#c:295"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:294"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:295"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:294"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:295"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:294"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:295"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_rec_succ"><span class="id" title="lemma">N.peano_rec_succ</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_alt"><span class="id" title="lemma">rev_alt</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_single_add" class="idref" href="#Raw.elementsZ_single_add"><span class="id" title="lemma">elementsZ_single_add</span></a> : <span class="id" title="keyword">∀</span> <a id="x:296" class="idref" href="#x:296"><span class="id" title="binder">x</span></a> <a id="c2:297" class="idref" href="#c2:297"><span class="id" title="binder">c2</span></a> <a id="c1:298" class="idref" href="#c1:298"><span class="id" title="binder">c1</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:296"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c1:298"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:297"><span class="id" title="variable">c2</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:296"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:298"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:296"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:298"><span class="id" title="variable">c1</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:297"><span class="id" title="variable">c2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c2</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c2'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">c1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.add_0_r"><span class="id" title="lemma">N.add_0_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">c1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.add_succ_r"><span class="id" title="lemma">N.add_succ_r</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> <span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_add"><span class="id" title="lemma">N2Z.inj_add</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_assoc"><span class="id" title="lemma">Z.add_assoc</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_single_succ_front" class="idref" href="#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a> : <span class="id" title="keyword">∀</span> <a id="x:299" class="idref" href="#x:299"><span class="id" title="binder">x</span></a> <a id="c:300" class="idref" href="#c:300"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:299"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:300"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#x:299"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:299"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#c:300"><span class="id" title="variable">c</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.add_1_l"><span class="id" title="lemma">N.add_1_l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_add"><span class="id" title="lemma">elementsZ_single_add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.one_succ"><span class="id" title="lemma">N.one_succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_0_r"><span class="id" title="lemma">Z.add_0_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_elementsZ_single" class="idref" href="#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> : <span class="id" title="keyword">∀</span> <a id="c:301" class="idref" href="#c:301"><span class="id" title="binder">c</span></a> <a id="y:302" class="idref" href="#y:302"><span class="id" title="binder">y</span></a> <a id="x:303" class="idref" href="#x:303"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:302"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:303"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:301"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:303"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:302"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:302"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:303"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:301"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_0_r"><span class="id" title="lemma">Z.add_0_r</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> <span class="id" title="var">IH</span> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_succ"><span class="id" title="lemma">N2Z.inj_succ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_succ_r"><span class="id" title="lemma">Z.add_succ_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [ | []] //. { <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_x_le</span> <span class="id" title="var">H_y_le</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_x_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_elementsZ_single1" class="idref" href="#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a> : <span class="id" title="keyword">∀</span> <a id="y:304" class="idref" href="#y:304"><span class="id" title="binder">y</span></a> <a id="x:305" class="idref" href="#x:305"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:304"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:305"><span class="id" title="variable">x</span></a> (1%<span class="id" title="var">N</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:305"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:304"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.length_elementsZ_single" class="idref" href="#Raw.length_elementsZ_single"><span class="id" title="lemma">length_elementsZ_single</span></a> : <span class="id" title="keyword">∀</span> <a id="cx:306" class="idref" href="#cx:306"><span class="id" title="binder">cx</span></a> <a id="x:307" class="idref" href="#x:307"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:307"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#cx:306"><span class="id" title="variable">cx</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.to_nat"><span class="id" title="definition">N.to_nat</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#cx:306"><span class="id" title="variable">cx</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">cx</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">cx'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.Nnat.html#N2Nat.inj_succ"><span class="id" title="lemma">N2Nat.inj_succ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_aux_irrel" class="idref" href="#Raw.fold_elementsZ_aux_irrel"><span class="id" title="lemma">fold_elementsZ_aux_irrel</span></a> {<a id="A:308" class="idref" href="#A:308"><span class="id" title="binder">A</span></a>} :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:309" class="idref" href="#f:309"><span class="id" title="binder">f</span></a> <a id="c:310" class="idref" href="#c:310"><span class="id" title="binder">c</span></a> (<a id="acc:311" class="idref" href="#acc:311"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:308"><span class="id" title="variable">A</span></a>) <a id="x:312" class="idref" href="#x:312"><span class="id" title="binder">x</span></a> <a id="H1:313" class="idref" href="#H1:313"><span class="id" title="binder">H1</span></a> <a id="H2:314" class="idref" href="#H2:314"><span class="id" title="binder">H2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:309"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:311"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:312"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:310"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#H1:313"><span class="id" title="variable">H1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:309"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:311"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:312"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:310"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#H2:314"><span class="id" title="variable">H2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">c</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Wf.html#well_founded_ind"><span class="id" title="lemma">well_founded_ind</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#NOP.lt_wf_0"><span class="id" title="lemma">lt_wf_0</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> <span class="id" title="var">c</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">H_c</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">case</span>; <span class="id" title="tactic">intro</span> <span class="id" title="var">H_H1</span>; <span class="id" title="tactic">case</span>; <span class="id" title="tactic">intro</span> <span class="id" title="var">H_H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">p</span> <span class="id" title="var">H_c</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>; <span class="id" title="tactic">case</span>; <span class="id" title="tactic">intro</span> <span class="id" title="var">H_H1</span>; <span class="id" title="tactic">case</span>; <span class="id" title="tactic">intro</span> <span class="id" title="var">H_H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a>; <span class="id" title="tactic">fold</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a> <span class="id" title="var">A</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>) ⇒ // <span class="id" title="var">acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt_pred_l"><span class="id" title="lemma">N.lt_pred_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">discriminate</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_single_pos" class="idref" href="#Raw.fold_elementsZ_single_pos"><span class="id" title="lemma">fold_elementsZ_single_pos</span></a> {<a id="A:315" class="idref" href="#A:315"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:316" class="idref" href="#f:316"><span class="id" title="binder">f</span></a> (<a id="acc:317" class="idref" href="#acc:317"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:315"><span class="id" title="variable">A</span></a>) <a id="x:318" class="idref" href="#x:318"><span class="id" title="binder">x</span></a> <a id="p:319" class="idref" href="#p:319"><span class="id" title="binder">p</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:316"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:317"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:318"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#p:319"><span class="id" title="variable">p</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:316"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:317"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:318"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">acc'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:316"><span class="id" title="variable">f</span></a> <span class="id" title="var">acc'</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:318"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.pred"><span class="id" title="definition">N.pred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.pos"><span class="id" title="abbreviation">N.pos</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#p:319"><span class="id" title="variable">p</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:317"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span> <span class="id" title="var">p</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span>: (<a class="idref" href="MSetsExtra.MSetIntervals.html#NOP.lt_wf_0"><span class="id" title="lemma">lt_wf_0</span></a> <span class="id" title="var">_</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">fold</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a> <span class="id" title="var">A</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>) ⇒ // <span class="id" title="var">acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux_irrel"><span class="id" title="lemma">fold_elementsZ_aux_irrel</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_single_zero" class="idref" href="#Raw.fold_elementsZ_single_zero"><span class="id" title="lemma">fold_elementsZ_single_zero</span></a> {<a id="A:320" class="idref" href="#A:320"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:321" class="idref" href="#f:321"><span class="id" title="binder">f</span></a> (<a id="acc:322" class="idref" href="#acc:322"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:320"><span class="id" title="variable">A</span></a>) <a id="x:323" class="idref" href="#x:323"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:321"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:322"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:323"><span class="id" title="variable">x</span></a> (0%<span class="id" title="var">N</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:322"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#NOP.lt_wf_0"><span class="id" title="lemma">lt_wf_0</span></a> (0%<span class="id" title="var">N</span>)); <span class="id" title="tactic">intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux"><span class="id" title="definition">fold_elementsZ_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_single_succ" class="idref" href="#Raw.fold_elementsZ_single_succ"><span class="id" title="lemma">fold_elementsZ_single_succ</span></a> {<a id="A:324" class="idref" href="#A:324"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:325" class="idref" href="#f:325"><span class="id" title="binder">f</span></a> (<a id="acc:326" class="idref" href="#acc:326"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:324"><span class="id" title="variable">A</span></a>) <a id="x:327" class="idref" href="#x:327"><span class="id" title="binder">x</span></a> <a id="c:328" class="idref" href="#c:328"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:325"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:326"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:327"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:328"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:325"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:326"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:327"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">acc'</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:325"><span class="id" title="variable">f</span></a> <span class="id" title="var">acc'</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:327"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#c:328"><span class="id" title="variable">c</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:326"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">c</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_pos"><span class="id" title="lemma">fold_elementsZ_single_pos</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">p</span>; <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_pos"><span class="id" title="lemma">fold_elementsZ_single_pos</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>) ⇒ // <span class="id" title="var">acc'</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.PArith.BinPos.html#Pos.pred_N_succ"><span class="id" title="lemma">Pos.pred_N_succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Raw.fold_opt" class="idref" href="#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> {<a id="A:329" class="idref" href="#A:329"><span class="id" title="binder">A</span></a> <a id="B:330" class="idref" href="#B:330"><span class="id" title="binder">B</span></a>} <a id="f:331" class="idref" href="#f:331"><span class="id" title="binder">f</span></a> (<a id="acc:332" class="idref" href="#acc:332"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:329"><span class="id" title="variable">A</span></a>) (<a id="bs:333" class="idref" href="#bs:333"><span class="id" title="binder">bs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:330"><span class="id" title="variable">B</span></a>) : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:329"><span class="id" title="variable">A</span></a>) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#bs:333"><span class="id" title="variable">bs</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:332"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| (<span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">bs'</span>) ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:331"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:332"><span class="id" title="variable">acc</span></a> <span class="id" title="var">b</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">acc'</span> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#fold_opt:334"><span class="id" title="definition">fold_opt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:331"><span class="id" title="variable">f</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">bs'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:332"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_opt_list_cons" class="idref" href="#Raw.fold_opt_list_cons"><span class="id" title="lemma">fold_opt_list_cons</span></a> : <span class="id" title="keyword">∀</span> {<a id="A:336" class="idref" href="#A:336"><span class="id" title="binder">A</span></a>} (<a id="bs:337" class="idref" href="#bs:337"><span class="id" title="binder">bs</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:336"><span class="id" title="variable">A</span></a>) (<a id="acc:338" class="idref" href="#acc:338"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:336"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> (<span class="id" title="keyword">fun</span> <a id="l:339" class="idref" href="#l:339"><span class="id" title="binder">l</span></a> <a id="x:340" class="idref" href="#x:340"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:340"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:339"><span class="id" title="variable">l</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:338"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#bs:337"><span class="id" title="variable">bs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#bs:337"><span class="id" title="variable">bs</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:338"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">bs</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">b</span> <span class="id" title="var">bs'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_opt_app" class="idref" href="#Raw.fold_opt_app"><span class="id" title="lemma">fold_opt_app</span></a> {<a id="A:341" class="idref" href="#A:341"><span class="id" title="binder">A</span></a> <a id="B:342" class="idref" href="#B:342"><span class="id" title="binder">B</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:343" class="idref" href="#f:343"><span class="id" title="binder">f</span></a> (<a id="acc:344" class="idref" href="#acc:344"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:341"><span class="id" title="variable">A</span></a>) (<a id="l1:345" class="idref" href="#l1:345"><span class="id" title="binder">l1</span></a> <a id="l2:346" class="idref" href="#l2:346"><span class="id" title="binder">l2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#B:342"><span class="id" title="variable">B</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:343"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:344"><span class="id" title="variable">acc</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#l1:345"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:346"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="keyword">let</span> (<a id="ab:347" class="idref" href="#ab:347"><span class="id" title="binder">ab</span></a>, <a id="acc':348" class="idref" href="#acc':348"><span class="id" title="binder">acc'</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:343"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:344"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l1:345"><span class="id" title="variable">l1</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ab:347"><span class="id" title="variable">ab</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc':348"><span class="id" title="variable">acc'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:343"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc':348"><span class="id" title="variable">acc'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:346"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l1</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">b</span> <span class="id" title="var">l1'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_l"><span class="id" title="lemma">app_nil_l</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">b</span>); <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_single_alt_def" class="idref" href="#Raw.fold_elementsZ_single_alt_def"><span class="id" title="lemma">fold_elementsZ_single_alt_def</span></a> {<a id="A:349" class="idref" href="#A:349"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:350" class="idref" href="#f:350"><span class="id" title="binder">f</span></a> <a id="c:351" class="idref" href="#c:351"><span class="id" title="binder">c</span></a> (<a id="acc:352" class="idref" href="#acc:352"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:349"><span class="id" title="variable">A</span></a>) <a id="x:353" class="idref" href="#x:353"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:350"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:352"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:353"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:351"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:350"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:352"><span class="id" title="variable">acc</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:353"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:351"><span class="id" title="variable">c</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_zero"><span class="id" title="lemma">fold_elementsZ_single_zero</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_succ"><span class="id" title="lemma">fold_elementsZ_single_succ</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_nil" class="idref" href="#Raw.fold_elementsZ_nil"><span class="id" title="lemma">fold_elementsZ_nil</span></a> {<a id="A:354" class="idref" href="#A:354"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:355" class="idref" href="#f:355"><span class="id" title="binder">f</span></a> (<a id="acc:356" class="idref" href="#acc:356"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:354"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:355"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:356"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:356"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_cons" class="idref" href="#Raw.fold_elementsZ_cons"><span class="id" title="lemma">fold_elementsZ_cons</span></a> {<a id="A:357" class="idref" href="#A:357"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:358" class="idref" href="#f:358"><span class="id" title="binder">f</span></a> (<a id="acc:359" class="idref" href="#acc:359"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:357"><span class="id" title="variable">A</span></a>) <a id="y:360" class="idref" href="#y:360"><span class="id" title="binder">y</span></a> <a id="c:361" class="idref" href="#c:361"><span class="id" title="binder">c</span></a> <a id="s:362" class="idref" href="#s:362"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:358"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:359"><span class="id" title="variable">acc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:360"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:361"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s:362"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><span class="id" title="keyword">let</span> (<a id="ab:363" class="idref" href="#ab:363"><span class="id" title="binder">ab</span></a>, <a id="acc':364" class="idref" href="#acc':364"><span class="id" title="binder">acc'</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single"><span class="id" title="definition">fold_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:358"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:359"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:360"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:361"><span class="id" title="variable">c</span></a> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ab:363"><span class="id" title="variable">ab</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc':364"><span class="id" title="variable">acc'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:358"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc':364"><span class="id" title="variable">acc'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:362"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_alt_def_aux" class="idref" href="#Raw.fold_elementsZ_alt_def_aux"><span class="id" title="lemma">fold_elementsZ_alt_def_aux</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:365" class="idref" href="#s:365"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="base:366" class="idref" href="#base:366"><span class="id" title="binder">base</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">fun</span> (<a id="l:367" class="idref" href="#l:367"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="x:368" class="idref" href="#x:368"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:368"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:367"><span class="id" title="variable">l</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#base:366"><span class="id" title="variable">base</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:365"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:365"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#base:366"><span class="id" title="variable">base</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">base</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_l"><span class="id" title="lemma">app_nil_l</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_cons"><span class="id" title="lemma">fold_elementsZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_alt_def"><span class="id" title="lemma">fold_elementsZ_single_alt_def</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_list_cons"><span class="id" title="lemma">fold_opt_list_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_elementsZ_alt_def" class="idref" href="#Raw.fold_elementsZ_alt_def"><span class="id" title="lemma">fold_elementsZ_alt_def</span></a> {<a id="A:369" class="idref" href="#A:369"><span class="id" title="binder">A</span></a>} : <span class="id" title="keyword">∀</span> <a id="f:370" class="idref" href="#f:370"><span class="id" title="binder">f</span></a> <a id="s:371" class="idref" href="#s:371"><span class="id" title="binder">s</span></a> (<a id="acc:372" class="idref" href="#acc:372"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:369"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ"><span class="id" title="definition">fold_elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:370"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:372"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:371"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:370"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:372"><span class="id" title="variable">acc</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:371"><span class="id" title="variable">s</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_cons"><span class="id" title="lemma">fold_elementsZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_alt_def"><span class="id" title="lemma">fold_elementsZ_single_alt_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_list_cons"><span class="id" title="lemma">fold_opt_list_cons</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def_aux"><span class="id" title="lemma">fold_elementsZ_alt_def_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_app_distr"><span class="id" title="lemma">rev_app_distr</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_involutive"><span class="id" title="lemma">rev_involutive</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_app"><span class="id" title="lemma">fold_opt_app</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt"><span class="id" title="definition">fold_opt</span></a> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_cons" class="idref" href="#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="x:373" class="idref" href="#x:373"><span class="id" title="binder">x</span></a> <a id="c:374" class="idref" href="#c:374"><span class="id" title="binder">c</span></a> <a id="s:375" class="idref" href="#s:375"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:373"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:374"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:375"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:375"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:373"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:374"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_cons"><span class="id" title="lemma">fold_elementsZ_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;!<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def"><span class="id" title="lemma">fold_elementsZ_alt_def</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_alt_def"><span class="id" title="lemma">fold_elementsZ_single_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_list_cons"><span class="id" title="lemma">fold_opt_list_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_involutive"><span class="id" title="lemma">rev_involutive</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def_aux"><span class="id" title="lemma">fold_elementsZ_alt_def_aux</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elements_cons" class="idref" href="#Raw.elements_cons"><span class="id" title="lemma">elements_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="x:376" class="idref" href="#x:376"><span class="id" title="binder">x</span></a> <a id="c:377" class="idref" href="#c:377"><span class="id" title="binder">c</span></a> <a id="s:378" class="idref" href="#s:378"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:376"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:377"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:378"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_single"><span class="id" title="definition">elements_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:376"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:377"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:378"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_single"><span class="id" title="definition">elements_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_app"><span class="id" title="lemma">map_app</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_app_distr"><span class="id" title="lemma">rev_app_distr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_rev"><span class="id" title="lemma">map_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_involutive"><span class="id" title="lemma">rev_involutive</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_app" class="idref" href="#Raw.elementsZ_app"><span class="id" title="lemma">elementsZ_app</span></a> : <span class="id" title="keyword">∀</span> (<a id="s1:379" class="idref" href="#s1:379"><span class="id" title="binder">s1</span></a> <a id="s2:380" class="idref" href="#s2:380"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:379"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:380"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:380"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:379"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1</span> <span class="id" title="var">IH1</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_comm_cons"><span class="id" title="lemma">app_comm_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> <span class="id" title="var">IH1</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_nil" class="idref" href="#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> : <span class="id" title="keyword">∀</span> <a id="y:381" class="idref" href="#y:381"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:381"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_cons" class="idref" href="#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="y:382" class="idref" href="#y:382"><span class="id" title="binder">y</span></a> <a id="x:383" class="idref" href="#x:383"><span class="id" title="binder">x</span></a> <a id="c:384" class="idref" href="#c:384"><span class="id" title="binder">c</span></a> <a id="s:385" class="idref" href="#s:385"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:382"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:383"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:384"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:385"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:382"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:383"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:384"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:382"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:385"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">firstorder</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_app" class="idref" href="#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:386" class="idref" href="#s1:386"><span class="id" title="binder">s1</span></a> <a id="s2:387" class="idref" href="#s2:387"><span class="id" title="binder">s2</span></a> <a id="y:388" class="idref" href="#y:388"><span class="id" title="binder">y</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:388"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:386"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:387"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:388"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:386"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:388"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:387"><span class="id" title="variable">s2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_app"><span class="id" title="lemma">elementsZ_app</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_rev" class="idref" href="#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> : <span class="id" title="keyword">∀</span> <a id="s:389" class="idref" href="#s:389"><span class="id" title="binder">s</span></a> <a id="y:390" class="idref" href="#y:390"><span class="id" title="binder">y</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:390"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:389"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:390"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:389"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_app"><span class="id" title="lemma">elementsZ_app</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;-!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_elementsZ_single_dec" class="idref" href="#Raw.In_elementsZ_single_dec"><span class="id" title="lemma">In_elementsZ_single_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="y:391" class="idref" href="#y:391"><span class="id" title="binder">y</span></a> <a id="x:392" class="idref" href="#x:392"><span class="id" title="binder">x</span></a> <a id="c:393" class="idref" href="#c:393"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:391"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:392"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:393"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:391"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:392"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:393"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.ZArith_dec.html#Z_le_dec"><span class="id" title="definition">Z_le_dec</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.ZArith_dec.html#Z_lt_dec"><span class="id" title="definition">Z_lt_dec</span></a> <span class="id" title="var">y</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_dec" class="idref" href="#Raw.InZ_dec"><span class="id" title="lemma">InZ_dec</span></a> : <span class="id" title="keyword">∀</span> <a id="y:394" class="idref" href="#y:394"><span class="id" title="binder">y</span></a> <a id="s:395" class="idref" href="#s:395"><span class="id" title="binder">s</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:394"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:395"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:394"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:395"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x</span> <span class="id" title="var">c</span>] <span class="id" title="var">s</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">IH</span> ⇒ [] <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_dec"><span class="id" title="lemma">In_elementsZ_single_dec</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_elementsZ_single_hd" class="idref" href="#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a> : <span class="id" title="keyword">∀</span> (<a id="c:396" class="idref" href="#c:396"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) <a id="x:397" class="idref" href="#x:397"><span class="id" title="binder">x</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#c:396"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:397"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:397"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:396"><span class="id" title="variable">c</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> <span class="id" title="var">H_c_neq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_add_pos_r"><span class="id" title="lemma">Z.lt_add_pos_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> (0%<span class="id" title="var">N</span>)  <span class="id" title="tactic">by</span> [].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_lt"><span class="id" title="lemma">N2Z.inj_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_0_lt_0"><span class="id" title="lemma">N.neq_0_lt_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab27"></a><h3 class="section">comparing intervals</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">Z_named_compare_cases</span> <span class="id" title="var">H</span> := <span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| [ |- <span class="id" title="keyword">context</span> [<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> ?<span class="id" title="var">z1</span> ?<span class="id" title="var">z2</span>] ] ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">z1</span> <span class="id" title="var">z2</span>); [<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_eq_iff"><span class="id" title="lemma">Z.compare_eq_iff</span></a> | <span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_lt_iff"><span class="id" title="lemma">Z.compare_lt_iff</span></a> | <span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_gt_iff"><span class="id" title="lemma">Z.compare_gt_iff</span></a>]; <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> //<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Ltac</span> <span class="id" title="var">Z_compare_cases</span> := <span class="id" title="keyword">let</span> <span class="id" title="var">H</span> := <span class="id" title="tactic">fresh</span> "H" <span class="id" title="tactic">in</span> <span class="id" title="var">Z_named_compare_cases</span> <span class="id" title="var">H</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_compare_elim" class="idref" href="#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> : <span class="id" title="keyword">∀</span> (<a id="y1:398" class="idref" href="#y1:398"><span class="id" title="binder">y1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c1:399" class="idref" href="#c1:399"><span class="id" title="binder">c1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="y2:400" class="idref" href="#y2:400"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:401" class="idref" href="#c2:401"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>         ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a>   ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>          ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>    ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a>          ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
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+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a>      ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a>      ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:400"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:401"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:398"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:399"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">repeat</span> <span class="id" title="var">Z_compare_cases</span>); <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">try</span> (<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq_le_incl"><span class="id" title="lemma">Z.eq_le_incl</span></a>); <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">try</span> (<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">try</span> (<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>); <span class="id" title="tactic">try</span> (<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_reg_l"><span class="id" title="lemma">Z.add_reg_l</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj"><span class="id" title="lemma">N2Z.inj</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_compare_swap" class="idref" href="#Raw.interval_compare_swap"><span class="id" title="lemma">interval_compare_swap</span></a> : <span class="id" title="keyword">∀</span> (<a id="y1:402" class="idref" href="#y1:402"><span class="id" title="binder">y1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c1:403" class="idref" href="#c1:403"><span class="id" title="binder">c1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="y2:404" class="idref" href="#y2:404"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:405" class="idref" href="#c2:405"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c1:403"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c2:405"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:404"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:405"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:402"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:403"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:402"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:403"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:404"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:405"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>         ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a>   ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>          ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>    ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a>          ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a>  ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a>      ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a>      ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">H_c12_neq_0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_antisym"><span class="id" title="lemma">Z.compare_antisym</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>) ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_antisym"><span class="id" title="lemma">Z.compare_antisym</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>)) ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_suff</span> : <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H1</span> <span class="id" title="var">H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">H_c12_neq_0</span> ⇒ <span class="id" title="var">H_c_neq_0</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="var">Z_compare_cases</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H_suff</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H0</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H_suff</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H_suff</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H0</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H_suff</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_1_compare_alt_def" class="idref" href="#Raw.interval_1_compare_alt_def"><span class="id" title="lemma">interval_1_compare_alt_def</span></a> : <span class="id" title="keyword">∀</span> (<a id="y:406" class="idref" href="#y:406"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="i:407" class="idref" href="#i:407"><span class="id" title="binder">i</span></a> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:406"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#i:407"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:406"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#i:407"><span class="id" title="variable">i</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">r</span> ⇒ <span class="id" title="var">r</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y1</span> [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> 1) <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">y1</span>); <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="var">Z_compare_cases</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zle_not_lt"><span class="id" title="lemma">Zle_not_lt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zlt_succ_le"><span class="id" title="lemma">Zlt_succ_le</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zle_not_lt"><span class="id" title="lemma">Zle_not_lt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zlt_succ_le"><span class="id" title="lemma">Zlt_succ_le</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_1_compare_elim" class="idref" href="#Raw.interval_1_compare_elim"><span class="id" title="lemma">interval_1_compare_elim</span></a> : <span class="id" title="keyword">∀</span> (<a id="y1:408" class="idref" href="#y1:408"><span class="id" title="binder">y1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="y2:409" class="idref" href="#y2:409"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:410" class="idref" href="#c2:410"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:410"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>         ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a>   ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>          ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:410"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>    ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:410"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a>          ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a>      ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c2:410"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:408"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:409"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:410"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a>      ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> (1%<span class="id" title="var">N</span>) <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_alt_def"><span class="id" title="lemma">interval_1_compare_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_succ</span>: <span class="id" title="keyword">∀</span> <a id="z:411" class="idref" href="#z:411"><span class="id" title="binder">z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#z:411"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:411"><span class="id" title="variable">z</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> 1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒ <span class="id" title="var">H_comp</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<span class="id" title="var">H_succ</span> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a> //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_lt</span>] [<span class="id" title="var">H_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zle_not_lt"><span class="id" title="lemma">Zle_not_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">H_lt</span>] <span class="id" title="var">H_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zle_not_lt"><span class="id" title="lemma">Zle_not_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [-&gt;] &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.zero_or_succ"><span class="id" title="lemma">N.zero_or_succ</span></a> <span class="id" title="var">c2</span>). { <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → <span class="id" title="var">_</span>; <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">c2'</span>] →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_succ"><span class="id" title="lemma">N2Z.inj_succ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_succ_r"><span class="id" title="lemma">Z.add_succ_r</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ_le_mono"><span class="id" title="lemma">Z.succ_le_mono</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] [<span class="id" title="var">H_le_y1</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y2</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_pre</span>; <span class="id" title="var">contradict</span> <span class="id" title="var">H_pre</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zle_not_lt"><span class="id" title="lemma">Zle_not_lt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_antisymm"><span class="id" title="lemma">Z.le_antisymm</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_le_y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab28"></a><h3 class="section">Alternative definition of addZ</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.addZ_aux_alt_def" class="idref" href="#Raw.addZ_aux_alt_def"><span class="id" title="lemma">addZ_aux_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="x:412" class="idref" href="#x:412"><span class="id" title="binder">x</span></a> <a id="s:413" class="idref" href="#s:413"><span class="id" title="binder">s</span></a> <a id="acc:414" class="idref" href="#acc:414"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:414"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:412"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:413"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:414"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:412"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:413"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> /= /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>); <span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> /= ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc_reverse"><span class="id" title="lemma">app_assoc_reverse</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">acc</span>)) (<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc_reverse"><span class="id" title="lemma">app_assoc_reverse</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.addZ_alt_def" class="idref" href="#Raw.addZ_alt_def"><span class="id" title="lemma">addZ_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="x:415" class="idref" href="#x:415"><span class="id" title="binder">x</span></a> <a id="s:416" class="idref" href="#s:416"><span class="id" title="binder">s</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:415"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:416"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:416"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:415"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:415"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:415"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s:416"><span class="id" title="variable">s</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:415"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:415"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span>) <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>); <span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux"><span class="id" title="definition">addZ_aux</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> /= ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc_reverse"><span class="id" title="lemma">app_assoc_reverse</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux_alt_def"><span class="id" title="lemma">addZ_aux_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab29"></a><h3 class="section">Auxiliary Lemmata about Invariant</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_cons" class="idref" href="#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="z:418" class="idref" href="#z:418"><span class="id" title="binder">z</span></a> <a id="x:419" class="idref" href="#x:419"><span class="id" title="binder">x</span></a> <a id="c:420" class="idref" href="#c:420"><span class="id" title="binder">c</span></a> <a id="s:421" class="idref" href="#s:421"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:418"><span class="id" title="variable">z</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:419"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:420"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:421"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#z:418"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:419"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">z</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_impl" class="idref" href="#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a> : <span class="id" title="keyword">∀</span> <a id="x:422" class="idref" href="#x:422"><span class="id" title="binder">x</span></a> <a id="y:423" class="idref" href="#y:423"><span class="id" title="binder">y</span></a> <a id="s:424" class="idref" href="#s:424"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:423"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:422"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:422"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:424"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:423"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:424"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">z</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_leq</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_nil" class="idref" href="#Raw.interval_list_invariant_nil"><span class="id" title="lemma">interval_list_invariant_nil</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> [].<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.Ok_nil" class="idref" href="#Raw.Ok_nil"><span class="id" title="lemma">Ok_nil</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.is_encoded_elems_list_app" class="idref" href="#Raw.is_encoded_elems_list_app"><span class="id" title="lemma">is_encoded_elems_list_app</span></a> : <span class="id" title="keyword">∀</span> <a id="l1:425" class="idref" href="#l1:425"><span class="id" title="binder">l1</span></a> <a id="l2:426" class="idref" href="#l2:426"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#l1:425"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:426"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l1:425"><span class="id" title="variable">l1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:426"><span class="id" title="variable">l2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">firstorder</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.is_encoded_elems_list_rev" class="idref" href="#Raw.is_encoded_elems_list_rev"><span class="id" title="lemma">is_encoded_elems_list_rev</span></a> : <span class="id" title="keyword">∀</span> <a id="l:427" class="idref" href="#l:427"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:427"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:427"><span class="id" title="variable">l</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">x</span> <span class="id" title="var">H_in</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> ⇒ //<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_cons" class="idref" href="#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="y:428" class="idref" href="#y:428"><span class="id" title="binder">y</span></a> <a id="c:429" class="idref" href="#c:429"><span class="id" title="binder">c</span></a> <a id="s':430" class="idref" href="#s':430"><span class="id" title="binder">s'</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:428"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:429"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':430"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#y:428"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:429"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s':430"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a>(<a class="idref" href="MSetsExtra.MSetIntervals.html#c:429"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':430"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> -/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">Bool.andb_true_iff</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#negb_true_iff"><span class="id" title="lemma">negb_true_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] [<span class="id" title="var">H_inf</span>] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_neq"><span class="id" title="lemma">N.eqb_neq</span></a> <span class="id" title="var">H_c</span> <span class="id" title="var">H_s'</span>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {                           <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inf</span>] [/<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_neq"><span class="id" title="lemma">N.eqb_neq</span></a> <span class="id" title="var">H_c</span>] <span class="id" title="var">H_s'</span>. <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_sing" class="idref" href="#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a> : <span class="id" title="keyword">∀</span> <a id="x:431" class="idref" href="#x:431"><span class="id" title="binder">x</span></a> <a id="c:432" class="idref" href="#c:432"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:431"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:432"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c:432"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.Ok_cons" class="idref" href="#Raw.Ok_cons"><span class="id" title="lemma">Ok_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="y:433" class="idref" href="#y:433"><span class="id" title="binder">y</span></a> <a id="c:434" class="idref" href="#c:434"><span class="id" title="binder">c</span></a> <a id="s':435" class="idref" href="#s':435"><span class="id" title="binder">s'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:433"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:434"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':435"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#y:433"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:434"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s':435"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a>(<a class="idref" href="MSetsExtra.MSetIntervals.html#c:434"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:433"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:434"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':435"><span class="id" title="variable">s'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list_app"><span class="id" title="lemma">is_encoded_elems_list_app</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list_rev"><span class="id" title="lemma">is_encoded_elems_list_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.Nin_elements_greater" class="idref" href="#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> : <span class="id" title="keyword">∀</span> <a id="s:436" class="idref" href="#s:436"><span class="id" title="binder">s</span></a> <a id="y:437" class="idref" href="#y:437"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:437"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:436"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:436"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:438" class="idref" href="#x:438"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#x:438"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:437"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:438"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:436"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">z</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">x</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a> <span class="id" title="var">H_y_lt</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c</span>] <span class="id" title="var">H_s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">x</span> <span class="id" title="var">H_x_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x_lt</span> : <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {              <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_z_leq</span>] <span class="id" title="var">_</span>; <span class="id" title="var">contradict</span> <span class="id" title="var">H_z_leq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">IH</span>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_le_add_r"><span class="id" title="lemma">Z_lt_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.Nin_elements_greater_equal" class="idref" href="#Raw.Nin_elements_greater_equal"><span class="id" title="lemma">Nin_elements_greater_equal</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:439" class="idref" href="#x:439"><span class="id" title="binder">x</span></a> <a id="s:440" class="idref" href="#s:440"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:439"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:440"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:440"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:439"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:440"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">H_inv</span> <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_alt_def" class="idref" href="#Raw.interval_list_elements_greater_alt_def"><span class="id" title="lemma">interval_list_elements_greater_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="s:441" class="idref" href="#s:441"><span class="id" title="binder">s</span></a> <a id="y:442" class="idref" href="#y:442"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:441"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:442"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:441"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:443" class="idref" href="#x:443"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#x:443"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:442"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:443"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:441"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">y</span> <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x2</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">H_c_neq</span>] <span class="id" title="var">_</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a> ⇒ <span class="id" title="var">H_ge</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<span class="id" title="var">H</span> <span class="id" title="var">x2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_alt2_def" class="idref" href="#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> : <span class="id" title="keyword">∀</span> <a id="s:444" class="idref" href="#s:444"><span class="id" title="binder">s</span></a> <a id="y:445" class="idref" href="#y:445"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:444"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:445"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:444"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:446" class="idref" href="#x:446"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:446"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:444"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:445"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:446"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">y</span> <span class="id" title="var">H</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt_def"><span class="id" title="lemma">interval_list_elements_greater_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_notInZ</span> <span class="id" title="var">x</span> <span class="id" title="var">H_inZ</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_notInZ</span> <span class="id" title="var">x</span> <span class="id" title="var">H_lt</span> <span class="id" title="var">H_inZ</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt</span> <span class="id" title="var">x</span> <span class="id" title="var">H_le</span> <span class="id" title="var">H_inZ</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_lt</span> <span class="id" title="var">x</span> <span class="id" title="var">H_inZ</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_intro" class="idref" href="#Raw.interval_list_elements_greater_intro"><span class="id" title="lemma">interval_list_elements_greater_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="s:447" class="idref" href="#s:447"><span class="id" title="binder">s</span></a> <a id="y:448" class="idref" href="#y:448"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:447"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:449" class="idref" href="#x:449"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:449"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:447"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:448"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:449"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:448"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:447"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">y</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_app_elim_1" class="idref" href="#Raw.interval_list_elements_greater_app_elim_1"><span class="id" title="lemma">interval_list_elements_greater_app_elim_1</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:450" class="idref" href="#s1:450"><span class="id" title="binder">s1</span></a> <a id="s2:451" class="idref" href="#s2:451"><span class="id" title="binder">s2</span></a> <a id="y:452" class="idref" href="#y:452"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:452"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:450"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:451"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:452"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:450"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s1</span> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_app_intro" class="idref" href="#Raw.interval_list_invariant_app_intro"><span class="id" title="lemma">interval_list_invariant_app_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:453" class="idref" href="#s1:453"><span class="id" title="binder">s1</span></a> <a id="s2:454" class="idref" href="#s2:454"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:453"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:454"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="x1:455" class="idref" href="#x1:455"><span class="id" title="binder">x1</span></a> <a id="x2:456" class="idref" href="#x2:456"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:455"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:453"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:456"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:454"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:455"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:456"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:453"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:454"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s2</span> <span class="id" title="var">_</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_comm_cons"><span class="id" title="lemma">app_comm_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c1_neq</span>] <span class="id" title="var">H_inv_s1'</span> <span class="id" title="var">H_inv_s2</span> <span class="id" title="var">H_inz_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr</span> <span class="id" title="var">H_inz_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s1'</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1'</span> <span class="id" title="var">c1'</span>] <span class="id" title="var">s1''</span>]; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">H_inz_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_l"><span class="id" title="lemma">app_nil_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_intro"><span class="id" title="lemma">interval_list_elements_greater_intro</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> <span class="id" title="var">H_x_in_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_inz</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.pred"><span class="id" title="definition">Z.pred</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>)) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_inz_s2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_inz</span> <span class="id" title="var">H_x_in_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ_pred"><span class="id" title="lemma">Z.succ_pred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_pred"><span class="id" title="lemma">Z.lt_le_pred</span></a>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_pred_l"><span class="id" title="lemma">Z.lt_pred_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_in_x1</span> <span class="id" title="var">H_in_x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_inz_s2</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_app_elim" class="idref" href="#Raw.interval_list_invariant_app_elim"><span class="id" title="lemma">interval_list_invariant_app_elim</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:457" class="idref" href="#s1:457"><span class="id" title="binder">s1</span></a> <a id="s2:458" class="idref" href="#s2:458"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:457"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:458"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:457"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:458"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="x1:459" class="idref" href="#x1:459"><span class="id" title="binder">x1</span></a> <a id="x2:460" class="idref" href="#x2:460"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:459"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:457"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:460"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:458"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:459"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:460"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>]; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_comm_cons"><span class="id" title="lemma">app_comm_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c1_neq_0</span>] /<span class="id" title="var">IH</span> [<span class="id" title="var">H_inv_s1'</span>] [<span class="id" title="var">H_inv_s2</span>] <span class="id" title="var">H_in_s1'_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">assumption</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s1'</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s1''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_s1'_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_y1_le</span> <span class="id" title="var">H_x1_lt</span> <span class="id" title="var">H_x2_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro"><span class="id" title="lemma">interval_list_invariant_app_intro</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_s12'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">x2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_s12'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zlt_le_succ"><span class="id" title="lemma">Zlt_le_succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_app_iff" class="idref" href="#Raw.interval_list_invariant_app_iff"><span class="id" title="lemma">interval_list_invariant_app_iff</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:461" class="idref" href="#s1:461"><span class="id" title="binder">s1</span></a> <a id="s2:462" class="idref" href="#s2:462"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:461"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:462"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:461"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:462"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="x1:463" class="idref" href="#x1:463"><span class="id" title="binder">x1</span></a> <a id="x2:464" class="idref" href="#x2:464"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:463"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:461"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:464"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:462"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:463"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:464"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_elim"><span class="id" title="lemma">interval_list_invariant_app_elim</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_inv_s1</span> [].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro"><span class="id" title="lemma">interval_list_invariant_app_intro</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_snoc_intro" class="idref" href="#Raw.interval_list_invariant_snoc_intro"><span class="id" title="lemma">interval_list_invariant_snoc_intro</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:465" class="idref" href="#s1:465"><span class="id" title="binder">s1</span></a> <a id="y2:466" class="idref" href="#y2:466"><span class="id" title="binder">y2</span></a> <a id="c2:467" class="idref" href="#c2:467"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:465"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#c2:467"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:468" class="idref" href="#x:468"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:468"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:465"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:468"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:466"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:465"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:466"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:467"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_c2_neq</span> <span class="id" title="var">H_in_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro"><span class="id" title="lemma">interval_list_invariant_app_intro</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>; <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_in_x1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab30"></a><h3 class="section">Properties of In and InZ</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.encode_decode_eq" class="idref" href="#Raw.encode_decode_eq"><span class="id" title="lemma">encode_decode_eq</span></a> : <span class="id" title="keyword">∀</span> <a id="x:469" class="idref" href="#x:469"><span class="id" title="binder">x</span></a> <a id="s:470" class="idref" href="#s:470"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:470"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:469"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:470"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:469"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:469"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_enc</span> <span class="id" title="var">H_in_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_enc</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in_x</span>) ⇒ [<span class="id" title="var">x'</span>] &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_alt_def" class="idref" href="#Raw.In_alt_def"><span class="id" title="lemma">In_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="x:471" class="idref" href="#x:471"><span class="id" title="binder">x</span></a> <a id="s:472" class="idref" href="#s:472"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:472"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:471"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:472"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:471"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:472"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span>] [<span class="id" title="var">H_y_eq</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_map_iff"><span class="id" title="lemma">in_map_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x'</span>] [<span class="id" title="var">H_y_eq'</span>] <span class="id" title="var">H_x'_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_x'_eq</span> :  (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">x'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_x'_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_enc_list</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> ⇒ [] [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_enc_list</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x'_in</span>) ⇒ [<span class="id" title="var">x''</span>] <span class="id" title="var">H_x'_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_y_eq'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -!<span class="id" title="var">H_x'_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> ⇒ <span class="id" title="var">H_y_eq'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">y</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_eq"><span class="id" title="axiom">Enc.encode_eq</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_enc_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq_equiv"><span class="id" title="axiom">Enc.E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_InZ" class="idref" href="#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> : <span class="id" title="keyword">∀</span> <a id="x:473" class="idref" href="#x:473"><span class="id" title="binder">x</span></a> <a id="s:474" class="idref" href="#s:474"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:474"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:473"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:474"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:473"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:474"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_alt_def"><span class="id" title="lemma">In_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_map_iff"><span class="id" title="lemma">in_map_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">x</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span>] [&lt;-] <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="z:475" class="idref" href="#z:475"><span class="id" title="binder">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:475"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">z</span>] <span class="id" title="var">H_y_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> -!<span class="id" title="var">H_y_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_enc_list</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc_list</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_In" class="idref" href="#Raw.InZ_In"><span class="id" title="lemma">InZ_In</span></a> : <span class="id" title="keyword">∀</span> <a id="x:476" class="idref" href="#x:476"><span class="id" title="binder">x</span></a> <a id="s:477" class="idref" href="#s:477"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:477"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:476"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:477"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:476"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:477"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">s</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_enc</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in</span>) ⇒ [<span class="id" title="var">e</span>] <span class="id" title="var">H_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab31"></a><h3 class="section">Membership specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.memZ_spec" class="idref" href="#Raw.memZ_spec"><span class="id" title="lemma">memZ_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:478" class="idref" href="#s:478"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:479" class="idref" href="#x:479"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="Hs:480" class="idref" href="#Hs:480"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:478"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.memZ"><span class="id" title="definition">memZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:479"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:478"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:479"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:478"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {    <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok_cons"><span class="id" title="lemma">Ok_cons</span></a> [<span class="id" title="var">H_inf</span>] [<span class="id" title="var">H_c</span>] [<span class="id" title="var">H_is_enc</span>] <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.memZ"><span class="id" title="definition">memZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> -/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.memZ"><span class="id" title="definition">memZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#f1746f02f141d025b1e53605943919bf"><span class="id" title="notation">&lt;?</span></a> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x_in</span>; <span class="id" title="var">contradict</span> <span class="id" title="var">H_x_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">y</span> := (<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)) ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_le_add_r"><span class="id" title="lemma">Z_lt_le_add_r</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y_le</span>]; <span class="id" title="var">contradict</span> <span class="id" title="var">H_y_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_ge"><span class="id" title="lemma">Z.ltb_ge</span></a> <span class="id" title="var">H_y_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#f1746f02f141d025b1e53605943919bf"><span class="id" title="notation">&lt;?</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_ge"><span class="id" title="lemma">Z.ltb_ge</span></a> <span class="id" title="var">H_yc_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_x_lt</span>; <span class="id" title="var">contradict</span> <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.mem_spec" class="idref" href="#Raw.mem_spec"><span class="id" title="lemma">mem_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:481" class="idref" href="#s:481"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:482" class="idref" href="#x:482"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:483" class="idref" href="#Hs:483"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:481"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:482"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:481"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:482"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:481"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.memZ_spec"><span class="id" title="lemma">memZ_spec</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.merge_interval_size_neq_0" class="idref" href="#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a> : <span class="id" title="keyword">∀</span> <a id="x1:484" class="idref" href="#x1:484"><span class="id" title="binder">x1</span></a> <a id="c1:485" class="idref" href="#c1:485"><span class="id" title="binder">c1</span></a> <a id="x2:486" class="idref" href="#x2:486"><span class="id" title="binder">x2</span></a> <a id="c2:487" class="idref" href="#c2:487"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c1:485"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:484"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:485"><span class="id" title="variable">c1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:486"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:487"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">c1</span> <span class="id" title="var">x2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_0_lt_0"><span class="id" title="lemma">N.neq_0_lt_0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.max_lt_iff"><span class="id" title="lemma">N.max_lt_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab32"></a><h3 class="section">insert if length not 0</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_invariant_insert_intervalZ_guarded" class="idref" href="#Raw.interval_list_invariant_insert_intervalZ_guarded"><span class="id" title="lemma">interval_list_invariant_insert_intervalZ_guarded</span></a> : <span class="id" title="keyword">∀</span> <a id="x:488" class="idref" href="#x:488"><span class="id" title="binder">x</span></a> <a id="c:489" class="idref" href="#c:489"><span class="id" title="binder">c</span></a> <a id="s:490" class="idref" href="#s:490"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:490"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:488"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:489"><span class="id" title="variable">c</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:490"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:488"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:489"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:490"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_neq"><span class="id" title="lemma">N.eqb_neq</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.interval_list_elements_greater_insert_intervalZ_guarded" class="idref" href="#Raw.interval_list_elements_greater_insert_intervalZ_guarded"><span class="id" title="lemma">interval_list_elements_greater_insert_intervalZ_guarded</span></a> : <span class="id" title="keyword">∀</span> <a id="x:491" class="idref" href="#x:491"><span class="id" title="binder">x</span></a> <a id="c:492" class="idref" href="#c:492"><span class="id" title="binder">c</span></a> <a id="y:493" class="idref" href="#y:493"><span class="id" title="binder">y</span></a> <a id="s:494" class="idref" href="#s:494"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:493"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:491"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:492"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:494"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c:492"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:493"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:494"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y:493"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:491"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">y</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.insert_intervalZ_guarded_app" class="idref" href="#Raw.insert_intervalZ_guarded_app"><span class="id" title="lemma">insert_intervalZ_guarded_app</span></a> : <span class="id" title="keyword">∀</span> <a id="x:495" class="idref" href="#x:495"><span class="id" title="binder">x</span></a> <a id="c:496" class="idref" href="#c:496"><span class="id" title="binder">c</span></a> <a id="s1:497" class="idref" href="#s1:497"><span class="id" title="binder">s1</span></a> <a id="s2:498" class="idref" href="#s2:498"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:495"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:496"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:497"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:498"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:495"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:496"><span class="id" title="variable">c</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#s1:497"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:498"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <span class="id" title="var">c</span> 0) ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.insert_intervalZ_guarded_rev_nil_app" class="idref" href="#Raw.insert_intervalZ_guarded_rev_nil_app"><span class="id" title="lemma">insert_intervalZ_guarded_rev_nil_app</span></a> : <span class="id" title="keyword">∀</span> <a id="x:499" class="idref" href="#x:499"><span class="id" title="binder">x</span></a> <a id="c:500" class="idref" href="#c:500"><span class="id" title="binder">c</span></a> <a id="s:501" class="idref" href="#s:501"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:499"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:500"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:501"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:499"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:500"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:501"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb"><span class="id" title="definition">N.eqb</span></a> <span class="id" title="var">c</span> 0) ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_insert_intervalZ_guarded" class="idref" href="#Raw.elementsZ_insert_intervalZ_guarded"><span class="id" title="lemma">elementsZ_insert_intervalZ_guarded</span></a> : <span class="id" title="keyword">∀</span> <a id="x:502" class="idref" href="#x:502"><span class="id" title="binder">x</span></a> <a id="c:503" class="idref" href="#c:503"><span class="id" title="binder">c</span></a> <a id="s:504" class="idref" href="#s:504"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:502"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:503"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:504"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:502"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:503"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:504"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_eq"><span class="id" title="lemma">N.eqb_eq</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_insert_intervalZ_guarded" class="idref" href="#Raw.InZ_insert_intervalZ_guarded"><span class="id" title="lemma">InZ_insert_intervalZ_guarded</span></a> : <span class="id" title="keyword">∀</span> <a id="y:505" class="idref" href="#y:505"><span class="id" title="binder">y</span></a> <a id="x:506" class="idref" href="#x:506"><span class="id" title="binder">x</span></a> <a id="c:507" class="idref" href="#c:507"><span class="id" title="binder">c</span></a> <a id="s:508" class="idref" href="#s:508"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:505"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:506"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:507"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:508"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:505"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:506"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:507"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:508"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_insert_intervalZ_guarded"><span class="id" title="lemma">elementsZ_insert_intervalZ_guarded</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab33"></a><h3 class="section">Merging intervals</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.merge_interval_size_add" class="idref" href="#Raw.merge_interval_size_add"><span class="id" title="lemma">merge_interval_size_add</span></a> : <span class="id" title="keyword">∀</span> <a id="x:509" class="idref" href="#x:509"><span class="id" title="binder">x</span></a> <a id="c1:510" class="idref" href="#c1:510"><span class="id" title="binder">c1</span></a> <a id="c2:511" class="idref" href="#c2:511"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:509"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:510"><span class="id" title="variable">c1</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:509"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:510"><span class="id" title="variable">c1</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:511"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c1:510"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:511"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">N</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_add"><span class="id" title="lemma">N2Z.inj_add</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.id"><span class="id" title="lemma">N2Z.id</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.max_r"><span class="id" title="lemma">N.max_r</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.le_add_r"><span class="id" title="lemma">N.le_add_r</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.merge_interval_size_eq_max" class="idref" href="#Raw.merge_interval_size_eq_max"><span class="id" title="lemma">merge_interval_size_eq_max</span></a> : <span class="id" title="keyword">∀</span> <a id="y1:512" class="idref" href="#y1:512"><span class="id" title="binder">y1</span></a> <a id="c1:513" class="idref" href="#c1:513"><span class="id" title="binder">c1</span></a> <a id="y2:514" class="idref" href="#y2:514"><span class="id" title="binder">y2</span></a> <a id="c2:515" class="idref" href="#c2:515"><span class="id" title="binder">c2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y1:512"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:514"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:515"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y1:512"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:512"><span class="id" title="variable">y1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:513"><span class="id" title="variable">c1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:514"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:515"><span class="id" title="variable">c2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.max"><span class="id" title="definition">Z.max</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#y1:512"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:513"><span class="id" title="variable">c1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#y2:514"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:515"><span class="id" title="variable">c2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">H_y1_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_max"><span class="id" title="lemma">N2Z.inj_max</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zle_minus_le_0"><span class="id" title="lemma">Zle_minus_le_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_max_distr_l"><span class="id" title="lemma">Z.add_max_distr_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.merge_interval_size_invariant" class="idref" href="#Raw.merge_interval_size_invariant"><span class="id" title="lemma">merge_interval_size_invariant</span></a> : <span class="id" title="keyword">∀</span> <a id="y1:516" class="idref" href="#y1:516"><span class="id" title="binder">y1</span></a> <a id="c1:517" class="idref" href="#c1:517"><span class="id" title="binder">c1</span></a> <a id="y2:518" class="idref" href="#y2:518"><span class="id" title="binder">y2</span></a> <a id="c2:519" class="idref" href="#c2:519"><span class="id" title="binder">c2</span></a> <a id="z:520" class="idref" href="#z:520"><span class="id" title="binder">z</span></a> <a id="s:521" class="idref" href="#s:521"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:521"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y1:516"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:517"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:518"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:519"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#y2:518"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:519"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:520"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:520"><span class="id" title="variable">z</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:521"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#c1:517"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:516"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y1:516"><span class="id" title="variable">y1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:517"><span class="id" title="variable">c1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:518"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:519"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:521"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">z</span> <span class="id" title="var">s</span> <span class="id" title="var">H_inv</span> <span class="id" title="var">H_le</span> <span class="id" title="var">H_le_z</span> <span class="id" title="var">H_gr</span> <span class="id" title="var">H_c1_neq_0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_eq_max"><span class="id" title="lemma">merge_interval_size_eq_max</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.max_r"><span class="id" title="lemma">Z.max_r</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_le_z</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.In_merge_interval" class="idref" href="#Raw.In_merge_interval"><span class="id" title="lemma">In_merge_interval</span></a> : <span class="id" title="keyword">∀</span> <a id="x1:522" class="idref" href="#x1:522"><span class="id" title="binder">x1</span></a> <a id="c1:523" class="idref" href="#c1:523"><span class="id" title="binder">c1</span></a> <a id="x2:524" class="idref" href="#x2:524"><span class="id" title="binder">x2</span></a> <a id="c2:525" class="idref" href="#c2:525"><span class="id" title="binder">c2</span></a> <a id="y:526" class="idref" href="#y:526"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#x1:522"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:524"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#x2:524"><span class="id" title="variable">x2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:522"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:523"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:526"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:522"><span class="id" title="variable">x1</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:522"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:523"><span class="id" title="variable">c1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:524"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:525"><span class="id" title="variable">c2</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:526"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:522"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:523"><span class="id" title="variable">c1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:526"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:524"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:525"><span class="id" title="variable">c2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">c1</span> <span class="id" title="var">x2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y</span> <span class="id" title="var">H_x1_le</span> <span class="id" title="var">H_x2_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_eq_max"><span class="id" title="lemma">merge_interval_size_eq_max</span></a>; <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.max_lt_iff"><span class="id" title="lemma">Z.max_lt_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_x_le</span>] [] <span class="id" title="var">H_y_lt</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.leb"><span class="id" title="definition">Z.leb</span></a> <span class="id" title="var">x2</span> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.leb_le"><span class="id" title="lemma">Z.leb_le</span></a> <span class="id" title="var">H_y'_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.leb_gt"><span class="id" title="lemma">Z.leb_gt</span></a> <span class="id" title="var">H_y_lt_x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_x2_le'</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.insert_interval_begin_spec" class="idref" href="#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="y:527" class="idref" href="#y:527"><span class="id" title="binder">y</span></a> <a id="s:528" class="idref" href="#s:528"><span class="id" title="binder">s</span></a> <a id="x:529" class="idref" href="#x:529"><span class="id" title="binder">x</span></a> <a id="c:530" class="idref" href="#c:530"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:528"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:529"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:528"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#c:530"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:529"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:530"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:528"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:527"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:529"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:530"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:528"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:527"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:529"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:530"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:527"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:528"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="var">H_c_neq</span> <span class="id" title="var">H_z_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c'_neq_0</span>] <span class="id" title="var">H_inv_s'</span> <span class="id" title="var">H_x_lt</span> <span class="id" title="var">H_c_neq_0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">Z_named_compare_cases</span> <span class="id" title="var">H_y'</span>; <span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_add"><span class="id" title="lemma">elementsZ_single_add</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_add"><span class="id" title="lemma">N2Z.inj_add</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_assoc"><span class="id" title="lemma">Z.add_assoc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eq_add_0"><span class="id" title="lemma">N.eq_add_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">c''</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">x</span> <span class="id" title="var">c</span> <span class="id" title="var">y'</span> <span class="id" title="var">c'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x_lt'</span> : <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y'</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_pre</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater"><span class="id" title="definition">interval_list_elements_greater</span></a> <span class="id" title="var">x</span> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_pre2</span> : <span class="id" title="var">c''</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">x</span> <span class="id" title="var">c''</span> <span class="id" title="var">H_inv_s'</span> <span class="id" title="var">H_pre</span> <span class="id" title="var">H_pre2</span>) ⇒ {<span class="id" title="var">IH</span>} {<span class="id" title="var">H_pre</span>} {<span class="id" title="var">H_pre2</span>} [-&gt;] →.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <span class="id" title="var">c''</span>; <span class="id" title="tactic">clear</span> <span class="id" title="var">c''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval"><span class="id" title="lemma">In_merge_interval</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab34"></a><h3 class="section">add specification</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.addZ_InZ" class="idref" href="#Raw.addZ_InZ"><span class="id" title="lemma">addZ_InZ</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:531" class="idref" href="#s:531"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:532" class="idref" href="#x:532"><span class="id" title="binder">x</span></a> <a id="y:533" class="idref" href="#y:533"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:531"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:533"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:532"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:531"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:532"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:533"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:533"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:531"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">z</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_alt_def"><span class="id" title="lemma">addZ_alt_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_l"><span class="id" title="lemma">app_nil_l</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">firstorder</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> [<span class="id" title="var">H_greater</span>] [<span class="id" title="var">H_c_neq_0</span>] <span class="id" title="var">H_inv_c'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">H_inv_c'</span>) ⇒ {} <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_alt_def"><span class="id" title="lemma">addZ_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_succ</span> : <span class="id" title="keyword">∀</span> <a id="z:534" class="idref" href="#z:534"><span class="id" title="binder">z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#z:534"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:534"><span class="id" title="variable">z</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_elim"><span class="id" title="lemma">interval_1_compare_elim</span></a> <span class="id" title="var">x</span> <span class="id" title="var">z</span> <span class="id" title="var">c</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> ?<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a> ?<span class="id" title="var">H_succ</span> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a> //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // <span class="id" title="var">H_x_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">z</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span>) <span class="id" title="var">s'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">z</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> <span class="id" title="var">s'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_inv_c'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_succ_0"><span class="id" title="lemma">N.neq_succ_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_x_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.addZ_invariant" class="idref" href="#Raw.addZ_invariant"><span class="id" title="lemma">addZ_invariant</span></a> : <span class="id" title="keyword">∀</span> <a id="s:535" class="idref" href="#s:535"><span class="id" title="binder">s</span></a> <a id="x:536" class="idref" href="#x:536"><span class="id" title="binder">x</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:535"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ"><span class="id" title="definition">addZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:536"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:535"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">z</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> [<span class="id" title="var">H_greater</span>] [<span class="id" title="var">H_c_neq_0</span>]<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H_inv_c'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">H_inv_c'</span>) ⇒ {} <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_alt_def"><span class="id" title="lemma">addZ_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_succ</span> : <span class="id" title="keyword">∀</span> <a id="z:537" class="idref" href="#z:537"><span class="id" title="binder">z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#z:537"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:537"><span class="id" title="variable">z</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_elim"><span class="id" title="lemma">interval_1_compare_elim</span></a> <span class="id" title="var">x</span> <span class="id" title="var">z</span> <span class="id" title="var">c</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare"><span class="id" title="definition">interval_1_compare</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒ <span class="id" title="var">H_comp</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> ?<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a> ?<span class="id" title="var">H_succ</span> ?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a> //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_z_gt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> /= !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">andb_true_iff</span></a> !<span class="id" title="var">H_succ</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#negb_true_iff"><span class="id" title="lemma">negb_true_iff</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_neq"><span class="id" title="lemma">N.eqb_neq</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ ?; <span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">andb_true_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_greater</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_succ_l"><span class="id" title="lemma">Z.add_succ_l</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_succ_r"><span class="id" title="lemma">Z.add_succ_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_succ"><span class="id" title="lemma">N2Z.inj_succ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#negb_true_iff"><span class="id" title="lemma">negb_true_iff</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_neq"><span class="id" title="lemma">N.eqb_neq</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_succ_0"><span class="id" title="lemma">N.neq_succ_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_intro"><span class="id" title="lemma">interval_list_elements_greater_intro</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">xx</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_InZ"><span class="id" title="lemma">addZ_InZ</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [&lt;- //|].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_succ_0"><span class="id" title="lemma">N.neq_succ_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.add_ok" class="idref" href="#Raw.add_ok"><span class="id" title="instance">add_ok</span></a> <a id="s:538" class="idref" href="#s:538"><span class="id" title="binder">s</span></a> <a id="x:539" class="idref" href="#x:539"><span class="id" title="binder">x</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:540" class="idref" href="#H:540"><span class="id" title="binder">Ok</span></a> <a id="H:540" class="idref" href="#H:540"><span class="id" title="binder">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:539"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:538"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok_s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok_s</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_isok_s</span>] <span class="id" title="var">H_pre</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_invariant"><span class="id" title="lemma">addZ_invariant</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_InZ"><span class="id" title="lemma">addZ_InZ</span></a> <span class="id" title="var">s</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">y</span> <span class="id" title="var">H_isok_s</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_pre</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.add_spec" class="idref" href="#Raw.add_spec"><span class="id" title="lemma">add_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:541" class="idref" href="#s:541"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:542" class="idref" href="#x:542"><span class="id" title="binder">x</span></a> <a id="y:543" class="idref" href="#y:543"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:544" class="idref" href="#Hs:544"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:541"><span class="id" title="variable">s</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:543"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:542"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:541"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:543"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:542"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:543"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:541"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">Hs'</span> := (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_ok"><span class="id" title="instance">add_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">Hs</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.addZ_InZ"><span class="id" title="lemma">addZ_InZ</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_eq"><span class="id" title="axiom">Enc.encode_eq</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">firstorder</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab35"></a><h3 class="section">empty specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.empty_ok" class="idref" href="#Raw.empty_ok"><span class="id" title="instance">empty_ok</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok_nil"><span class="id" title="lemma">Ok_nil</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.empty_spec'" class="idref" href="#Raw.empty_spec'"><span class="id" title="lemma">empty_spec'</span></a> : <span class="id" title="keyword">∀</span> <a id="x:545" class="idref" href="#x:545"><span class="id" title="binder">x</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:545"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">a</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.empty_spec" class="idref" href="#Raw.empty_spec"><span class="id" title="lemma">empty_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> ⇒ <span class="id" title="var">a</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty_spec'"><span class="id" title="lemma">empty_spec'</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab36"></a><h3 class="section">is_empty specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.is_empty_spec" class="idref" href="#Raw.is_empty_spec"><span class="id" title="lemma">is_empty_spec</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:546" class="idref" href="#s:546"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="Hs:547" class="idref" href="#Hs:547"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:546"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_empty"><span class="id" title="definition">is_empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:546"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:546"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> [ | [<span class="id" title="var">x</span> <span class="id" title="var">c</span>] <span class="id" title="var">s</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ // <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty_spec"><span class="id" title="lemma">empty_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok_cons"><span class="id" title="lemma">Ok_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">H_c_neq</span>] [<span class="id" title="var">H_enc</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradiction</span> (<span class="id" title="var">H</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>)) ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_cons"><span class="id" title="lemma">elements_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq_equiv"><span class="id" title="axiom">Enc.E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_single"><span class="id" title="definition">elements_single</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_map_iff"><span class="id" title="lemma">in_map_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab37"></a><h3 class="section">singleton specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.singleton_ok" class="idref" href="#Raw.singleton_ok"><span class="id" title="instance">singleton_ok</span></a> <a id="x:548" class="idref" href="#x:548"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:548"><span class="id" title="variable">x</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.singleton_alt_def"><span class="id" title="lemma">singleton_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_ok"><span class="id" title="instance">add_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty_ok"><span class="id" title="instance">empty_ok</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.singleton_spec" class="idref" href="#Raw.singleton_spec"><span class="id" title="lemma">singleton_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:549" class="idref" href="#x:549"><span class="id" title="binder">x</span></a> <a id="y:550" class="idref" href="#y:550"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:550"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:549"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:550"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:549"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.singleton_alt_def"><span class="id" title="lemma">singleton_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_spec"><span class="id" title="lemma">add_spec</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.empty"><span class="id" title="definition">empty</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab38"></a><h3 class="section">add_list specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.add_list_ok" class="idref" href="#Raw.add_list_ok"><span class="id" title="lemma">add_list_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="l:551" class="idref" href="#l:551"><span class="id" title="binder">l</span></a> <a id="s:552" class="idref" href="#s:552"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:552"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_list"><span class="id" title="definition">add_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:551"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:552"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">l'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">H_s_ok</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_ok"><span class="id" title="instance">add_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.add_list_spec" class="idref" href="#Raw.add_list_spec"><span class="id" title="lemma">add_list_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:553" class="idref" href="#x:553"><span class="id" title="binder">x</span></a> <a id="l:554" class="idref" href="#l:554"><span class="id" title="binder">l</span></a> <a id="s:555" class="idref" href="#s:555"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:555"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:553"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_list"><span class="id" title="definition">add_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:554"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:555"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">SetoidList.InA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:553"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:554"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:553"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:555"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span> <span class="id" title="var">l'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">H_ok</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add_spec"><span class="id" title="lemma">add_spec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab39"></a><h3 class="section">union specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.union_aux_flatten_alt_def" class="idref" href="#Raw.union_aux_flatten_alt_def"><span class="id" title="lemma">union_aux_flatten_alt_def</span></a> : <span class="id" title="keyword">∀</span> (<a id="s1:556" class="idref" href="#s1:556"><span class="id" title="binder">s1</span></a> <a id="s2:557" class="idref" href="#s2:557"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="acc:558" class="idref" href="#acc:558"><span class="id" title="binder">acc</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:556"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:557"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:556"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:557"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:557"><span class="id" title="variable">s2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:556"><span class="id" title="variable">s1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:557"><span class="id" title="variable">s2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> ((<span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c2</span>)%<span class="id" title="var">N</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:556"><span class="id" title="variable">s1</span></a> <span class="id" title="var">l2</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> ((<span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c2</span>)%<span class="id" title="var">N</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>) <span class="id" title="var">l2</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:557"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:557"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:556"><span class="id" title="variable">s1</span></a> <span class="id" title="var">l2</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:558"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s1</span>, <span class="id" title="var">s2</span> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.union_aux_alt_def" class="idref" href="#Raw.union_aux_alt_def"><span class="id" title="lemma">union_aux_alt_def</span></a> : <span class="id" title="keyword">∀</span> (<a id="s1:559" class="idref" href="#s1:559"><span class="id" title="binder">s1</span></a> <a id="s2:560" class="idref" href="#s2:560"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="acc:561" class="idref" href="#acc:561"><span class="id" title="binder">acc</span></a>, <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux"><span class="id" title="definition">union_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:559"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:560"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:561"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:561"><span class="id" title="variable">acc</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:559"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:560"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">acc</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">l1</span> <span class="id" title="var">IH1</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux_flatten_alt_def"><span class="id" title="lemma">union_aux_flatten_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span> <span class="id" title="var">s2</span>; <span class="id" title="tactic">move</span> : <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">l2</span> <span class="id" title="var">IH2</span>]; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux_flatten_alt_def"><span class="id" title="lemma">union_aux_flatten_alt_def</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l1</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">l2</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?(<span class="id" title="var">IH1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>)) ?(<span class="id" title="var">IH1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH2</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>)) ?(<span class="id" title="var">IH2</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH1</span> <span class="id" title="var">acc</span>) //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.union_alt_def" class="idref" href="#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a> : <span class="id" title="keyword">∀</span> (<a id="s1:562" class="idref" href="#s1:562"><span class="id" title="binder">s1</span></a> <a id="s2:563" class="idref" href="#s2:563"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:562"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:563"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:562"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:563"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:563"><span class="id" title="variable">s2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:562"><span class="id" title="variable">s1</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:563"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> ((<span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c2</span>)%<span class="id" title="var">N</span>) <span class="id" title="var">l2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:562"><span class="id" title="variable">s1</span></a> <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> ((<span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a><span class="id" title="var">c2</span>)%<span class="id" title="var">N</span>) <span class="id" title="var">l2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <span class="id" title="var">l2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>) <span class="id" title="var">l2</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:563"><span class="id" title="variable">s2</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:563"><span class="id" title="variable">s2</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:562"><span class="id" title="variable">s1</span></a> <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux_flatten_alt_def"><span class="id" title="lemma">union_aux_flatten_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">l1</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">l2</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_aux_alt_def"><span class="id" title="lemma">union_aux_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.union_InZ" class="idref" href="#Raw.union_InZ"><span class="id" title="lemma">union_InZ</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:564" class="idref" href="#s1:564"><span class="id" title="binder">s1</span></a> <a id="s2:565" class="idref" href="#s2:565"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:564"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:565"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:566" class="idref" href="#y:566"><span class="id" title="binder">y</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:566"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:564"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:565"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:566"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:564"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:566"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:565"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">l1</span> <span class="id" title="var">IH1</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">l2</span> <span class="id" title="var">IH2</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_inv_s1</span>) (<span class="id" title="var">H_inv_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_l1</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_l1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_l2</span>] [<span class="id" title="var">H_c2_neq_0</span>] <span class="id" title="var">H_inv_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_l2</span>) ⇒ {} <span class="id" title="var">IH2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> :  <span class="id" title="keyword">∀</span> <a id="s2:567" class="idref" href="#s2:567"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:567"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:568" class="idref" href="#y:568"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:568"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:567"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:568"><span class="id" title="variable">y</span></a> <span class="id" title="var">l1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:568"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:567"><span class="id" title="variable">s2</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {} <span class="id" title="var">IH1</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <span class="id" title="var">IH1</span> // <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y2_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <span class="id" title="var">c2</span>)%<span class="id" title="var">N</span> <span class="id" title="keyword">with</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y2_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_add"><span class="id" title="lemma">merge_interval_size_add</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">c''</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_inv_insert</span> <span class="id" title="var">H_InZ_insert</span>] : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c''</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y2_eq</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_assoc"><span class="id" title="lemma">Z.add_assoc</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_add"><span class="id" title="lemma">N2Z.inj_add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_InZ_insert</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> /<span class="id" title="var">c''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y2_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval"><span class="id" title="lemma">In_merge_interval</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_lt</span>] [<span class="id" title="var">H_y2_lt</span>] <span class="id" title="var">H_y1_c1_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">c''</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_inv_insert</span> <span class="id" title="var">H_InZ_insert</span>] : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c''</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_le_add_r"><span class="id" title="lemma">Z_lt_le_add_r</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_InZ_insert</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> /<span class="id" title="var">c''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval"><span class="id" title="lemma">In_merge_interval</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_lt</span>] [<span class="id" title="var">H_y1_lt</span>] <span class="id" title="var">H_y2_c_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">c''</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_inv_insert</span> <span class="id" title="var">H_InZ_insert</span>] : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c''</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_InZ_insert</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> /<span class="id" title="var">c''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval"><span class="id" title="lemma">In_merge_interval</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [? ?]; <span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le</span>] [<span class="id" title="var">H_y1_c1_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span> <span class="id" title="var">H_y_lt</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] [<span class="id" title="var">H_y2_c2_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le</span> <span class="id" title="var">H_y_lt</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <span class="id" title="var">IH2</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <span class="id" title="var">c2</span>)%<span class="id" title="var">N</span> <span class="id" title="keyword">with</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y1_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_add"><span class="id" title="lemma">merge_interval_size_add</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.add_comm"><span class="id" title="lemma">N.add_comm</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">c''</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size"><span class="id" title="definition">merge_interval_size</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_inv_insert</span> <span class="id" title="var">H_InZ_insert</span>] : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin"><span class="id" title="definition">insert_interval_begin</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c''</span> <span class="id" title="var">l2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c''</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_InZ_insert</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> /<span class="id" title="var">c''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y1_eq</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval"><span class="id" title="lemma">In_merge_interval</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.union_invariant" class="idref" href="#Raw.union_invariant"><span class="id" title="lemma">union_invariant</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:569" class="idref" href="#s1:569"><span class="id" title="binder">s1</span></a> <a id="s2:570" class="idref" href="#s2:570"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:569"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:570"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:569"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:570"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">l1</span> <span class="id" title="var">IH1</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s2</span> <span class="id" title="var">_</span> <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">l2</span> <span class="id" title="var">IH2</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_inv_s1</span>) (<span class="id" title="var">H_inv_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_l1</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_l1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_l2</span>] [<span class="id" title="var">H_c2_neq_0</span>] <span class="id" title="var">H_inv_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_l2</span>) ⇒ {} <span class="id" title="var">IH2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> :  <span class="id" title="keyword">∀</span> <a id="s2:571" class="idref" href="#s2:571"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:571"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:571"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {} <span class="id" title="var">IH1</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def"><span class="id" title="lemma">union_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_inv'</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <span class="id" title="var">l1</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_intro"><span class="id" title="lemma">interval_list_elements_greater_intro</span></a> ⇒ // <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_InZ"><span class="id" title="lemma">union_InZ</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y2_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y2_eq</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_assoc"><span class="id" title="lemma">Z.add_assoc</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_add"><span class="id" title="lemma">N2Z.inj_add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eq_add_0"><span class="id" title="lemma">N.eq_add_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_lt</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_le_add_r"><span class="id" title="lemma">Z_lt_le_add_r</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_lt</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0"><span class="id" title="lemma">merge_interval_size_neq_0</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [? ?]; <span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH2</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_intro"><span class="id" title="lemma">interval_list_elements_greater_intro</span></a> ⇒ // <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_InZ"><span class="id" title="lemma">union_InZ</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec"><span class="id" title="lemma">insert_interval_begin_spec</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eq_add_0"><span class="id" title="lemma">N.eq_add_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.union_ok" class="idref" href="#Raw.union_ok"><span class="id" title="instance">union_ok</span></a> <a id="s1:572" class="idref" href="#s1:572"><span class="id" title="binder">s1</span></a> <a id="s2:573" class="idref" href="#s2:573"><span class="id" title="binder">s2</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:574" class="idref" href="#H:574"><span class="id" title="binder">Ok</span></a> <a id="H:574" class="idref" href="#H:574"><span class="id" title="binder">s1</span></a>, <a id="H0:575" class="idref" href="#H0:575"><span class="id" title="binder">Ok</span></a> <a id="H0:575" class="idref" href="#H0:575"><span class="id" title="binder">s2</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:572"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:573"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok_s1</span> <span class="id" title="var">H_ok_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok_s1</span>) (<span class="id" title="var">H_ok_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s1</span>] <span class="id" title="var">H_pre1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s2</span>] <span class="id" title="var">H_pre2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_invariant"><span class="id" title="lemma">union_invariant</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_InZ"><span class="id" title="lemma">union_InZ</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_pre1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_pre2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.union_spec" class="idref" href="#Raw.union_spec"><span class="id" title="lemma">union_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:576" class="idref" href="#s:576"><span class="id" title="binder">s</span></a> <a id="s':577" class="idref" href="#s':577"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:578" class="idref" href="#x:578"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:579" class="idref" href="#Hs:579"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:576"><span class="id" title="variable">s</span></a>) (<a id="Hs':580" class="idref" href="#Hs':580"><span class="id" title="binder">Hs'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':577"><span class="id" title="variable">s'</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:578"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:576"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':577"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:578"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:576"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:578"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':577"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span> <span class="id" title="var">H_ok</span> <span class="id" title="var">H_ok'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.union_InZ"><span class="id" title="lemma">union_InZ</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab40"></a><h3 class="section">inter specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.inter_aux_alt_def" class="idref" href="#Raw.inter_aux_alt_def"><span class="id" title="lemma">inter_aux_alt_def</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="y2:581" class="idref" href="#y2:581"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:582" class="idref" href="#c2:582"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="s:583" class="idref" href="#s:583"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="acc:584" class="idref" href="#acc:584"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:581"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:582"><span class="id" title="variable">c2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:584"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:583"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:581"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:582"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:583"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:584"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_l"><span class="id" title="lemma">app_nil_l</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">s'</span>) ⇒ <span class="id" title="var">acc''</span> <span class="id" title="var">s''</span> <span class="id" title="var">H_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?(<span class="id" title="var">IH</span> <span class="id" title="var">acc</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y2</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y2</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?<span class="id" title="var">H_eq</span> -?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> -?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_comm_cons"><span class="id" title="lemma">app_comm_cons</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.inter_aux_props" class="idref" href="#Raw.inter_aux_props"><span class="id" title="lemma">inter_aux_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="y2:585" class="idref" href="#y2:585"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:586" class="idref" href="#c2:586"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="s:587" class="idref" href="#s:587"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="acc:588" class="idref" href="#acc:588"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:588"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:587"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:589" class="idref" href="#x1:589"><span class="id" title="binder">x1</span></a> <a id="x2:590" class="idref" href="#x2:590"><span class="id" title="binder">x2</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:589"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:588"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:590"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:587"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:590"><span class="id" title="variable">x2</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:585"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:586"><span class="id" title="variable">c2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:589"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:590"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c2:586"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:585"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:586"><span class="id" title="variable">c2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:588"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:587"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:591" class="idref" href="#y:591"><span class="id" title="binder">y</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:591"><span class="id" title="variable">y</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:591"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:588"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:591"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:587"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:591"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:585"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:586"><span class="id" title="variable">c2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:592" class="idref" href="#y:592"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:592"><span class="id" title="variable">y</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:592"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:587"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:593" class="idref" href="#y:593"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:593"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:587"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:585"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:586"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:593"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:593"><span class="id" title="variable">y</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <span class="id" title="var">acc'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_acc</span> [<span class="id" title="var">H_gr_s1'</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_s1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_acc_lt</span> <span class="id" title="var">H_c2_neq_0</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux_alt_def"><span class="id" title="lemma">inter_aux_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span> <span class="id" title="var">H_inter_aux_eq</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P1</span> := <span class="id" title="keyword">∀</span> <a id="y:596" class="idref" href="#y:596"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:594"><span class="id" title="variable">y</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">((</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:594"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:594"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">))</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P2</span> := (<span class="id" title="keyword">∀</span> <a id="y:599" class="idref" href="#y:599"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:597"><span class="id" title="variable">y</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:597"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:597"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:597"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:597"><span class="id" title="variable">y</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P3</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <span class="id" title="var">acc'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P4</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">P1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P3</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P4</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_P1</span>] [<span class="id" title="var">H_P2</span>] [<span class="id" title="var">H_P3</span>] <span class="id" title="var">H_P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_P1</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_P2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_P2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_app_distr"><span class="id" title="lemma">rev_app_distr</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro"><span class="id" title="lemma">interval_list_invariant_app_intro</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x1_in</span> /<span class="id" title="var">H_P1</span> [<span class="id" title="var">H_x2_in1</span>] <span class="id" title="var">H_x2_in2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_acc_lt</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_gr_s1'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ // ⇒ <span class="id" title="var">H_gr_s1'_alt</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <span class="id" title="keyword">∀</span> (<a id="acc:600" class="idref" href="#acc:600"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:600"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:601" class="idref" href="#y:601"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:601"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:600"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:601"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:601"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>        <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:600"><span class="id" title="variable">acc</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">P1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P3</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P4</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc0</span> <span class="id" title="var">H_inv_acc0</span> <span class="id" title="var">H_in_acc0</span> <span class="id" title="var">H_y2c_lt</span> <span class="id" title="var">H_inter_aux_eq0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_acc0_lt</span> : (<span class="id" title="keyword">∀</span> <a id="x1:602" class="idref" href="#x1:602"><span class="id" title="binder">x1</span></a> <a id="x2:603" class="idref" href="#x2:603"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:602"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:603"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:603"><span class="id" title="variable">x2</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:602"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:603"><span class="id" title="variable">x2</span></a>). {<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_x1_in_acc0</span> <span class="id" title="var">H_x2_in_s1'</span> <span class="id" title="var">H_x2_in_yc2</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_yc1_lt_x2</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">x1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_s1'_alt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_x1_in_acc0</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_acc0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}          <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">acc0</span> <span class="id" title="var">H_inv_acc0</span> <span class="id" title="var">H_inv_s1'</span> <span class="id" title="var">H_in_acc0_lt</span> <span class="id" title="var">H_c2_neq_0</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_inter_aux_eq0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H1</span>] [<span class="id" title="var">H2</span>] [<span class="id" title="var">H3</span>] [<span class="id" title="var">H4</span>] <span class="id" title="var">H5</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">H1</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H_in_acc0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_y_lt</span> <span class="id" title="var">H_lt_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y</span>) <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) ⇒ //.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {} <span class="id" title="var">IH</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">H_in_acc_lt</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <span class="id" title="var">H_inter_aux_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒ <span class="id" title="var">H_comp</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}          <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_eq_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_eq_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}          <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_lt_y2</span>] [<span class="id" title="var">H_y2_lt_yc1</span>] <span class="id" title="var">H_yc1_lt_yc2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_to_N_minus_neq_0"><span class="id" title="lemma">Z_to_N_minus_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">last</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">y2</span> <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span> /<span class="id" title="var">P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_lt</span>] [<span class="id" title="var">H_y1_lt</span>] <span class="id" title="var">H_yc1_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">P4</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span> <span class="id" title="var">s'</span> <span class="id" title="var">acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {          <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc2_intro</span> : <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}       <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <span class="id" title="var">H_yc2_intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_in_s</span>] [<span class="id" title="var">H_y2_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in_s</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {              <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s1'_alt</span> <span class="id" title="var">H_lt_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_to_N_minus_neq_0"><span class="id" title="lemma">Z_to_N_minus_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span> /<span class="id" title="var">P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y12_eq</span>] <span class="id" title="var">H_c12_eq</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">P4</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>. {         <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_y_lt</span> <span class="id" title="var">H_lt_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) <span class="id" title="tactic">by</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}        <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le_y1</span>] [<span class="id" title="var">H_yc1_le_yc2</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [?] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}          <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span> /<span class="id" title="var">P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] [<span class="id" title="var">H_yc2_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">P4</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {          <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_in_s</span>] [<span class="id" title="var">H_y2_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span> /<span class="id" title="var">P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_yc2_lt</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">P4</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le_y</span>] <span class="id" title="var">H_y_lt_yc1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le_y</span>] <span class="id" title="var">H_y_lt_yc2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s1'_alt</span> <span class="id" title="var">H_lt_y</span> [<span class="id" title="var">_</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}         <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span> /<span class="id" title="var">P4</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y1_eq</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">P4</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span> <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le_y</span>] <span class="id" title="var">H_y_lt_yc1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le_y</span>] <span class="id" title="var">H_y_lt_yc2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s1'_alt</span> <span class="id" title="var">H_lt_y</span> [<span class="id" title="var">_</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.inter_aux2_props" class="idref" href="#Raw.inter_aux2_props"><span class="id" title="lemma">inter_aux2_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s2:604" class="idref" href="#s2:604"><span class="id" title="binder">s2</span></a> <a id="s1:605" class="idref" href="#s1:605"><span class="id" title="binder">s1</span></a> <a id="acc:606" class="idref" href="#acc:606"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:606"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:605"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:604"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:607" class="idref" href="#x1:607"><span class="id" title="binder">x1</span></a> <a id="x2:608" class="idref" href="#x2:608"><span class="id" title="binder">x2</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:607"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:606"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:608"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:605"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:608"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:604"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:607"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:608"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:609" class="idref" href="#y:609"><span class="id" title="binder">y</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:609"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2"><span class="id" title="definition">inter_aux2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:606"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:605"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:604"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:609"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:606"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:609"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:605"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:609"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:604"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2"><span class="id" title="definition">inter_aux2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:606"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:605"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:604"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s2'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s1</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2"><span class="id" title="definition">inter_aux2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="keyword">match</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> <span class="id" title="var">acc</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> <span class="id" title="var">acc</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> <span class="id" title="var">acc</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">case</span> <span class="id" title="var">s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">acc</span> <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">H_inv_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_s2'</span>] [<span class="id" title="var">H_c2_neq_0</span>] <span class="id" title="var">H_inv_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_acc_s12</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_gr_s2'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2"><span class="id" title="definition">inter_aux2</span></a>; <span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2"><span class="id" title="definition">inter_aux2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> <span class="id" title="var">s1</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_s1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span> <span class="id" title="var">_</span>] <span class="id" title="var">_</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux"><span class="id" title="definition">inter_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">acc</span> <span class="id" title="var">s1</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">s1'</span> <span class="id" title="var">H_inter_aux_eq</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_acc_s1_yc2</span> : <span class="id" title="keyword">∀</span> <a id="x1:612" class="idref" href="#x1:612"><span class="id" title="binder">x1</span></a> <a id="x2:613" class="idref" href="#x2:613"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:612"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:613"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:613"><span class="id" title="variable">x2</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:612"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:613"><span class="id" title="variable">x2</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_x1_in</span> <span class="id" title="var">H_x2_in1</span> <span class="id" title="var">H_x2_in2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_acc_s12</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux_props"><span class="id" title="lemma">inter_aux_props</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">s1</span> <span class="id" title="var">acc</span> <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_acc_s1_yc2</span> <span class="id" title="var">H_c2_neq_0</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_inter_aux_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_in_acc'</span>] [<span class="id" title="var">H_in_s1'_elim</span>] [<span class="id" title="var">H_in_s1'_intro</span>]<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[<span class="id" title="var">H_inv_acc'</span>] <span class="id" title="var">H_inv_s1'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_acc'_s2'</span> : <span class="id" title="keyword">∀</span> <a id="x1:614" class="idref" href="#x1:614"><span class="id" title="binder">x1</span></a> <a id="x2:615" class="idref" href="#x2:615"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:614"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:615"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:615"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s2'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:614"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:615"><span class="id" title="variable">x2</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> /<span class="id" title="var">H_in_acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_acc</span> <span class="id" title="var">H_in_s1'</span> <span class="id" title="var">H_in_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_acc_s12</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_s1'_elim</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_in_s1</span>] [<span class="id" title="var">_</span>] <span class="id" title="var">H_x1_lt</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s2'</span> <span class="id" title="var">H_lt_x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">s1'</span> <span class="id" title="var">acc'</span> <span class="id" title="var">H_inv_acc'</span> <span class="id" title="var">H_inv_s1'</span> <span class="id" title="var">H_inv_s2'</span> <span class="id" title="var">H_in_acc'_s2'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inZ_res</span>] <span class="id" title="var">H_inv_res</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_inZ_res</span> <span class="id" title="var">H_in_acc'</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y_in_s1'</span> <span class="id" title="var">H_y_in_s2'</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_s1'_elim</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_in_acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y_in_s1</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_yc2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>; <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_s1'_intro</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.inter_InZ" class="idref" href="#Raw.inter_InZ"><span class="id" title="lemma">inter_InZ</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:616" class="idref" href="#s1:616"><span class="id" title="binder">s1</span></a> <a id="s2:617" class="idref" href="#s2:617"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:616"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:617"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:618" class="idref" href="#y:618"><span class="id" title="binder">y</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:618"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:616"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:617"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:618"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:616"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:618"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:617"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2_props"><span class="id" title="lemma">inter_aux2_props</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_inter</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_inter</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.inter_invariant" class="idref" href="#Raw.inter_invariant"><span class="id" title="lemma">inter_invariant</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:619" class="idref" href="#s1:619"><span class="id" title="binder">s1</span></a> <a id="s2:620" class="idref" href="#s2:620"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:619"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:620"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:619"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:620"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2_props"><span class="id" title="lemma">inter_aux2_props</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.inter_ok" class="idref" href="#Raw.inter_ok"><span class="id" title="instance">inter_ok</span></a> <a id="s1:621" class="idref" href="#s1:621"><span class="id" title="binder">s1</span></a> <a id="s2:622" class="idref" href="#s2:622"><span class="id" title="binder">s2</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:623" class="idref" href="#H:623"><span class="id" title="binder">Ok</span></a> <a id="H:623" class="idref" href="#H:623"><span class="id" title="binder">s1</span></a>, <a id="H0:624" class="idref" href="#H0:624"><span class="id" title="binder">Ok</span></a> <a id="H0:624" class="idref" href="#H0:624"><span class="id" title="binder">s2</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:621"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:622"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok_s1</span> <span class="id" title="var">H_ok_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok_s1</span>) (<span class="id" title="var">H_ok_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s1</span>] <span class="id" title="var">H_pre1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s2</span>] <span class="id" title="var">H_pre2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_invariant"><span class="id" title="lemma">inter_invariant</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_InZ"><span class="id" title="lemma">inter_InZ</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_pre1</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.inter_spec" class="idref" href="#Raw.inter_spec"><span class="id" title="lemma">inter_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:625" class="idref" href="#s:625"><span class="id" title="binder">s</span></a> <a id="s':626" class="idref" href="#s':626"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:627" class="idref" href="#x:627"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:628" class="idref" href="#Hs:628"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:625"><span class="id" title="variable">s</span></a>) (<a id="Hs':629" class="idref" href="#Hs':629"><span class="id" title="binder">Hs'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':626"><span class="id" title="variable">s'</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:627"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:625"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':626"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:627"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:625"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:627"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':626"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span> <span class="id" title="var">H_ok</span> <span class="id" title="var">H_ok'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.inter_InZ"><span class="id" title="lemma">inter_InZ</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab41"></a><h3 class="section">diff specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.diff_aux_alt_def" class="idref" href="#Raw.diff_aux_alt_def"><span class="id" title="lemma">diff_aux_alt_def</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="y2:630" class="idref" href="#y2:630"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:631" class="idref" href="#c2:631"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="s:632" class="idref" href="#s:632"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="acc:633" class="idref" href="#acc:633"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:630"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:631"><span class="id" title="variable">c2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:633"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:632"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:630"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:631"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:632"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:633"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">acc'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_l"><span class="id" title="lemma">app_nil_l</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">acc'</span>) ⇒ <span class="id" title="var">acc''</span> <span class="id" title="var">s''</span> <span class="id" title="var">H_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>);<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">acc</span>)) ?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> <span class="id" title="var">acc</span>) ?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>))<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;?(<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)) ?<span class="id" title="var">H_eq</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> ?<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded_app"><span class="id" title="lemma">insert_intervalZ_guarded_app</span></a> -?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> -?<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_comm_cons"><span class="id" title="lemma">app_comm_cons</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.diff_aux_props" class="idref" href="#Raw.diff_aux_props"><span class="id" title="lemma">diff_aux_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="y2:634" class="idref" href="#y2:634"><span class="id" title="binder">y2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="c2:635" class="idref" href="#c2:635"><span class="id" title="binder">c2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) (<a id="s:636" class="idref" href="#s:636"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) <a id="acc:637" class="idref" href="#acc:637"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:637"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:636"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:638" class="idref" href="#x1:638"><span class="id" title="binder">x1</span></a> <a id="x2:639" class="idref" href="#x2:639"><span class="id" title="binder">x2</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:638"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:637"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:639"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:636"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:638"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:639"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:640" class="idref" href="#x:640"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:640"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:637"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:640"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:634"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c2:635"><span class="id" title="variable">c2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:634"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:635"><span class="id" title="variable">c2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:637"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:636"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:641" class="idref" href="#y:641"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:641"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:641"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:637"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:636"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:641"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:634"><span class="id" title="variable">y2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:635"><span class="id" title="variable">c2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:642" class="idref" href="#x:642"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:642"><span class="id" title="variable">x</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:642"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:634"><span class="id" title="variable">y2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:635"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_alt"><span class="id" title="lemma">rev_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in_acc</span> <span class="id" title="var">H_c2_neq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>; <span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in</span>. <span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> ⇒ [] [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a> <span class="id" title="var">H_neq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_neq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">H_in_acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_acc</span> [<span class="id" title="var">H_gr_s1'</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_s1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_s1</span> <span class="id" title="var">H_in_acc</span> <span class="id" title="var">H_c2_neq_0</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_alt_def"><span class="id" title="lemma">diff_aux_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span> <span class="id" title="var">H_diff_aux_eq</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P1</span> := <span class="id" title="keyword">∀</span> <a id="y:645" class="idref" href="#y:645"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:643"><span class="id" title="variable">y</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:643"><span class="id" title="variable">y</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:643"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s1'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:643"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P2</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <span class="id" title="var">s'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">P3</span> := <span class="id" title="keyword">∀</span> <a id="x:648" class="idref" href="#x:648"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:646"><span class="id" title="variable">x</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:646"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<span class="id" title="var">P1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P3</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_P1</span>] [<span class="id" title="var">H_P2</span>] <span class="id" title="var">H_P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_P1</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_app_distr"><span class="id" title="lemma">rev_app_distr</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;!<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y</span> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_in_acc</span> <span class="id" title="var">H_y_lt</span> [<span class="id" title="var">H_y_ge</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_y_ge</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Zorder.html#Zlt_not_le"><span class="id" title="lemma">Zlt_not_le</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_app_distr"><span class="id" title="lemma">rev_app_distr</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro"><span class="id" title="lemma">interval_list_invariant_app_intro</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_acc'</span> <span class="id" title="var">H_x2_in_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">x2</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s1'</span>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_P1</span> <span class="id" title="var">x2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_in_acc</span> <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_gr_s1'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ // ⇒ <span class="id" title="var">H_gr_s1'_alt</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <span class="id" title="keyword">∀</span> (<a id="acc:649" class="idref" href="#acc:649"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:649"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:650" class="idref" href="#x:650"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:650"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:649"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:650"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:650"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:649"><span class="id" title="variable">acc</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">P1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">P3</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc0</span> <span class="id" title="var">H_inv_acc0</span> <span class="id" title="var">H_in_acc0</span> <span class="id" title="var">H_c1_before</span> <span class="id" title="var">H_diff_aux_eq0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_s1'</span> : (<span class="id" title="keyword">∀</span> <a id="x1:651" class="idref" href="#x1:651"><span class="id" title="binder">x1</span></a> <a id="x2:652" class="idref" href="#x2:652"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:651"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:652"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:651"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:652"><span class="id" title="variable">x2</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_x1_in_acc0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s1'_alt</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_x1_in_acc0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a> <span class="id" title="var">H_in_acc0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_acc0'</span> : (<span class="id" title="keyword">∀</span> <a id="x:653" class="idref" href="#x:653"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:653"><span class="id" title="variable">x</span></a> <span class="id" title="var">acc0</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:653"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_acc0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">acc0</span> <span class="id" title="var">H_inv_acc0</span> <span class="id" title="var">H_inv_s1'</span> <span class="id" title="var">H_in_s1'</span> <span class="id" title="var">H_in_acc0'</span> <span class="id" title="var">H_c2_neq_0</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_diff_aux_eq0</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H1</span>] [<span class="id" title="var">H2</span>] <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H1</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_acc0</span> -(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_acc0</span> -(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_in</span> <span class="id" title="var">H_nin_i2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in</span> ⇒ [] <span class="id" title="var">H_in</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {} <span class="id" title="var">IH</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">H_in_s1</span> <span class="id" title="var">H_in_acc</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>) <span class="id" title="var">H_diff_aux_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒ <span class="id" title="var">H_comp</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span> <span class="id" title="var">H_x_lt</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_eq_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> -<span class="id" title="var">H_eq_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_eq_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_lt_y2</span>] [<span class="id" title="var">H_y2_lt_yc1</span>] <span class="id" title="var">H_yc1_lt_yc2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing"><span class="id" title="lemma">interval_list_invariant_sing</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_to_N_minus_neq_0"><span class="id" title="lemma">Z_to_N_minus_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">last</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">y2</span> <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_lt</span>] [<span class="id" title="var">H_y1_lt</span>] <span class="id" title="var">H_yc1_lt</span> [<span class="id" title="var">H_acc'</span>] <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc1_intro</span> : <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_nin_yc2</span> : <span class="id" title="keyword">∀</span> <a id="y:654" class="idref" href="#y:654"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:654"><span class="id" title="variable">y</span></a> <span class="id" title="var">s1'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">≤</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:654"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> /<span class="id" title="var">H_gr_s1'_alt</span> <span class="id" title="var">H_lt_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le_y</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_ngt"><span class="id" title="lemma">Z.le_ngt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <span class="id" title="var">H_yc1_intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {             <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_le_y</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_nin_yc2</span> <span class="id" title="var">y</span>). <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] []; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>; <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y_ge</span>] <span class="id" title="var">H_y_lt_yc1</span> <span class="id" title="var">H_nin_yc2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> <span class="id" title="var">H_yc1_intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_to_N_minus_neq_0"><span class="id" title="lemma">Z_to_N_minus_neq_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y12_eq</span>] <span class="id" title="var">H_c12_eq</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>. {         <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>, <span class="id" title="var">H_gr_s1'_alt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_inv_s1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le_y1</span>] [<span class="id" title="var">H_yc1_le_yc2</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] [<span class="id" title="var">H_yc2_le_yc1</span>] <span class="id" title="var">_</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc1_intro</span> : <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}       <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y1_intro</span> : <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y2</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}              <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_insert_intervalZ_guarded"><span class="id" title="lemma">InZ_insert_intervalZ_guarded</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;!<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H_yc1_intro</span> <span class="id" title="var">H_y1_intro</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#or_assoc"><span class="id" title="lemma">or_assoc</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [[[]|]|] //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le_y</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_ngt"><span class="id" title="lemma">Z.le_ngt</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2c_le_y</span>] <span class="id" title="var">H_y_lt_yc1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">_</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_nge"><span class="id" title="lemma">Z.lt_nge</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_in_s1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_suff</span> : <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#306329b0eca7a2b86c198702f594ad8e"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">_</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_nge"><span class="id" title="lemma">Z.lt_nge</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_s1'_alt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] []; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le_y</span>] <span class="id" title="var">H_y_lt</span> <span class="id" title="var">H_neq_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Logic.Decidable.html#not_and"><span class="id" title="lemma">not_and</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H_neq_y2</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_decidable"><span class="id" title="lemma">Z.le_decidable</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">H_neq_y2</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nle_gt"><span class="id" title="lemma">Z.nle_gt</span></a> <span class="id" title="var">H_y_lt'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>; <span class="id" title="tactic">left</span>; <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a> <span class="id" title="var">H_le_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>; <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded_rev_nil_app"><span class="id" title="lemma">insert_intervalZ_guarded_rev_nil_app</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_insert_intervalZ_guarded"><span class="id" title="lemma">interval_list_invariant_insert_intervalZ_guarded</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_yc1_intro</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> ((<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">)</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y1_intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_eq"><span class="id" title="lemma">N.eqb_eq</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_iff"><span class="id" title="lemma">N2Z.inj_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Z2N.id"><span class="id" title="lemma">Z2N.id</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_0_sub"><span class="id" title="lemma">Z.le_0_sub</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Zminus_eq"><span class="id" title="lemma">Zminus_eq</span></a> <span class="id" title="var">H_yc1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl"><span class="id" title="lemma">interval_list_elements_greater_impl</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_s1'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_yc1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H_y1_intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_insert_intervalZ_guarded"><span class="id" title="lemma">InZ_insert_intervalZ_guarded</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <span class="id" title="var">H_y1_intro</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_x_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_yc2_lt</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>. {         <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_y2_le</span> <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">H_y_in</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s1'_alt</span> <span class="id" title="var">H_lt_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ ?. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<span class="id" title="var">P1</span> /<span class="id" title="var">P2</span> /<span class="id" title="var">P3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_yc2_eq</span> [] <span class="id" title="var">H_acc'</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">IH</span> <span class="id" title="var">P1</span> <span class="id" title="var">P2</span> <span class="id" title="var">P3</span> <span class="id" title="var">H_comp</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>. {         <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_y2_le</span> <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">H_y_in</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_gr_s1'_alt</span> <span class="id" title="var">H_lt_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ ?. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.diff_aux2_props" class="idref" href="#Raw.diff_aux2_props"><span class="id" title="lemma">diff_aux2_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s2:655" class="idref" href="#s2:655"><span class="id" title="binder">s2</span></a> <a id="s1:656" class="idref" href="#s1:656"><span class="id" title="binder">s1</span></a> <a id="acc:657" class="idref" href="#acc:657"><span class="id" title="binder">acc</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:657"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:656"><span class="id" title="variable">s1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:655"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:658" class="idref" href="#x1:658"><span class="id" title="binder">x1</span></a> <a id="x2:659" class="idref" href="#x2:659"><span class="id" title="binder">x2</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:658"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:657"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:659"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:655"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:658"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:659"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:660" class="idref" href="#y:660"><span class="id" title="binder">y</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:660"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2"><span class="id" title="definition">diff_aux2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:657"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:656"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:655"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:660"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:657"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:660"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:656"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:660"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:655"><span class="id" title="variable">s2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2"><span class="id" title="definition">diff_aux2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:657"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:656"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:655"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s2'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s1</span> <span class="id" title="var">acc</span> <span class="id" title="var">H_inv_acc_s1</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2"><span class="id" title="definition">diff_aux2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="keyword">match</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <span class="id" title="var">acc</span> <span class="id" title="var">s1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <span class="id" title="var">acc</span> <span class="id" title="var">s1</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>) <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <span class="id" title="var">acc</span> <span class="id" title="var">s1</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">case</span> <span class="id" title="var">s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}        <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">acc</span> <span class="id" title="var">H_inv_acc_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_s2'</span>] [<span class="id" title="var">H_c2_neq_0</span>] <span class="id" title="var">H_inv_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_acc_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2"><span class="id" title="definition">diff_aux2</span></a>; <span class="id" title="tactic">fold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2"><span class="id" title="definition">diff_aux2</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> <span class="id" title="var">s1</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_s1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y_in'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_acc_s2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_y_in</span> <span class="id" title="var">H_y_in'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_succ_diag_l"><span class="id" title="lemma">Z.nlt_succ_diag_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_inv_acc_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_s1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span> <span class="id" title="var">_</span>] <span class="id" title="var">_</span> &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">acc</span> <span class="id" title="var">s1</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">s1'</span> <span class="id" title="var">H_diff_aux_eq</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_acc_lt_y2</span> : (<span class="id" title="keyword">∀</span> <a id="x:663" class="idref" href="#x:663"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:663"><span class="id" title="variable">x</span></a> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:663"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> <span class="id" title="var">H_x_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_in</span>: (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y2</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">s2'</span>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_acc_s2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x_in</span> <span class="id" title="var">H_y2_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_diag_r"><span class="id" title="lemma">Z.lt_succ_diag_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">H_inv_acc</span> [<span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_acc_s1</span>]] :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <span class="id" title="var">acc</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:664" class="idref" href="#x1:664"><span class="id" title="binder">x1</span></a> <a id="x2:665" class="idref" href="#x2:665"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:664"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:665"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:664"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:665"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>. {<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_inv_acc_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_elim"><span class="id" title="lemma">interval_list_invariant_app_elim</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [?] [?] <span class="id" title="var">H_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_in_x1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_props"><span class="id" title="lemma">diff_aux_props</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">s1</span> <span class="id" title="var">acc</span> <span class="id" title="var">H_inv_acc</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_acc_s1</span> <span class="id" title="var">H_acc_lt_y2</span> <span class="id" title="var">H_c2_neq_0</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<span class="id" title="var">H_diff_aux_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inZ_res</span>] [<span class="id" title="var">H_inv_res</span>] <span class="id" title="var">H_inZ_acc'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_acc'_s2'</span> : (<span class="id" title="keyword">∀</span> <a id="x1:666" class="idref" href="#x1:666"><span class="id" title="binder">x1</span></a> <a id="x2:667" class="idref" href="#x2:667"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:666"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:667"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s2'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:666"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:667"><span class="id" title="variable">x2</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_x1_in</span> <span class="id" title="var">H_x2_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_inZ_acc'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_gr_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr_s2'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">s1'</span> <span class="id" title="var">acc'</span> <span class="id" title="var">H_inv_res</span> <span class="id" title="var">H_inv_s2'</span> <span class="id" title="var">H_acc'_s2'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_inZ_diff_res</span> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_inZ_diff_res</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_inZ_res</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.diff_InZ" class="idref" href="#Raw.diff_InZ"><span class="id" title="lemma">diff_InZ</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:668" class="idref" href="#s1:668"><span class="id" title="binder">s1</span></a> <a id="s2:669" class="idref" href="#s2:669"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:668"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:669"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="y:670" class="idref" href="#y:670"><span class="id" title="binder">y</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:670"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:668"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:669"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:670"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:668"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:670"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:669"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2_props"><span class="id" title="lemma">diff_aux2_props</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_in_diff</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_diff</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.diff_invariant" class="idref" href="#Raw.diff_invariant"><span class="id" title="lemma">diff_invariant</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s1:671" class="idref" href="#s1:671"><span class="id" title="binder">s1</span></a> <a id="s2:672" class="idref" href="#s2:672"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:671"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:672"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:671"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:672"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2_props"><span class="id" title="lemma">diff_aux2_props</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.diff_ok" class="idref" href="#Raw.diff_ok"><span class="id" title="instance">diff_ok</span></a> <a id="s1:673" class="idref" href="#s1:673"><span class="id" title="binder">s1</span></a> <a id="s2:674" class="idref" href="#s2:674"><span class="id" title="binder">s2</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:675" class="idref" href="#H:675"><span class="id" title="binder">Ok</span></a> <a id="H:675" class="idref" href="#H:675"><span class="id" title="binder">s1</span></a>, <a id="H0:676" class="idref" href="#H0:676"><span class="id" title="binder">Ok</span></a> <a id="H0:676" class="idref" href="#H0:676"><span class="id" title="binder">s2</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:673"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:674"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok_s1</span> <span class="id" title="var">H_ok_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok_s1</span>) (<span class="id" title="var">H_ok_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.add"><span class="id" title="definition">add</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s1</span>] <span class="id" title="var">H_pre1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s2</span>] <span class="id" title="var">H_pre2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_invariant"><span class="id" title="lemma">diff_invariant</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_InZ"><span class="id" title="lemma">diff_InZ</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<span class="id" title="var">H_pre1</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.diff_spec" class="idref" href="#Raw.diff_spec"><span class="id" title="lemma">diff_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:677" class="idref" href="#s:677"><span class="id" title="binder">s</span></a> <a id="s':678" class="idref" href="#s':678"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:679" class="idref" href="#x:679"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:680" class="idref" href="#Hs:680"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:677"><span class="id" title="variable">s</span></a>) (<a id="Hs':681" class="idref" href="#Hs':681"><span class="id" title="binder">Hs'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':678"><span class="id" title="variable">s'</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:679"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:677"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':678"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:679"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:677"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:679"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':678"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span> <span class="id" title="var">H_ok</span> <span class="id" title="var">H_ok'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_InZ"><span class="id" title="lemma">diff_InZ</span></a> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab42"></a><h3 class="section">remove specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.removeZ_alt_def" class="idref" href="#Raw.removeZ_alt_def"><span class="id" title="lemma">removeZ_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="x:682" class="idref" href="#x:682"><span class="id" title="binder">x</span></a> <a id="s:683" class="idref" href="#s:683"><span class="id" title="binder">s</span></a> <a id="acc:684" class="idref" href="#acc:684"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:683"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>  <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ_aux"><span class="id" title="definition">removeZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:684"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:682"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:683"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:682"><span class="id" title="variable">x</span></a> (1%<span class="id" title="var">N</span>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:684"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:683"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">rev_append</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_elim"><span class="id" title="lemma">interval_1_compare_elim</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_alt_def"><span class="id" title="lemma">interval_1_compare_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_swap"><span class="id" title="lemma">interval_compare_swap</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>. <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> (1%<span class="id" title="var">N</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a>1%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a>) ⇒ <span class="id" title="var">H_eq</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y2</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y2_y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_yc2_nlt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_nlt</span>) (<span class="id" title="var">H_yc2_nlt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> → /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y2</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_lt_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y2_y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_yc2_nlt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_nlt</span>) (<span class="id" title="var">H_yc2_nlt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> → /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_eq</span>] <span class="id" title="var">H_c1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // .<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_lt_y2</span>] <span class="id" title="var">H_y2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_nlt</span>) (<span class="id" title="var">H_y2_lt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> → /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">y2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>)%<span class="id" title="var">N</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y1_eq</span> <span class="id" title="var">H_c1_eq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.sub_diag"><span class="id" title="lemma">Z.sub_diag</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>)%<span class="id" title="var">N</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y1_eq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.sub_diag"><span class="id" title="lemma">Z.sub_diag</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le</span>] [<span class="id" title="var">H_yc1_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] <span class="id" title="var">H_y2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_nlt</span>) (<span class="id" title="var">H_y2_lt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> → /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> →.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y1_eq</span> : (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y2</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc1_eq</span> : (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_antisymm"><span class="id" title="lemma">Z.le_antisymm</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_yc1_le</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">y2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>)%<span class="id" title="var">N</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_yc1_eq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.sub_diag"><span class="id" title="lemma">Z.sub_diag</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>)%<span class="id" title="var">N</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y1_eq</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.sub_diag"><span class="id" title="lemma">Z.sub_diag</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">y2</span> (1%<span class="id" title="var">N</span>) <span class="id" title="var">acc</span> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_yc1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>] <span class="id" title="var">s''</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a> /=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#1718f1bc306f7edd524650b393f464a3"><span class="id" title="notation">?=</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_lt_iff"><span class="id" title="lemma">Z.compare_lt_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} { <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] [<span class="id" title="var">H_yc2_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_y2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_nlt</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_nlt</span>) (<span class="id" title="var">H_y2_lt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_nlt"><span class="id" title="lemma">Z.ltb_nlt</span></a> → /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded"><span class="id" title="definition">insert_intervalZ_guarded</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> ((<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <span class="id" title="var">y1</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span>),  (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> (<span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">y2</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#59a5ff4a6cbb127625aba1a377a243d1"><span class="id" title="notation">=?</span></a> 0)%<span class="id" title="var">N</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">H_y2_lt'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_lt</span> : (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">y2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_diag_r"><span class="id" title="lemma">Z.lt_succ_diag_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_lt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y2_lt</span> : (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y1</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_diag_r"><span class="id" title="lemma">Z.lt_succ_diag_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_y2_lt</span>) ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.removeZ_interval_list_invariant" class="idref" href="#Raw.removeZ_interval_list_invariant"><span class="id" title="lemma">removeZ_interval_list_invariant</span></a> : <span class="id" title="keyword">∀</span> <a id="s:685" class="idref" href="#s:685"><span class="id" title="binder">s</span></a> <a id="x:686" class="idref" href="#x:686"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:685"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:686"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:685"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ_alt_def"><span class="id" title="lemma">removeZ_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_props"><span class="id" title="lemma">diff_aux_props</span></a> <span class="id" title="var">x</span> (1%<span class="id" title="var">N</span>) <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">x</span> 1%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">s</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span> <span class="id" title="var">H_eq</span> [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> [] //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.removeZ_spec" class="idref" href="#Raw.removeZ_spec"><span class="id" title="lemma">removeZ_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:687" class="idref" href="#s:687"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:688" class="idref" href="#x:688"><span class="id" title="binder">x</span></a> <a id="y:689" class="idref" href="#y:689"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="Hs:690" class="idref" href="#Hs:690"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:687"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:689"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:688"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:687"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:689"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:687"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:689"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:688"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ"><span class="id" title="definition">removeZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ_alt_def"><span class="id" title="lemma">removeZ_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_props"><span class="id" title="lemma">diff_aux_props</span></a> <span class="id" title="var">x</span> (1%<span class="id" title="var">N</span>) <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.diff_aux"><span class="id" title="definition">diff_aux</span></a> <span class="id" title="var">x</span> 1%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">s</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">s'</span> <span class="id" title="var">H_eq</span> [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_app"><span class="id" title="lemma">InZ_app</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H1</span> <span class="id" title="var">H2</span>]; <span class="id" title="tactic">split</span> ⇒ //;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H3</span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">H2</span>; <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.remove_ok" class="idref" href="#Raw.remove_ok"><span class="id" title="instance">remove_ok</span></a> <a id="s:691" class="idref" href="#s:691"><span class="id" title="binder">s</span></a> <a id="x:692" class="idref" href="#x:692"><span class="id" title="binder">x</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:693" class="idref" href="#H:693"><span class="id" title="binder">Ok</span></a> <a id="H:693" class="idref" href="#H:693"><span class="id" title="binder">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:692"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:691"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove"><span class="id" title="definition">remove</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_is_ok_s</span> <span class="id" title="var">H_enc_s</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ_interval_list_invariant"><span class="id" title="lemma">removeZ_interval_list_invariant</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ_spec"><span class="id" title="lemma">removeZ_spec</span></a> <span class="id" title="var">s</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">y</span> <span class="id" title="var">H_is_ok_s</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> ⇒ → [] <span class="id" title="var">H_y_in</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc_s</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.remove_spec" class="idref" href="#Raw.remove_spec"><span class="id" title="lemma">remove_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:694" class="idref" href="#s:694"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:695" class="idref" href="#x:695"><span class="id" title="binder">x</span></a> <a id="y:696" class="idref" href="#y:696"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:697" class="idref" href="#Hs:697"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:694"><span class="id" title="variable">s</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:696"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:695"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:694"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:696"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:694"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:696"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:695"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_rs</span> := (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove_ok"><span class="id" title="instance">remove_ok</span></a> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">Hs</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove"><span class="id" title="definition">remove</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.removeZ_spec"><span class="id" title="lemma">removeZ_spec</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_eq"><span class="id" title="axiom">Enc.encode_eq</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab43"></a><h3 class="section">remove_list specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.remove_list_ok" class="idref" href="#Raw.remove_list_ok"><span class="id" title="lemma">remove_list_ok</span></a> : <span class="id" title="keyword">∀</span> <a id="l:698" class="idref" href="#l:698"><span class="id" title="binder">l</span></a> <a id="s:699" class="idref" href="#s:699"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:699"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove_list"><span class="id" title="definition">remove_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:698"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:699"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">l'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">H_s_ok</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove_ok"><span class="id" title="instance">remove_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.remove_list_spec" class="idref" href="#Raw.remove_list_spec"><span class="id" title="lemma">remove_list_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:700" class="idref" href="#x:700"><span class="id" title="binder">x</span></a> <a id="l:701" class="idref" href="#l:701"><span class="id" title="binder">l</span></a> <a id="s:702" class="idref" href="#s:702"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:702"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:700"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove_list"><span class="id" title="definition">remove_list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:701"><span class="id" title="variable">l</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:702"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:700"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:701"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:700"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:702"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span> <span class="id" title="var">l'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">H_ok</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.remove_spec"><span class="id" title="lemma">remove_spec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab44"></a><h3 class="section">subset specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.subset_flatten_alt_def" class="idref" href="#Raw.subset_flatten_alt_def"><span class="id" title="lemma">subset_flatten_alt_def</span></a> : <span class="id" title="keyword">∀</span> (<a id="s1:703" class="idref" href="#s1:703"><span class="id" title="binder">s1</span></a> <a id="s2:704" class="idref" href="#s2:704"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:703"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:704"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:703"><span class="id" title="variable">s1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:704"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">_</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before"><span class="id" title="constructor">ICR_before</span></a>       ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_before_touch"><span class="id" title="constructor">ICR_before_touch</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after"><span class="id" title="constructor">ICR_after</span></a>        ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:703"><span class="id" title="variable">s1</span></a> <span class="id" title="var">l2</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_after_touch"><span class="id" title="constructor">ICR_after_touch</span></a>  ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_before"><span class="id" title="constructor">ICR_overlap_before</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_overlap_after"><span class="id" title="constructor">ICR_overlap_after</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_equal"><span class="id" title="constructor">ICR_equal</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <span class="id" title="var">l1</span> <span class="id" title="var">l2</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_1"><span class="id" title="constructor">ICR_subsume_1</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:704"><span class="id" title="variable">s2</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.ICR_subsume_2"><span class="id" title="constructor">ICR_subsume_2</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s1</span>, <span class="id" title="var">s2</span> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.subset_props_aux" class="idref" href="#Raw.subset_props_aux"><span class="id" title="lemma">subset_props_aux</span></a> : <span class="id" title="keyword">∀</span> <a id="y1:705" class="idref" href="#y1:705"><span class="id" title="binder">y1</span></a> <a id="c1:706" class="idref" href="#c1:706"><span class="id" title="binder">c1</span></a> <a id="l1:707" class="idref" href="#l1:707"><span class="id" title="binder">l1</span></a> <a id="y2:708" class="idref" href="#y2:708"><span class="id" title="binder">y2</span></a> <a id="c2:709" class="idref" href="#c2:709"><span class="id" title="binder">c2</span></a> <a id="l2:710" class="idref" href="#l2:710"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:711" class="idref" href="#y:711"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:711"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:705"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:706"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l1:707"><span class="id" title="variable">l1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:711"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:708"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:709"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:710"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:712" class="idref" href="#y:712"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:712"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:705"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:706"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l1:707"><span class="id" title="variable">l1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:712"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:708"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:709"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:710"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">l1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span>] [<span class="id" title="var">H_y_in</span> <span class="id" title="var">H_y_nin</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_y_nin</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.subset_props_aux_before" class="idref" href="#Raw.subset_props_aux_before"><span class="id" title="lemma">subset_props_aux_before</span></a> : <span class="id" title="keyword">∀</span> <a id="y1:713" class="idref" href="#y1:713"><span class="id" title="binder">y1</span></a> <a id="c1:714" class="idref" href="#c1:714"><span class="id" title="binder">c1</span></a> <a id="l1:715" class="idref" href="#l1:715"><span class="id" title="binder">l1</span></a> <a id="y2:716" class="idref" href="#y2:716"><span class="id" title="binder">y2</span></a> <a id="c2:717" class="idref" href="#c2:717"><span class="id" title="binder">c2</span></a> <a id="l2:718" class="idref" href="#l2:718"><span class="id" title="binder">l2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c1:714"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> 0%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:716"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:717"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:718"><span class="id" title="variable">l2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:713"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:716"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:719" class="idref" href="#y:719"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:719"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:713"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:714"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l1:715"><span class="id" title="variable">l1</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:719"><span class="id" title="variable">y</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:716"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:717"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l2:718"><span class="id" title="variable">l2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">l1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span> <span class="id" title="var">l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_c1_neq_0</span> [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_inv_l2</span>] <span class="id" title="var">H_c2_neq_0</span> <span class="id" title="var">H_y1_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux"><span class="id" title="lemma">subset_props_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y1</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">l2</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_ngt"><span class="id" title="lemma">Z.le_ngt</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.subset_props" class="idref" href="#Raw.subset_props"><span class="id" title="lemma">subset_props</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:720" class="idref" href="#s1:720"><span class="id" title="binder">s1</span></a> <a id="s2:721" class="idref" href="#s2:721"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:720"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:721"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:720"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:721"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:722" class="idref" href="#y:722"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:722"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:720"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:722"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:721"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">l1</span> <span class="id" title="var">IH1</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s2</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def"><span class="id" title="lemma">subset_flatten_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">l2</span> <span class="id" title="var">IH2</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def"><span class="id" title="lemma">subset_flatten_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>; <span class="id" title="tactic">move</span> : (<span class="id" title="var">H</span> <span class="id" title="var">y1</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> ⇒ {} <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_inv_s1</span>) (<span class="id" title="var">H_inv_s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_l1</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_l1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_l2</span>] [<span class="id" title="var">H_c2_neq_0</span>] <span class="id" title="var">H_inv_l2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH2</span> <span class="id" title="var">H_inv_s1</span> <span class="id" title="var">H_inv_l2</span>) ⇒ {} <span class="id" title="var">IH2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> :  <span class="id" title="keyword">∀</span> <a id="s2:723" class="idref" href="#s2:723"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:723"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <span class="id" title="var">l1</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:723"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y:724" class="idref" href="#y:724"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:724"><span class="id" title="variable">y</span></a> <span class="id" title="var">l1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:724"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:723"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {} <span class="id" title="var">IH1</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_yc2_nin</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def"><span class="id" title="lemma">subset_flatten_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">c1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_lt_y2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux_before"><span class="id" title="lemma">subset_props_aux_before</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c1</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y2_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux_before"><span class="id" title="lemma">subset_props_aux_before</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_y2_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_lt</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux_before"><span class="id" title="lemma">subset_props_aux_before</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_lt</span>] [<span class="id" title="var">H_y1_lt</span>] <span class="id" title="var">H_yc2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux"><span class="id" title="lemma">subset_props_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_eq</span>] <span class="id" title="var">H_c1_eq</span>; <span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">y</span>; <span class="id" title="tactic">move</span> : (<span class="id" title="var">H_pre</span> <span class="id" title="var">y</span>) ⇒ {<span class="id" title="var">H_pre</span>};<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">H_y_in_l1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <span class="id" title="var">y</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <span class="id" title="var">y2</span> <span class="id" title="var">c2</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr_l1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;// <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>; <span class="id" title="tactic">move</span> : (<span class="id" title="var">H</span> <span class="id" title="var">y</span> <span class="id" title="var">H_y_in_l1</span>) ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_ngt"><span class="id" title="abbreviation">Z.lt_ngt</span></a> <span class="id" title="var">H_neq</span> [<span class="id" title="var">_</span>] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_lt_y1</span>] [<span class="id" title="var">H_yc1_le</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">y</span>; <span class="id" title="tactic">move</span> : (<span class="id" title="var">H_pre</span> <span class="id" title="var">y</span>) ⇒ {<span class="id" title="var">H_pre</span>};<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> []; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">move</span> ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">H_y_in_l1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_pre</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] [<span class="id" title="var">_</span>] []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux_before"><span class="id" title="lemma">subset_props_aux_before</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_yc2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux"><span class="id" title="lemma">subset_props_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> (<span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_yc2_lt_y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">y</span>; <span class="id" title="tactic">move</span> : (<span class="id" title="var">H_pre</span> <span class="id" title="var">y</span>) ⇒ {<span class="id" title="var">H_pre</span>};<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">≤</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#bdb3edde5f841a282ce8fc1b00e3d211"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y2_le</span> <span class="id" title="var">H_y_lt</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_y_in</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_le</span>] <span class="id" title="var">H_y_lt'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_trans"><span class="id" title="lemma">Z.le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y1</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c2</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux"><span class="id" title="lemma">subset_props_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">y1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.subset_spec" class="idref" href="#Raw.subset_spec"><span class="id" title="lemma">subset_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:725" class="idref" href="#s:725"><span class="id" title="binder">s</span></a> <a id="s':726" class="idref" href="#s':726"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="Hs:727" class="idref" href="#Hs:727"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:725"><span class="id" title="variable">s</span></a>) (<a id="Hs':728" class="idref" href="#Hs':728"><span class="id" title="binder">Hs'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':726"><span class="id" title="variable">s'</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:725"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':726"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Subset"><span class="id" title="definition">Subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:725"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':726"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">Hs</span> <span class="id" title="var">Hs'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">Hs</span>) (<span class="id" title="var">Hs'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_s</span> <span class="id" title="var">H_enc_s</span>] [<span class="id" title="var">H_inv_s'</span> <span class="id" title="var">H_enc_s'</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_props"><span class="id" title="lemma">subset_props</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">H_inv_s</span> <span class="id" title="var">H_inv_s'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Subset"><span class="id" title="definition">Subset</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">enc_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_pre</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_pre</span> <span class="id" title="var">y</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_enc_s</span> <span class="id" title="var">_</span> <span class="id" title="var">H_y_in</span>) ⇒ [<span class="id" title="var">e</span>] <span class="id" title="var">H_e</span>. <span class="id" title="tactic">subst</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_pre</span> <span class="id" title="var">e</span>) <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab45"></a><h3 class="section">elements and elementsZ specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elements_spec1" class="idref" href="#Raw.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:729" class="idref" href="#s:729"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:730" class="idref" href="#x:730"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:731" class="idref" href="#Hs:731"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:729"><span class="id" title="variable">s</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:730"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:729"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:730"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:729"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_alt_def"><span class="id" title="lemma">In_alt_def</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.NoDupA_elementsZ_single" class="idref" href="#Raw.NoDupA_elementsZ_single"><span class="id" title="lemma">NoDupA_elementsZ_single</span></a>: <span class="id" title="keyword">∀</span> <a id="c:732" class="idref" href="#c:732"><span class="id" title="binder">c</span></a> <a id="x:733" class="idref" href="#x:733"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:733"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:732"><span class="id" title="variable">c</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_app"><span class="id" title="lemma">NoDupA_app</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq_equiv"><span class="id" title="definition">Z.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_singleton"><span class="id" title="lemma">NoDupA_singleton</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [&lt;-] <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [&lt;-] <span class="id" title="var">H_y_in'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_y_in</span> <span class="id" title="var">H_y_in'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_x_le</span>] <span class="id" title="var">H_y_lt</span> [] // <span class="id" title="var">H_y_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_spec2w" class="idref" href="#Raw.elementsZ_spec2w"><span class="id" title="lemma">elementsZ_spec2w</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:734" class="idref" href="#s:734"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="Hs:735" class="idref" href="#Hs:735"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:734"><span class="id" title="variable">s</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:734"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_nil"><span class="id" title="constructor">NoDupA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok_s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok_s</span>) ⇒ /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok_cons"><span class="id" title="lemma">Ok_cons</span></a> [<span class="id" title="var">H_interval_list_elements_greater</span>] [<span class="id" title="var">H_c</span>] [<span class="id" title="var">H_enc</span>] <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_app"><span class="id" title="lemma">NoDupA_app</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq_equiv"><span class="id" title="definition">Z.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_rev"><span class="id" title="lemma">NoDupA_rev</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq_equiv"><span class="id" title="definition">Z.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.NoDupA_elementsZ_single"><span class="id" title="lemma">NoDupA_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [&lt;-] <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [&lt;-] <span class="id" title="var">H_y_in'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_y_in'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_x_le</span>] <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> <span class="id" title="var">s'</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)) ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>, <span class="id" title="var">H_y_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_y_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elements_spec2w" class="idref" href="#Raw.elements_spec2w"><span class="id" title="lemma">elements_spec2w</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:736" class="idref" href="#s:736"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="Hs:737" class="idref" href="#Hs:737"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:736"><span class="id" title="variable">s</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:736"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_rev"><span class="id" title="lemma">NoDupA_rev</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq_equiv"><span class="id" title="axiom">Enc.E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#NoDupA_map"><span class="id" title="lemma">NoDupA_map</span></a>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_spec2w"><span class="id" title="lemma">elementsZ_spec2w</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span> <span class="id" title="var">H_x1_in</span> <span class="id" title="var">H_x2_in</span> <span class="id" title="var">H_dec_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_is_enc</span>: <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">Hs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_is_enc</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x1_in</span>) ⇒ [<span class="id" title="var">y1</span> <span class="id" title="var">H_x1_eq</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_is_enc</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x2_in</span>) ⇒ [<span class="id" title="var">y2</span> <span class="id" title="var">H_x2_eq</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_dec_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_x1_eq</span> -<span class="id" title="var">H_x2_eq</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_eq"><span class="id" title="axiom">Enc.encode_eq</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab46"></a><h3 class="section">equal specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.equal_alt_def" class="idref" href="#Raw.equal_alt_def"><span class="id" title="lemma">equal_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:738" class="idref" href="#s1:738"><span class="id" title="binder">s1</span></a> <a id="s2:739" class="idref" href="#s2:739"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:738"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:739"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:738"><span class="id" title="variable">s1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:739"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x</span> <span class="id" title="var">cx</span>] <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span> <span class="id" title="var">cy</span>] <span class="id" title="var">ys</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">andb_true_iff</span></a> <span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_eq"><span class="id" title="lemma">N.eqb_eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eqb_eq"><span class="id" title="lemma">Z.eqb_eq</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [-&gt;] [-&gt;] → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [-&gt;] → → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.equal_elementsZ" class="idref" href="#Raw.equal_elementsZ"><span class="id" title="lemma">equal_elementsZ</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:740" class="idref" href="#s:740"><span class="id" title="binder">s</span></a> <a id="s':741" class="idref" href="#s':741"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) {<a id="Hs:742" class="idref" href="#Hs:742"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:740"><span class="id" title="variable">s</span></a>} {<a id="Hs':743" class="idref" href="#Hs':743"><span class="id" title="binder">Hs'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':741"><span class="id" title="variable">s'</span></a>},<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:744" class="idref" href="#x:744"><span class="id" title="binder">x</span></a>, (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:744"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:740"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:744"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':741"><span class="id" title="variable">s'</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s:740"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':741"><span class="id" title="variable">s'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok_s</span> <span class="id" title="var">H_ok_s'</span> <span class="id" title="var">H_InZ_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [] : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset"><span class="id" title="definition">subset</span></a> <span class="id" title="var">s'</span> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_spec"><span class="id" title="lemma">subset_spec</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Subset"><span class="id" title="definition">Subset</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ <span class="id" title="var">x</span>; <span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> <span class="id" title="var">H_InZ_eq</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok_s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H_ok_s</span> <span class="id" title="var">H_ok_s'</span> <span class="id" title="var">H_InZ_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">s</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1</span> <span class="id" title="var">IH</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s2</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_s1</span>] [<span class="id" title="var">H_c1_neq_0</span>] <span class="id" title="var">H_inv_s1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr_s2</span>] [<span class="id" title="var">H_c2_neq_0</span>] <span class="id" title="var">H_inv_s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def"><span class="id" title="lemma">subset_flatten_alt_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def"><span class="id" title="lemma">subset_flatten_alt_def</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s2</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_swap"><span class="id" title="lemma">interval_compare_swap</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">c1</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim"><span class="id" title="lemma">interval_compare_elim</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">c1</span> <span class="id" title="var">x2</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_compare"><span class="id" title="definition">interval_compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c1</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">x2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [-&gt;] → <span class="id" title="var">H_sub_s12</span> <span class="id" title="var">H_sub_s21</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">s2</span> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.equal_spec" class="idref" href="#Raw.equal_spec"><span class="id" title="lemma">equal_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:745" class="idref" href="#s:745"><span class="id" title="binder">s</span></a> <a id="s':746" class="idref" href="#s':746"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) {<a id="Hs:747" class="idref" href="#Hs:747"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:745"><span class="id" title="variable">s</span></a>} {<a id="Hs':748" class="idref" href="#Hs':748"><span class="id" title="binder">Hs'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':746"><span class="id" title="variable">s'</span></a>},<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:745"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':746"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:745"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':746"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">Hs</span> <span class="id" title="var">Hs'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal_alt_def"><span class="id" title="lemma">equal_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal_elementsZ"><span class="id" title="lemma">equal_elementsZ</span></a> ⇒ // <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_ex</span> : (<span class="id" title="keyword">∀</span> <a id="s'':749" class="idref" href="#s'':749"><span class="id" title="binder">s''</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s'':749"><span class="id" title="variable">s''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#s'':749"><span class="id" title="variable">s''</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="z:750" class="idref" href="#z:750"><span class="id" title="binder">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:750"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">HH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">HH</span> (<span class="id" title="var">H3</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ex</span> <span class="id" title="var">s</span> <span class="id" title="var">Hs</span> <span class="id" title="var">H3</span>) ⇒ [<span class="id" title="var">z</span>] &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> ⇒ &lt;- //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">HH</span> (<span class="id" title="var">H3</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ex</span> <span class="id" title="var">s'</span> <span class="id" title="var">Hs'</span> <span class="id" title="var">H3</span>) ⇒ [<span class="id" title="var">z</span>] &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> ⇒ &lt;- //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s''</span> <span class="id" title="var">H_ok</span> <span class="id" title="var">H_in_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_enc</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s''</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab47"></a><h3 class="section">compare</h3>
+
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.lt" class="idref" href="#Raw.lt"><span class="id" title="definition">lt</span></a> (<a id="s1:751" class="idref" href="#s1:751"><span class="id" title="binder">s1</span></a> <a id="s2:752" class="idref" href="#s2:752"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) : <span class="id" title="keyword">Prop</span> := (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:751"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:752"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.compare_eq_Eq" class="idref" href="#Raw.compare_eq_Eq"><span class="id" title="lemma">compare_eq_Eq</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:753" class="idref" href="#s1:753"><span class="id" title="binder">s1</span></a> <a id="s2:754" class="idref" href="#s2:754"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:753"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:754"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:753"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:754"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s2'</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">andb_true_iff</span></a> -<span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eqb_eq"><span class="id" title="lemma">Z.eqb_eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.eqb_eq"><span class="id" title="lemma">N.eqb_eq</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_eq_iff"><span class="id" title="lemma">Z.compare_eq_iff</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : <span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y2</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">clear</span> <span class="id" title="var">H</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare_eq_iff"><span class="id" title="lemma">N.compare_eq_iff</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare"><span class="id" title="definition">N.compare</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : <span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">c2</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_neq</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H</span> ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {        <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_neq</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">c1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">c2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H</span> ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_neq</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H</span> ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_neq</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><span class="id" title="var">y1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H</span> ⇒ {<span class="id" title="var">H</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.compare_eq_Lt_nil_l" class="idref" href="#Raw.compare_eq_Lt_nil_l"><span class="id" title="lemma">compare_eq_Lt_nil_l</span></a> : <span class="id" title="keyword">∀</span> <a id="s:755" class="idref" href="#s:755"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:755"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:755"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;&gt;'_x"><span class="id" title="notation">≠</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> ⇒ //=.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.compare_eq_Lt_nil_r" class="idref" href="#Raw.compare_eq_Lt_nil_r"><span class="id" title="lemma">compare_eq_Lt_nil_r</span></a> : <span class="id" title="keyword">∀</span> <a id="s:756" class="idref" href="#s:756"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:756"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s'</span>] ⇒ //=.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.compare_eq_Lt_cons" class="idref" href="#Raw.compare_eq_Lt_cons"><span class="id" title="lemma">compare_eq_Lt_cons</span></a> : <span class="id" title="keyword">∀</span> <a id="y1:757" class="idref" href="#y1:757"><span class="id" title="binder">y1</span></a> <a id="y2:758" class="idref" href="#y2:758"><span class="id" title="binder">y2</span></a> <a id="c1:759" class="idref" href="#c1:759"><span class="id" title="binder">c1</span></a> <a id="c2:760" class="idref" href="#c2:760"><span class="id" title="binder">c2</span></a> <a id="s1:761" class="idref" href="#s1:761"><span class="id" title="binder">s1</span></a> <a id="s2:762" class="idref" href="#s2:762"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:757"><span class="id" title="variable">y1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c1:759"><span class="id" title="variable">c1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s1:761"><span class="id" title="variable">s1</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y2:758"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:760"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#s2:762"><span class="id" title="variable">s2</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:757"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:758"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:757"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:758"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c1:759"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#::N_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:760"><span class="id" title="variable">c2</span></a>)%<span class="id" title="var">N</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#y1:757"><span class="id" title="variable">y1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y2:758"><span class="id" title="variable">y2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#c1:759"><span class="id" title="variable">c1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c2:760"><span class="id" title="variable">c2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:761"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:762"><span class="id" title="variable">s2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_eq_iff"><span class="id" title="lemma">Z.compare_eq_iff</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare"><span class="id" title="definition">N.compare</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare_eq_iff"><span class="id" title="lemma">N.compare_eq_iff</span></a> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [| []]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt_irrefl"><span class="id" title="lemma">N.lt_irrefl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">_</span>] → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare_lt_iff"><span class="id" title="lemma">N.compare_lt_iff</span></a> <span class="id" title="var">H_c1_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>. <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare_gt_iff"><span class="id" title="lemma">N.compare_gt_iff</span></a> <span class="id" title="var">H_c2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [| []]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_irrefl"><span class="id" title="lemma">Z.lt_irrefl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt_asymm"><span class="id" title="lemma">N.lt_asymm</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [] <span class="id" title="var">H_c1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_c2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">subst</span> <span class="id" title="var">c1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt_irrefl"><span class="id" title="lemma">N.lt_irrefl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_lt_iff"><span class="id" title="lemma">Z.compare_lt_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_gt_iff"><span class="id" title="lemma">Z.compare_gt_iff</span></a> <span class="id" title="var">H_y2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [| []]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_asymm"><span class="id" title="lemma">Z.lt_asymm</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_y1_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>. <span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.compare_antisym" class="idref" href="#Raw.compare_antisym"><span class="id" title="lemma">compare_antisym</span></a>: <span class="id" title="keyword">∀</span> (<a id="s1:763" class="idref" href="#s1:763"><span class="id" title="binder">s1</span></a> <a id="s2:764" class="idref" href="#s2:764"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:763"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:764"><span class="id" title="variable">s2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompOpp"><span class="id" title="definition">CompOpp</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:764"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:763"><span class="id" title="variable">s1</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s1</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s2</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s2'</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare_antisym"><span class="id" title="lemma">Z.compare_antisym</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare_antisym"><span class="id" title="lemma">N.compare_antisym</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.compare"><span class="id" title="definition">Z.compare</span></a> <span class="id" title="var">y1</span> <span class="id" title="var">y2</span>) ⇒ //=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.compare"><span class="id" title="definition">N.compare</span></a> <span class="id" title="var">c1</span> <span class="id" title="var">c2</span>) ⇒ //=.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.compare_spec" class="idref" href="#Raw.compare_spec"><span class="id" title="lemma">compare_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s1:765" class="idref" href="#s1:765"><span class="id" title="binder">s1</span></a> <a id="s2:766" class="idref" href="#s2:766"><span class="id" title="binder">s2</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:765"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:766"><span class="id" title="variable">s2</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:765"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:766"><span class="id" title="variable">s2</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.lt"><span class="id" title="definition">lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare_antisym"><span class="id" title="lemma">compare_antisym</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">s1</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Eq"><span class="id" title="lemma">compare_eq_Eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal_alt_def"><span class="id" title="lemma">equal_alt_def</span></a> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompEq"><span class="id" title="constructor">CompEq</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompLt"><span class="id" title="constructor">CompLt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompGt"><span class="id" title="constructor">CompGt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.lt_Irreflexive" class="idref" href="#Raw.lt_Irreflexive"><span class="id" title="lemma">lt_Irreflexive</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Irreflexive"><span class="id" title="class">Irreflexive</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Irreflexive"><span class="id" title="class">Irreflexive</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Reflexive"><span class="id" title="class">Reflexive</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#complement"><span class="id" title="definition">complement</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare"><span class="id" title="definition">compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Eq"><span class="id" title="lemma">compare_eq_Eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal_alt_def"><span class="id" title="lemma">equal_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.lt_Transitive" class="idref" href="#Raw.lt_Transitive"><span class="id" title="lemma">lt_Transitive</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">x</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y1</span> <span class="id" title="var">c1</span>] <span class="id" title="var">s1'</span> <span class="id" title="var">IH</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">y</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y2</span> <span class="id" title="var">c2</span>] <span class="id" title="var">s2'</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">z</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y3</span> <span class="id" title="var">c3</span>] <span class="id" title="var">s3'</span>] ⇒ //.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_cons"><span class="id" title="lemma">compare_eq_Lt_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_y1_lt</span> | [[-&gt;] <span class="id" title="var">H_c1_lt</span> | [-&gt;] [-&gt;] <span class="id" title="var">H_comp</span>]]<br/>
+            [<span class="id" title="var">H_y2_lt</span> | [[&lt;-] <span class="id" title="var">H_c2_lt</span> | [&lt;-] [&lt;-] <span class="id" title="var">H_comp'</span>]]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.lt_trans"><span class="id" title="lemma">N.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">c2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>; <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> (<span class="id" title="var">IH</span> <span class="id" title="var">s2'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab48"></a><h3 class="section">elements is sorted</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_single_sorted" class="idref" href="#Raw.elementsZ_single_sorted"><span class="id" title="lemma">elementsZ_single_sorted</span></a> : <span class="id" title="keyword">∀</span> <a id="c:767" class="idref" href="#c:767"><span class="id" title="binder">c</span></a> <a id="x:768" class="idref" href="#x:768"><span class="id" title="binder">x</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#sort"><span class="id" title="abbreviation">sort</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:768"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:767"><span class="id" title="variable">c</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_nil"><span class="id" title="constructor">Sorted_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_cons"><span class="id" title="constructor">Sorted_cons</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.zero_or_succ"><span class="id" title="lemma">N.zero_or_succ</span></a> <span class="id" title="var">c'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base"><span class="id" title="lemma">elementsZ_single_base</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">c''</span>] →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">constructor</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_diag_r"><span class="id" title="lemma">Z.lt_succ_diag_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elementsZ_sorted" class="idref" href="#Raw.elementsZ_sorted"><span class="id" title="lemma">elementsZ_sorted</span></a> : <span class="id" title="keyword">∀</span> <a id="s:769" class="idref" href="#s:769"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:769"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#sort"><span class="id" title="abbreviation">sort</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:769"><span class="id" title="variable">s</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil"><span class="id" title="lemma">elementsZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_nil"><span class="id" title="constructor">Sorted_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_app_distr"><span class="id" title="lemma">rev_app_distr</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_involutive"><span class="id" title="lemma">rev_involutive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c_neq_0</span>] <span class="id" title="var">H_inv_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#SortA_app"><span class="id" title="lemma">SortA_app</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">eqA</span> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#eq_equivalence"><span class="id" title="instance">eq_equivalence</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_sorted"><span class="id" title="lemma">elementsZ_single_sorted</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> [<span class="id" title="var">_</span>] [&lt;-] /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a> [<span class="id" title="var">_</span> <span class="id" title="var">H_x1_lt</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> [<span class="id" title="var">_</span>] [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In_rev"><span class="id" title="abbreviation">In_rev</span></a> ⇒ <span class="id" title="var">H_x2_in</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := (<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>)) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.elements_sorted" class="idref" href="#Raw.elements_sorted"><span class="id" title="lemma">elements_sorted</span></a> : <span class="id" title="keyword">∀</span> <a id="s:770" class="idref" href="#s:770"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:770"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#sort"><span class="id" title="abbreviation">sort</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.lt"><span class="id" title="axiom">Enc.E.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:770"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> [<span class="id" title="var">H_inv</span>] <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_rev"><span class="id" title="lemma">map_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : (<span class="id" title="keyword">∀</span> <a id="x:771" class="idref" href="#x:771"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:771"><span class="id" title="variable">x</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="e:772" class="idref" href="#e:772"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.t"><span class="id" title="axiom">Enc.E.t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:772"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:771"><span class="id" title="variable">x</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_enc</span> <span class="id" title="var">x</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In_rev"><span class="id" title="abbreviation">In_rev</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_sorted"><span class="id" title="lemma">elementsZ_sorted</span></a> <span class="id" title="var">s</span> <span class="id" title="var">H_inv</span>) ⇒ {<span class="id" title="var">H_enc</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= ⇒ <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_nil"><span class="id" title="constructor">Sorted_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_sort</span> <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_inv"><span class="id" title="lemma">Sorted_inv</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H_sort</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">H_sort</span> <span class="id" title="var">H_hd_rel</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_cons"><span class="id" title="constructor">Sorted_cons</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">xx</span> <span class="id" title="var">H_xx_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_cons"><span class="id" title="lemma">in_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_hd_rel</span> <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">xs</span> ⇒ //=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x'</span> <span class="id" title="var">xs'</span> <span class="id" title="var">H_hd_rel</span> <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#HdRel_inv"><span class="id" title="lemma">HdRel_inv</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H_hd_rel</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#HdRel_cons"><span class="id" title="constructor">HdRel_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_lt"><span class="id" title="axiom">Enc.encode_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">y</span> <span class="id" title="var">H_y</span>] : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:773" class="idref" href="#y:773"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:773"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">x</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> [<span class="id" title="var">y'</span> <span class="id" title="var">H_y'</span>] : (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y':774" class="idref" href="#y':774"><span class="id" title="binder">y'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':774"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">x'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc</span>. <span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>; <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_hd_rel</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -!<span class="id" title="var">H_y</span> -!<span class="id" title="var">H_y'</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab49"></a><h3 class="section">choose specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.min_eltZ_spec1" class="idref" href="#Raw.min_eltZ_spec1"><span class="id" title="definition">min_eltZ_spec1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:775" class="idref" href="#s:775"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:776" class="idref" href="#x:776"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:775"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:775"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:776"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:776"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:775"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x'</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">H_c_neq</span>] <span class="id" title="var">_</span> [-&gt;].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.min_eltZ_spec2" class="idref" href="#Raw.min_eltZ_spec2"><span class="id" title="lemma">min_eltZ_spec2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:777" class="idref" href="#s:777"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:778" class="idref" href="#x:778"><span class="id" title="binder">x</span></a> <a id="y:779" class="idref" href="#y:779"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) (<a id="Hs:780" class="idref" href="#Hs:780"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:777"><span class="id" title="variable">s</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:777"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:778"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:779"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:777"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:779"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:778"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">H_ok</span> <span class="id" title="var">H_min</span> <span class="id" title="var">H_in</span> <span class="id" title="var">H_y_lt_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater"><span class="id" title="lemma">Nin_elements_greater</span></a> <span class="id" title="var">s</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.pred"><span class="id" title="definition">Z.pred</span></a> <span class="id" title="var">x</span>)) ⇒ //; <span class="id" title="var">last</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_ok</span> <span class="id" title="var">H_min</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">z</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">_</span> [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons"><span class="id" title="lemma">interval_list_elements_greater_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_pred_l"><span class="id" title="lemma">Z.lt_pred_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_pred"><span class="id" title="lemma">Z.lt_le_pred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.min_eltZ_spec3" class="idref" href="#Raw.min_eltZ_spec3"><span class="id" title="definition">min_eltZ_spec3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:781" class="idref" href="#s:781"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:781"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="x:782" class="idref" href="#x:782"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:782"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:781"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x'</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">x</span> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.min_elt_spec1" class="idref" href="#Raw.min_elt_spec1"><span class="id" title="definition">min_elt_spec1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:783" class="idref" href="#s:783"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:784" class="idref" href="#x:784"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:785" class="idref" href="#Hs:785"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:783"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt"><span class="id" title="definition">min_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:783"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:784"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:784"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:783"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt"><span class="id" title="definition">min_elt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">z</span> <span class="id" title="var">H_min_elt</span> [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_In"><span class="id" title="lemma">InZ_In</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec1"><span class="id" title="definition">min_eltZ_spec1</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.min_elt_spec2" class="idref" href="#Raw.min_elt_spec2"><span class="id" title="definition">min_elt_spec2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:786" class="idref" href="#s:786"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:787" class="idref" href="#x:787"><span class="id" title="binder">x</span></a> <a id="y:788" class="idref" href="#y:788"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:789" class="idref" href="#Hs:789"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:786"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt"><span class="id" title="definition">min_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:786"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:787"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:788"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:786"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.lt"><span class="id" title="axiom">Enc.E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:788"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:787"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt"><span class="id" title="definition">min_elt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">z</span> <span class="id" title="var">H_min_elt</span> [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> ⇒ <span class="id" title="var">H_inZ</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_y_eq</span> : <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_y_eq</span> -<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_lt"><span class="id" title="axiom">Enc.encode_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec2"><span class="id" title="lemma">min_eltZ_spec2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_ok</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode_encode_ok"><span class="id" title="axiom">Enc.decode_encode_ok</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">z</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.encode_decode_eq"><span class="id" title="lemma">encode_decode_eq</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">s</span> := <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec1"><span class="id" title="definition">min_eltZ_spec1</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.min_elt_spec3" class="idref" href="#Raw.min_elt_spec3"><span class="id" title="definition">min_elt_spec3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:790" class="idref" href="#s:790"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt"><span class="id" title="definition">min_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:790"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:790"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt"><span class="id" title="definition">min_elt</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ"><span class="id" title="definition">min_eltZ</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x'</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.choose_spec1" class="idref" href="#Raw.choose_spec1"><span class="id" title="definition">choose_spec1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:791" class="idref" href="#s:791"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:792" class="idref" href="#x:792"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:793" class="idref" href="#Hs:793"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:791"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:791"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:792"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:792"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:791"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec1"><span class="id" title="definition">min_elt_spec1</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.choose_spec2" class="idref" href="#Raw.choose_spec2"><span class="id" title="definition">choose_spec2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:794" class="idref" href="#s:794"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:794"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:794"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec3"><span class="id" title="definition">min_elt_spec3</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.choose_spec3" class="idref" href="#Raw.choose_spec3"><span class="id" title="lemma">choose_spec3</span></a>: <span class="id" title="keyword">∀</span> <a id="s:795" class="idref" href="#s:795"><span class="id" title="binder">s</span></a> <a id="s':796" class="idref" href="#s':796"><span class="id" title="binder">s'</span></a> <a id="x:797" class="idref" href="#x:797"><span class="id" title="binder">x</span></a> <a id="x':798" class="idref" href="#x':798"><span class="id" title="binder">x'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:795"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':796"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:795"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:797"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':796"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x':798"><span class="id" title="variable">x'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:795"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':796"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:797"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x':798"><span class="id" title="variable">x'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span> <span class="id" title="var">x'</span> <span class="id" title="var">Hs</span> <span class="id" title="var">Hs'</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hx'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal_spec"><span class="id" title="lemma">equal_spec</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.equal_alt_def"><span class="id" title="lemma">equal_alt_def</span></a> ⇒ <span class="id" title="var">H_s_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">Hx</span> <span class="id" title="var">Hx'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_s_eq</span> ⇒ → [] //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.max_eltZ_spec1" class="idref" href="#Raw.max_eltZ_spec1"><span class="id" title="definition">max_eltZ_spec1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:799" class="idref" href="#s:799"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:800" class="idref" href="#x:800"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:799"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:799"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:800"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:800"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:799"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x'</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">H_c_neq</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>] <span class="id" title="var">s''</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_pred"><span class="id" title="lemma">Z.lt_le_pred</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_pred_l"><span class="id" title="lemma">Z.lt_pred_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv</span> <span class="id" title="var">H_max_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.max_eltZ_spec2" class="idref" href="#Raw.max_eltZ_spec2"><span class="id" title="lemma">max_eltZ_spec2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:801" class="idref" href="#s:801"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:802" class="idref" href="#x:802"><span class="id" title="binder">x</span></a> <a id="y:803" class="idref" href="#y:803"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:801"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:801"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:802"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:803"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:801"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:802"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:803"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> <span class="id" title="var">x'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c_neq_0</span>] <span class="id" title="var">H_inv_s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_gr'</span> : (<span class="id" title="keyword">∀</span> <a id="xx:804" class="idref" href="#xx:804"><span class="id" title="binder">xx</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#xx:804"><span class="id" title="variable">xx</span></a> (<span class="id" title="var">s'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#xx:804"><span class="id" title="variable">xx</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>] <span class="id" title="var">s''</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">lia</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_max_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] <span class="id" title="var">H_x'_lt</span> <span class="id" title="var">H_lt_x'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_x_in</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <span class="id" title="var">x</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">s''</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec1"><span class="id" title="definition">max_eltZ_spec1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_gr'</span> <span class="id" title="var">_</span> <span class="id" title="var">H_x_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.nlt_ge"><span class="id" title="lemma">Z.nlt_ge</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_incl"><span class="id" title="lemma">Z.lt_le_incl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">x'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.max_eltZ_eq_None" class="idref" href="#Raw.max_eltZ_eq_None"><span class="id" title="lemma">max_eltZ_eq_None</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:805" class="idref" href="#s:805"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:805"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:805"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x'</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>] ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">s'</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>] <span class="id" title="var">s''</span>] ⇒ //=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">H_pre</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H</span> <span class="id" title="var">H_pre</span>) ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.max_eltZ_spec3" class="idref" href="#Raw.max_eltZ_spec3"><span class="id" title="definition">max_eltZ_spec3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:806" class="idref" href="#s:806"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:806"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="x:807" class="idref" href="#x:807"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:807"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:806"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_eq_None"><span class="id" title="lemma">max_eltZ_eq_None</span></a> → <span class="id" title="var">x</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.max_elt_spec1" class="idref" href="#Raw.max_elt_spec1"><span class="id" title="definition">max_elt_spec1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:808" class="idref" href="#s:808"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:809" class="idref" href="#x:809"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:810" class="idref" href="#Hs:810"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:808"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_elt"><span class="id" title="definition">max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:808"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:809"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:809"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:808"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_elt"><span class="id" title="definition">max_elt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">z</span> <span class="id" title="var">H_max_elt</span> [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_In"><span class="id" title="lemma">InZ_In</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec1"><span class="id" title="definition">max_eltZ_spec1</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.max_elt_spec2" class="idref" href="#Raw.max_elt_spec2"><span class="id" title="definition">max_elt_spec2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:811" class="idref" href="#s:811"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:812" class="idref" href="#x:812"><span class="id" title="binder">x</span></a> <a id="y:813" class="idref" href="#y:813"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="Hs:814" class="idref" href="#Hs:814"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:811"><span class="id" title="variable">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_elt"><span class="id" title="definition">max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:811"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:812"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:813"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:811"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">~(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.lt"><span class="id" title="axiom">Enc.E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:812"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:813"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_elt"><span class="id" title="definition">max_elt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok</span>) ⇒ [<span class="id" title="var">H_inv</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">z</span> <span class="id" title="var">H_max_elt</span> [&lt;-].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_InZ"><span class="id" title="lemma">In_InZ</span></a> ⇒ <span class="id" title="var">H_inZ</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode_lt"><span class="id" title="axiom">Enc.encode_lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec2"><span class="id" title="lemma">max_eltZ_spec2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H_inv</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.encode"><span class="id" title="axiom">Enc.encode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">z</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">z</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.encode_decode_eq"><span class="id" title="lemma">encode_decode_eq</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">s</span> := <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec1"><span class="id" title="definition">max_eltZ_spec1</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.max_elt_spec3" class="idref" href="#Raw.max_elt_spec3"><span class="id" title="definition">max_elt_spec3</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:815" class="idref" href="#s:815"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_elt"><span class="id" title="definition">max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:815"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:815"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_elt"><span class="id" title="definition">max_elt</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Empty"><span class="id" title="definition">Empty</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ"><span class="id" title="definition">max_eltZ</span></a> <span class="id" title="var">s</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_eq_None"><span class="id" title="lemma">max_eltZ_eq_None</span></a> → <span class="id" title="var">_</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab50"></a><h3 class="section">fold specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.fold_spec" class="idref" href="#Raw.fold_spec"><span class="id" title="lemma">fold_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:816" class="idref" href="#s:816"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="A:817" class="idref" href="#A:817"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:818" class="idref" href="#i:818"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:817"><span class="id" title="variable">A</span></a>) (<a id="f:819" class="idref" href="#f:819"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:817"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:817"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:819"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:816"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#i:818"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:819"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:816"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#i:818"><span class="id" title="variable">i</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">A</span> <span class="id" title="var">i</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def"><span class="id" title="lemma">fold_elementsZ_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_rev"><span class="id" title="lemma">map_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">i</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">i</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab51"></a><h3 class="section">cardinal specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.cardinalN_spec" class="idref" href="#Raw.cardinalN_spec"><span class="id" title="lemma">cardinalN_spec</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:820" class="idref" href="#s:820"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="c:821" class="idref" href="#c:821"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>), <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.cardinalN"><span class="id" title="definition">cardinalN</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:821"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:820"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#c:821"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#64708cc1fcff61c206ed571fe7828ea8"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.of_nat"><span class="id" title="definition">N.of_nat</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:820"><span class="id" title="variable">s</span></a>)))%<span class="id" title="var">N</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">x</span> <span class="id" title="var">cx</span>] <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements_nil"><span class="id" title="lemma">elements_nil</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.add_0_r"><span class="id" title="lemma">N.add_0_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_length"><span class="id" title="lemma">rev_length</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_length"><span class="id" title="lemma">map_length</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons"><span class="id" title="lemma">elementsZ_cons</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_length"><span class="id" title="lemma">app_length</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.Nnat.html#Nat2N.inj_add"><span class="id" title="lemma">Nat2N.inj_add</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_length"><span class="id" title="lemma">rev_length</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.length_elementsZ_single"><span class="id" title="lemma">length_elementsZ_single</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.Nnat.html#N2Nat.id"><span class="id" title="lemma">N2Nat.id</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">ring</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.cardinal_spec" class="idref" href="#Raw.cardinal_spec"><span class="id" title="lemma">cardinal_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:822" class="idref" href="#s:822"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:822"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:822"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.cardinalN_spec"><span class="id" title="lemma">cardinalN_spec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.add_0_l"><span class="id" title="lemma">N.add_0_l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.Nnat.html#Nat2N.id"><span class="id" title="lemma">Nat2N.id</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab52"></a><h3 class="section">for_all specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.for_all_spec" class="idref" href="#Raw.for_all_spec"><span class="id" title="lemma">for_all_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:823" class="idref" href="#s:823"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="f:824" class="idref" href="#f:824"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="Hs:825" class="idref" href="#Hs:825"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:823"><span class="id" title="variable">s</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#f:824"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:824"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:823"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <a id="x:826" class="idref" href="#x:826"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#f:824"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:826"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:823"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span> <span class="id" title="var">Hs</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.for_all"><span class="id" title="definition">for_all</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.For_all"><span class="id" title="definition">For_all</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def"><span class="id" title="lemma">fold_elementsZ_alt_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_rev"><span class="id" title="lemma">map_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {      <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ // <span class="id" title="var">_</span> <span class="id" title="var">x</span> /= /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>)) ⇒ <span class="id" title="var">H_f_eq</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">HH</span> <span class="id" title="var">x'</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">move</span> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">HH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">HH</span> <span class="id" title="var">x'</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">HH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">HH</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">HH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">xs</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">exfalso</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in</span>: (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">xs</span>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq_equiv"><span class="id" title="axiom">Enc.E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">HH</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_f_eq</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab53"></a><h3 class="section">exists specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.exists_spec" class="idref" href="#Raw.exists_spec"><span class="id" title="lemma">exists_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:827" class="idref" href="#s:827"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="f:828" class="idref" href="#f:828"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="Hs:829" class="idref" href="#Hs:829"><span class="id" title="binder">Hs</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:827"><span class="id" title="variable">s</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq"><span class="id" title="axiom">Enc.E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#f:828"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:828"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:827"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <a id="x:830" class="idref" href="#x:830"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#f:828"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:830"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:827"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span> <span class="id" title="var">Hs</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.exists_"><span class="id" title="definition">exists_</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Exists"><span class="id" title="definition">Exists</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def"><span class="id" title="lemma">fold_elementsZ_alt_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#map_rev"><span class="id" title="lemma">map_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ"><span class="id" title="definition">elementsZ</span></a> <span class="id" title="var">s</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {      <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x</span>] /= [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>)) ⇒ <span class="id" title="var">H_f_eq</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ <span class="id" title="var">_</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <span class="id" title="var">x</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E.eq_equiv"><span class="id" title="axiom">Enc.E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">xs</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x0</span>] [<span class="id" title="var">H_in</span>] <span class="id" title="var">H_f_x0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x0</span>] [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> <span class="id" title="var">H_in</span> <span class="id" title="var">H_f_x0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_in</span> ⇒ [] // <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_f_x0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in</span> <span class="id" title="var">H_f_eq</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab54"></a><h3 class="section">filter specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.partitionZ_aux_invariant" class="idref" href="#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> (<a id="x:831" class="idref" href="#x:831"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) <a id="acc:832" class="idref" href="#acc:832"><span class="id" title="binder">acc</span></a> <a id="c:833" class="idref" href="#c:833"><span class="id" title="binder">c</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:832"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:833"><span class="id" title="variable">c</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:833"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ (<span class="id" title="keyword">∀</span> <a id="y':835" class="idref" href="#y':835"><span class="id" title="binder">y'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':835"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:832"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':835"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:831"><span class="id" title="variable">x</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ (<a class="idref" href="MSetsExtra.MSetIntervals.html#x:831"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c'</span>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partitionZ_aux_invariant_insert" class="idref" href="#Raw.partitionZ_aux_invariant_insert"><span class="id" title="lemma">partitionZ_aux_invariant_insert</span></a> : <span class="id" title="keyword">∀</span> <a id="x:836" class="idref" href="#x:836"><span class="id" title="binder">x</span></a> <a id="acc:837" class="idref" href="#acc:837"><span class="id" title="binder">acc</span></a> <a id="c:838" class="idref" href="#c:838"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:836"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:837"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:838"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:836"><span class="id" title="variable">x</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:837"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:838"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:836"><span class="id" title="variable">x</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">c</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv</span>] <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_iff"><span class="id" title="lemma">interval_list_invariant_app_iff</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_1_r"><span class="id" title="lemma">Z.add_1_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x1_in</span> [] // &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span> <span class="id" title="var">c'</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_iff"><span class="id" title="lemma">interval_list_invariant_app_iff</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_succ"><span class="id" title="lemma">N2Z.inj_succ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.add_succ_r"><span class="id" title="lemma">Z.add_succ_r</span></a> .<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_nil"><span class="id" title="lemma">interval_list_invariant_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] [<span class="id" title="var">H_inv_acc</span>] [] [] <span class="id" title="var">_</span> [<span class="id" title="var">H_c_neq_0</span>] <span class="id" title="var">_</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H_in_c</span> →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.neq_succ_0"><span class="id" title="lemma">N.neq_succ_0</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x1_in</span> [] // [<span class="id" title="var">H_y_le</span>] <span class="id" title="var">H_x2_lt</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_le_trans"><span class="id" title="lemma">Z.lt_le_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_c</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_refl"><span class="id" title="lemma">Z.le_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_lt_add_r"><span class="id" title="lemma">Z_lt_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partitionZ_aux_invariant_skip" class="idref" href="#Raw.partitionZ_aux_invariant_skip"><span class="id" title="lemma">partitionZ_aux_invariant_skip</span></a> : <span class="id" title="keyword">∀</span> <a id="x:839" class="idref" href="#x:839"><span class="id" title="binder">x</span></a> <a id="acc:840" class="idref" href="#acc:840"><span class="id" title="binder">acc</span></a> <a id="c:841" class="idref" href="#c:841"><span class="id" title="binder">c</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:839"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:840"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:841"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:839"><span class="id" title="variable">x</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:840"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:841"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">c</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv</span>] <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y'</span> <span class="id" title="var">H_y'_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">x</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_diag_r"><span class="id" title="lemma">Z.lt_succ_diag_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span> <span class="id" title="var">c'</span>] [<span class="id" title="var">H_inv</span>] →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ_lt_mono"><span class="id" title="lemma">Z.succ_lt_mono</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_iff"><span class="id" title="lemma">interval_list_invariant_app_iff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [] [<span class="id" title="var">_</span>] [<span class="id" title="var">H_c'_neq</span>] <span class="id" title="var">_</span> <span class="id" title="var">H_pre</span> <span class="id" title="var">H_y'_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_trans"><span class="id" title="lemma">Z.lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_pre</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Z_le_add_r"><span class="id" title="lemma">Z_le_add_r</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Raw.partitionZ_fold_current" class="idref" href="#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a> (<a id="c:842" class="idref" href="#c:842"><span class="id" title="binder">c</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>)) :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:842"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">yc</span> ⇒ <span class="id" title="var">yc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_partitionZ_fold_current_Some" class="idref" href="#Raw.InZ_partitionZ_fold_current_Some"><span class="id" title="lemma">InZ_partitionZ_fold_current_Some</span></a> : <span class="id" title="keyword">∀</span> <a id="yc:844" class="idref" href="#yc:844"><span class="id" title="binder">yc</span></a> <a id="y:845" class="idref" href="#y:845"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:845"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#yc:844"><span class="id" title="variable">yc</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:845"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#yc:844"><span class="id" title="variable">yc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_partitionZ_fold_insert" class="idref" href="#Raw.InZ_partitionZ_fold_insert"><span class="id" title="lemma">InZ_partitionZ_fold_insert</span></a> : <span class="id" title="keyword">∀</span> <a id="c:846" class="idref" href="#c:846"><span class="id" title="binder">c</span></a> <a id="x:847" class="idref" href="#x:847"><span class="id" title="binder">x</span></a> <a id="y:848" class="idref" href="#y:848"><span class="id" title="binder">y</span></a> <a id="l:849" class="idref" href="#l:849"><span class="id" title="binder">l</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:846"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#x:847"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:848"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c'</span><br/>
+&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a><br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:848"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:846"><span class="id" title="variable">c</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:847"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:849"><span class="id" title="variable">l</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#x:847"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:848"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:848"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:846"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:848"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#l:849"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">l</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_insert"><span class="id" title="definition">partitionZ_fold_insert</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">c</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>] →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ"><span class="id" title="lemma">elementsZ_single_succ</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_app_iff"><span class="id" title="lemma">in_app_iff</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1"><span class="id" title="lemma">In_elementsZ_single1</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.InZ_partitionZ_fold_skip" class="idref" href="#Raw.InZ_partitionZ_fold_skip"><span class="id" title="lemma">InZ_partitionZ_fold_skip</span></a> : <span class="id" title="keyword">∀</span> <a id="c:851" class="idref" href="#c:851"><span class="id" title="binder">c</span></a> <a id="acc:852" class="idref" href="#acc:852"><span class="id" title="binder">acc</span></a> <a id="y:853" class="idref" href="#y:853"><span class="id" title="binder">y</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:853"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:852"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:851"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:853"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:851"><span class="id" title="variable">c</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:853"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:852"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">acc</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;/<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">c</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y'</span> <span class="id" title="var">c'</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.filterZ_single_aux_props" class="idref" href="#Raw.filterZ_single_aux_props"><span class="id" title="lemma">filterZ_single_aux_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:854" class="idref" href="#f:854"><span class="id" title="binder">f</span></a> <a id="c:855" class="idref" href="#c:855"><span class="id" title="binder">c</span></a> <a id="x:856" class="idref" href="#x:856"><span class="id" title="binder">x</span></a> <a id="acc:857" class="idref" href="#acc:857"><span class="id" title="binder">acc</span></a> <a id="cur:858" class="idref" href="#cur:858"><span class="id" title="binder">cur</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:856"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:857"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#cur:858"><span class="id" title="variable">cur</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:854"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc:857"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#cur:858"><span class="id" title="variable">cur</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:856"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:855"><span class="id" title="variable">c</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="r:859" class="idref" href="#r:859"><span class="id" title="binder">r</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <span class="id" title="var">acc'</span> <span class="id" title="var">c'</span> <span class="id" title="tactic">in</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#r:859"><span class="id" title="variable">r</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y':860" class="idref" href="#y':860"><span class="id" title="binder">y'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':860"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#r:859"><span class="id" title="variable">r</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':860"><span class="id" title="variable">y'</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:857"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#cur:858"><span class="id" title="variable">cur</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:854"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':860"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':860"><span class="id" title="variable">y'</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:856"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:855"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span> <span class="id" title="var">cur</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv</span>] <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_zero"><span class="id" title="lemma">fold_elementsZ_single_zero</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span> <span class="id" title="var">cur</span> <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → :  <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">cur</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">x</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.succ"><span class="id" title="definition">N.succ</span></a> <span class="id" title="var">c'</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <span class="id" title="var">f</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">cur</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">x</span>) (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">c'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_succ"><span class="id" title="lemma">fold_elementsZ_single_succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">cur</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">x</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">cur'</span> <span class="id" title="var">H_fold_eq</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">cur'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">c'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc''</span> <span class="id" title="var">cur''</span> <span class="id" title="var">H_succ_eq</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_inv'</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">acc'</span> <span class="id" title="var">cur'</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_fold_eq</span> <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span>); <span class="id" title="tactic">move</span> ⇒ [&lt;-] &lt;-. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant_insert"><span class="id" title="lemma">partitionZ_aux_invariant_insert</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant_skip"><span class="id" title="lemma">partitionZ_aux_invariant_skip</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <span class="id" title="var">x</span>) <span class="id" title="var">acc'</span> <span class="id" title="var">cur'</span> <span class="id" title="var">H_inv'</span>) ⇒ {<span class="id" title="var">IH</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_succ_eq</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">r</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_skip"><span class="id" title="definition">partitionZ_fold_skip</span></a> <span class="id" title="var">acc''</span> <span class="id" title="var">cur''</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_r</span>] <span class="id" title="var">H_in_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_fold_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_r</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span>) ⇒ <span class="id" title="var">H_fx</span> [&lt;-] &lt;-. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_skip"><span class="id" title="lemma">InZ_partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_current_Some"><span class="id" title="lemma">InZ_partitionZ_fold_current_Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_skip"><span class="id" title="lemma">InZ_partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_insert"><span class="id" title="lemma">InZ_partitionZ_fold_insert</span></a>; <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_inv</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> ⇒ [[<span class="id" title="var">_</span>]].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">cur</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="var">last</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ &lt;-.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_skip"><span class="id" title="lemma">InZ_partitionZ_fold_skip</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current"><span class="id" title="definition">partitionZ_fold_current</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_skip"><span class="id" title="lemma">InZ_partitionZ_fold_skip</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front"><span class="id" title="lemma">elementsZ_single_succ_front</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H_fy'</span> []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x_eq</span>; <span class="id" title="tactic">subst</span> <span class="id" title="var">y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H_fy'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_fx</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.filterZ_single_props" class="idref" href="#Raw.filterZ_single_props"><span class="id" title="lemma">filterZ_single_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:861" class="idref" href="#f:861"><span class="id" title="binder">f</span></a> <a id="c:862" class="idref" href="#c:862"><span class="id" title="binder">c</span></a> <a id="x:863" class="idref" href="#x:863"><span class="id" title="binder">x</span></a> <a id="acc:864" class="idref" href="#acc:864"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:864"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y':865" class="idref" href="#y':865"><span class="id" title="binder">y'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':865"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:864"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':865"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:863"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>     <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single"><span class="id" title="definition">filterZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:861"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:864"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:863"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:862"><span class="id" title="variable">c</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">r</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <span class="id" title="var">r</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y':866" class="idref" href="#y':866"><span class="id" title="binder">y'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':866"><span class="id" title="variable">y'</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':866"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:864"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:861"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':866"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':866"><span class="id" title="variable">y'</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single"><span class="id" title="definition">elementsZ_single</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:863"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:862"><span class="id" title="variable">c</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_inv</span> <span class="id" title="var">H_acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single"><span class="id" title="definition">filterZ_single</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_inv'</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> <span class="id" title="var">x</span> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant"><span class="id" title="definition">partitionZ_aux_invariant</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux_props"><span class="id" title="lemma">filterZ_single_aux_props</span></a> <span class="id" title="var">f</span> <span class="id" title="var">c</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <span class="id" title="var">H_inv'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">x</span> <span class="id" title="var">c</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">acc'</span> <span class="id" title="var">cur'</span> /= <span class="id" title="var">H_res</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.filterZ_aux_props" class="idref" href="#Raw.filterZ_aux_props"><span class="id" title="lemma">filterZ_aux_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:867" class="idref" href="#f:867"><span class="id" title="binder">f</span></a> <a id="s:868" class="idref" href="#s:868"><span class="id" title="binder">s</span></a> <a id="acc:869" class="idref" href="#acc:869"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:868"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:869"><span class="id" title="variable">acc</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x1:870" class="idref" href="#x1:870"><span class="id" title="binder">x1</span></a> <a id="x2:871" class="idref" href="#x2:871"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:870"><span class="id" title="variable">x1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:869"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:871"><span class="id" title="variable">x2</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:868"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:870"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:871"><span class="id" title="variable">x2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>   <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux"><span class="id" title="definition">filterZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:869"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:867"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:868"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">r</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y':872" class="idref" href="#y':872"><span class="id" title="binder">y'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':872"><span class="id" title="variable">y'</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':872"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc:869"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:867"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':872"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':872"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:868"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">H_inv</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux"><span class="id" title="definition">filterZ_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y'</span>; <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev"><span class="id" title="lemma">InZ_rev</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons"><span class="id" title="lemma">interval_list_invariant_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_gr</span>] [<span class="id" title="var">H_c_neq_0</span>] <span class="id" title="var">H_inv_s'</span> <span class="id" title="var">H_inv</span> <span class="id" title="var">H_in_acc</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def"><span class="id" title="lemma">interval_list_elements_greater_alt2_def</span></a> ⇒ // <span class="id" title="var">H_gr</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_pre</span> : (<span class="id" title="keyword">∀</span> <a id="y':873" class="idref" href="#y':873"><span class="id" title="binder">y'</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':873"><span class="id" title="variable">y'</span></a> <span class="id" title="var">acc</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':873"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">H_x1_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_acc</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd"><span class="id" title="lemma">In_elementsZ_single_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_props"><span class="id" title="lemma">filterZ_single_props</span></a> <span class="id" title="var">f</span> <span class="id" title="var">c</span> <span class="id" title="var">y</span> <span class="id" title="var">acc</span> <span class="id" title="var">H_inv</span> <span class="id" title="var">H_pre</span>) ⇒ {<span class="id" title="var">H_pre</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">acc'</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single"><span class="id" title="definition">filterZ_single</span></a> <span class="id" title="var">f</span> <span class="id" title="var">acc</span> <span class="id" title="var">y</span> <span class="id" title="var">c</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv'</span>] <span class="id" title="var">H_in_acc'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_pre</span> : (<span class="id" title="keyword">∀</span> <a id="x1:874" class="idref" href="#x1:874"><span class="id" title="binder">x1</span></a> <a id="x2:875" class="idref" href="#x2:875"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:874"><span class="id" title="variable">x1</span></a> <span class="id" title="var">acc'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:875"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:874"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:875"><span class="id" title="variable">x2</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x1</span> <span class="id" title="var">x2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_acc'</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single"><span class="id" title="lemma">In_elementsZ_single</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_x1_in</span> <span class="id" title="var">H_x2_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H_in_acc</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">_</span>] [<span class="id" title="var">_</span>] <span class="id" title="var">H_x1_lt</span> <span class="id" title="var">H_x2_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_lt_trans"><span class="id" title="lemma">Z.le_lt_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> := <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#46584eddd5fdb16176a10a2843177d3a"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <span class="id" title="var">c</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.le_succ_l"><span class="id" title="lemma">Z.le_succ_l</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_gr</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">acc'</span> <span class="id" title="var">H_inv_s'</span> <span class="id" title="var">H_inv'</span> <span class="id" title="var">H_pre</span>) ⇒ {<span class="id" title="var">H_pre</span>}.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv_r</span>] <span class="id" title="var">H_in_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_r</span> <span class="id" title="var">H_in_acc'</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons"><span class="id" title="lemma">InZ_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.filterZ_props" class="idref" href="#Raw.filterZ_props"><span class="id" title="lemma">filterZ_props</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="f:876" class="idref" href="#f:876"><span class="id" title="binder">f</span></a> <a id="s:877" class="idref" href="#s:877"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:877"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ"><span class="id" title="definition">filterZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:876"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:877"><span class="id" title="variable">s</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">r</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="y':878" class="idref" href="#y':878"><span class="id" title="binder">y'</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':878"><span class="id" title="variable">y'</span></a> <span class="id" title="var">r</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#f:876"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':878"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y':878"><span class="id" title="variable">y'</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:877"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <span class="id" title="var">H_inv_s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ"><span class="id" title="definition">filterZ</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_pre_1</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant"><span class="id" title="definition">interval_list_invariant</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_pre_2</span> : (<span class="id" title="keyword">∀</span> <a id="x1:879" class="idref" href="#x1:879"><span class="id" title="binder">x1</span></a> <a id="x2:880" class="idref" href="#x2:880"><span class="id" title="binder">x2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:879"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:880"><span class="id" title="variable">x2</span></a> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Z.succ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x1:879"><span class="id" title="variable">x1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#::Z_scope:x_'&lt;'_x"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x2:880"><span class="id" title="variable">x2</span></a>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux_props"><span class="id" title="lemma">filterZ_aux_props</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <span class="id" title="var">H_inv_s</span> <span class="id" title="var">H_pre_1</span> <span class="id" title="var">H_pre_2</span>) ⇒ {<span class="id" title="var">H_pre_1</span>} {<span class="id" title="var">H_pre_2</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv'</span>] <span class="id" title="var">H_in_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_r</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil"><span class="id" title="lemma">InZ_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.filter_ok" class="idref" href="#Raw.filter_ok"><span class="id" title="instance">filter_ok</span></a> <a id="s:881" class="idref" href="#s:881"><span class="id" title="binder">s</span></a> <a id="f:882" class="idref" href="#f:882"><span class="id" title="binder">f</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:883" class="idref" href="#H:883"><span class="id" title="binder">Ok</span></a> <a id="H:883" class="idref" href="#H:883"><span class="id" title="binder">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:882"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:881"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv</span> <span class="id" title="var">H_enc</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">f'</span> := (<span class="id" title="keyword">fun</span> <a id="z:886" class="idref" href="#z:886"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ⇒ <span class="id" title="var">f</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:884"><span class="id" title="variable">z</span></a>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_props"><span class="id" title="lemma">filterZ_props</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">s</span> <span class="id" title="var">H_inv</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv'</span>] <span class="id" title="var">H_in_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.IsOk"><span class="id" title="definition">IsOk</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list"><span class="id" title="definition">is_encoded_elems_list</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> /<span class="id" title="var">H_in_r</span> [<span class="id" title="var">_</span>] <span class="id" title="var">H_x_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H_enc</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.filter_spec" class="idref" href="#Raw.filter_spec"><span class="id" title="lemma">filter_spec</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:887" class="idref" href="#s:887"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="x:888" class="idref" href="#x:888"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a>) (<a id="f:889" class="idref" href="#f:889"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:887"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>      <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:888"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:889"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:887"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:888"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:887"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:889"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:888"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">f</span> <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_suff</span> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">∀</span> <a id="x:890" class="idref" href="#x:890"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:890"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <span class="id" title="var">f</span> <span class="id" title="var">s</span>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:890"><span class="id" title="variable">x</span></a> <span class="id" title="var">s</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <span class="id" title="var">f</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:890"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.In_alt_def"><span class="id" title="lemma">In_alt_def</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elements"><span class="id" title="definition">elements</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#rev_map_alt_def"><span class="id" title="lemma">rev_map_alt_def</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;-!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_rev"><span class="id" title="lemma">in_rev</span></a> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#in_map_iff"><span class="id" title="lemma">in_map_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <span class="id" title="var">H_suff</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.InZ"><span class="id" title="definition">InZ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y</span>] [&lt;-] [?] ?.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] [<span class="id" title="var">y</span>] [&lt;-] ? ?.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">set</span> <span class="id" title="var">f'</span> := (<span class="id" title="keyword">fun</span> <a id="z:893" class="idref" href="#z:893"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ⇒ <span class="id" title="var">f</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.decode"><span class="id" title="axiom">Enc.decode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:891"><span class="id" title="variable">z</span></a>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_ok</span>) ⇒ [<span class="id" title="var">H_inv</span> <span class="id" title="var">_</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_props"><span class="id" title="lemma">filterZ_props</span></a> <span class="id" title="var">f'</span> <span class="id" title="var">s</span> <span class="id" title="var">H_inv</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H_inv'</span>] <span class="id" title="var">H_in_r</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_in_r</span>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab55"></a><h3 class="section">partition specification</h3>
+
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partitionZ_single_aux_alt_def" class="idref" href="#Raw.partitionZ_single_aux_alt_def"><span class="id" title="lemma">partitionZ_single_aux_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="f:894" class="idref" href="#f:894"><span class="id" title="binder">f</span></a> <a id="c:895" class="idref" href="#c:895"><span class="id" title="binder">c</span></a> <a id="y:896" class="idref" href="#y:896"><span class="id" title="binder">y</span></a> <a id="acc_t:897" class="idref" href="#acc_t:897"><span class="id" title="binder">acc_t</span></a> <a id="c_t:898" class="idref" href="#c_t:898"><span class="id" title="binder">c_t</span></a> <a id="acc_f:899" class="idref" href="#acc_f:899"><span class="id" title="binder">acc_f</span></a> <a id="c_f:900" class="idref" href="#c_f:900"><span class="id" title="binder">c_f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_single_aux"><span class="id" title="definition">partitionZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:894"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">((</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:897"><span class="id" title="variable">acc_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c_t:898"><span class="id" title="variable">c_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">),</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:899"><span class="id" title="variable">acc_f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c_f:900"><span class="id" title="variable">c_f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:896"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:895"><span class="id" title="variable">c</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:894"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:897"><span class="id" title="variable">acc_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c_t:898"><span class="id" title="variable">c_t</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:896"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:895"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> (<span class="id" title="keyword">fun</span> <a id="x:901" class="idref" href="#x:901"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:894"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:901"><span class="id" title="variable">x</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:899"><span class="id" title="variable">acc_f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c_f:900"><span class="id" title="variable">c_f</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#y:896"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#c:895"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_single_aux"><span class="id" title="definition">partitionZ_single_aux</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">c</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">c'</span> <span class="id" title="var">IH</span>] <span class="id" title="keyword">using</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N.peano_ind"><span class="id" title="definition">N.peano_ind</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">acc_t</span> <span class="id" title="var">c_t</span> <span class="id" title="var">acc_f</span> <span class="id" title="var">c_f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_zero"><span class="id" title="lemma">fold_elementsZ_single_zero</span></a> //. } {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">y</span> <span class="id" title="var">acc_t</span> <span class="id" title="var">c_t</span> <span class="id" title="var">acc_f</span> <span class="id" title="var">c_f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_succ"><span class="id" title="lemma">fold_elementsZ_single_succ</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_fun"><span class="id" title="definition">partitionZ_fold_fun</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">))</span></a> <span class="id" title="var">y</span>) ⇒ [] [<span class="id" title="var">acc_t'</span> <span class="id" title="var">c_t'</span>] [<span class="id" title="var">acc_f'</span> <span class="id" title="var">c_f'</span>] <span class="id" title="var">H_fold_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> ⇒ {<span class="id" title="var">IH</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_t'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a> (<span class="id" title="keyword">fun</span> <a id="x:902" class="idref" href="#x:902"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<span class="id" title="var">f</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:902"><span class="id" title="variable">x</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <span class="id" title="var">c_f'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [-&gt;] → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_fold_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_fun"><span class="id" title="definition">partitionZ_fold_fun</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_fold_fun"><span class="id" title="definition">filterZ_fold_fun</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">y</span>); <span class="id" title="tactic">move</span> ⇒ [&lt;-] &lt;- &lt;- &lt;- //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partitionZ_aux_alt_def" class="idref" href="#Raw.partitionZ_aux_alt_def"><span class="id" title="lemma">partitionZ_aux_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="f:903" class="idref" href="#f:903"><span class="id" title="binder">f</span></a> <a id="s:904" class="idref" href="#s:904"><span class="id" title="binder">s</span></a> <a id="acc_t:905" class="idref" href="#acc_t:905"><span class="id" title="binder">acc_t</span></a> <a id="acc_f:906" class="idref" href="#acc_f:906"><span class="id" title="binder">acc_f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux"><span class="id" title="definition">partitionZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:905"><span class="id" title="variable">acc_t</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:906"><span class="id" title="variable">acc_f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:903"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:904"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux"><span class="id" title="definition">filterZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_t:905"><span class="id" title="variable">acc_t</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:903"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:904"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux"><span class="id" title="definition">filterZ_aux</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#acc_f:906"><span class="id" title="variable">acc_f</span></a> (<span class="id" title="keyword">fun</span> <a id="x:907" class="idref" href="#x:907"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:903"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:907"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:904"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| [<span class="id" title="var">y</span> <span class="id" title="var">c</span>] <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">acc_t</span> <span class="id" title="var">acc_f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_single"><span class="id" title="definition">partitionZ_single</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single"><span class="id" title="definition">filterZ_single</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_single_aux_alt_def"><span class="id" title="lemma">partitionZ_single_aux_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_t</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">y</span> <span class="id" title="var">c</span>) ⇒ <span class="id" title="var">acc_t'</span> <span class="id" title="var">c_t'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux"><span class="id" title="definition">filterZ_single_aux</span></a> (<span class="id" title="keyword">fun</span> <a id="x:909" class="idref" href="#x:909"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<span class="id" title="var">f</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:908"><span class="id" title="variable">x</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><span class="id" title="var">acc_f</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a> <span class="id" title="var">y</span> <span class="id" title="var">c</span>) ⇒ <span class="id" title="var">acc_f'</span> <span class="id" title="var">c_f'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partitionZ_alt_def" class="idref" href="#Raw.partitionZ_alt_def"><span class="id" title="lemma">partitionZ_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="f:910" class="idref" href="#f:910"><span class="id" title="binder">f</span></a> <a id="s:911" class="idref" href="#s:911"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ"><span class="id" title="definition">partitionZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:910"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:911"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ"><span class="id" title="definition">filterZ</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:910"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:911"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ"><span class="id" title="definition">filterZ</span></a> (<span class="id" title="keyword">fun</span> <a id="x:912" class="idref" href="#x:912"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:910"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:912"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:911"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ"><span class="id" title="definition">partitionZ</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filterZ"><span class="id" title="definition">filterZ</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_alt_def"><span class="id" title="lemma">partitionZ_aux_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partition_alt_def" class="idref" href="#Raw.partition_alt_def"><span class="id" title="lemma">partition_alt_def</span></a> : <span class="id" title="keyword">∀</span> <a id="f:913" class="idref" href="#f:913"><span class="id" title="binder">f</span></a> <a id="s:914" class="idref" href="#s:914"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:913"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:914"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:913"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:914"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <a id="x:915" class="idref" href="#x:915"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:913"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:915"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:914"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">f</span> <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_alt_def"><span class="id" title="lemma">partitionZ_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.partition_ok1" class="idref" href="#Raw.partition_ok1"><span class="id" title="instance">partition_ok1</span></a> <a id="s:916" class="idref" href="#s:916"><span class="id" title="binder">s</span></a> <a id="f:917" class="idref" href="#f:917"><span class="id" title="binder">f</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:918" class="idref" href="#H:918"><span class="id" title="binder">Ok</span></a> <a id="H:918" class="idref" href="#H:918"><span class="id" title="binder">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:917"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:916"><span class="id" title="variable">s</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition_alt_def"><span class="id" title="lemma">partition_alt_def</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter_ok"><span class="id" title="instance">filter_ok</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Global Instance</span> <a id="Raw.partition_ok2" class="idref" href="#Raw.partition_ok2"><span class="id" title="instance">partition_ok2</span></a> <a id="s:919" class="idref" href="#s:919"><span class="id" title="binder">s</span></a> <a id="f:920" class="idref" href="#f:920"><span class="id" title="binder">f</span></a> : <span class="id" title="keyword">∀</span> `(<a id="H:921" class="idref" href="#H:921"><span class="id" title="binder">Ok</span></a> <a id="H:921" class="idref" href="#H:921"><span class="id" title="binder">s</span></a>), <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:920"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:919"><span class="id" title="variable">s</span></a>)).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_ok</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition_alt_def"><span class="id" title="lemma">partition_alt_def</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter_ok"><span class="id" title="instance">filter_ok</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partition_spec1" class="idref" href="#Raw.partition_spec1"><span class="id" title="lemma">partition_spec1</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:922" class="idref" href="#s:922"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="f:923" class="idref" href="#f:923"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:923"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:922"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:923"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:922"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition_alt_def"><span class="id" title="lemma">partition_alt_def</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Raw.partition_spec2" class="idref" href="#Raw.partition_spec2"><span class="id" title="lemma">partition_spec2</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="s:924" class="idref" href="#s:924"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.t"><span class="id" title="definition">t</span></a>) (<a id="f:925" class="idref" href="#f:925"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Ok"><span class="id" title="class">Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:924"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a>    <br/>
+&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:925"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:924"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <a id="x:926" class="idref" href="#x:926"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#f:925"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:926"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#s:924"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.partition_alt_def"><span class="id" title="lemma">partition_alt_def</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#Raw.Equal"><span class="id" title="definition">Equal</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw"><span class="id" title="module">Raw</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab56"></a><h2 class="section">Main Module</h2>
+
+
+<div class="paragraph"> </div>
+
+    We can now build the invariant into the set type to obtain an instantiation
+    of module type <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="MSetIntervals" class="idref" href="#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> (<span class="id" title="var">Enc</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>) &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#SetsOn"><span class="id" title="module">SetsOn</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E"><span class="id" title="module">Enc.E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="MSetIntervals.E" class="idref" href="#MSetIntervals.E"><span class="id" title="module">E</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc.E"><span class="id" title="module">Enc.E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="MSetIntervals.Raw" class="idref" href="#MSetIntervals.Raw"><span class="id" title="module">Raw</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#Raw"><span class="id" title="module">Raw</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#Enc"><span class="id" title="module">Enc</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Local</span> <span class="id" title="keyword">Unset</span> <span class="id" title="var">Elimination</span> <span class="id" title="var">Schemes</span>.<br/>
+&nbsp;<span class="id" title="keyword">Local</span> <span class="id" title="keyword">Unset</span> <span class="id" title="var">Case</span> <span class="id" title="var">Analysis</span> <span class="id" title="var">Schemes</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.elt" class="idref" href="#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">Raw.elt</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Record</span> <a id="MSetIntervals.t_" class="idref" href="#MSetIntervals.t_"><span class="id" title="record">t_</span></a> := <span class="id" title="var">Mkt</span> {<a id="MSetIntervals.this" class="idref" href="#MSetIntervals.this"><span class="id" title="projection">this</span></a> :&gt; <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">Raw.t</span></a>; <a id="MSetIntervals.is_ok" class="idref" href="#MSetIntervals.is_ok"><span class="id" title="projection">is_ok</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Ok"><span class="id" title="class">Raw.Ok</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#this:928"><span class="id" title="method">this</span></a>}.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.t" class="idref" href="#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t_"><span class="id" title="record">t_</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Arguments</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> <span class="id" title="var">this</span> {<span class="id" title="var">is_ok</span>}.<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">is_ok</span> : <span class="id" title="var">typeclass_instances</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.In" class="idref" href="#MSetIntervals.In"><span class="id" title="definition">In</span></a> (<a id="x:930" class="idref" href="#x:930"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<a id="s:931" class="idref" href="#s:931"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:930"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:931"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.Equal" class="idref" href="#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> (<a id="s:932" class="idref" href="#s:932"><span class="id" title="binder">s</span></a> <a id="s':933" class="idref" href="#s':933"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">∀</span> <a id="a:934" class="idref" href="#a:934"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:934"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:932"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:934"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':933"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.Subset" class="idref" href="#MSetIntervals.Subset"><span class="id" title="definition">Subset</span></a> (<a id="s:935" class="idref" href="#s:935"><span class="id" title="binder">s</span></a> <a id="s':936" class="idref" href="#s':936"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">∀</span> <a id="a:937" class="idref" href="#a:937"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:937"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:935"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:937"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':936"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.Empty" class="idref" href="#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> (<a id="s:938" class="idref" href="#s:938"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">∀</span> <a id="a:939" class="idref" href="#a:939"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:939"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:938"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.For_all" class="idref" href="#MSetIntervals.For_all"><span class="id" title="definition">For_all</span></a> (<a id="P:940" class="idref" href="#P:940"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)(<a id="s:941" class="idref" href="#s:941"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">∀</span> <a id="x:942" class="idref" href="#x:942"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:942"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:941"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#P:940"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:942"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.Exists" class="idref" href="#MSetIntervals.Exists"><span class="id" title="definition">Exists</span></a> (<a id="P:943" class="idref" href="#P:943"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>)(<a id="s:944" class="idref" href="#s:944"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:945" class="idref" href="#x:945"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:945"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:944"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#P:943"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:945"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.mem" class="idref" href="#MSetIntervals.mem"><span class="id" title="definition">mem</span></a> (<a id="x:946" class="idref" href="#x:946"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<a id="s:947" class="idref" href="#s:947"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem"><span class="id" title="definition">Raw.mem</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:946"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:947"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.add" class="idref" href="#MSetIntervals.add"><span class="id" title="definition">add</span></a> (<a id="x:948" class="idref" href="#x:948"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<a id="s:949" class="idref" href="#s:949"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.add"><span class="id" title="definition">Raw.add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:948"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:949"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>)).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.remove" class="idref" href="#MSetIntervals.remove"><span class="id" title="definition">remove</span></a> (<a id="x:950" class="idref" href="#x:950"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>)(<a id="s:951" class="idref" href="#s:951"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove"><span class="id" title="definition">Raw.remove</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:950"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:951"><span class="id" title="variable">s</span></a>.(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>)).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.singleton" class="idref" href="#MSetIntervals.singleton"><span class="id" title="definition">singleton</span></a> (<a id="x:952" class="idref" href="#x:952"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton"><span class="id" title="definition">Raw.singleton</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:952"><span class="id" title="variable">x</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.union" class="idref" href="#MSetIntervals.union"><span class="id" title="definition">union</span></a> (<a id="s:953" class="idref" href="#s:953"><span class="id" title="binder">s</span></a> <a id="s':954" class="idref" href="#s':954"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.union"><span class="id" title="definition">Raw.union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:953"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':954"><span class="id" title="variable">s'</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.inter" class="idref" href="#MSetIntervals.inter"><span class="id" title="definition">inter</span></a> (<a id="s:955" class="idref" href="#s:955"><span class="id" title="binder">s</span></a> <a id="s':956" class="idref" href="#s':956"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter"><span class="id" title="definition">Raw.inter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:955"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':956"><span class="id" title="variable">s'</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.diff" class="idref" href="#MSetIntervals.diff"><span class="id" title="definition">diff</span></a> (<a id="s:957" class="idref" href="#s:957"><span class="id" title="binder">s</span></a> <a id="s':958" class="idref" href="#s':958"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff"><span class="id" title="definition">Raw.diff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:957"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':958"><span class="id" title="variable">s'</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.equal" class="idref" href="#MSetIntervals.equal"><span class="id" title="definition">equal</span></a> (<a id="s:959" class="idref" href="#s:959"><span class="id" title="binder">s</span></a> <a id="s':960" class="idref" href="#s':960"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal"><span class="id" title="definition">Raw.equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:959"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':960"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.subset" class="idref" href="#MSetIntervals.subset"><span class="id" title="definition">subset</span></a> (<a id="s:961" class="idref" href="#s:961"><span class="id" title="binder">s</span></a> <a id="s':962" class="idref" href="#s':962"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset"><span class="id" title="definition">Raw.subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:961"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':962"><span class="id" title="variable">s'</span></a>.(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.this"><span class="id" title="projection">this</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.empty" class="idref" href="#MSetIntervals.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty"><span class="id" title="definition">Raw.empty</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.is_empty" class="idref" href="#MSetIntervals.is_empty"><span class="id" title="definition">is_empty</span></a> (<a id="s:963" class="idref" href="#s:963"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty"><span class="id" title="definition">Raw.is_empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:963"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.elements" class="idref" href="#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> (<a id="s:964" class="idref" href="#s:964"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">Raw.elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:964"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.min_elt" class="idref" href="#MSetIntervals.min_elt"><span class="id" title="definition">min_elt</span></a> (<a id="s:965" class="idref" href="#s:965"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt"><span class="id" title="definition">Raw.min_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:965"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.max_elt" class="idref" href="#MSetIntervals.max_elt"><span class="id" title="definition">max_elt</span></a> (<a id="s:966" class="idref" href="#s:966"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt"><span class="id" title="definition">Raw.max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:966"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.choose" class="idref" href="#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> (<a id="s:967" class="idref" href="#s:967"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">Raw.choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:967"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.compare" class="idref" href="#MSetIntervals.compare"><span class="id" title="definition">compare</span></a> (<a id="s1:968" class="idref" href="#s1:968"><span class="id" title="binder">s1</span></a> <a id="s2:969" class="idref" href="#s2:969"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#comparison"><span class="id" title="inductive">comparison</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare"><span class="id" title="definition">Raw.compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s1:968"><span class="id" title="variable">s1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s2:969"><span class="id" title="variable">s2</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.fold" class="idref" href="#MSetIntervals.fold"><span class="id" title="definition">fold</span></a> {<a id="A:970" class="idref" href="#A:970"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>}(<a id="f:971" class="idref" href="#f:971"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:970"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:970"><span class="id" title="variable">A</span></a>)(<a id="s:972" class="idref" href="#s:972"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:970"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:970"><span class="id" title="variable">A</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold"><span class="id" title="definition">Raw.fold</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:971"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:972"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.cardinal" class="idref" href="#MSetIntervals.cardinal"><span class="id" title="definition">cardinal</span></a> (<a id="s:973" class="idref" href="#s:973"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal"><span class="id" title="definition">Raw.cardinal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:973"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.filter" class="idref" href="#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> (<a id="f:974" class="idref" href="#f:974"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<a id="s:975" class="idref" href="#s:975"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">Raw.filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:974"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:975"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.for_all" class="idref" href="#MSetIntervals.for_all"><span class="id" title="definition">for_all</span></a> (<a id="f:976" class="idref" href="#f:976"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<a id="s:977" class="idref" href="#s:977"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all"><span class="id" title="definition">Raw.for_all</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:976"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:977"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.exists_" class="idref" href="#MSetIntervals.exists_"><span class="id" title="definition">exists_</span></a> (<a id="f:978" class="idref" href="#f:978"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<a id="s:979" class="idref" href="#s:979"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_"><span class="id" title="definition">Raw.exists_</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:978"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:979"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.partition" class="idref" href="#MSetIntervals.partition"><span class="id" title="definition">partition</span></a> (<a id="f:980" class="idref" href="#f:980"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>)(<a id="s:981" class="idref" href="#s:981"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="p:982" class="idref" href="#p:982"><span class="id" title="binder">p</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition"><span class="id" title="definition">Raw.partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:980"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:981"><span class="id" title="variable">s</span></a> <span class="id" title="tactic">in</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#p:982"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Mkt"><span class="id" title="constructor">Mkt</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#p:982"><span class="id" title="variable">p</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>.<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="MSetIntervals.In_compat" class="idref" href="#MSetIntervals.In_compat"><span class="id" title="instance">In_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">red</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">H_eq_xy</span> <span class="id" title="var">x'</span> <span class="id" title="var">y'</span> →.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <span class="id" title="var">H_eq_xy</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.eq" class="idref" href="#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="MSetIntervals.eq_equiv" class="idref" href="#MSetIntervals.eq_equiv"><span class="id" title="instance">eq_equiv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">firstorder</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.eq_dec" class="idref" href="#MSetIntervals.eq_dec"><span class="id" title="definition">eq_dec</span></a> : <span class="id" title="keyword">∀</span> (<a id="s:983" class="idref" href="#s:983"><span class="id" title="binder">s</span></a> <a id="s':984" class="idref" href="#s':984"><span class="id" title="binder">s'</span></a>:<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:983"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':984"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s:983"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#s':984"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">intros</span> (<span class="id" title="var">s</span>,<span class="id" title="var">Hs</span>) (<span class="id" title="var">s'</span>,<span class="id" title="var">Hs'</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">change</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Raw.Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Raw.Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s'</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="tactic">destruct</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal"><span class="id" title="definition">Raw.equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">s'</span>) <span class="id" title="var">eqn</span>:<span class="id" title="var">H</span>; [<span class="id" title="tactic">left</span>|<span class="id" title="tactic">right</span>];<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_spec"><span class="id" title="lemma">Raw.equal_spec</span></a>; <span class="id" title="tactic">congruence</span>.<br/>
+&nbsp;<span class="id" title="keyword">Defined</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Definition</span> <a id="MSetIntervals.lt" class="idref" href="#MSetIntervals.lt"><span class="id" title="definition">lt</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">Raw.lt</span></a>.<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="MSetIntervals.lt_strorder" class="idref" href="#MSetIntervals.lt_strorder"><span class="id" title="instance">lt_strorder</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#StrictOrder"><span class="id" title="class">StrictOrder</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">constructor</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt_Irreflexive"><span class="id" title="lemma">Raw.lt_Irreflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Irreflexive"><span class="id" title="class">Irreflexive</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#complement"><span class="id" title="definition">complement</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Reflexive"><span class="id" title="class">Reflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt_Transitive"><span class="id" title="lemma">Raw.lt_Transitive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Transitive"><span class="id" title="class">Transitive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="MSetIntervals.lt_compat" class="idref" href="#MSetIntervals.lt_compat"><span class="id" title="instance">lt_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">lt</span></a>.<br/>
+&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">red</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x1</span> <span class="id" title="var">H_x1_ok</span>] [<span class="id" title="var">y1</span> <span class="id" title="var">H_y1_ok</span>] <span class="id" title="var">H_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x2</span> <span class="id" title="var">H_x2_ok</span>] [<span class="id" title="var">y2</span> <span class="id" title="var">H_y2_ok</span>].<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">lt</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="keyword">∀</span> <a id="a:986" class="idref" href="#a:986"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:985"><span class="id" title="variable">a</span></a> <span class="id" title="var">x1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:985"><span class="id" title="variable">a</span></a> <span class="id" title="var">y1</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Raw.Equal</span></a> <span class="id" title="var">x1</span> <span class="id" title="var">y1</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="keyword">∀</span> <a id="a:988" class="idref" href="#a:988"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:987"><span class="id" title="variable">a</span></a> <span class="id" title="var">x2</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:987"><span class="id" title="variable">a</span></a> <span class="id" title="var">y2</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Raw.Equal</span></a> <span class="id" title="var">x2</span> <span class="id" title="var">y2</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -!<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_spec"><span class="id" title="lemma">Raw.equal_spec</span></a> !<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_alt_def"><span class="id" title="lemma">Raw.equal_alt_def</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → → //.<br/>
+&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">Section</span> <a id="MSetIntervals.Spec" class="idref" href="#MSetIntervals.Spec"><span class="id" title="section">Spec</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="MSetIntervals.Spec.s" class="idref" href="#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a id="MSetIntervals.Spec.s'" class="idref" href="#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.t"><span class="id" title="definition">t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="MSetIntervals.Spec.x" class="idref" href="#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a id="MSetIntervals.Spec.y" class="idref" href="#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="MSetIntervals.Spec.f" class="idref" href="#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="MSetIntervals.compatb" class="idref" href="#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.mem_spec" class="idref" href="#MSetIntervals.mem_spec"><span class="id" title="lemma">mem_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem_spec"><span class="id" title="lemma">Raw.mem_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.equal_spec" class="idref" href="#MSetIntervals.equal_spec"><span class="id" title="lemma">equal_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_spec"><span class="id" title="lemma">Raw.equal_spec</span></a> //. <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.subset_spec" class="idref" href="#MSetIntervals.subset_spec"><span class="id" title="lemma">subset_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Subset"><span class="id" title="definition">Subset</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset_spec"><span class="id" title="lemma">Raw.subset_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.empty_spec" class="idref" href="#MSetIntervals.empty_spec"><span class="id" title="lemma">empty_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty"><span class="id" title="definition">empty</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty_spec"><span class="id" title="lemma">Raw.empty_spec</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.is_empty_spec" class="idref" href="#MSetIntervals.is_empty_spec"><span class="id" title="lemma">is_empty_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty"><span class="id" title="definition">is_empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty_spec"><span class="id" title="lemma">Raw.is_empty_spec</span></a> //. <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.add_spec" class="idref" href="#MSetIntervals.add_spec"><span class="id" title="lemma">add_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.add_spec"><span class="id" title="lemma">Raw.add_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.remove_spec" class="idref" href="#MSetIntervals.remove_spec"><span class="id" title="lemma">remove_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove_spec"><span class="id" title="lemma">Raw.remove_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.singleton_spec" class="idref" href="#MSetIntervals.singleton_spec"><span class="id" title="lemma">singleton_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton_spec"><span class="id" title="lemma">Raw.singleton_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.union_spec" class="idref" href="#MSetIntervals.union_spec"><span class="id" title="lemma">union_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.union_spec"><span class="id" title="lemma">Raw.union_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.inter_spec" class="idref" href="#MSetIntervals.inter_spec"><span class="id" title="lemma">inter_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter_spec"><span class="id" title="lemma">Raw.inter_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.diff_spec" class="idref" href="#MSetIntervals.diff_spec"><span class="id" title="lemma">diff_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff_spec"><span class="id" title="lemma">Raw.diff_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.fold_spec" class="idref" href="#MSetIntervals.fold_spec"><span class="id" title="lemma">fold_spec</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:994" class="idref" href="#A:994"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:995" class="idref" href="#i:995"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#A:994"><span class="id" title="variable">A</span></a>) (<a id="f:996" class="idref" href="#f:996"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:994"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#A:994"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#f:996"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#i:995"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<span class="id" title="keyword">fun</span> <a id="a:997" class="idref" href="#a:997"><span class="id" title="binder">a</span></a> <a id="e:998" class="idref" href="#e:998"><span class="id" title="binder">e</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#f:996"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:998"><span class="id" title="variable">e</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:997"><span class="id" title="variable">a</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#i:995"><span class="id" title="variable">i</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold_spec"><span class="id" title="lemma">Raw.fold_spec</span></a> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.cardinal_spec" class="idref" href="#MSetIntervals.cardinal_spec"><span class="id" title="lemma">cardinal_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal_spec"><span class="id" title="lemma">Raw.cardinal_spec</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.filter_spec" class="idref" href="#MSetIntervals.filter_spec"><span class="id" title="lemma">filter_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>; <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter_spec"><span class="id" title="lemma">Raw.filter_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.for_all_spec" class="idref" href="#MSetIntervals.for_all_spec"><span class="id" title="lemma">for_all_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <a id="x:999" class="idref" href="#x:999"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:999"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all_spec"><span class="id" title="lemma">Raw.for_all_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.exists_spec" class="idref" href="#MSetIntervals.exists_spec"><span class="id" title="lemma">exists_spec</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <a id="x:1000" class="idref" href="#x:1000"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:1000"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_spec"><span class="id" title="lemma">Raw.exists_spec</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.partition_spec1" class="idref" href="#MSetIntervals.partition_spec1"><span class="id" title="lemma">partition_spec1</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>; <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition_spec1"><span class="id" title="lemma">Raw.partition_spec1</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.partition_spec2" class="idref" href="#MSetIntervals.partition_spec2"><span class="id" title="lemma">partition_spec2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>)) (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <a id="x:1001" class="idref" href="#x:1001"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#x:1001"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>; <span class="id" title="tactic">exact</span> (@<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition_spec2"><span class="id" title="lemma">Raw.partition_spec2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.elements_spec1" class="idref" href="#MSetIntervals.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> //. <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.elements_spec2w" class="idref" href="#MSetIntervals.elements_spec2w"><span class="id" title="lemma">elements_spec2w</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec2w"><span class="id" title="lemma">Raw.elements_spec2w</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.elements_spec2" class="idref" href="#MSetIntervals.elements_spec2"><span class="id" title="lemma">elements_spec2</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#sort"><span class="id" title="abbreviation">sort</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="axiom">E.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_sorted"><span class="id" title="lemma">Raw.elements_sorted</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.choose_spec1" class="idref" href="#MSetIntervals.choose_spec1"><span class="id" title="lemma">choose_spec1</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec1"><span class="id" title="definition">Raw.choose_spec1</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.choose_spec2" class="idref" href="#MSetIntervals.choose_spec2"><span class="id" title="lemma">choose_spec2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec2"><span class="id" title="definition">Raw.choose_spec2</span></a> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.choose_spec3" class="idref" href="#MSetIntervals.choose_spec3"><span class="id" title="lemma">choose_spec3</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> → : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec3"><span class="id" title="lemma">Raw.choose_spec3</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.min_elt_spec1" class="idref" href="#MSetIntervals.min_elt_spec1"><span class="id" title="lemma">min_elt_spec1</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec1"><span class="id" title="definition">Raw.min_elt_spec1</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.min_elt_spec2" class="idref" href="#MSetIntervals.min_elt_spec2"><span class="id" title="lemma">min_elt_spec2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt"><span class="id" title="definition">min_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec2"><span class="id" title="definition">Raw.min_elt_spec2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.min_elt_spec3" class="idref" href="#MSetIntervals.min_elt_spec3"><span class="id" title="lemma">min_elt_spec3</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec3"><span class="id" title="definition">Raw.min_elt_spec3</span></a> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.max_elt_spec1" class="idref" href="#MSetIntervals.max_elt_spec1"><span class="id" title="lemma">max_elt_spec1</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt"><span class="id" title="definition">max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec1"><span class="id" title="definition">Raw.max_elt_spec1</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.max_elt_spec2" class="idref" href="#MSetIntervals.max_elt_spec2"><span class="id" title="lemma">max_elt_spec2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt"><span class="id" title="definition">max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="axiom">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec2"><span class="id" title="definition">Raw.max_elt_spec2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.max_elt_spec3" class="idref" href="#MSetIntervals.max_elt_spec3"><span class="id" title="lemma">max_elt_spec3</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt"><span class="id" title="definition">max_elt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">exact</span> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec3"><span class="id" title="definition">Raw.max_elt_spec3</span></a> <span class="id" title="var">_</span>). <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="MSetIntervals.compare_spec" class="idref" href="#MSetIntervals.compare_spec"><span class="id" title="lemma">compare_spec</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare"><span class="id" title="definition">compare</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">s1</span> <span class="id" title="var">H_ok_s1</span>] [<span class="id" title="var">s2</span> <span class="id" title="var">H_ok_s2</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare_spec"><span class="id" title="lemma">Raw.compare_spec</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#CompSpec"><span class="id" title="definition">CompSpec</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq"><span class="id" title="definition">eq</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Equal</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">In</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt"><span class="id" title="definition">lt</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare"><span class="id" title="definition">compare</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">replace</span> (<span class="id" title="keyword">∀</span> <a id="a:1003" class="idref" href="#a:1003"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:1002"><span class="id" title="variable">a</span></a> <span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.In"><span class="id" title="definition">Raw.In</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#a:1002"><span class="id" title="variable">a</span></a> <span class="id" title="var">s2</span>) <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Raw.Equal</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> <span class="id" title="var">H_eq</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal"><span class="id" title="definition">Raw.Equal</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">s1</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] <span class="id" title="var">H</span>; <span class="id" title="tactic">constructor</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_spec"><span class="id" title="lemma">Raw.equal_spec</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_alt_def"><span class="id" title="lemma">Raw.equal_alt_def</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec"><span class="id" title="section">Spec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab57"></a><h2 class="section">Instantiations</h2>
+
+
+<div class="paragraph"> </div>
+
+    It remains to provide instantiations for commonly used datatypes. 
+<div class="paragraph"> </div>
+
+<a id="lab58"></a><h3 class="section">Z</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="ElementEncodeZ" class="idref" href="#ElementEncodeZ"><span class="id" title="module">ElementEncodeZ</span></a> &lt;: <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="ElementEncodeZ.E" class="idref" href="#ElementEncodeZ.E"><span class="id" title="module">E</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z"><span class="id" title="module">Z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ElementEncodeZ.encode" class="idref" href="#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> (<a id="z:1004" class="idref" href="#z:1004"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#z:1004"><span class="id" title="variable">z</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ElementEncodeZ.decode" class="idref" href="#ElementEncodeZ.decode"><span class="id" title="definition">decode</span></a> (<a id="z:1005" class="idref" href="#z:1005"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="MSetsExtra.MSetIntervals.html#z:1005"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeZ.decode_encode_ok" class="idref" href="#ElementEncodeZ.decode_encode_ok"><span class="id" title="lemma">decode_encode_ok</span></a>: <span class="id" title="keyword">∀</span> (<a id="e:1006" class="idref" href="#e:1006"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.decode"><span class="id" title="definition">decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:1006"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:1006"><span class="id" title="variable">e</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">by</span> []. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeZ.encode_eq" class="idref" href="#ElementEncodeZ.encode_eq"><span class="id" title="lemma">encode_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:1007" class="idref" href="#e1:1007"><span class="id" title="binder">e1</span></a> <a id="e2:1008" class="idref" href="#e2:1008"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="definition">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1007"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1008"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.eq"><span class="id" title="definition">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1007"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1008"><span class="id" title="variable">e2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">by</span> []. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeZ.encode_lt" class="idref" href="#ElementEncodeZ.encode_lt"><span class="id" title="lemma">encode_lt</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:1009" class="idref" href="#e1:1009"><span class="id" title="binder">e1</span></a> <a id="e2:1010" class="idref" href="#e2:1010"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="definition">Z.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1009"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1010"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.lt"><span class="id" title="definition">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1009"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1010"><span class="id" title="variable">e2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">by</span> []. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ"><span class="id" title="module">ElementEncodeZ</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="MSetIntervalsZ" class="idref" href="#MSetIntervalsZ"><span class="id" title="module">MSetIntervalsZ</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#SetsOn"><span class="id" title="module">SetsOn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z"><span class="id" title="module">Z</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeZ"><span class="id" title="module">ElementEncodeZ</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab59"></a><h3 class="section">N</h3>
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="ElementEncodeN" class="idref" href="#ElementEncodeN"><span class="id" title="module">ElementEncodeN</span></a> &lt;: <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="ElementEncodeN.E" class="idref" href="#ElementEncodeN.E"><span class="id" title="module">E</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N"><span class="id" title="module">N</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ElementEncodeN.encode" class="idref" href="#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> (<a id="n:1011" class="idref" href="#n:1011"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#N"><span class="id" title="inductive">N</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_N"><span class="id" title="definition">Z.of_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:1011"><span class="id" title="variable">n</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ElementEncodeN.decode" class="idref" href="#ElementEncodeN.decode"><span class="id" title="definition">decode</span></a> (<a id="z:1012" class="idref" href="#z:1012"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_N"><span class="id" title="definition">Z.to_N</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:1012"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeN.decode_encode_ok" class="idref" href="#ElementEncodeN.decode_encode_ok"><span class="id" title="lemma">decode_encode_ok</span></a>: <span class="id" title="keyword">∀</span> (<a id="e:1013" class="idref" href="#e:1013"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.decode"><span class="id" title="definition">decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:1013"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:1013"><span class="id" title="variable">e</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.decode"><span class="id" title="definition">decode</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.id"><span class="id" title="lemma">N2Z.id</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeN.encode_eq" class="idref" href="#ElementEncodeN.encode_eq"><span class="id" title="lemma">encode_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:1014" class="idref" href="#e1:1014"><span class="id" title="binder">e1</span></a> <a id="e2:1015" class="idref" href="#e2:1015"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="axiom">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1014"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1015"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.eq"><span class="id" title="definition">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1014"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1015"><span class="id" title="variable">e2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e1</span> <span class="id" title="var">e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="axiom">Z.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_iff"><span class="id" title="lemma">N2Z.inj_iff</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.eq"><span class="id" title="definition">E.eq</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeN.encode_lt" class="idref" href="#ElementEncodeN.encode_lt"><span class="id" title="lemma">encode_lt</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:1016" class="idref" href="#e1:1016"><span class="id" title="binder">e1</span></a> <a id="e2:1017" class="idref" href="#e2:1017"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="axiom">Z.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1016"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1017"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.lt"><span class="id" title="definition">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1016"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1017"><span class="id" title="variable">e2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e1</span> <span class="id" title="var">e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode"><span class="id" title="definition">encode</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#N2Z.inj_lt"><span class="id" title="lemma">N2Z.inj_lt</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN"><span class="id" title="module">ElementEncodeN</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="MSetIntervalsN" class="idref" href="#MSetIntervalsN"><span class="id" title="module">MSetIntervalsN</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#SetsOn"><span class="id" title="module">SetsOn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.NArith.BinNat.html#N"><span class="id" title="module">N</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeN"><span class="id" title="module">ElementEncodeN</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab60"></a><h3 class="section">nat</h3>
+
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="ElementEncodeNat" class="idref" href="#ElementEncodeNat"><span class="id" title="module">ElementEncodeNat</span></a> &lt;: <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncode"><span class="id" title="module">ElementEncode</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="ElementEncodeNat.E" class="idref" href="#ElementEncodeNat.E"><span class="id" title="module">E</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">Nat</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ElementEncodeNat.encode" class="idref" href="#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> (<a id="n:1018" class="idref" href="#n:1018"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.of_nat"><span class="id" title="definition">Z.of_nat</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#n:1018"><span class="id" title="variable">n</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="ElementEncodeNat.decode" class="idref" href="#ElementEncodeNat.decode"><span class="id" title="definition">decode</span></a> (<a id="z:1019" class="idref" href="#z:1019"><span class="id" title="binder">z</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.to_nat"><span class="id" title="definition">Z.to_nat</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#z:1019"><span class="id" title="variable">z</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeNat.decode_encode_ok" class="idref" href="#ElementEncodeNat.decode_encode_ok"><span class="id" title="lemma">decode_encode_ok</span></a>: <span class="id" title="keyword">∀</span> (<a id="e:1020" class="idref" href="#e:1020"><span class="id" title="binder">e</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.decode"><span class="id" title="definition">decode</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:1020"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e:1020"><span class="id" title="variable">e</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.decode"><span class="id" title="definition">decode</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Nat2Z.id"><span class="id" title="lemma">Nat2Z.id</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeNat.encode_eq" class="idref" href="#ElementEncodeNat.encode_eq"><span class="id" title="lemma">encode_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:1021" class="idref" href="#e1:1021"><span class="id" title="binder">e1</span></a> <a id="e2:1022" class="idref" href="#e2:1022"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="axiom">Z.eq</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1021"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1022"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.eq"><span class="id" title="definition">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1021"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1022"><span class="id" title="variable">e2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e1</span> <span class="id" title="var">e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.eq"><span class="id" title="axiom">Z.eq</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Nat2Z.inj_iff"><span class="id" title="lemma">Nat2Z.inj_iff</span></a> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.eq"><span class="id" title="definition">E.eq</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="ElementEncodeNat.encode_lt" class="idref" href="#ElementEncodeNat.encode_lt"><span class="id" title="lemma">encode_lt</span></a> : <span class="id" title="keyword">∀</span> (<a id="e1:1023" class="idref" href="#e1:1023"><span class="id" title="binder">e1</span></a> <a id="e2:1024" class="idref" href="#e2:1024"><span class="id" title="binder">e2</span></a> : <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.t"><span class="id" title="definition">E.t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.BinInt.html#Z.lt"><span class="id" title="axiom">Z.lt</span></a> (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1023"><span class="id" title="variable">e1</span></a>) (<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1024"><span class="id" title="variable">e2</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.lt"><span class="id" title="definition">E.lt</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e1:1023"><span class="id" title="variable">e1</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#e2:1024"><span class="id" title="variable">e2</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">e1</span> <span class="id" title="var">e2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode"><span class="id" title="definition">encode</span></a> -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ZArith.Znat.html#Nat2Z.inj_lt"><span class="id" title="lemma">Nat2Z.inj_lt</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat"><span class="id" title="module">ElementEncodeNat</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="MSetIntervalsNat" class="idref" href="#MSetIntervalsNat"><span class="id" title="module">MSetIntervalsNat</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#SetsOn"><span class="id" title="module">SetsOn</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Arith.PeanoNat.html#Nat"><span class="id" title="module">Nat</span></a> := <a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals"><span class="id" title="module">MSetIntervals</span></a> <a class="idref" href="MSetsExtra.MSetIntervals.html#ElementEncodeNat"><span class="id" title="module">ElementEncodeNat</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>MSetsExtra.MSetListWithDups</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library MSetsExtra.MSetListWithDups</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab61"></a><h1 class="section">Weak sets implemented as lists with duplicates</h1>
+
+
+<div class="paragraph"> </div>
+
+    This file contains an implementation of the weak set interface
+    <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a></span>. As a datatype unsorted lists are used
+    that might contain duplicates. 
+
+<div class="paragraph"> </div>
+
+    This implementation is useful, if one needs very efficient
+    insert and union operation, and can guarantee that one does not
+    add too many duplicates. The operation <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a></span> is implemented
+    by sorting the list first. Therefore this instantiation can only
+    be used if the element type is ordered. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Export</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#"><span class="id" title="library">MSetInterface</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">mathcomp.ssreflect.ssreflect</span>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#"><span class="id" title="library">List</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.OrdersFacts.html#"><span class="id" title="library">OrdersFacts</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.OrdersLists.html#"><span class="id" title="library">OrdersLists</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorting.html#"><span class="id" title="library">Sorting</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Permutation.html#"><span class="id" title="library">Permutation</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#"><span class="id" title="library">MSetWithDups</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab62"></a><h2 class="section">Removing duplicates from sorted lists</h2>
+
+
+<div class="paragraph"> </div>
+
+    The following module <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a></span> defines an operation
+    <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a></span> that removes duplicates from a sorted
+    list. In order to talk about sorted lists, the element type needs
+    to be ordered.
+
+<div class="paragraph"> </div>
+
+    This function is combined with a sort function to get a function
+    <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA"><span class="id" title="definition">remove_dups_by_sortingA</span></a></span> to sort unsorted lists and then remove
+    duplicates.  
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="RemoveDupsFromSorted" class="idref" href="#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a> (<span class="id" title="keyword">Import</span> <span class="id" title="var">X</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+First, we need some infrastructure for our ordered type 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a id="RemoveDupsFromSorted.MX" class="idref" href="#RemoveDupsFromSorted.MX"><span class="id" title="module">MX</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.OrdersFacts.html#OrderedTypeFacts"><span class="id" title="module">OrderedTypeFacts</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a id="RemoveDupsFromSorted.XTotalLeBool" class="idref" href="#RemoveDupsFromSorted.XTotalLeBool"><span class="id" title="module">XTotalLeBool</span></a> &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#TotalLeBool"><span class="id" title="module">TotalLeBool</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="RemoveDupsFromSorted.XTotalLeBool.t" class="idref" href="#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="RemoveDupsFromSorted.XTotalLeBool.leb" class="idref" href="#RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a> <a id="x:1" class="idref" href="#x:1"><span class="id" title="binder">x</span></a> <a id="y:2" class="idref" href="#y:2"><span class="id" title="binder">y</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:1"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:2"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Infix</span> <a id="c679ee482e47aa89e9818d26cb26e383" class="idref" href="#c679ee482e47aa89e9818d26cb26e383"><span class="id" title="notation">&quot;</span></a>&lt;=?" := <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 35).<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Theorem</span> <a id="RemoveDupsFromSorted.XTotalLeBool.leb_total" class="idref" href="#RemoveDupsFromSorted.XTotalLeBool.leb_total"><span class="id" title="lemma">leb_total</span></a> : <span class="id" title="keyword">∀</span> (<a id="a1:3" class="idref" href="#a1:3"><span class="id" title="binder">a1</span></a> <a id="a2:4" class="idref" href="#a2:4"><span class="id" title="binder">a2</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t"><span class="id" title="definition">t</span></a>), <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:3"><span class="id" title="variable">a1</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#c679ee482e47aa89e9818d26cb26e383"><span class="id" title="notation">&lt;=?</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:4"><span class="id" title="variable">a2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:4"><span class="id" title="variable">a2</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#c679ee482e47aa89e9818d26cb26e383"><span class="id" title="notation">&lt;=?</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:3"><span class="id" title="variable">a1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.MX.compare_antisym"><span class="id" title="lemma">compare_antisym</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span>); <span class="id" title="tactic">rewrite</span> /=; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="RemoveDupsFromSorted.XTotalLeBool.le" class="idref" href="#RemoveDupsFromSorted.XTotalLeBool.le"><span class="id" title="definition">le</span></a> <a id="x:5" class="idref" href="#x:5"><span class="id" title="binder">x</span></a> <a id="y:6" class="idref" href="#y:6"><span class="id" title="binder">y</span></a> := (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb"><span class="id" title="definition">leb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:5"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:6"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool"><span class="id" title="module">XTotalLeBool</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="RemoveDupsFromSorted.eqb_eq_alt" class="idref" href="#RemoveDupsFromSorted.eqb_eq_alt"><span class="id" title="lemma">eqb_eq_alt</span></a> : <span class="id" title="keyword">∀</span> <a id="x:7" class="idref" href="#x:7"><span class="id" title="binder">x</span></a> <a id="y:8" class="idref" href="#y:8"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:8"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:7"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:8"><span class="id" title="variable">y</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_alt"><span class="id" title="lemma">eqb_alt</span></a> -<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) ⇒ //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Now we can define our main function 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="RemoveDupsFromSorted.remove_dups_from_sortedA_aux" class="idref" href="#RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> (<a id="acc:9" class="idref" href="#acc:9"><span class="id" title="binder">acc</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>) (<a id="l:10" class="idref" href="#l:10"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:10"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:9"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">List.rev'</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:9"><span class="id" title="variable">acc</span></a>)<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#remove_dups_from_sortedA_aux:11"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:9"><span class="id" title="variable">acc</span></a> <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#remove_dups_from_sortedA_aux:11"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> (<span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:9"><span class="id" title="variable">acc</span></a>) <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="RemoveDupsFromSorted.remove_dups_from_sortedA" class="idref" href="#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::core_scope:x_':'_x"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+We can prove some technical lemmata 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt" class="idref" href="#RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt"><span class="id" title="lemma">remove_dups_from_sortedA_aux_alt</span></a> : <span class="id" title="keyword">∀</span> (<a id="l:14" class="idref" href="#l:14"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>) <a id="acc:15" class="idref" href="#acc:15"><span class="id" title="binder">acc</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:15"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:14"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev"><span class="id" title="definition">List.rev</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:15"><span class="id" title="variable">acc</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">++</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:14"><span class="id" title="variable">l</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bc347c51eaf667706ae54503b26d52c6"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">acc</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux"><span class="id" title="definition">remove_dups_from_sortedA_aux</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_alt"><span class="id" title="lemma">rev_alt</span></a> /= <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_nil_r"><span class="id" title="lemma">app_nil_r</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> <span class="id" title="var">xs</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> -!<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_alt"><span class="id" title="lemma">rev_alt</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">ys</span> <span class="id" title="var">H_xs_eq</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -!<span class="id" title="var">H_xs_eq</span> !(<span class="id" title="var">IH</span> <span class="id" title="var">acc</span>) !(<span class="id" title="var">IH</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">acc</span>)) (<span class="id" title="var">IH</span> (<span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= -<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#app_assoc"><span class="id" title="lemma">app_assoc</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="RemoveDupsFromSorted.remove_dups_from_sortedA_alt" class="idref" href="#RemoveDupsFromSorted.remove_dups_from_sortedA_alt"><span class="id" title="lemma">remove_dups_from_sortedA_alt</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:16" class="idref" href="#l:16"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:16"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:16"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">with</span> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:16"><span class="id" title="variable">l</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys</span> ⇒ <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> /= /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev'"><span class="id" title="definition">rev'</span></a> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">xs</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">ys</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt"><span class="id" title="lemma">remove_dups_from_sortedA_aux_alt</span></a> /= //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="RemoveDupsFromSorted.remove_dups_from_sortedA_hd" class="idref" href="#RemoveDupsFromSorted.remove_dups_from_sortedA_hd"><span class="id" title="lemma">remove_dups_from_sortedA_hd</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="x:19" class="idref" href="#x:19"><span class="id" title="binder">x</span></a> <a id="xs:20" class="idref" href="#xs:20"><span class="id" title="binder">xs</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">(</span></a><a id="x':21" class="idref" href="#x':21"><span class="id" title="binder">x'</span></a>:<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>) <a id="xs':22" class="idref" href="#xs':22"><span class="id" title="binder">xs'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#xs:20"><span class="id" title="variable">xs</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#x':21"><span class="id" title="variable">x'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#xs':22"><span class="id" title="variable">xs'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:19"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x':21"><span class="id" title="variable">x'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">xs</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">x</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">xs</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span> <span class="id" title="var">ys</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">x</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt"><span class="id" title="lemma">remove_dups_from_sortedA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_alt"><span class="id" title="lemma">eqb_alt</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_refl"><span class="id" title="lemma">compare_refl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt"><span class="id" title="lemma">remove_dups_from_sortedA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>); <span class="id" title="var">last</span> <span class="id" title="tactic">first</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>, (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> (<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">reflexivity</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_alt"><span class="id" title="lemma">eqb_alt</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_refl"><span class="id" title="lemma">compare_refl</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_eqb_xy</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">y</span>) ⇒ {<span class="id" title="var">IH</span>} [<span class="id" title="var">x'</span>] [<span class="id" title="var">xs'</span>] [-&gt;] <span class="id" title="var">H_eqb_yx'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x'</span>, <span class="id" title="var">xs'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_eqb_xy</span> <span class="id" title="var">H_eqb_yx'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt"><span class="id" title="lemma">eqb_eq_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eq_trans"><span class="id" title="definition">MX.eq_trans</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Finally we get our main result for removing duplicates from sorted lists 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="RemoveDupsFromSorted.remove_dups_from_sortedA_spec" class="idref" href="#RemoveDupsFromSorted.remove_dups_from_sortedA_spec"><span class="id" title="lemma">remove_dups_from_sortedA_spec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:23" class="idref" href="#l:23"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.le"><span class="id" title="definition">le</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:23"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="l':24" class="idref" href="#l':24"><span class="id" title="binder">l'</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:23"><span class="id" title="variable">l</span></a> <span class="id" title="tactic">in</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l':24"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l':24"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:25" class="idref" href="#x:25"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:23"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:25"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l':24"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">l</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. { <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt"><span class="id" title="lemma">remove_dups_from_sortedA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt"><span class="id" title="lemma">remove_dups_from_sortedA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> <span class="id" title="var">xs</span> ⇒ {<span class="id" title="var">xs</span>}. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_cons"><span class="id" title="constructor">Sorted_cons</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_singleton"><span class="id" title="lemma">NoDupA_singleton</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">y</span> <span class="id" title="var">ys</span> <span class="id" title="var">IH</span> <span class="id" title="var">H_sorted_x_y_ys</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_inv"><span class="id" title="lemma">Sorted_inv</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H_sorted_x_y_ys</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">H_sorted_y_ys</span> <span class="id" title="var">H_hd_rel</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#HdRel_inv"><span class="id" title="lemma">HdRel_inv</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">H_hd_rel</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> :  <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y':26" class="idref" href="#y':26"><span class="id" title="binder">y'</span></a> <a id="ys':27" class="idref" href="#ys':27"><span class="id" title="binder">ys'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> (<span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y':26"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#ys':27"><span class="id" title="variable">ys'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">y</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y':26"><span class="id" title="variable">y'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_hd"><span class="id" title="lemma">remove_dups_from_sortedA_hd</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">y'</span>] [<span class="id" title="var">ys'</span>] [<span class="id" title="var">H_yys'_intro</span>] /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt"><span class="id" title="lemma">eqb_eq_alt</span></a> <span class="id" title="var">H_eq_y_y'</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">IH</span> <span class="id" title="var">H_sorted_y_ys</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<span class="id" title="var">H_yys'_intro</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ {<span class="id" title="var">IH</span>} [<span class="id" title="var">IH1</span>] [<span class="id" title="var">IH2</span>] <span class="id" title="var">IH3</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb"><span class="id" title="definition">eqb</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt"><span class="id" title="lemma">eqb_eq_alt</span></a> ⇒ <span class="id" title="var">H_eq_x_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> <span class="id" title="var">IH3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_eq_x'_x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eq_trans"><span class="id" title="definition">eq_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">y</span> := <span class="id" title="var">x</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eq_trans"><span class="id" title="definition">eq_trans</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">y</span> := <span class="id" title="var">y</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_neqb_x_y</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_sorted</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.lt"><span class="id" title="axiom">lt</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_cons"><span class="id" title="constructor">Sorted_cons</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#HdRel_cons"><span class="id" title="constructor">HdRel_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_lt_iff"><span class="id" title="lemma">compare_lt_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : (<a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_compat"><span class="id" title="instance">compare_compat</span></a>; <span class="id" title="tactic">eauto</span>;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eq_refl"><span class="id" title="definition">eq_refl</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_hd_rel</span> <span class="id" title="var">H_neqb_x_y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_alt"><span class="id" title="lemma">eqb_alt</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.le"><span class="id" title="definition">le</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.leb"><span class="id" title="definition">leb</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="var">last</span> <span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">assumption</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA_cons"><span class="id" title="constructor">NoDupA_cons</span></a> ⇒ //.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> [<span class="id" title="var">x'</span>] [<span class="id" title="var">H_eq_xx'</span>] <span class="id" title="var">H_in_x'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#Forall"><span class="id" title="inductive">Forall</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#X.lt"><span class="id" title="axiom">lt</span></a> <span class="id" title="var">x</span>) (<span class="id" title="var">y'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">ys'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted_extends"><span class="id" title="lemma">Sorted_extends</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sets.Relations_1.html#Transitive"><span class="id" title="definition">Relations_1.Transitive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.lt_trans"><span class="id" title="definition">lt_trans</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#Forall_forall"><span class="id" title="lemma">Forall_forall</span></a> ⇒ <span class="id" title="var">H_forall</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_forall</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in_x'</span>) ⇒ {<span class="id" title="var">H_forall</span>}.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">H_eq_xx'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_lt_iff"><span class="id" title="lemma">compare_lt_iff</span></a> -<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !(<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <span class="id" title="var">x0</span> <span class="id" title="var">x</span>) <span class="id" title="var">IH3</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Next, we combine it with sorting 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a id="RemoveDupsFromSorted.XSort" class="idref" href="#RemoveDupsFromSorted.XSort"><span class="id" title="module">XSort</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Mergesort.html#Sort"><span class="id" title="module">Sort</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool"><span class="id" title="module">XTotalLeBool</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="RemoveDupsFromSorted.remove_dups_by_sortingA" class="idref" href="#RemoveDupsFromSorted.remove_dups_by_sortingA"><span class="id" title="definition">remove_dups_by_sortingA</span></a> (<a id="l:28" class="idref" href="#l:28"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA"><span class="id" title="definition">remove_dups_from_sortedA</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.sort"><span class="id" title="definition">XSort.sort</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:28"><span class="id" title="variable">l</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="RemoveDupsFromSorted.remove_dups_by_sortingA_spec" class="idref" href="#RemoveDupsFromSorted.remove_dups_by_sortingA_spec"><span class="id" title="lemma">remove_dups_by_sortingA_spec</span></a> : <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="l:29" class="idref" href="#l:29"><span class="id" title="binder">l</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.t"><span class="id" title="definition">t</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <a id="l':30" class="idref" href="#l':30"><span class="id" title="binder">l'</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA"><span class="id" title="definition">remove_dups_by_sortingA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:29"><span class="id" title="variable">l</span></a> <span class="id" title="tactic">in</span> (<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.lt"><span class="id" title="axiom">lt</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l':30"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l':30"><span class="id" title="variable">l'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:31" class="idref" href="#x:31"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:31"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:29"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:31"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l':30"><span class="id" title="variable">l'</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intro</span> <span class="id" title="var">l</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:32" class="idref" href="#x:32"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:32"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.sort"><span class="id" title="definition">sort</span></a> <span class="id" title="var">l</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.eq"><span class="id" title="axiom">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:32"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.le"><span class="id" title="definition">le</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.sort"><span class="id" title="definition">sort</span></a> <span class="id" title="var">l</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">unfold</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA"><span class="id" title="definition">remove_dups_by_sortingA</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_spec"><span class="id" title="lemma">remove_dups_from_sortedA_spec</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.sort"><span class="id" title="definition">sort</span></a> <span class="id" title="var">l</span>)).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H_spec</span> [<span class="id" title="var">H_in_sort</span> <span class="id" title="var">H_sorted_sort</span>].<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H_spec</span> <span class="id" title="var">H_sorted_sort</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H1</span>] [<span class="id" title="var">H2</span>] <span class="id" title="var">H3</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<span class="id" title="var">H_in_sort</span> <span class="id" title="var">H3</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_in_sort</span> : <span class="id" title="keyword">∀</span> <a id="x:33" class="idref" href="#x:33"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:33"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.sort"><span class="id" title="definition">XSort.sort</span></a> <span class="id" title="var">l</span>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#In"><span class="id" title="definition">List.In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:33"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_perm</span> := (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.Permuted_sort"><span class="id" title="lemma">XSort.Permuted_sort</span></a> <span class="id" title="var">l</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Permutation.html#Permutation_in"><span class="id" title="lemma">Permutation_in</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Permutation.html#Permutation_sym"><span class="id" title="lemma">Permutation_sym</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_rewrite</span> <span class="id" title="var">H_in_sort</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.Sorted_sort"><span class="id" title="lemma">Sorted_sort</span></a> <span class="id" title="var">l</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#is_true"><span class="id" title="definition">is_true</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.le"><span class="id" title="definition">le</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.leb"><span class="id" title="definition">leb</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab63"></a><h2 class="section">Operations Module</h2>
+
+
+<div class="paragraph"> </div>
+
+With removing duplicates defined, we can implement
+the operations for our set implementation easily. 
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="Ops" class="idref" href="#Ops"><span class="id" title="module">Ops</span></a> (<span class="id" title="var">X</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#WOps"><span class="id" title="module">WOps</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <a id="Ops.RDFS" class="idref" href="#Ops.RDFS"><span class="id" title="module">RDFS</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a id="Ops.MX" class="idref" href="#Ops.MX"><span class="id" title="module">MX</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.OrdersFacts.html#OrderedTypeFacts"><span class="id" title="module">OrderedTypeFacts</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X"><span class="id" title="module">X</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.elt" class="idref" href="#Ops.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.t"><span class="id" title="axiom">X.t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.t" class="idref" href="#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.empty" class="idref" href="#Ops.empty"><span class="id" title="definition">empty</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.is_empty" class="idref" href="#Ops.is_empty"><span class="id" title="definition">is_empty</span></a> (<a id="l:34" class="idref" href="#l:34"><span class="id" title="binder">l</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) := <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#l:34"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.mem" class="idref" href="#Ops.mem"><span class="id" title="definition">mem</span></a> (<a id="x:36" class="idref" href="#x:36"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a>) (<a id="s:37" class="idref" href="#s:37"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:37"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:36"><span class="id" title="variable">x</span></a> <span class="id" title="var">y</span> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#mem:38"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:36"><span class="id" title="variable">x</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.add" class="idref" href="#Ops.add"><span class="id" title="definition">add</span></a> <a id="x:40" class="idref" href="#x:40"><span class="id" title="binder">x</span></a> (<a id="s:41" class="idref" href="#s:41"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:40"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:41"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.singleton" class="idref" href="#Ops.singleton"><span class="id" title="definition">singleton</span></a> (<a id="x:42" class="idref" href="#x:42"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a>) := <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:42"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.rev_filter_aux" class="idref" href="#Ops.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a id="acc:43" class="idref" href="#acc:43"><span class="id" title="binder">acc</span></a> (<a id="f:44" class="idref" href="#f:44"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <a id="s:45" class="idref" href="#s:45"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:45"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:43"><span class="id" title="variable">acc</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">xs</span>) ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#rev_filter_aux:46"><span class="id" title="definition">rev_filter_aux</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#f:44"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:43"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:43"><span class="id" title="variable">acc</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:44"><span class="id" title="variable">f</span></a> <span class="id" title="var">xs</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.rev_filter" class="idref" href="#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.filter" class="idref" href="#Ops.filter"><span class="id" title="definition">filter</span></a> (<a id="f:48" class="idref" href="#f:48"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:49" class="idref" href="#s:49"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:48"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:49"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.remove" class="idref" href="#Ops.remove"><span class="id" title="definition">remove</span></a> <a id="x:50" class="idref" href="#x:50"><span class="id" title="binder">x</span></a> <a id="s:51" class="idref" href="#s:51"><span class="id" title="binder">s</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> (<span class="id" title="keyword">fun</span> <a id="y:52" class="idref" href="#y:52"><span class="id" title="binder">y</span></a> ⇒ <span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#X.compare"><span class="id" title="axiom">X.compare</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:50"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:52"><span class="id" title="variable">y</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> | <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <span class="id" title="keyword">end</span>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:51"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.union" class="idref" href="#Ops.union"><span class="id" title="definition">union</span></a> (<a id="s1:53" class="idref" href="#s1:53"><span class="id" title="binder">s1</span></a> <a id="s2:54" class="idref" href="#s2:54"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append"><span class="id" title="definition">List.rev_append</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s2:54"><span class="id" title="variable">s2</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s1:53"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.inter" class="idref" href="#Ops.inter"><span class="id" title="definition">inter</span></a> (<a id="s1:55" class="idref" href="#s1:55"><span class="id" title="binder">s1</span></a> <a id="s2:56" class="idref" href="#s2:56"><span class="id" title="binder">s2</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter"><span class="id" title="definition">rev_filter</span></a> (<span class="id" title="keyword">fun</span> <a id="y:57" class="idref" href="#y:57"><span class="id" title="binder">y</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:57"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s2:56"><span class="id" title="variable">s2</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s1:55"><span class="id" title="variable">s1</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.elements" class="idref" href="#Ops.elements"><span class="id" title="definition">elements</span></a> (<a id="x:58" class="idref" href="#x:58"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:58"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.elements_dist" class="idref" href="#Ops.elements_dist"><span class="id" title="definition">elements_dist</span></a> (<a id="x:59" class="idref" href="#x:59"><span class="id" title="binder">x</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.remove_dups_by_sortingA"><span class="id" title="definition">RDFS.remove_dups_by_sortingA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:59"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.fold" class="idref" href="#Ops.fold"><span class="id" title="definition">fold</span></a> {<a id="B:60" class="idref" href="#B:60"><span class="id" title="binder">B</span></a> : <span class="id" title="keyword">Type</span>} (<a id="f:61" class="idref" href="#f:61"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#B:60"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#B:60"><span class="id" title="variable">B</span></a>) (<a id="s:62" class="idref" href="#s:62"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) (<a id="i:63" class="idref" href="#i:63"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#B:60"><span class="id" title="variable">B</span></a>) : <a class="idref" href="MSetsExtra.MSetListWithDups.html#B:60"><span class="id" title="variable">B</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:61"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:62"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#i:63"><span class="id" title="variable">i</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.diff" class="idref" href="#Ops.diff"><span class="id" title="definition">diff</span></a> (<a id="s:64" class="idref" href="#s:64"><span class="id" title="binder">s</span></a> <a id="s':65" class="idref" href="#s':65"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':65"><span class="id" title="variable">s'</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:64"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.subset" class="idref" href="#Ops.subset"><span class="id" title="definition">subset</span></a> (<a id="s:66" class="idref" href="#s:66"><span class="id" title="binder">s</span></a> <a id="s':67" class="idref" href="#s':67"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#forallb"><span class="id" title="definition">List.forallb</span></a> (<span class="id" title="keyword">fun</span> <a id="x:68" class="idref" href="#x:68"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:68"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':67"><span class="id" title="variable">s'</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:66"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.equal" class="idref" href="#Ops.equal"><span class="id" title="definition">equal</span></a> (<a id="s:69" class="idref" href="#s:69"><span class="id" title="binder">s</span></a> <a id="s':70" class="idref" href="#s':70"><span class="id" title="binder">s'</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#andb"><span class="id" title="definition">andb</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:69"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':70"><span class="id" title="variable">s'</span></a>) (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':70"><span class="id" title="variable">s'</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:69"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.for_all" class="idref" href="#Ops.for_all"><span class="id" title="definition">for_all</span></a> (<a id="f:71" class="idref" href="#f:71"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:72" class="idref" href="#s:72"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:72"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:71"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#for_all:73"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:71"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.exists_" class="idref" href="#Ops.exists_"><span class="id" title="definition">exists_</span></a> (<a id="f:75" class="idref" href="#f:75"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:76" class="idref" href="#s:76"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:76"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:75"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#exists_:77"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:75"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Fixpoint</span> <a id="Ops.partition_aux" class="idref" href="#Ops.partition_aux"><span class="id" title="definition">partition_aux</span></a> (<a id="a1:79" class="idref" href="#a1:79"><span class="id" title="binder">a1</span></a> <a id="a2:80" class="idref" href="#a2:80"><span class="id" title="binder">a2</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) (<a id="f:81" class="idref" href="#f:81"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="s:82" class="idref" href="#s:82"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#11c698c8685bb8ab1cf725545c085ac4"><span class="id" title="notation">×</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:82"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:79"><span class="id" title="variable">a1</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:80"><span class="id" title="variable">a2</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">l</span> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">if</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:81"><span class="id" title="variable">f</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">then</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#partition_aux:83"><span class="id" title="definition">partition_aux</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:79"><span class="id" title="variable">a1</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:80"><span class="id" title="variable">a2</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:81"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.ssr.ssreflect.html#::boolean_if_scope:'if'_x_'then'_x_'else'_x"><span class="id" title="notation">else</span></a> <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#partition_aux:83"><span class="id" title="definition">partition_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:79"><span class="id" title="variable">a1</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:80"><span class="id" title="variable">a2</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:81"><span class="id" title="variable">f</span></a> <span class="id" title="var">l</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.partition" class="idref" href="#Ops.partition"><span class="id" title="definition">partition</span></a> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.partition_aux"><span class="id" title="definition">partition_aux</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.cardinal" class="idref" href="#Ops.cardinal"><span class="id" title="definition">cardinal</span></a> (<a id="s:85" class="idref" href="#s:85"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:85"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Ops.choose" class="idref" href="#Ops.choose"><span class="id" title="definition">choose</span></a> (<a id="s:86" class="idref" href="#s:86"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.t"><span class="id" title="definition">t</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elt"><span class="id" title="definition">elt</span></a> :=<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:86"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#nil"><span class="id" title="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#::list_scope:x_'::'_x"><span class="id" title="notation">::</span></a><span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops"><span class="id" title="module">Ops</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab64"></a><h2 class="section">Main Module</h2>
+
+
+<div class="paragraph"> </div>
+
+    Using these operations, we can define the main functor. For this,
+    we need to prove that the provided operations do indeed satisfy
+    the weak set interface. This is mostly straightforward and
+    unsurprising. The only interesting part is that removing
+    duplicates from a sorted list behaves as expected. This has
+    however already been proved in module <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted"><span class="id" title="module">RemoveDupsFromSorted</span></a></span>.
+  
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <a id="Make" class="idref" href="#Make"><span class="id" title="module">Make</span></a> (<span class="id" title="var">E</span>:<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Orders.html#OrderedType"><span class="id" title="module">OrderedType</span></a>) &lt;: <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops"><span class="id" title="module">Ops</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Import</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.MX"><span class="id" title="module">MX</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab65"></a><h2 class="section">Proofs of set operation specifications.</h2>
+ Logical predicates 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.In" class="idref" href="#Make.In"><span class="id" title="definition">In</span></a> <a id="x:88" class="idref" href="#x:88"><span class="id" title="binder">x</span></a> (<a id="s:89" class="idref" href="#s:89"><span class="id" title="binder">s</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">SetoidList.InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:88"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:89"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="keyword">Instance</span> <a id="Make.In_compat" class="idref" href="#Make.In_compat"><span class="id" title="instance">In_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">repeat</span> <span class="id" title="tactic">red</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="var">H0</span>. <span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.Equal" class="idref" href="#Make.Equal"><span class="id" title="definition">Equal</span></a> <a id="s:90" class="idref" href="#s:90"><span class="id" title="binder">s</span></a> <a id="s':91" class="idref" href="#s':91"><span class="id" title="binder">s'</span></a> := <span class="id" title="keyword">∀</span> <a id="a:92" class="idref" href="#a:92"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a:92"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:90"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a:92"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':91"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.Subset" class="idref" href="#Make.Subset"><span class="id" title="definition">Subset</span></a> <a id="s:93" class="idref" href="#s:93"><span class="id" title="binder">s</span></a> <a id="s':94" class="idref" href="#s':94"><span class="id" title="binder">s'</span></a> := <span class="id" title="keyword">∀</span> <a id="a:95" class="idref" href="#a:95"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a:95"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:93"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a:95"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':94"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.Empty" class="idref" href="#Make.Empty"><span class="id" title="definition">Empty</span></a> <a id="s:96" class="idref" href="#s:96"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">∀</span> <a id="a:97" class="idref" href="#a:97"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a:97"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:96"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.For_all" class="idref" href="#Make.For_all"><span class="id" title="definition">For_all</span></a> (<a id="P:98" class="idref" href="#P:98"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="s:99" class="idref" href="#s:99"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">∀</span> <a id="x:100" class="idref" href="#x:100"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:100"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:99"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#P:98"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:100"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.Exists" class="idref" href="#Make.Exists"><span class="id" title="definition">Exists</span></a> (<a id="P:101" class="idref" href="#P:101"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="s:102" class="idref" href="#s:102"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:103" class="idref" href="#x:103"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:103"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:102"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#P:101"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:103"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="0782b721c578548b54e05a1c77141c22" class="idref" href="#0782b721c578548b54e05a1c77141c22"><span class="id" title="notation">&quot;</span></a>s  [=]  t" := (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="fd7d5d3d921683174e91bdd5e1caa682" class="idref" href="#fd7d5d3d921683174e91bdd5e1caa682"><span class="id" title="notation">&quot;</span></a>s  [&lt;=]  t" := (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="Make.eq" class="idref" href="#Make.eq"><span class="id" title="definition">eq</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.eq_equiv" class="idref" href="#Make.eq_equiv"><span class="id" title="lemma">eq_equiv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence"><span class="id" title="class">Equivalence</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.eq"><span class="id" title="definition">eq</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">constructor</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">done</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">constructor</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">constructor</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="var">H0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Specifications of set operators 
+</div>
+<div class="code">
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="Make.compatb" class="idref" href="#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.mem_spec" class="idref" href="#Make.mem_spec"><span class="id" title="lemma">mem_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:104" class="idref" href="#s:104"><span class="id" title="binder">s</span></a> <a id="x:105" class="idref" href="#x:105"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:105"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:104"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:105"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:104"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span> <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compare_eq_iff"><span class="id" title="lemma">MX.compare_eq_iff</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#E.compare"><span class="id" title="axiom">E.compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>; <span class="id" title="tactic">intuition</span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>; <span class="id" title="tactic">intuition</span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.subset_spec" class="idref" href="#Make.subset_spec"><span class="id" title="lemma">subset_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:106" class="idref" href="#s:106"><span class="id" title="binder">s</span></a> <a id="s':107" class="idref" href="#s':107"><span class="id" title="binder">s'</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:106"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':107"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:106"><span class="id" title="variable">s</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#fd7d5d3d921683174e91bdd5e1caa682"><span class="id" title="notation">[&lt;=]</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#s':107"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.subset"><span class="id" title="definition">subset</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#forallb_forall"><span class="id" title="lemma">forallb_forall</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Subset"><span class="id" title="definition">Subset</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">z</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_alt"><span class="id" title="lemma">InA_alt</span></a> [] <span class="id" title="var">x</span> [<span class="id" title="var">H_z_eq</span>] <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<span class="id" title="var">H</span> <span class="id" title="var">_</span> <span class="id" title="var">H_in</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem_spec"><span class="id" title="lemma">mem_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_replace</span> <span class="id" title="var">z</span> <span class="id" title="keyword">with</span> <span class="id" title="var">x</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">z</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem_spec"><span class="id" title="lemma">mem_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#In_InA"><span class="id" title="lemma">In_InA</span></a> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.equal_spec" class="idref" href="#Make.equal_spec"><span class="id" title="lemma">equal_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:108" class="idref" href="#s:108"><span class="id" title="binder">s</span></a> <a id="s':109" class="idref" href="#s':109"><span class="id" title="binder">s'</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:108"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':109"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:108"><span class="id" title="variable">s</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#0782b721c578548b54e05a1c77141c22"><span class="id" title="notation">[=]</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#s':109"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.equal"><span class="id" title="definition">equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">Bool.andb_true_iff</span></a> !<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.subset_spec"><span class="id" title="lemma">subset_spec</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Subset"><span class="id" title="definition">Subset</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">H1</span> <span class="id" title="var">H2</span>] <span class="id" title="var">a</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">H2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">a</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.eq_dec" class="idref" href="#Make.eq_dec"><span class="id" title="lemma">eq_dec</span></a> :  <span class="id" title="keyword">∀</span> <a id="x:110" class="idref" href="#x:110"><span class="id" title="binder">x</span></a> <a id="y:111" class="idref" href="#y:111"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.t"><span class="id" title="definition">t</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:110"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:111"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.eq"><span class="id" title="definition">eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:110"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:111"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">change</span> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">{</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}+{</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Specif.html#87727981cdc1579fef00b9d9c1d3b9da"><span class="id" title="notation">}</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">destruct</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.equal"><span class="id" title="definition">equal</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>) <span class="id" title="var">eqn</span>:<span class="id" title="var">H</span>; [<span class="id" title="tactic">left</span>|<span class="id" title="tactic">right</span>];<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> &lt;- <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.equal_spec"><span class="id" title="lemma">equal_spec</span></a>; <span class="id" title="tactic">congruence</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.empty_spec" class="idref" href="#Make.empty_spec"><span class="id" title="lemma">empty_spec</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.empty"><span class="id" title="definition">empty</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.empty"><span class="id" title="definition">empty</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>. <span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">a</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.is_empty_spec" class="idref" href="#Make.is_empty_spec"><span class="id" title="lemma">is_empty_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:112" class="idref" href="#s:112"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.is_empty"><span class="id" title="definition">is_empty</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:112"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:112"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.is_empty"><span class="id" title="definition">is_empty</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span>; <span class="id" title="tactic">split</span> ⇒ //. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">a</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>; <span class="id" title="var">contradiction</span> (<span class="id" title="var">H</span> <span class="id" title="var">a</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.add_spec" class="idref" href="#Make.add_spec"><span class="id" title="lemma">add_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:113" class="idref" href="#s:113"><span class="id" title="binder">s</span></a> <a id="x:114" class="idref" href="#x:114"><span class="id" title="binder">x</span></a> <a id="y:115" class="idref" href="#y:115"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:115"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.add"><span class="id" title="definition">add</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:114"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:113"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:115"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:114"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:115"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:113"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.add"><span class="id" title="definition">add</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.singleton_spec" class="idref" href="#Make.singleton_spec"><span class="id" title="lemma">singleton_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="x:116" class="idref" href="#x:116"><span class="id" title="binder">x</span></a> <a id="y:117" class="idref" href="#y:117"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:117"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.singleton"><span class="id" title="definition">singleton</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:116"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:117"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:116"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.singleton"><span class="id" title="definition">singleton</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.rev_filter_aux_spec" class="idref" href="#Make.rev_filter_aux_spec"><span class="id" title="lemma">rev_filter_aux_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:118" class="idref" href="#s:118"><span class="id" title="binder">s</span></a> <a id="acc:119" class="idref" href="#acc:119"><span class="id" title="binder">acc</span></a> <a id="x:120" class="idref" href="#x:120"><span class="id" title="binder">x</span></a> <a id="f:121" class="idref" href="#f:121"><span class="id" title="binder">f</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:121"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:120"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:119"><span class="id" title="variable">acc</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:121"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:118"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:120"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:118"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:121"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:120"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:120"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#acc:119"><span class="id" title="variable">acc</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">acc</span> <span class="id" title="var">x</span> <span class="id" title="var">f</span> <span class="id" title="var">H_compat</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">x</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span> <span class="id" title="var">s'</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">acc</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f</span> <span class="id" title="var">y</span>) ⇒ <span class="id" title="var">H_fy</span>; <span class="id" title="tactic">rewrite</span> !<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>; <span class="id" title="tactic">intuition</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">first</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">left</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_replace</span> <span class="id" title="var">x</span> <span class="id" title="keyword">with</span> <span class="id" title="var">y</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">contradict</span> <span class="id" title="var">H1</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_replace</span> <span class="id" title="var">x</span> <span class="id" title="keyword">with</span> <span class="id" title="var">y</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_fy</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.filter_spec" class="idref" href="#Make.filter_spec"><span class="id" title="lemma">filter_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:122" class="idref" href="#s:122"><span class="id" title="binder">s</span></a> <a id="x:123" class="idref" href="#x:123"><span class="id" title="binder">x</span></a> <a id="f:124" class="idref" href="#f:124"><span class="id" title="binder">f</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:124"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:123"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:124"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:122"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:123"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:122"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:124"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:123"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">f</span> <span class="id" title="var">H_compat</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter"><span class="id" title="definition">rev_filter</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux_spec"><span class="id" title="lemma">rev_filter_aux_spec</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.remove_spec" class="idref" href="#Make.remove_spec"><span class="id" title="lemma">remove_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:125" class="idref" href="#s:125"><span class="id" title="binder">s</span></a> <a id="x:126" class="idref" href="#x:126"><span class="id" title="binder">x</span></a> <a id="y:127" class="idref" href="#y:127"><span class="id" title="binder">y</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:127"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.remove"><span class="id" title="definition">remove</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:126"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:125"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:127"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:125"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:127"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:126"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.remove"><span class="id" title="definition">remove</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter"><span class="id" title="definition">rev_filter</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_compat</span> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> ((<span class="id" title="keyword">fun</span> <a id="y0:128" class="idref" href="#y0:128"><span class="id" title="binder">y0</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> ⇒<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.compare"><span class="id" title="axiom">E.compare</span></a> <span class="id" title="var">x</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y0:128"><span class="id" title="variable">y0</span></a> <span class="id" title="keyword">with</span><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>)). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_replace</span> <span class="id" title="var">x0</span> <span class="id" title="keyword">with</span> <span class="id" title="var">y0</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux_spec"><span class="id" title="lemma">rev_filter_aux_spec</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> → : (<a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <span class="id" title="var">y</span> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>; <span class="id" title="tactic">move</span> ⇒ ?; <span class="id" title="tactic">by</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Symmetric"><span class="id" title="method">Equivalence_Symmetric</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> -<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compare_eq_iff"><span class="id" title="lemma">compare_eq_iff</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#E.compare"><span class="id" title="axiom">E.compare</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">inversion</span> <span class="id" title="var">H0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">inversion</span> <span class="id" title="var">H0</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.union_spec" class="idref" href="#Make.union_spec"><span class="id" title="lemma">union_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:129" class="idref" href="#s:129"><span class="id" title="binder">s</span></a> <a id="s':130" class="idref" href="#s':130"><span class="id" title="binder">s'</span></a> <a id="x:131" class="idref" href="#x:131"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:131"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.union"><span class="id" title="definition">union</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:129"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':130"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:131"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:129"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:131"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':130"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.union"><span class="id" title="definition">union</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#rev_append_rev"><span class="id" title="lemma">rev_append_rev</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_app_iff"><span class="id" title="lemma">InA_app_iff</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_rev"><span class="id" title="lemma">InA_rev</span></a>; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.inter_spec" class="idref" href="#Make.inter_spec"><span class="id" title="lemma">inter_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:132" class="idref" href="#s:132"><span class="id" title="binder">s</span></a> <a id="s':133" class="idref" href="#s':133"><span class="id" title="binder">s'</span></a> <a id="x:134" class="idref" href="#x:134"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:134"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:132"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':133"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:134"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:132"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:134"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':133"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">have</span> <span class="id" title="var">H_compat</span> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> (<span class="id" title="keyword">fun</span> <a id="y:135" class="idref" href="#y:135"><span class="id" title="binder">y</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#y:135"><span class="id" title="variable">y</span></a> <span class="id" title="var">s'</span>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">red</span>; <span class="id" title="tactic">intros</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : ( <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">x0</span> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">y</span> <span class="id" title="var">s'</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>). {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">y</span> <span class="id" title="var">s'</span>), (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem"><span class="id" title="definition">mem</span></a> <span class="id" title="var">x0</span> <span class="id" title="var">s'</span>); <span class="id" title="tactic">intuition</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> !<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem_spec"><span class="id" title="lemma">mem_spec</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">setoid_replace</span> <span class="id" title="var">x0</span> <span class="id" title="keyword">with</span> <span class="id" title="var">y</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.inter"><span class="id" title="definition">inter</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux_spec"><span class="id" title="lemma">rev_filter_aux_spec</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.mem_spec"><span class="id" title="lemma">mem_spec</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.fold_spec" class="idref" href="#Make.fold_spec"><span class="id" title="lemma">fold_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:136" class="idref" href="#s:136"><span class="id" title="binder">s</span></a> (<a id="A:137" class="idref" href="#A:137"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:138" class="idref" href="#i:138"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#A:137"><span class="id" title="variable">A</span></a>) (<a id="f:139" class="idref" href="#f:139"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#A:137"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#A:137"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.fold"><span class="id" title="definition">fold</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:139"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:136"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#i:138"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:139"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:136"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#i:138"><span class="id" title="variable">i</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.elements_spec1" class="idref" href="#Make.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:140" class="idref" href="#s:140"><span class="id" title="binder">s</span></a> <a id="x:141" class="idref" href="#x:141"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:141"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:140"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:141"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:140"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.diff_spec" class="idref" href="#Make.diff_spec"><span class="id" title="lemma">diff_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:142" class="idref" href="#s:142"><span class="id" title="binder">s</span></a> <a id="s':143" class="idref" href="#s':143"><span class="id" title="binder">s'</span></a> <a id="x:144" class="idref" href="#x:144"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:144"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:142"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':143"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:144"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:142"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:144"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s':143"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.diff"><span class="id" title="definition">diff</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.fold_spec"><span class="id" title="lemma">fold_spec</span></a> -(<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_spec1"><span class="id" title="lemma">elements_spec1</span></a> <span class="id" title="var">s'</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <span class="id" title="var">s'</span>) <span class="id" title="keyword">as</span> [| <span class="id" title="var">y</span> <span class="id" title="var">ys</span> <span class="id" title="var">IH</span>] ⇒ <span class="id" title="var">s</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> /=; <span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.remove_spec"><span class="id" title="lemma">remove_spec</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">tauto</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.cardinal_spec" class="idref" href="#Make.cardinal_spec"><span class="id" title="lemma">cardinal_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:145" class="idref" href="#s:145"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:145"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:145"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.cardinal"><span class="id" title="definition">cardinal</span></a> //. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.for_all_spec" class="idref" href="#Make.for_all_spec"><span class="id" title="lemma">for_all_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:146" class="idref" href="#s:146"><span class="id" title="binder">s</span></a> <a id="f:147" class="idref" href="#f:147"><span class="id" title="binder">f</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:147"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.for_all"><span class="id" title="definition">for_all</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:147"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:146"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <a id="x:148" class="idref" href="#x:148"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:147"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:148"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:146"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span> <span class="id" title="var">H_compat</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.For_all"><span class="id" title="definition">For_all</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">_</span> <span class="id" title="var">x</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span>) ⇒ <span class="id" title="var">H_fx</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">x'</span> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ → //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span> <span class="id" title="var">x'</span> <span class="id" title="var">H_in</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">suff</span> : <span class="id" title="var">f</span> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_fx</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (@<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.exists_spec" class="idref" href="#Make.exists_spec"><span class="id" title="lemma">exists_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:149" class="idref" href="#s:149"><span class="id" title="binder">s</span></a> <a id="f:150" class="idref" href="#f:150"><span class="id" title="binder">f</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:150"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.exists_"><span class="id" title="definition">exists_</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:150"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:149"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <a id="x:151" class="idref" href="#x:151"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:150"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:151"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:149"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span> <span class="id" title="var">H_compat</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Exists"><span class="id" title="definition">Exists</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x</span>] [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /=.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">case_eq</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span>) ⇒ <span class="id" title="var">H_fx</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ // <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> (@<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq_equiv"><span class="id" title="axiom">E.eq_equiv</span></a>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span>. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x'</span>] [<span class="id" title="var">H_in</span>] <span class="id" title="var">H_fx'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">∃</span> <span class="id" title="var">x'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">split</span> ⇒ //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">right</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [<span class="id" title="var">x'</span>] [] /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons"><span class="id" title="lemma">InA_cons</span></a> []. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ →.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H_fx</span> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">by</span> <span class="id" title="tactic">∃</span> <span class="id" title="var">x'</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} <br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.partition_aux_spec" class="idref" href="#Make.partition_aux_spec"><span class="id" title="lemma">partition_aux_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="a1:152" class="idref" href="#a1:152"><span class="id" title="binder">a1</span></a> <a id="a2:153" class="idref" href="#a2:153"><span class="id" title="binder">a2</span></a> <a id="s:154" class="idref" href="#s:154"><span class="id" title="binder">s</span></a> <a id="f:155" class="idref" href="#f:155"><span class="id" title="binder">f</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition_aux"><span class="id" title="definition">partition_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:152"><span class="id" title="variable">a1</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:153"><span class="id" title="variable">a2</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:155"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:154"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a1:152"><span class="id" title="variable">a1</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:155"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:154"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#a2:153"><span class="id" title="variable">a2</span></a> (<span class="id" title="keyword">fun</span> <a id="x:156" class="idref" href="#x:156"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#f:155"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:156"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:154"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#e6756e10c36f149b18b4a8741ed83079"><span class="id" title="notation">)</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">a1</span> <span class="id" title="var">a2</span> <span class="id" title="var">s</span> <span class="id" title="var">f</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : <span class="id" title="var">a1</span> <span class="id" title="var">a2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">s</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">x</span> <span class="id" title="var">xs</span> <span class="id" title="var">IH</span>]. {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition_aux"><span class="id" title="definition">partition_aux</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux"><span class="id" title="definition">rev_filter_aux</span></a> //.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;} {<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">a1</span> <span class="id" title="var">a2</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /= <span class="id" title="var">IH</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">case</span> (<span class="id" title="var">f</span> <span class="id" title="var">x</span>) ⇒ //.<br/>
+
+<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;}<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.partition_spec1" class="idref" href="#Make.partition_spec1"><span class="id" title="lemma">partition_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:157" class="idref" href="#s:157"><span class="id" title="binder">s</span></a> <a id="f:158" class="idref" href="#f:158"><span class="id" title="binder">f</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:158"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:158"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:157"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#0782b721c578548b54e05a1c77141c22"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:158"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:157"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">f</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition_aux_spec"><span class="id" title="lemma">partition_aux_spec</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter"><span class="id" title="definition">rev_filter</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.partition_spec2" class="idref" href="#Make.partition_spec2"><span class="id" title="lemma">partition_spec2</span></a> : <span class="id" title="keyword">∀</span> <a id="s:159" class="idref" href="#s:159"><span class="id" title="binder">s</span></a> <a id="f:160" class="idref" href="#f:160"><span class="id" title="binder">f</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:159"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetListWithDups.html#0782b721c578548b54e05a1c77141c22"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> (<span class="id" title="keyword">fun</span> <a id="x:161" class="idref" href="#x:161"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#f:160"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:161"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:159"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span> <span class="id" title="var">f</span> <span class="id" title="var">_</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition"><span class="id" title="definition">partition</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.partition_aux_spec"><span class="id" title="lemma">partition_aux_spec</span></a> /<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.filter"><span class="id" title="definition">filter</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.rev_filter"><span class="id" title="definition">rev_filter</span></a> //.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.choose_spec1" class="idref" href="#Make.choose_spec1"><span class="id" title="lemma">choose_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="s:162" class="idref" href="#s:162"><span class="id" title="binder">s</span></a> <a id="x:163" class="idref" href="#x:163"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:162"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:163"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:163"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:162"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ [] // <span class="id" title="var">y</span> <span class="id" title="var">s'</span> <span class="id" title="var">x</span> [-&gt;].<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_cons_hd"><span class="id" title="constructor">InA_cons_hd</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.RelationClasses.html#Equivalence_Reflexive"><span class="id" title="method">Equivalence_Reflexive</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.choose_spec2" class="idref" href="#Make.choose_spec2"><span class="id" title="lemma">choose_spec2</span></a> : <span class="id" title="keyword">∀</span> <a id="s:164" class="idref" href="#s:164"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:164"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:164"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">move</span> ⇒ [] // <span class="id" title="var">_</span> <span class="id" title="var">a</span>. <span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.In"><span class="id" title="definition">In</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA_nil"><span class="id" title="lemma">InA_nil</span></a> //. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.elements_dist_spec_full" class="idref" href="#Make.elements_dist_spec_full"><span class="id" title="lemma">elements_dist_spec_full</span></a> :<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">∀</span> <a id="s:165" class="idref" href="#s:165"><span class="id" title="binder">s</span></a>,<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Sorting.Sorted.html#Sorted"><span class="id" title="inductive">Sorted</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.lt"><span class="id" title="axiom">E.lt</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:165"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:165"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:166" class="idref" href="#x:166"><span class="id" title="binder">x</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:166"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:165"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:166"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:165"><span class="id" title="variable">s</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> ⇒ <span class="id" title="var">s</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> /<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">move</span> : (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.remove_dups_by_sortingA_spec"><span class="id" title="lemma">RDFS.remove_dups_by_sortingA_spec</span></a> <span class="id" title="var">s</span>).<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">simpl</span>.<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">firstorder</span>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.elements_dist_spec1" class="idref" href="#Make.elements_dist_spec1"><span class="id" title="lemma">elements_dist_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="x:167" class="idref" href="#x:167"><span class="id" title="binder">x</span></a> <a id="s:168" class="idref" href="#s:168"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:167"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:168"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#x:167"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements"><span class="id" title="definition">elements</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:168"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec_full"><span class="id" title="lemma">elements_dist_spec_full</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="Make.elements_dist_spec2w" class="idref" href="#Make.elements_dist_spec2w"><span class="id" title="lemma">elements_dist_spec2w</span></a> : <span class="id" title="keyword">∀</span> <a id="s:169" class="idref" href="#s:169"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetListWithDups.html#s:169"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">intros</span>; <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec_full"><span class="id" title="lemma">elements_dist_spec_full</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetListWithDups.html#Make"><span class="id" title="module">Make</span></a>.<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/html/MSetsExtra.MSetWithDups.html b/html/MSetsExtra.MSetWithDups.html
new file mode 100644
index 0000000..85666d1
--- /dev/null
+++ b/html/MSetsExtra.MSetWithDups.html
@@ -0,0 +1,254 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>MSetsExtra.MSetWithDups</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<h1 class="libtitle">Library MSetsExtra.MSetWithDups</h1>
+
+<div class="code">
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab66"></a><h1 class="section">Signature for weak sets which may contain duplicates</h1>
+
+
+<div class="paragraph"> </div>
+
+    The interface <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span> demands that <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a></span> returns a list
+    without duplicates and that the fold function iterates over this
+    result. Another potential problem is that the function <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.cardinal"><span class="id" title="definition">cardinal</span></a></span>
+    is supposed to return the length of the elements list.
+
+<div class="paragraph"> </div>
+
+    Therefore, implementations that store duplicates internally and for
+    which the fold function would visit elements multiple times are
+    ruled out.  Such implementations might be desirable for performance
+    reasons, though. One such (sometimes useful) example are unsorted
+    lists with duplicates. They have a very efficient insert and union
+    operation. If they are used in such a way that not too many
+    membership tests happen and that not too many duplicates
+    accumulate, it might be a very efficient datastructure.
+
+<div class="paragraph"> </div>
+
+    In order to allow efficient weak set implementations that use
+    duplicates internally, we provide new module types in this
+    file. There is <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a></span>, which is a proper subset of
+    <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>. It just removes the problematic properties of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a></span>
+    and <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.cardinal"><span class="id" title="definition">cardinal</span></a></span>.
+
+<div class="paragraph"> </div>
+
+    Since one is of course interested in specifying the cardinality
+    and in computing a list of elements without duplicates, there is
+    also an extension <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a></span> of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a></span>. This
+    extension introduces a new operation <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a></span>, which is a
+    version of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a></span> without duplicates. This allows to
+    specify <span class="inlinecode"><span class="id" title="var">cardinality</span></span> with respect to <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a></span>.
+
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#"><span class="id" title="library">Coq.MSets.MSetInterface</span></a>.<br/>
+<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <span class="id" title="library">mathcomp.ssreflect.ssreflect</span>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab67"></a><h2 class="section">WSetsOnWithDups</h2>
+
+
+<div class="paragraph"> </div>
+
+    The module type <span class="inlinecode"><span class="id" title="var">WSetOnWithDups</span></span> is a proper subset of <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>;
+    the problematic parameters <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.cardinal_spec"><span class="id" title="axiom">cardinal_spec</span></a></span> and <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec2w"><span class="id" title="lemma">elements_spec2w</span></a></span>
+    are missing. 
+
+<div class="paragraph"> </div>
+
+    We use this approach to be as noninvasive as possible. If we had the
+    liberty to modify the existing MSet library, it might be better to
+    define WSetsOnWithDups as below and define WSetOn by adding the two
+    extra parameters.
+ 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="WSetsOnWithDups" class="idref" href="#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#WOps"><span class="id" title="module">WOps</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.In" class="idref" href="#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/>
+&nbsp;&nbsp;#[<span class="id" title="var">local</span>] <span class="id" title="var">Declare</span> <span class="id" title="keyword">Instance</span> <a id="WSetsOnWithDups.In_compat" class="idref" href="#WSetsOnWithDups.In_compat"><span class="id" title="instance">In_compat</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#iff"><span class="id" title="definition">iff</span></a>) <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOnWithDups.Equal" class="idref" href="#WSetsOnWithDups.Equal"><span class="id" title="definition">Equal</span></a> <a id="s:2" class="idref" href="#s:2"><span class="id" title="binder">s</span></a> <a id="s':3" class="idref" href="#s':3"><span class="id" title="binder">s'</span></a> := <span class="id" title="keyword">∀</span> <a id="a:4" class="idref" href="#a:4"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#a:4"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:2"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#a:4"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s':3"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOnWithDups.Subset" class="idref" href="#WSetsOnWithDups.Subset"><span class="id" title="definition">Subset</span></a> <a id="s:5" class="idref" href="#s:5"><span class="id" title="binder">s</span></a> <a id="s':6" class="idref" href="#s':6"><span class="id" title="binder">s'</span></a> := <span class="id" title="keyword">∀</span> <a id="a:7" class="idref" href="#a:7"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:5"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#a:7"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s':6"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOnWithDups.Empty" class="idref" href="#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a id="s:8" class="idref" href="#s:8"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">∀</span> <a id="a:9" class="idref" href="#a:9"><span class="id" title="binder">a</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#a:9"><span class="id" title="variable">a</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:8"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOnWithDups.For_all" class="idref" href="#WSetsOnWithDups.For_all"><span class="id" title="definition">For_all</span></a> (<a id="P:10" class="idref" href="#P:10"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="s:11" class="idref" href="#s:11"><span class="id" title="binder">s</span></a> := <span class="id" title="keyword">∀</span> <a id="x:12" class="idref" href="#x:12"><span class="id" title="binder">x</span></a>, <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:12"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:11"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#P:10"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:12"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOnWithDups.Exists" class="idref" href="#WSetsOnWithDups.Exists"><span class="id" title="definition">Exists</span></a> (<a id="P:13" class="idref" href="#P:13"><span class="id" title="binder">P</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) <a id="s:14" class="idref" href="#s:14"><span class="id" title="binder">s</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="x:15" class="idref" href="#x:15"><span class="id" title="binder">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:15"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:14"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#P:13"><span class="id" title="variable">P</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:15"><span class="id" title="variable">x</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="7475f07315dc73e72314c0ce774a3805" class="idref" href="#7475f07315dc73e72314c0ce774a3805"><span class="id" title="notation">&quot;</span></a>s  [=]  t" := (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Equal"><span class="id" title="definition">Equal</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="872688c46747533055320201b074ca6f" class="idref" href="#872688c46747533055320201b074ca6f"><span class="id" title="notation">&quot;</span></a>s  [&lt;=]  t" := (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Subset"><span class="id" title="definition">Subset</span></a> <span class="id" title="var">s</span> <span class="id" title="var">t</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOnWithDups.eq" class="idref" href="#WSetsOnWithDups.eq"><span class="id" title="definition">eq</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> := <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Equal"><span class="id" title="definition">Equal</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Equalities.html#IsEq"><span class="id" title="module">IsEq</span></a>. </div>
+
+<div class="doc">
+<span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.eq"><span class="id" title="definition">eq</span></a></span> is obviously an equivalence, for subtyping only 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Equalities.html#HasEqDec"><span class="id" title="module">HasEqDec</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Section</span> <a id="WSetsOnWithDups.Spec" class="idref" href="#WSetsOnWithDups.Spec"><span class="id" title="section">Spec</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="WSetsOnWithDups.Spec.s" class="idref" href="#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a id="WSetsOnWithDups.Spec.s'" class="idref" href="#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>: <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.t"><span class="id" title="axiom">t</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="WSetsOnWithDups.Spec.x" class="idref" href="#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a id="WSetsOnWithDups.Spec.y" class="idref" href="#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Variable</span> <a id="WSetsOnWithDups.Spec.f" class="idref" href="#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Notation</span> <a id="WSetsOnWithDups.compatb" class="idref" href="#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#Proper"><span class="id" title="class">Proper</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Classes.Morphisms.html#8dc5652698a6e16f72dd37bd17d3b973"><span class="id" title="notation">==&gt;</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#eq"><span class="id" title="inductive">Logic.eq</span></a>)) (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.mem_spec" class="idref" href="#WSetsOnWithDups.mem_spec"><span class="id" title="axiom">mem_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.mem"><span class="id" title="axiom">mem</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.equal_spec" class="idref" href="#WSetsOnWithDups.equal_spec"><span class="id" title="axiom">equal_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.equal"><span class="id" title="axiom">equal</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#7475f07315dc73e72314c0ce774a3805"><span class="id" title="notation">[=]</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.subset_spec" class="idref" href="#WSetsOnWithDups.subset_spec"><span class="id" title="axiom">subset_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.subset"><span class="id" title="axiom">subset</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#872688c46747533055320201b074ca6f"><span class="id" title="notation">[&lt;=]</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.empty_spec" class="idref" href="#WSetsOnWithDups.empty_spec"><span class="id" title="axiom">empty_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.empty"><span class="id" title="axiom">empty</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.is_empty_spec" class="idref" href="#WSetsOnWithDups.is_empty_spec"><span class="id" title="axiom">is_empty_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.is_empty"><span class="id" title="axiom">is_empty</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.add_spec" class="idref" href="#WSetsOnWithDups.add_spec"><span class="id" title="axiom">add_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.add"><span class="id" title="axiom">add</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.remove_spec" class="idref" href="#WSetsOnWithDups.remove_spec"><span class="id" title="axiom">remove_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.remove"><span class="id" title="axiom">remove</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.singleton_spec" class="idref" href="#WSetsOnWithDups.singleton_spec"><span class="id" title="axiom">singleton_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.singleton"><span class="id" title="axiom">singleton</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y"><span class="id" title="variable">y</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.union_spec" class="idref" href="#WSetsOnWithDups.union_spec"><span class="id" title="axiom">union_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.union"><span class="id" title="axiom">union</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e25"><span class="id" title="notation">∨</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.inter_spec" class="idref" href="#WSetsOnWithDups.inter_spec"><span class="id" title="axiom">inter_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.inter"><span class="id" title="axiom">inter</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.diff_spec" class="idref" href="#WSetsOnWithDups.diff_spec"><span class="id" title="axiom">diff_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.diff"><span class="id" title="axiom">diff</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'"><span class="id" title="variable">s'</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.fold_spec" class="idref" href="#WSetsOnWithDups.fold_spec"><span class="id" title="axiom">fold_spec</span></a> : <span class="id" title="keyword">∀</span> (<a id="A:32" class="idref" href="#A:32"><span class="id" title="binder">A</span></a> : <span class="id" title="keyword">Type</span>) (<a id="i:33" class="idref" href="#i:33"><span class="id" title="binder">i</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#A:32"><span class="id" title="variable">A</span></a>) (<a id="f:34" class="idref" href="#f:34"><span class="id" title="binder">f</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elt"><span class="id" title="definition">elt</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#A:32"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#A:32"><span class="id" title="variable">A</span></a>),<br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.fold"><span class="id" title="axiom">fold</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#f:34"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#i:33"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.List.html#fold_left"><span class="id" title="definition">fold_left</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Program.Basics.html#flip"><span class="id" title="definition">flip</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#f:34"><span class="id" title="variable">f</span></a>) (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetWithDups.html#i:33"><span class="id" title="variable">i</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.filter_spec" class="idref" href="#WSetsOnWithDups.filter_spec"><span class="id" title="axiom">filter_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.filter"><span class="id" title="axiom">filter</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.for_all_spec" class="idref" href="#WSetsOnWithDups.for_all_spec"><span class="id" title="axiom">for_all_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.for_all"><span class="id" title="axiom">for_all</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.For_all"><span class="id" title="definition">For_all</span></a> (<span class="id" title="keyword">fun</span> <a id="x:37" class="idref" href="#x:37"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:37"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.exists_spec" class="idref" href="#WSetsOnWithDups.exists_spec"><span class="id" title="axiom">exists_spec</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.exists_"><span class="id" title="axiom">exists_</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Exists"><span class="id" title="definition">Exists</span></a> (<span class="id" title="keyword">fun</span> <a id="x:39" class="idref" href="#x:39"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:39"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.partition_spec1" class="idref" href="#WSetsOnWithDups.partition_spec1"><span class="id" title="axiom">partition_spec1</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.partition"><span class="id" title="axiom">partition</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetWithDups.html#7475f07315dc73e72314c0ce774a3805"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.filter"><span class="id" title="axiom">filter</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.partition_spec2" class="idref" href="#WSetsOnWithDups.partition_spec2"><span class="id" title="axiom">partition_spec2</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.compatb"><span class="id" title="abbreviation">compatb</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.partition"><span class="id" title="axiom">partition</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="MSetsExtra.MSetWithDups.html#7475f07315dc73e72314c0ce774a3805"><span class="id" title="notation">[=]</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.filter"><span class="id" title="axiom">filter</span></a> (<span class="id" title="keyword">fun</span> <a id="x:42" class="idref" href="#x:42"><span class="id" title="binder">x</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f"><span class="id" title="variable">f</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:42"><span class="id" title="variable">x</span></a>)) <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.elements_spec1" class="idref" href="#WSetsOnWithDups.elements_spec1"><span class="id" title="axiom">elements_spec1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.choose_spec1" class="idref" href="#WSetsOnWithDups.choose_spec1"><span class="id" title="axiom">choose_spec1</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.choose"><span class="id" title="axiom">choose</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In"><span class="id" title="axiom">In</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x"><span class="id" title="variable">x</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDups.choose_spec2" class="idref" href="#WSetsOnWithDups.choose_spec2"><span class="id" title="axiom">choose_spec2</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.choose"><span class="id" title="axiom">choose</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Empty"><span class="id" title="definition">Empty</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s"><span class="id" title="variable">s</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec"><span class="id" title="section">Spec</span></a>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab68"></a><h2 class="section">WSetsOnWithDupsExtra</h2>
+
+
+<div class="paragraph"> </div>
+
+    <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a></span> introduces <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a></span> in order to
+    specify cardinality and in order to get an operation similar to
+    the original behavior of <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetListWithDups.html#Ops.elements"><span class="id" title="definition">elements</span></a></span>. 
+</div>
+<div class="code">
+<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Type</span> <a id="WSetsOnWithDupsExtra" class="idref" href="#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups"><span class="id" title="module">WSetsOnWithDups</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+An operation for getting an elements list without duplicates 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDupsExtra.elements_dist" class="idref" href="#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> : <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.t"><span class="id" title="axiom">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elt"><span class="id" title="definition">elt</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDupsExtra.elements_dist_spec1" class="idref" href="#WSetsOnWithDupsExtra.elements_dist_spec1"><span class="id" title="axiom">elements_dist_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="x:48" class="idref" href="#x:48"><span class="id" title="binder">x</span></a> <a id="s:49" class="idref" href="#s:49"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:48"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:49"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:48"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:49"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDupsExtra.elements_dist_spec2w" class="idref" href="#WSetsOnWithDupsExtra.elements_dist_spec2w"><span class="id" title="axiom">elements_dist_spec2w</span></a> : <span class="id" title="keyword">∀</span> <a id="s:51" class="idref" href="#s:51"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:51"><span class="id" title="variable">s</span></a>).<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+Cardinality can then be specified with respect to <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a></span>. 
+</div>
+<div class="code">
+&nbsp;&nbsp;<span class="id" title="keyword">Parameter</span> <a id="WSetsOnWithDupsExtra.cardinal_spec" class="idref" href="#WSetsOnWithDupsExtra.cardinal_spec"><span class="id" title="axiom">cardinal_spec</span></a> : <span class="id" title="keyword">∀</span> <a id="s:53" class="idref" href="#s:53"><span class="id" title="binder">s</span></a>, <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.cardinal"><span class="id" title="axiom">cardinal</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:53"><span class="id" title="variable">s</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Datatypes.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist"><span class="id" title="axiom">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:53"><span class="id" title="variable">s</span></a>).<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a>.<br/>
+
+<br/>
+</div>
+
+<div class="doc">
+<a id="lab69"></a><h2 class="section">WSetOn to WSetsOnWithDupsExtra</h2>
+
+
+<div class="paragraph"> </div>
+
+    Since <span class="inlinecode"><a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a></span> is morally a weaker version of <span class="inlinecode"><span class="id" title="var">WSetsOn</span></span>
+    that allows the fold operation to visit elements multiple time, we can write then
+    following conversion. 
+</div>
+<div class="code">
+
+<br/>
+<span class="id" title="keyword">Module</span> <a id="WSetsOn_TO_WSetsOnWithDupsExtra" class="idref" href="#WSetsOn_TO_WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOn_TO_WSetsOnWithDupsExtra</span></a> (<span class="id" title="var">E</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Structures.Equalities.html#DecidableType"><span class="id" title="module">DecidableType</span></a>) (<span class="id" title="var">W</span> : <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.MSets.MSetInterface.html#WSetsOn"><span class="id" title="module">WSetsOn</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E"><span class="id" title="module">E</span></a>) &lt;: <br/>
+&nbsp;&nbsp;<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOnWithDupsExtra</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E"><span class="id" title="module">E</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Include</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#W"><span class="id" title="module">W</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Definition</span> <a id="WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist" class="idref" href="#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a> := <a class="idref" href="MSetsExtra.MSetWithDups.html#W.elements"><span class="id" title="axiom">W.elements</span></a>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1" class="idref" href="#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1"><span class="id" title="lemma">elements_dist_spec1</span></a> : <span class="id" title="keyword">∀</span> <a id="x:55" class="idref" href="#x:55"><span class="id" title="binder">x</span></a> <a id="s:56" class="idref" href="#s:56"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:55"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:56"><span class="id" title="variable">s</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Init.Logic.html#::type_scope:x_'&lt;-&gt;'_x"><span class="id" title="notation">↔</span></a><br/>
+&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#InA"><span class="id" title="inductive">InA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#x:55"><span class="id" title="variable">x</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements"><span class="id" title="axiom">elements</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:56"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="var">done</span>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Lemma</span> <a id="WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w" class="idref" href="#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w"><span class="id" title="lemma">elements_dist_spec2w</span></a> : <span class="id" title="keyword">∀</span> <a id="s:57" class="idref" href="#s:57"><span class="id" title="binder">s</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/V8.16.1/stdlib//Coq.Lists.SetoidList.html#NoDupA"><span class="id" title="inductive">NoDupA</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#E.eq"><span class="id" title="axiom">E.eq</span></a> (<a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist"><span class="id" title="definition">elements_dist</span></a> <a class="idref" href="MSetsExtra.MSetWithDups.html#s:57"><span class="id" title="variable">s</span></a>).<br/>
+&nbsp;&nbsp;<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_spec2w"><span class="id" title="axiom">elements_spec2w</span></a>. <span class="id" title="keyword">Qed</span>.<br/>
+
+<br/>
+<span class="id" title="keyword">End</span> <a class="idref" href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra"><span class="id" title="module">WSetsOn_TO_WSetsOnWithDupsExtra</span></a>.<br/>
+
+<br/>
+</div>
+</div>
+
+<div id="footer">
+<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
\ No newline at end of file
diff --git a/coqdoc/coqdoc.css b/html/coqdoc.css
similarity index 97%
rename from coqdoc/coqdoc.css
rename to html/coqdoc.css
index dbc930f..48096e5 100644
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+++ b/html/coqdoc.css
@@ -230,6 +230,10 @@ tr.infrulemiddle hr {
     color: rgb(40%,0%,40%);
 }
 
+.id[title="binder"] {
+    color: rgb(40%,0%,40%);
+}
+
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 }
@@ -327,3 +331,8 @@ ul.doclist {
     margin-top: 0em;
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+
+.code :target {
+    border: 2px solid #D4D4D4;
+    background-color: #e5eecc;
+}
diff --git a/html/index.html b/html/index.html
new file mode 100644
index 0000000..f5b604f
--- /dev/null
+++ b/html/index.html
@@ -0,0 +1,5391 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<table>
+<tr>
+<td>Global Index</td>
+<td><a href="index.html#global_A">A</a></td>
+<td><a href="index.html#global_B">B</a></td>
+<td><a href="index.html#global_C">C</a></td>
+<td>D</td>
+<td><a href="index.html#global_E">E</a></td>
+<td><a href="index.html#global_F">F</a></td>
+<td>G</td>
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+<td>J</td>
+<td>K</td>
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+<td>U</td>
+<td>V</td>
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+<td><a href="index.html#global_X">X</a></td>
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+<td>_</td>
+<td>other</td>
+<td>(2110 entries)</td>
+</tr>
+<tr>
+<td>Notation Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
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+<td>F</td>
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+<td>(5 entries)</td>
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+<tr>
+<td>Binder Index</td>
+<td><a href="index.html#binder_A">A</a></td>
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+<td>_</td>
+<td>other</td>
+<td>(1515 entries)</td>
+</tr>
+<tr>
+<td>Module Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#module_E">E</a></td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#module_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
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+<td>P</td>
+<td>Q</td>
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+<td>U</td>
+<td>V</td>
+<td><a href="index.html#module_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(43 entries)</td>
+</tr>
+<tr>
+<td>Variable Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
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+<td>(10 entries)</td>
+</tr>
+<tr>
+<td>Library Index</td>
+<td>A</td>
+<td>B</td>
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+<td>F</td>
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+<td>(4 entries)</td>
+</tr>
+<tr>
+<td>Lemma Index</td>
+<td><a href="index.html#lemma_A">A</a></td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
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+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#lemma_M">M</a></td>
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+<td><a href="index.html#lemma_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
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+<td><a href="index.html#lemma_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td><a href="index.html#lemma_Z">Z</a></td>
+<td>_</td>
+<td>other</td>
+<td>(257 entries)</td>
+</tr>
+<tr>
+<td>Axiom Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#axiom_E">E</a></td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#axiom_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td>M</td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td>R</td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#axiom_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(32 entries)</td>
+</tr>
+<tr>
+<td>Constructor Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td>M</td>
+<td>N</td>
+<td><a href="index.html#constructor_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#constructor_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(10 entries)</td>
+</tr>
+<tr>
+<td>Projection Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#projection_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#projection_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#projection_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(7 entries)</td>
+</tr>
+<tr>
+<td>Inductive Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td>M</td>
+<td>N</td>
+<td><a href="index.html#inductive_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#inductive_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(2 entries)</td>
+</tr>
+<tr>
+<td>Section Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#section_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#section_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#section_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(3 entries)</td>
+</tr>
+<tr>
+<td>Instance Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#instance_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#instance_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#instance_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(17 entries)</td>
+</tr>
+<tr>
+<td>Abbreviation Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#abbreviation_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td>R</td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#abbreviation_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(3 entries)</td>
+</tr>
+<tr>
+<td>Definition Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#definition_E">E</a></td>
+<td><a href="index.html#definition_F">F</a></td>
+<td>G</td>
+<td><a href="index.html#definition_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#definition_M">M</a></td>
+<td>N</td>
+<td><a href="index.html#definition_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#definition_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#definition_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(199 entries)</td>
+</tr>
+<tr>
+<td>Record Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#record_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#record_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#record_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(3 entries)</td>
+</tr>
+</table>
+<hr/>
+<h1>Global Index</h1>
+<a id="global_A"></a><h2>A </h2>
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+<a href="MSetsExtra.MSetListWithDups.html#A:137">A:137</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:140">A:140</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:147">A:147</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:158">A:158</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:175">A:175</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:190">A:190</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:20">A:20</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:201">A:201</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:218">A:218</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:27">A:27</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:275">a:275</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:278">a:278</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:280">a:280</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:3">A:3</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:30">A:30</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:302">A:302</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:308">A:308</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:31">A:31</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:315">A:315</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:316">A:316</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#A:32">A:32</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:320">A:320</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:324">A:324</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:329">A:329</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:336">A:336</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:341">A:341</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:343">A:343</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:349">A:349</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:35">A:35</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:351">A:351</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:354">A:354</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:357">A:357</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:359">A:359</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:36">A:36</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:369">A:369</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:37">A:37</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:371">A:371</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:382">A:382</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:39">A:39</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:394">A:394</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#a:4">a:4</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:54">A:54</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:58">A:58</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:63">A:63</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:63">A:63</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:65">A:65</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:67">A:67</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:68">A:68</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:7">A:7</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#a:7">a:7</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:817">A:817</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:9">A:9</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#a:9">a:9</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:91">A:91</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#a:92">a:92</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:934">a:934</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:937">a:937</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:939">a:939</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#a:95">a:95</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:97">A:97</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#a:97">a:97</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:970">A:970</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:985">a:985</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:986">a:986</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:987">a:987</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:988">a:988</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#A:99">A:99</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#A:994">A:994</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#a:997">a:997</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_B"></a><h2>B </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#base:309">base:309</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#base:350">base:350</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#base:358">base:358</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#base:366">base:366</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#bs:333">bs:333</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#bs:337">bs:337</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:100">B:100</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:103">B:103</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:127">B:127</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:13">B:13</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:159">B:159</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:21">B:21</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:212">B:212</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:227">B:227</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:228">B:228</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#b:232">b:232</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#b:236">b:236</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:237">B:237</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#b:240">b:240</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:253">B:253</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:268">B:268</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:28">B:28</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:287">B:287</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:290">B:290</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:294">B:294</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:297">B:297</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:303">B:303</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:317">B:317</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:32">B:32</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:330">B:330</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:342">B:342</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:344">B:344</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:352">B:352</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:360">B:360</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#B:37">B:37</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:372">B:372</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:383">B:383</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:395">B:395</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#B:60">B:60</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#B:92">B:92</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="global_C"></a><h2>C </h2>
+<a href="MSetsExtra.MSetIntervals.html#cur:178">cur:178</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#cur:182">cur:182</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#cur:858">cur:858</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#cx:306">cx:306</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#c_f:900">c_f:900</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#c_t:898">c_t:898</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#c1:298">c1:298</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#c1:399">c1:399</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#c1:403">c1:403</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
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+<a href="MSetsExtra.MSetListWithDups.html#f:44">f:44</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#f:48">f:48</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:55">f:55</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#f:61">f:61</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:64">f:64</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:69">f:69</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#f:70">f:70</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#f:71">f:71</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#f:75">f:75</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#f:81">f:81</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:819">f:819</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:824">f:824</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:828">f:828</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:854">f:854</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:861">f:861</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:867">f:867</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:876">f:876</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:882">f:882</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:889">f:889</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:894">f:894</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:903">f:903</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:910">f:910</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:913">f:913</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:917">f:917</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:920">f:920</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:923">f:923</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:925">f:925</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:971">f:971</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:974">f:974</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:976">f:976</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:978">f:978</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:980">f:980</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#f:996">f:996</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort">HasFoldWithAbort</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps">HasFoldWithAbortOps</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort_spec">HasFoldWithAbortOps.choose_with_abort_spec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort">HasFoldWithAbortOps.choose_with_abort</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort_spec">HasFoldWithAbortOps.exists_with_abort_spec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort">HasFoldWithAbortOps.exists_with_abort</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort_spec">HasFoldWithAbortOps.filter_with_abort_spec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort">HasFoldWithAbortOps.filter_with_abort</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort">HasFoldWithAbortOps.foldWithAbort</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt">HasFoldWithAbortOps.foldWithAbortGt</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt">HasFoldWithAbortOps.foldWithAbortGtLt</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec">HasFoldWithAbortOps.foldWithAbortGtLtSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec">HasFoldWithAbortOps.foldWithAbortGtSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtSpec_Equal</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal">HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec">HasFoldWithAbortOps.foldWithAbortSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg">HasFoldWithAbortOps.FoldWithAbortSpecArg</a> [record, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred">HasFoldWithAbortOps.foldWithAbortSpecArgsForPred</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec_Equal">HasFoldWithAbortOps.foldWithAbortSpec_Equal</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort_spec">HasFoldWithAbortOps.forall_with_abort_spec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort">HasFoldWithAbortOps.forall_with_abort</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P'">HasFoldWithAbortOps.fwasa_P'</a> [projection, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt">HasFoldWithAbortOps.fwasa_f_gt</a> [projection, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt">HasFoldWithAbortOps.fwasa_f_lt</a> [projection, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre">HasFoldWithAbortOps.fwasa_f_pre</a> [projection, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecompute">HasFoldWithAbort.foldWithAbortPrecompute</a> [axiom, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecomputeSpec">HasFoldWithAbort.foldWithAbortPrecomputeSpec</a> [axiom, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Heq:62">Heq:62</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs':580">Hs':580</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs':629">Hs':629</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs':681">Hs':681</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs':728">Hs':728</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs':743">Hs':743</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs':748">Hs':748</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:268">Hs:268</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:480">Hs:480</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:483">Hs:483</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:544">Hs:544</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:547">Hs:547</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:579">Hs:579</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:628">Hs:628</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:680">Hs:680</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:690">Hs:690</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:697">Hs:697</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:727">Hs:727</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:731">Hs:731</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:735">Hs:735</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:737">Hs:737</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:742">Hs:742</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:747">Hs:747</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:780">Hs:780</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:785">Hs:785</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:789">Hs:789</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:793">Hs:793</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:810">Hs:810</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:814">Hs:814</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:825">Hs:825</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Hs:829">Hs:829</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H0:575">H0:575</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H0:624">H0:624</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H0:676">H0:676</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H1:313">H1:313</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H2:314">H2:314</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:540">H:540</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:574">H:574</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:59">H:59</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:623">H:623</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:675">H:675</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:693">H:693</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:883">H:883</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:918">H:918</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:921">H:921</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_I"></a><h2>I </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:34">In:34</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:6">In:6</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:62">In:62</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:96">In:96</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':105">i':105</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':13">i':13</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':319">i':319</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':41">i':41</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':69">i':69</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i1:86">i1:86</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i2:87">i2:87</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:104">i:104</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:12">i:12</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#i:138">i:138</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:160">i:160</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:176">i:176</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:191">i:191</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:202">i:202</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:231">i:231</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:318">i:318</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#i:33">i:33</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:40">i:40</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:407">i:407</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#i:63">i:63</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:68">i:68</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:818">i:818</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:91">i:91</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:995">i:995</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_L"></a><h2>L </h2>
+<a href="MSetsExtra.MSetIntervals.html#ls:290">ls:290</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#lt:33">lt:33</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#lt:61">lt:61</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#lt:95">lt:95</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l':24">l':24</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l':30">l':30</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:345">l1:345</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:425">l1:425</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:707">l1:707</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:715">l1:715</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:346">l2:346</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:426">l2:426</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:710">l2:710</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:718">l2:718</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:10">l:10</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:103">l:103</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:115">l:115</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:119">l:119</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:134">l:134</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:14">l:14</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:16">l:16</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:17">l:17</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:23">l:23</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:24">l:24</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:254">l:254</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:258">l:258</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:260">l:260</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:28">l:28</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:29">l:29</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:30">l:30</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:339">l:339</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:34">l:34</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:34">l:34</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:367">l:367</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:39">l:39</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:427">l:427</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:50">l:50</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:551">l:551</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:554">l:554</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:698">l:698</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:701">l:701</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:75">l:75</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:849">l:849</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Make">Make</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA">MakeAVLSetsWithFoldA</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute">MakeAVLSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA">MakeGenTreeFoldA</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_RawSpec">MakeGenTreeFoldA.foldWithAbort_RawSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw">MakeGenTreeFoldA.foldWithAbort_Raw</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA">MakeListSetsWithFoldA</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute">MakeListSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortRaw">MakeListSetsWithFoldA.foldWithAbortRaw</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA">MakeRBTSetsWithFoldA</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute">MakeRBTSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA">MakeWeakListSetsWithFoldA</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.E">MakeWeakListSetsWithFoldA.E</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute">MakeWeakListSetsWithFoldA.foldWithAbortPrecompute</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [lemma, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortRaw">MakeWeakListSetsWithFoldA.foldWithAbortRaw</a> [definition, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.Raw">MakeWeakListSetsWithFoldA.Raw</a> [module, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.add_spec">Make.add_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.cardinal_spec">Make.cardinal_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.choose_spec2">Make.choose_spec2</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.choose_spec1">Make.choose_spec1</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.compatb">Make.compatb</a> [abbreviation, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.diff_spec">Make.diff_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec2w">Make.elements_dist_spec2w</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec1">Make.elements_dist_spec1</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec_full">Make.elements_dist_spec_full</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_spec1">Make.elements_spec1</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Empty">Make.Empty</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.empty_spec">Make.empty_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.eq">Make.eq</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Equal">Make.Equal</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.equal_spec">Make.equal_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.eq_dec">Make.eq_dec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.eq_equiv">Make.eq_equiv</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Exists">Make.Exists</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.exists_spec">Make.exists_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.filter_spec">Make.filter_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.fold_spec">Make.fold_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.for_all_spec">Make.for_all_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.For_all">Make.For_all</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.In">Make.In</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.inter_spec">Make.inter_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.In_compat">Make.In_compat</a> [instance, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.is_empty_spec">Make.is_empty_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.mem_spec">Make.mem_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.partition_spec2">Make.partition_spec2</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.partition_spec1">Make.partition_spec1</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.partition_aux_spec">Make.partition_aux_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.remove_spec">Make.remove_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux_spec">Make.rev_filter_aux_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.singleton_spec">Make.singleton_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Subset">Make.Subset</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.subset_spec">Make.subset_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.union_spec">Make.union_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#fd7d5d3d921683174e91bdd5e1caa682">_ [&lt;=] _</a> [notation, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#0782b721c578548b54e05a1c77141c22">_ [=] _</a> [notation, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html">MSetFoldWithAbort</a> [library]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals">MSetIntervals</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html">MSetIntervals</a> [library]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervalsN">MSetIntervalsN</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervalsNat">MSetIntervalsNat</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervalsZ">MSetIntervalsZ</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.add">MSetIntervals.add</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.add_spec">MSetIntervals.add_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal">MSetIntervals.cardinal</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal_spec">MSetIntervals.cardinal_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose">MSetIntervals.choose</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec3">MSetIntervals.choose_spec3</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec2">MSetIntervals.choose_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec1">MSetIntervals.choose_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare">MSetIntervals.compare</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare_spec">MSetIntervals.compare_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb">MSetIntervals.compatb</a> [abbreviation, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff">MSetIntervals.diff</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff_spec">MSetIntervals.diff_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.E">MSetIntervals.E</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements">MSetIntervals.elements</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec2">MSetIntervals.elements_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec2w">MSetIntervals.elements_spec2w</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec1">MSetIntervals.elements_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt">MSetIntervals.elt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty">MSetIntervals.empty</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty">MSetIntervals.Empty</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty_spec">MSetIntervals.empty_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq">MSetIntervals.eq</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal">MSetIntervals.equal</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal">MSetIntervals.Equal</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_spec">MSetIntervals.equal_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq_dec">MSetIntervals.eq_dec</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq_equiv">MSetIntervals.eq_equiv</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Exists">MSetIntervals.Exists</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_spec">MSetIntervals.exists_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_">MSetIntervals.exists_</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter">MSetIntervals.filter</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter_spec">MSetIntervals.filter_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold">MSetIntervals.fold</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold_spec">MSetIntervals.fold_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all_spec">MSetIntervals.for_all_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all">MSetIntervals.for_all</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.For_all">MSetIntervals.For_all</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.In">MSetIntervals.In</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter">MSetIntervals.inter</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter_spec">MSetIntervals.inter_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.In_compat">MSetIntervals.In_compat</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty_spec">MSetIntervals.is_empty_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty">MSetIntervals.is_empty</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_ok">MSetIntervals.is_ok</a> [projection, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt">MSetIntervals.lt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt_compat">MSetIntervals.lt_compat</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt_strorder">MSetIntervals.lt_strorder</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec3">MSetIntervals.max_elt_spec3</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec2">MSetIntervals.max_elt_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec1">MSetIntervals.max_elt_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt">MSetIntervals.max_elt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem">MSetIntervals.mem</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem_spec">MSetIntervals.mem_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec3">MSetIntervals.min_elt_spec3</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec2">MSetIntervals.min_elt_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec1">MSetIntervals.min_elt_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt">MSetIntervals.min_elt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition">MSetIntervals.partition</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition_spec2">MSetIntervals.partition_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition_spec1">MSetIntervals.partition_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Raw">MSetIntervals.Raw</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove">MSetIntervals.remove</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove_spec">MSetIntervals.remove_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton">MSetIntervals.singleton</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton_spec">MSetIntervals.singleton_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec">MSetIntervals.Spec</a> [section, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f">MSetIntervals.Spec.f</a> [variable, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s">MSetIntervals.Spec.s</a> [variable, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'">MSetIntervals.Spec.s'</a> [variable, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x">MSetIntervals.Spec.x</a> [variable, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y">MSetIntervals.Spec.y</a> [variable, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset">MSetIntervals.subset</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Subset">MSetIntervals.Subset</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset_spec">MSetIntervals.subset_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.t">MSetIntervals.t</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.this">MSetIntervals.this</a> [projection, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.t_">MSetIntervals.t_</a> [record, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.union">MSetIntervals.union</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.union_spec">MSetIntervals.union_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html">MSetListWithDups</a> [library]<br/>
+<a href="MSetsExtra.MSetWithDups.html">MSetWithDups</a> [library]<br/>
+<br/><br/><a id="global_N"></a><h2>N </h2>
+<a href="MSetsExtra.MSetIntervals.html#NoDupA_map">NoDupA_map</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#NOP">NOP</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:1011">n:1011</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:1018">n:1018</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:2">n:2</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:289">n:289</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:4">n:4</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:52">n:52</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_O"></a><h2>O </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Ops">Ops</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops">Ops</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.acc_pred">Ops.acc_pred</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.add">Ops.add</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.add">Ops.add</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.addZ">Ops.addZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.addZ_aux">Ops.addZ_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.add_list">Ops.add_list</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.cardinal">Ops.cardinal</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.cardinal">Ops.cardinal</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.cardinalN">Ops.cardinalN</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.choose">Ops.choose</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.choose">Ops.choose</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.compare">Ops.compare</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.diff">Ops.diff</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.diff">Ops.diff</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.diff_aux2">Ops.diff_aux2</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.diff_aux">Ops.diff_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.elements">Ops.elements</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.elements">Ops.elements</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.elementsZ">Ops.elementsZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.elements_dist">Ops.elements_dist</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.elt">Ops.elt</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.elt">Ops.elt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.empty">Ops.empty</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.empty">Ops.empty</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.equal">Ops.equal</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.equal">Ops.equal</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.exists_">Ops.exists_</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.exists_">Ops.exists_</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.filter">Ops.filter</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filter">Ops.filter</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ">Ops.filterZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_aux">Ops.filterZ_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_single">Ops.filterZ_single</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_single_aux">Ops.filterZ_single_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_fold_fun">Ops.filterZ_fold_fun</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.fold">Ops.fold</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold">Ops.fold</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ">Ops.fold_elementsZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_single">Ops.fold_elementsZ_single</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_aux">Ops.fold_elementsZ_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.for_all">Ops.for_all</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.for_all">Ops.for_all</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.from_elements">Ops.from_elements</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch">Ops.ICR_after_touch</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_after">Ops.ICR_after</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2">Ops.ICR_subsume_2</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1">Ops.ICR_subsume_1</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_equal">Ops.ICR_equal</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_after">Ops.ICR_overlap_after</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_before">Ops.ICR_overlap_before</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch">Ops.ICR_before_touch</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_before">Ops.ICR_before</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.insert_intervalZ_guarded">Ops.insert_intervalZ_guarded</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin">Ops.insert_interval_begin</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.inter">Ops.inter</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.inter">Ops.inter</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.interval_1_compare">Ops.interval_1_compare</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.interval_compare">Ops.interval_compare</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.interval_compare_result">Ops.interval_compare_result</a> [inductive, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.inter_aux2">Ops.inter_aux2</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.inter_aux">Ops.inter_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.is_empty">Ops.is_empty</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.is_empty">Ops.is_empty</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.max_elt">Ops.max_elt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.max_eltZ">Ops.max_eltZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.mem">Ops.mem</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.mem">Ops.mem</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.memZ">Ops.memZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.merge_interval_size">Ops.merge_interval_size</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.min_elt">Ops.min_elt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.min_eltZ">Ops.min_eltZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.MX">Ops.MX</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.partition">Ops.partition</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partition">Ops.partition</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ">Ops.partitionZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_aux">Ops.partitionZ_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_single">Ops.partitionZ_single</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_single_aux">Ops.partitionZ_single_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_fun">Ops.partitionZ_fold_fun</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip">Ops.partitionZ_fold_skip</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_insert">Ops.partitionZ_fold_insert</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.partition_aux">Ops.partition_aux</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.RDFS">Ops.RDFS</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.remove">Ops.remove</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.remove">Ops.remove</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.removeZ">Ops.removeZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.removeZ_aux">Ops.removeZ_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.remove_list">Ops.remove_list</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter">Ops.rev_filter</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter_aux">Ops.rev_filter_aux</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.singleton">Ops.singleton</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.singleton">Ops.singleton</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.singleton_alt_def">Ops.singleton_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.subset">Ops.subset</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.subset">Ops.subset</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.t">Ops.t</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.t">Ops.t</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.union">Ops.union</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.union">Ops.union</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.union_aux">Ops.union_aux</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_P"></a><h2>P </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#pre_x:392">pre_x:392</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#pre_x:369">pre_x:369</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#P:10">P:10</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#P:101">P:101</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:112">P:112</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#P:13">P:13</a> [binder, in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:18">P:18</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:220">P:220</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:231">p:231</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:236">p:236</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:239">P:239</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:242">p:242</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:247">p:247</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:257">p:257</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:265">p:265</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:270">P:270</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:281">P:281</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:284">P:284</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:292">P:292</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:300">P:300</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#p:319">p:319</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:326">P:326</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:46">P:46</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#p:53">p:53</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:75">P:75</a> [binder, in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:940">P:940</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:943">P:943</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#P:98">P:98</a> [binder, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#p:982">p:982</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="global_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw">Raw</a> [module, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_invariant">Raw.addZ_invariant</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_InZ">Raw.addZ_InZ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_alt_def">Raw.addZ_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux_alt_def">Raw.addZ_aux_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_list_spec">Raw.add_list_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_list_ok">Raw.add_list_ok</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_spec">Raw.add_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_ok">Raw.add_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.cardinalN_spec">Raw.cardinalN_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.cardinal_spec">Raw.cardinal_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.choose_spec3">Raw.choose_spec3</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.choose_spec2">Raw.choose_spec2</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.choose_spec1">Raw.choose_spec1</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_spec">Raw.compare_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_antisym">Raw.compare_antisym</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_cons">Raw.compare_eq_Lt_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_nil_r">Raw.compare_eq_Lt_nil_r</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_nil_l">Raw.compare_eq_Lt_nil_l</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Eq">Raw.compare_eq_Eq</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_spec">Raw.diff_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_ok">Raw.diff_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_invariant">Raw.diff_invariant</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_InZ">Raw.diff_InZ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2_props">Raw.diff_aux2_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_props">Raw.diff_aux_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_alt_def">Raw.diff_aux_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_sorted">Raw.elementsZ_sorted</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_sorted">Raw.elementsZ_single_sorted</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_spec2w">Raw.elementsZ_spec2w</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_insert_intervalZ_guarded">Raw.elementsZ_insert_intervalZ_guarded</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_app">Raw.elementsZ_app</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons">Raw.elementsZ_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front">Raw.elementsZ_single_succ_front</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_add">Raw.elementsZ_single_add</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ">Raw.elementsZ_single_succ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base">Raw.elementsZ_single_base</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single">Raw.elementsZ_single</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil">Raw.elementsZ_nil</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_sorted">Raw.elements_sorted</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_spec2w">Raw.elements_spec2w</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_spec1">Raw.elements_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_cons">Raw.elements_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_single">Raw.elements_single</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_nil">Raw.elements_nil</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Empty">Raw.Empty</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.empty_spec">Raw.empty_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.empty_spec'">Raw.empty_spec'</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.empty_ok">Raw.empty_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.encode_decode_eq">Raw.encode_decode_eq</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Equal">Raw.Equal</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.equal_spec">Raw.equal_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.equal_elementsZ">Raw.equal_elementsZ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.equal_alt_def">Raw.equal_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Exists">Raw.Exists</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.exists_spec">Raw.exists_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_props">Raw.filterZ_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux_props">Raw.filterZ_aux_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_props">Raw.filterZ_single_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux_props">Raw.filterZ_single_aux_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filter_spec">Raw.filter_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filter_ok">Raw.filter_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_spec">Raw.fold_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def">Raw.fold_elementsZ_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def_aux">Raw.fold_elementsZ_alt_def_aux</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_cons">Raw.fold_elementsZ_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_nil">Raw.fold_elementsZ_nil</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_alt_def">Raw.fold_elementsZ_single_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_app">Raw.fold_opt_app</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_list_cons">Raw.fold_opt_list_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_opt">Raw.fold_opt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_succ">Raw.fold_elementsZ_single_succ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_zero">Raw.fold_elementsZ_single_zero</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_pos">Raw.fold_elementsZ_single_pos</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux_irrel">Raw.fold_elementsZ_aux_irrel</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.ForNotations">Raw.ForNotations</a> [section, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.for_all_spec">Raw.for_all_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.For_all">Raw.For_all</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In">Raw.In</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec">Raw.insert_interval_begin_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded_rev_nil_app">Raw.insert_intervalZ_guarded_rev_nil_app</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded_app">Raw.insert_intervalZ_guarded_app</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_insert_intervalZ_guarded">Raw.interval_list_elements_greater_insert_intervalZ_guarded</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_insert_intervalZ_guarded">Raw.interval_list_invariant_insert_intervalZ_guarded</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_snoc_intro">Raw.interval_list_invariant_snoc_intro</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_iff">Raw.interval_list_invariant_app_iff</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_elim">Raw.interval_list_invariant_app_elim</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro">Raw.interval_list_invariant_app_intro</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_app_elim_1">Raw.interval_list_elements_greater_app_elim_1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_intro">Raw.interval_list_elements_greater_intro</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def">Raw.interval_list_elements_greater_alt2_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt_def">Raw.interval_list_elements_greater_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing">Raw.interval_list_invariant_sing</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons">Raw.interval_list_invariant_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_nil">Raw.interval_list_invariant_nil</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl">Raw.interval_list_elements_greater_impl</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons">Raw.interval_list_elements_greater_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_elim">Raw.interval_1_compare_elim</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_alt_def">Raw.interval_1_compare_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_swap">Raw.interval_compare_swap</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim">Raw.interval_compare_elim</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant">Raw.interval_list_invariant</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater">Raw.interval_list_elements_greater</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_spec">Raw.inter_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_ok">Raw.inter_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_invariant">Raw.inter_invariant</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_InZ">Raw.inter_InZ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2_props">Raw.inter_aux2_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_aux_props">Raw.inter_aux_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_aux_alt_def">Raw.inter_aux_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ">Raw.InZ</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_skip">Raw.InZ_partitionZ_fold_skip</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_insert">Raw.InZ_partitionZ_fold_insert</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_current_Some">Raw.InZ_partitionZ_fold_current_Some</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_insert_intervalZ_guarded">Raw.InZ_insert_intervalZ_guarded</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_In">Raw.InZ_In</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_dec">Raw.InZ_dec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev">Raw.InZ_rev</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_app">Raw.InZ_app</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons">Raw.InZ_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil">Raw.InZ_nil</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval">Raw.In_merge_interval</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_InZ">Raw.In_InZ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_alt_def">Raw.In_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd">Raw.In_elementsZ_single_hd</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_dec">Raw.In_elementsZ_single_dec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1">Raw.In_elementsZ_single1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single">Raw.In_elementsZ_single</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.IsOk">Raw.IsOk</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.IsOk_Ok">Raw.IsOk_Ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_empty_spec">Raw.is_empty_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list_rev">Raw.is_encoded_elems_list_rev</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list_app">Raw.is_encoded_elems_list_app</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list">Raw.is_encoded_elems_list</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.length_elementsZ_single">Raw.length_elementsZ_single</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.lt">Raw.lt</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.lt_Transitive">Raw.lt_Transitive</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.lt_Irreflexive">Raw.lt_Irreflexive</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_elt_spec3">Raw.max_elt_spec3</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_elt_spec2">Raw.max_elt_spec2</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_elt_spec1">Raw.max_elt_spec1</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec3">Raw.max_eltZ_spec3</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_eq_None">Raw.max_eltZ_eq_None</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec2">Raw.max_eltZ_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec1">Raw.max_eltZ_spec1</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.memZ_spec">Raw.memZ_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.mem_spec">Raw.mem_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_invariant">Raw.merge_interval_size_invariant</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_eq_max">Raw.merge_interval_size_eq_max</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_add">Raw.merge_interval_size_add</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0">Raw.merge_interval_size_neq_0</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec3">Raw.min_elt_spec3</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec2">Raw.min_elt_spec2</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec1">Raw.min_elt_spec1</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec3">Raw.min_eltZ_spec3</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec2">Raw.min_eltZ_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec1">Raw.min_eltZ_spec1</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater_equal">Raw.Nin_elements_greater_equal</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater">Raw.Nin_elements_greater</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.NoDupA_elementsZ_single">Raw.NoDupA_elementsZ_single</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.ok">Raw.ok</a> [projection, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok">Raw.Ok</a> [record, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.ok">Raw.ok</a> [constructor, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok">Raw.Ok</a> [inductive, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok_cons">Raw.Ok_cons</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok_nil">Raw.Ok_nil</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_alt_def">Raw.partitionZ_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_alt_def">Raw.partitionZ_aux_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_single_aux_alt_def">Raw.partitionZ_single_aux_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current">Raw.partitionZ_fold_current</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant_skip">Raw.partitionZ_aux_invariant_skip</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant_insert">Raw.partitionZ_aux_invariant_insert</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant">Raw.partitionZ_aux_invariant</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_spec2">Raw.partition_spec2</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_spec1">Raw.partition_spec1</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_ok2">Raw.partition_ok2</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_ok1">Raw.partition_ok1</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_alt_def">Raw.partition_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.removeZ_spec">Raw.removeZ_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.removeZ_interval_list_invariant">Raw.removeZ_interval_list_invariant</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.removeZ_alt_def">Raw.removeZ_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_list_spec">Raw.remove_list_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_list_ok">Raw.remove_list_ok</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_spec">Raw.remove_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_ok">Raw.remove_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.singleton_spec">Raw.singleton_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.singleton_ok">Raw.singleton_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Subset">Raw.Subset</a> [definition, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_spec">Raw.subset_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_props">Raw.subset_props</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux_before">Raw.subset_props_aux_before</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux">Raw.subset_props_aux</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def">Raw.subset_flatten_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_spec">Raw.union_spec</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_ok">Raw.union_ok</a> [instance, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_invariant">Raw.union_invariant</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_InZ">Raw.union_InZ</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def">Raw.union_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_aux_alt_def">Raw.union_aux_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_aux_flatten_alt_def">Raw.union_aux_flatten_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted">RemoveDupsFromSorted</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt">RemoveDupsFromSorted.eqb_eq_alt</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.MX">RemoveDupsFromSorted.MX</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA_spec">RemoveDupsFromSorted.remove_dups_by_sortingA_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA">RemoveDupsFromSorted.remove_dups_by_sortingA</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_spec">RemoveDupsFromSorted.remove_dups_from_sortedA_spec</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_hd">RemoveDupsFromSorted.remove_dups_from_sortedA_hd</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_alt</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA">RemoveDupsFromSorted.remove_dups_from_sortedA</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux">RemoveDupsFromSorted.remove_dups_from_sortedA_aux</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XSort">RemoveDupsFromSorted.XSort</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool">RemoveDupsFromSorted.XTotalLeBool</a> [module, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.le">RemoveDupsFromSorted.XTotalLeBool.le</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb">RemoveDupsFromSorted.XTotalLeBool.leb</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb_total">RemoveDupsFromSorted.XTotalLeBool.leb_total</a> [lemma, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t">RemoveDupsFromSorted.XTotalLeBool.t</a> [definition, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#c679ee482e47aa89e9818d26cb26e383">_ &lt;=? _</a> [notation, in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#rev_map_alt_def">rev_map_alt_def</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
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+<br/><br/><a id="global_T"></a><h2>T </h2>
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+<br/><br/><a id="global_W"></a><h2>W </h2>
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+<a href="MSetsExtra.MSetIntervals.html#Z_le_add_r">Z_le_add_r</a> [lemma, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:1">z:1</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:1004">z:1004</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:1005">z:1005</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:1012">z:1012</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:1019">z:1019</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:206">z:206</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:227">z:227</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:233">z:233</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:237">z:237</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:241">z:241</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:3">z:3</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:411">z:411</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:418">z:418</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:475">z:475</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:520">z:520</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:534">z:534</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:537">z:537</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:750">z:750</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:884">z:884</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:885">z:885</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:886">z:886</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:891">z:891</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:892">z:892</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:893">z:893</a> [binder, in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/>
+<h1>Notation Index</h1>
+<a id="notation_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetListWithDups.html#fd7d5d3d921683174e91bdd5e1caa682">_ [&lt;=] _</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#0782b721c578548b54e05a1c77141c22">_ [=] _</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<br/><br/><a id="notation_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetListWithDups.html#c679ee482e47aa89e9818d26cb26e383">_ &lt;=? _</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<br/><br/><a id="notation_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#872688c46747533055320201b074ca6f">_ [&lt;=] _</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
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+<br/><br/><hr/>
+<h1>Binder Index</h1>
+<a id="binder_A"></a><h2>A </h2>
+<a href="MSetsExtra.MSetIntervals.html#ab:347">ab:347</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ab:363">ab:363</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
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+<a href="MSetsExtra.MSetIntervals.html#acc_f:200">acc_f:200</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#acc_t:199">acc_t:199</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#acc_f:196">acc_f:196</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#acc_t:195">acc_t:195</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
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+<a href="MSetsExtra.MSetIntervals.html#acc':364">acc':364</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#acc:106">acc:106</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
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+<a href="MSetsExtra.MSetIntervals.html#acc:141">acc:141</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
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+<a href="MSetsExtra.MSetIntervals.html#H:918">H:918</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#H:921">H:921</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="binder_I"></a><h2>I </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:34">In:34</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:6">In:6</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:62">In:62</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#In:96">In:96</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':105">i':105</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':13">i':13</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':319">i':319</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':41">i':41</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i':69">i':69</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i1:86">i1:86</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i2:87">i2:87</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:104">i:104</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:12">i:12</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#i:138">i:138</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:160">i:160</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:176">i:176</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:191">i:191</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:202">i:202</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:231">i:231</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:318">i:318</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#i:33">i:33</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:40">i:40</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:407">i:407</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#i:63">i:63</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#i:68">i:68</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:818">i:818</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:91">i:91</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#i:995">i:995</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="binder_L"></a><h2>L </h2>
+<a href="MSetsExtra.MSetIntervals.html#ls:290">ls:290</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#lt:33">lt:33</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#lt:61">lt:61</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#lt:95">lt:95</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l':24">l':24</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l':30">l':30</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:345">l1:345</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:425">l1:425</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:707">l1:707</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l1:715">l1:715</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:346">l2:346</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:426">l2:426</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:710">l2:710</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l2:718">l2:718</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:10">l:10</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:103">l:103</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:115">l:115</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:119">l:119</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:134">l:134</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:14">l:14</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:16">l:16</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:17">l:17</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:23">l:23</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:24">l:24</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:254">l:254</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:258">l:258</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:260">l:260</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:28">l:28</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:29">l:29</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:30">l:30</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:339">l:339</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#l:34">l:34</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:34">l:34</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:367">l:367</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:39">l:39</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:427">l:427</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:50">l:50</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:551">l:551</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:554">l:554</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:698">l:698</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:701">l:701</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:75">l:75</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#l:849">l:849</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="binder_N"></a><h2>N </h2>
+<a href="MSetsExtra.MSetIntervals.html#n:1011">n:1011</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:1018">n:1018</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:2">n:2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:289">n:289</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:4">n:4</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#n:52">n:52</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="binder_P"></a><h2>P </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#pre_x:392">pre_x:392</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#pre_x:369">pre_x:369</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#P:10">P:10</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#P:101">P:101</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:112">P:112</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#P:13">P:13</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:18">P:18</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:220">P:220</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:231">p:231</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:236">p:236</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:239">P:239</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:242">p:242</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:247">p:247</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:257">p:257</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#p:265">p:265</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:270">P:270</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:281">P:281</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:284">P:284</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:292">P:292</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:300">P:300</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#p:319">p:319</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:326">P:326</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:46">P:46</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#p:53">p:53</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#P:75">P:75</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:940">P:940</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#P:943">P:943</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#P:98">P:98</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#p:982">p:982</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="binder_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#r:859">r:859</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="binder_S"></a><h2>S </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':114">st':114</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':120">st':120</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':124">st':124</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':20">st':20</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':26">st':26</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':328">st':328</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':334">st':334</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':338">st':338</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':48">st':48</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':54">st':54</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':77">st':77</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#st':83">st':83</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
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+<a href="MSetsExtra.MSetIntervals.html#z:1012">z:1012</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:1019">z:1019</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:206">z:206</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:227">z:227</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:233">z:233</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:237">z:237</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:241">z:241</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:3">z:3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:411">z:411</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:418">z:418</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:475">z:475</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:520">z:520</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:534">z:534</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:537">z:537</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:750">z:750</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:884">z:884</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:885">z:885</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:886">z:886</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:891">z:891</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:892">z:892</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#z:893">z:893</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/>
+<h1>Module Index</h1>
+<a id="module_E"></a><h2>E </h2>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode">ElementEncode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN">ElementEncodeN</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat">ElementEncodeNat</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.E">ElementEncodeNat.E</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN.E">ElementEncodeN.E</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ">ElementEncodeZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.E">ElementEncodeZ.E</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode.E">ElementEncode.E</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="module_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort">HasFoldWithAbort</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps">HasFoldWithAbortOps</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="module_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Make">Make</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA">MakeAVLSetsWithFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA">MakeGenTreeFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA">MakeListSetsWithFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA">MakeRBTSetsWithFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA">MakeWeakListSetsWithFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.E">MakeWeakListSetsWithFoldA.E</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.Raw">MakeWeakListSetsWithFoldA.Raw</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals">MSetIntervals</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervalsN">MSetIntervalsN</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervalsNat">MSetIntervalsNat</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervalsZ">MSetIntervalsZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.E">MSetIntervals.E</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Raw">MSetIntervals.Raw</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="module_N"></a><h2>N </h2>
+<a href="MSetsExtra.MSetIntervals.html#NOP">NOP</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="module_O"></a><h2>O </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Ops">Ops</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops">Ops</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.MX">Ops.MX</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.RDFS">Ops.RDFS</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<br/><br/><a id="module_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw">Raw</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted">RemoveDupsFromSorted</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.MX">RemoveDupsFromSorted.MX</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XSort">RemoveDupsFromSorted.XSort</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool">RemoveDupsFromSorted.XTotalLeBool</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<br/><br/><a id="module_S"></a><h2>S </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA">SetsWithFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#SetsWithFoldA.E">SetsWithFoldA.E</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="module_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups">WSetsOnWithDups</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra">WSetsOnWithDupsExtra</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra">WSetsOn_TO_WSetsOnWithDupsExtra</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithDupsFoldA">WSetsWithDupsFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithDupsFoldA.E">WSetsWithDupsFoldA.E</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA">WSetsWithFoldA</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#WSetsWithFoldA.E">WSetsWithFoldA.E</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><hr/>
+<h1>Variable Index</h1>
+<a id="variable_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.f">MSetIntervals.Spec.f</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s">MSetIntervals.Spec.s</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.s'">MSetIntervals.Spec.s'</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.x">MSetIntervals.Spec.x</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec.y">MSetIntervals.Spec.y</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="variable_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.f">WSetsOnWithDups.Spec.f</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s">WSetsOnWithDups.Spec.s</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.s'">WSetsOnWithDups.Spec.s'</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.x">WSetsOnWithDups.Spec.x</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec.y">WSetsOnWithDups.Spec.y</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><hr/>
+<h1>Library Index</h1>
+<a id="library_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html">MSetFoldWithAbort</a> <br/>
+<a href="MSetsExtra.MSetIntervals.html">MSetIntervals</a> <br/>
+<a href="MSetsExtra.MSetListWithDups.html">MSetListWithDups</a> <br/>
+<a href="MSetsExtra.MSetWithDups.html">MSetWithDups</a> <br/>
+<br/><br/><hr/>
+<h1>Lemma Index</h1>
+<a id="lemma_A"></a><h2>A </h2>
+<a href="MSetsExtra.MSetIntervals.html#add_add_sub_eq">add_add_sub_eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="lemma_E"></a><h2>E </h2>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.decode_encode_ok">ElementEncodeNat.decode_encode_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode_lt">ElementEncodeNat.encode_lt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode_eq">ElementEncodeNat.encode_eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN.decode_encode_ok">ElementEncodeN.decode_encode_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode_lt">ElementEncodeN.encode_lt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode_eq">ElementEncodeN.encode_eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.decode_encode_ok">ElementEncodeZ.decode_encode_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode_lt">ElementEncodeZ.encode_lt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode_eq">ElementEncodeZ.encode_eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="lemma_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort_spec">HasFoldWithAbortOps.choose_with_abort_spec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort_spec">HasFoldWithAbortOps.exists_with_abort_spec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort_spec">HasFoldWithAbortOps.filter_with_abort_spec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec">HasFoldWithAbortOps.foldWithAbortGtLtSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtLtSpec_Equal</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec">HasFoldWithAbortOps.foldWithAbortGtSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtSpec_Equal">HasFoldWithAbortOps.foldWithAbortGtSpec_Equal</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal">HasFoldWithAbortOps.foldWithAbortPrecomputeSpec_Equal</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec">HasFoldWithAbortOps.foldWithAbortSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpec_Equal">HasFoldWithAbortOps.foldWithAbortSpec_Equal</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort_spec">HasFoldWithAbortOps.forall_with_abort_spec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="lemma_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeAVLSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_RawSpec">MakeGenTreeFoldA.foldWithAbort_RawSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeRBTSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec">MakeWeakListSetsWithFoldA.foldWithAbortPrecomputeSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.add_spec">Make.add_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.cardinal_spec">Make.cardinal_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.choose_spec2">Make.choose_spec2</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.choose_spec1">Make.choose_spec1</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.diff_spec">Make.diff_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec2w">Make.elements_dist_spec2w</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec1">Make.elements_dist_spec1</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_dist_spec_full">Make.elements_dist_spec_full</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.elements_spec1">Make.elements_spec1</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.empty_spec">Make.empty_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.equal_spec">Make.equal_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.eq_dec">Make.eq_dec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.eq_equiv">Make.eq_equiv</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.exists_spec">Make.exists_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.filter_spec">Make.filter_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.fold_spec">Make.fold_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.for_all_spec">Make.for_all_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.inter_spec">Make.inter_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.is_empty_spec">Make.is_empty_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.mem_spec">Make.mem_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.partition_spec2">Make.partition_spec2</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.partition_spec1">Make.partition_spec1</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.partition_aux_spec">Make.partition_aux_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.remove_spec">Make.remove_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.rev_filter_aux_spec">Make.rev_filter_aux_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.singleton_spec">Make.singleton_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.subset_spec">Make.subset_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.union_spec">Make.union_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.add_spec">MSetIntervals.add_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal_spec">MSetIntervals.cardinal_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec3">MSetIntervals.choose_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec2">MSetIntervals.choose_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose_spec1">MSetIntervals.choose_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare_spec">MSetIntervals.compare_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff_spec">MSetIntervals.diff_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec2">MSetIntervals.elements_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec2w">MSetIntervals.elements_spec2w</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements_spec1">MSetIntervals.elements_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty_spec">MSetIntervals.empty_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal_spec">MSetIntervals.equal_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_spec">MSetIntervals.exists_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter_spec">MSetIntervals.filter_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold_spec">MSetIntervals.fold_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all_spec">MSetIntervals.for_all_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter_spec">MSetIntervals.inter_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty_spec">MSetIntervals.is_empty_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec3">MSetIntervals.max_elt_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec2">MSetIntervals.max_elt_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt_spec1">MSetIntervals.max_elt_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem_spec">MSetIntervals.mem_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec3">MSetIntervals.min_elt_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec2">MSetIntervals.min_elt_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt_spec1">MSetIntervals.min_elt_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition_spec2">MSetIntervals.partition_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition_spec1">MSetIntervals.partition_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove_spec">MSetIntervals.remove_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton_spec">MSetIntervals.singleton_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset_spec">MSetIntervals.subset_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.union_spec">MSetIntervals.union_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="lemma_N"></a><h2>N </h2>
+<a href="MSetsExtra.MSetIntervals.html#NoDupA_map">NoDupA_map</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="lemma_O"></a><h2>O </h2>
+<a href="MSetsExtra.MSetIntervals.html#Ops.acc_pred">Ops.acc_pred</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.singleton_alt_def">Ops.singleton_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="lemma_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_invariant">Raw.addZ_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_InZ">Raw.addZ_InZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_alt_def">Raw.addZ_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.addZ_aux_alt_def">Raw.addZ_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_list_spec">Raw.add_list_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_list_ok">Raw.add_list_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_spec">Raw.add_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.cardinalN_spec">Raw.cardinalN_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.cardinal_spec">Raw.cardinal_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.choose_spec3">Raw.choose_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_spec">Raw.compare_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_antisym">Raw.compare_antisym</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_cons">Raw.compare_eq_Lt_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_nil_r">Raw.compare_eq_Lt_nil_r</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Lt_nil_l">Raw.compare_eq_Lt_nil_l</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.compare_eq_Eq">Raw.compare_eq_Eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_spec">Raw.diff_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_invariant">Raw.diff_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_InZ">Raw.diff_InZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_aux2_props">Raw.diff_aux2_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_props">Raw.diff_aux_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_aux_alt_def">Raw.diff_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_sorted">Raw.elementsZ_sorted</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_sorted">Raw.elementsZ_single_sorted</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_spec2w">Raw.elementsZ_spec2w</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_insert_intervalZ_guarded">Raw.elementsZ_insert_intervalZ_guarded</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_app">Raw.elementsZ_app</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_cons">Raw.elementsZ_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ_front">Raw.elementsZ_single_succ_front</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_add">Raw.elementsZ_single_add</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_succ">Raw.elementsZ_single_succ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single_base">Raw.elementsZ_single_base</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_nil">Raw.elementsZ_nil</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_sorted">Raw.elements_sorted</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_spec2w">Raw.elements_spec2w</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_spec1">Raw.elements_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_cons">Raw.elements_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_nil">Raw.elements_nil</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.empty_spec">Raw.empty_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.empty_spec'">Raw.empty_spec'</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.encode_decode_eq">Raw.encode_decode_eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.equal_spec">Raw.equal_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.equal_elementsZ">Raw.equal_elementsZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.equal_alt_def">Raw.equal_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.exists_spec">Raw.exists_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_props">Raw.filterZ_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_aux_props">Raw.filterZ_aux_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_props">Raw.filterZ_single_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filterZ_single_aux_props">Raw.filterZ_single_aux_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filter_spec">Raw.filter_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_spec">Raw.fold_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def">Raw.fold_elementsZ_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_alt_def_aux">Raw.fold_elementsZ_alt_def_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_cons">Raw.fold_elementsZ_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_nil">Raw.fold_elementsZ_nil</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_alt_def">Raw.fold_elementsZ_single_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_app">Raw.fold_opt_app</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_opt_list_cons">Raw.fold_opt_list_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_succ">Raw.fold_elementsZ_single_succ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_zero">Raw.fold_elementsZ_single_zero</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_single_pos">Raw.fold_elementsZ_single_pos</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_elementsZ_aux_irrel">Raw.fold_elementsZ_aux_irrel</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.for_all_spec">Raw.for_all_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.insert_interval_begin_spec">Raw.insert_interval_begin_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded_rev_nil_app">Raw.insert_intervalZ_guarded_rev_nil_app</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.insert_intervalZ_guarded_app">Raw.insert_intervalZ_guarded_app</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_insert_intervalZ_guarded">Raw.interval_list_elements_greater_insert_intervalZ_guarded</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_insert_intervalZ_guarded">Raw.interval_list_invariant_insert_intervalZ_guarded</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_snoc_intro">Raw.interval_list_invariant_snoc_intro</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_iff">Raw.interval_list_invariant_app_iff</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_elim">Raw.interval_list_invariant_app_elim</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_app_intro">Raw.interval_list_invariant_app_intro</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_app_elim_1">Raw.interval_list_elements_greater_app_elim_1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_intro">Raw.interval_list_elements_greater_intro</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt2_def">Raw.interval_list_elements_greater_alt2_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_alt_def">Raw.interval_list_elements_greater_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_sing">Raw.interval_list_invariant_sing</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_cons">Raw.interval_list_invariant_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant_nil">Raw.interval_list_invariant_nil</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_impl">Raw.interval_list_elements_greater_impl</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater_cons">Raw.interval_list_elements_greater_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_elim">Raw.interval_1_compare_elim</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_1_compare_alt_def">Raw.interval_1_compare_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_swap">Raw.interval_compare_swap</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_compare_elim">Raw.interval_compare_elim</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_spec">Raw.inter_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_invariant">Raw.inter_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_InZ">Raw.inter_InZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_aux2_props">Raw.inter_aux2_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_aux_props">Raw.inter_aux_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_aux_alt_def">Raw.inter_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_skip">Raw.InZ_partitionZ_fold_skip</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_insert">Raw.InZ_partitionZ_fold_insert</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_partitionZ_fold_current_Some">Raw.InZ_partitionZ_fold_current_Some</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_insert_intervalZ_guarded">Raw.InZ_insert_intervalZ_guarded</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_In">Raw.InZ_In</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_dec">Raw.InZ_dec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_rev">Raw.InZ_rev</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_app">Raw.InZ_app</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_cons">Raw.InZ_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ_nil">Raw.InZ_nil</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_merge_interval">Raw.In_merge_interval</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_InZ">Raw.In_InZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_alt_def">Raw.In_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_hd">Raw.In_elementsZ_single_hd</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single_dec">Raw.In_elementsZ_single_dec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single1">Raw.In_elementsZ_single1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In_elementsZ_single">Raw.In_elementsZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_empty_spec">Raw.is_empty_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list_rev">Raw.is_encoded_elems_list_rev</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list_app">Raw.is_encoded_elems_list_app</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.length_elementsZ_single">Raw.length_elementsZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.lt_Transitive">Raw.lt_Transitive</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.lt_Irreflexive">Raw.lt_Irreflexive</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_eq_None">Raw.max_eltZ_eq_None</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec2">Raw.max_eltZ_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.memZ_spec">Raw.memZ_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.mem_spec">Raw.mem_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_invariant">Raw.merge_interval_size_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_eq_max">Raw.merge_interval_size_eq_max</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_add">Raw.merge_interval_size_add</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.merge_interval_size_neq_0">Raw.merge_interval_size_neq_0</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec2">Raw.min_eltZ_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater_equal">Raw.Nin_elements_greater_equal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Nin_elements_greater">Raw.Nin_elements_greater</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.NoDupA_elementsZ_single">Raw.NoDupA_elementsZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok_cons">Raw.Ok_cons</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok_nil">Raw.Ok_nil</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_alt_def">Raw.partitionZ_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_alt_def">Raw.partitionZ_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_single_aux_alt_def">Raw.partitionZ_single_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant_skip">Raw.partitionZ_aux_invariant_skip</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant_insert">Raw.partitionZ_aux_invariant_insert</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_spec2">Raw.partition_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_spec1">Raw.partition_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_alt_def">Raw.partition_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.removeZ_spec">Raw.removeZ_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.removeZ_interval_list_invariant">Raw.removeZ_interval_list_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.removeZ_alt_def">Raw.removeZ_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_list_spec">Raw.remove_list_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_list_ok">Raw.remove_list_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_spec">Raw.remove_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.singleton_spec">Raw.singleton_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_spec">Raw.subset_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_props">Raw.subset_props</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux_before">Raw.subset_props_aux_before</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_props_aux">Raw.subset_props_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.subset_flatten_alt_def">Raw.subset_flatten_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_spec">Raw.union_spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_invariant">Raw.union_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_InZ">Raw.union_InZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_alt_def">Raw.union_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_aux_alt_def">Raw.union_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_aux_flatten_alt_def">Raw.union_aux_flatten_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.eqb_eq_alt">RemoveDupsFromSorted.eqb_eq_alt</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA_spec">RemoveDupsFromSorted.remove_dups_by_sortingA_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_spec">RemoveDupsFromSorted.remove_dups_from_sortedA_spec</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_hd">RemoveDupsFromSorted.remove_dups_from_sortedA_hd</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_alt</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt">RemoveDupsFromSorted.remove_dups_from_sortedA_aux_alt</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb_total">RemoveDupsFromSorted.XTotalLeBool.leb_total</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#rev_map_alt_def">rev_map_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#rev_map_aux_alt_def">rev_map_aux_alt_def</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="lemma_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec2w</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist_spec1</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><a id="lemma_Z"></a><h2>Z </h2>
+<a href="MSetsExtra.MSetIntervals.html#Z_to_N_minus_neq_0">Z_to_N_minus_neq_0</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Z_lt_le_add_r">Z_lt_le_add_r</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Z_lt_add_r">Z_lt_add_r</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Z_le_add_r">Z_le_add_r</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/>
+<h1>Axiom Index</h1>
+<a id="axiom_E"></a><h2>E </h2>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode.decode">ElementEncode.decode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode.decode_encode_ok">ElementEncode.decode_encode_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode.encode">ElementEncode.encode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode.encode_lt">ElementEncode.encode_lt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncode.encode_eq">ElementEncode.encode_eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="axiom_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecompute">HasFoldWithAbort.foldWithAbortPrecompute</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbort.foldWithAbortPrecomputeSpec">HasFoldWithAbort.foldWithAbortPrecomputeSpec</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="axiom_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.cardinal_spec">WSetsOnWithDupsExtra.cardinal_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist_spec2w">WSetsOnWithDupsExtra.elements_dist_spec2w</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist_spec1">WSetsOnWithDupsExtra.elements_dist_spec1</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDupsExtra.elements_dist">WSetsOnWithDupsExtra.elements_dist</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.add_spec">WSetsOnWithDups.add_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.choose_spec2">WSetsOnWithDups.choose_spec2</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.choose_spec1">WSetsOnWithDups.choose_spec1</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.diff_spec">WSetsOnWithDups.diff_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.elements_spec1">WSetsOnWithDups.elements_spec1</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.empty_spec">WSetsOnWithDups.empty_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.equal_spec">WSetsOnWithDups.equal_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.exists_spec">WSetsOnWithDups.exists_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.filter_spec">WSetsOnWithDups.filter_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.fold_spec">WSetsOnWithDups.fold_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.for_all_spec">WSetsOnWithDups.for_all_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In">WSetsOnWithDups.In</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.inter_spec">WSetsOnWithDups.inter_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.is_empty_spec">WSetsOnWithDups.is_empty_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.mem_spec">WSetsOnWithDups.mem_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.partition_spec2">WSetsOnWithDups.partition_spec2</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.partition_spec1">WSetsOnWithDups.partition_spec1</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.remove_spec">WSetsOnWithDups.remove_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.singleton_spec">WSetsOnWithDups.singleton_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.subset_spec">WSetsOnWithDups.subset_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.union_spec">WSetsOnWithDups.union_spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><hr/>
+<h1>Constructor Index</h1>
+<a id="constructor_O"></a><h2>O </h2>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_after_touch">Ops.ICR_after_touch</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_after">Ops.ICR_after</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_2">Ops.ICR_subsume_2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_subsume_1">Ops.ICR_subsume_1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_equal">Ops.ICR_equal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_after">Ops.ICR_overlap_after</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_overlap_before">Ops.ICR_overlap_before</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_before_touch">Ops.ICR_before_touch</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.ICR_before">Ops.ICR_before</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="constructor_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.ok">Raw.ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/>
+<h1>Projection Index</h1>
+<a id="projection_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_P'">HasFoldWithAbortOps.fwasa_P'</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_gt">HasFoldWithAbortOps.fwasa_f_gt</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_lt">HasFoldWithAbortOps.fwasa_f_lt</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.fwasa_f_pre">HasFoldWithAbortOps.fwasa_f_pre</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="projection_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_ok">MSetIntervals.is_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.this">MSetIntervals.this</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="projection_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.ok">Raw.ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/>
+<h1>Inductive Index</h1>
+<a id="inductive_O"></a><h2>O </h2>
+<a href="MSetsExtra.MSetIntervals.html#Ops.interval_compare_result">Ops.interval_compare_result</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="inductive_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok">Raw.Ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/>
+<h1>Section Index</h1>
+<a id="section_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Spec">MSetIntervals.Spec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="section_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.ForNotations">Raw.ForNotations</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="section_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Spec">WSetsOnWithDups.Spec</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><hr/>
+<h1>Instance Index</h1>
+<a id="instance_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Make.In_compat">Make.In_compat</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq_equiv">MSetIntervals.eq_equiv</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.In_compat">MSetIntervals.In_compat</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt_compat">MSetIntervals.lt_compat</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt_strorder">MSetIntervals.lt_strorder</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="instance_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.add_ok">Raw.add_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.diff_ok">Raw.diff_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.empty_ok">Raw.empty_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.filter_ok">Raw.filter_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.inter_ok">Raw.inter_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.IsOk_Ok">Raw.IsOk_Ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_ok2">Raw.partition_ok2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partition_ok1">Raw.partition_ok1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.remove_ok">Raw.remove_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.singleton_ok">Raw.singleton_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.union_ok">Raw.union_ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="instance_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.In_compat">WSetsOnWithDups.In_compat</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><hr/>
+<h1>Abbreviation Index</h1>
+<a id="abbreviation_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Make.compatb">Make.compatb</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.compatb">MSetIntervals.compatb</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="abbreviation_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.compatb">WSetsOnWithDups.compatb</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><hr/>
+<h1>Definition Index</h1>
+<a id="definition_E"></a><h2>E </h2>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.decode">ElementEncodeNat.decode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeNat.encode">ElementEncodeNat.encode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN.decode">ElementEncodeN.decode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeN.encode">ElementEncodeN.encode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.decode">ElementEncodeZ.decode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#ElementEncodeZ.encode">ElementEncodeZ.encode</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="definition_F"></a><h2>F </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtLtSpecPred">foldWithAbortGtLtSpecPred</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtLtType">foldWithAbortGtLtType</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtSpecPred">foldWithAbortGtSpecPred</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortGtType">foldWithAbortGtType</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeSpecPred">foldWithAbortPrecomputeSpecPred</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortPrecomputeType">foldWithAbortPrecomputeType</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortSpecPred">foldWithAbortSpecPred</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#foldWithAbortType">foldWithAbortType</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="definition_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.choose_with_abort">HasFoldWithAbortOps.choose_with_abort</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.exists_with_abort">HasFoldWithAbortOps.exists_with_abort</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.filter_with_abort">HasFoldWithAbortOps.filter_with_abort</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbort">HasFoldWithAbortOps.foldWithAbort</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGt">HasFoldWithAbortOps.foldWithAbortGt</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortGtLt">HasFoldWithAbortOps.foldWithAbortGtLt</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.foldWithAbortSpecArgsForPred">HasFoldWithAbortOps.foldWithAbortSpecArgsForPred</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.forall_with_abort">HasFoldWithAbortOps.forall_with_abort</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="definition_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeAVLSetsWithFoldA.foldWithAbortPrecompute">MakeAVLSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeGenTreeFoldA.foldWithAbort_Raw">MakeGenTreeFoldA.foldWithAbort_Raw</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortPrecompute">MakeListSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeListSetsWithFoldA.foldWithAbortRaw">MakeListSetsWithFoldA.foldWithAbortRaw</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeRBTSetsWithFoldA.foldWithAbortPrecompute">MakeRBTSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortPrecompute">MakeWeakListSetsWithFoldA.foldWithAbortPrecompute</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetFoldWithAbort.html#MakeWeakListSetsWithFoldA.foldWithAbortRaw">MakeWeakListSetsWithFoldA.foldWithAbortRaw</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Empty">Make.Empty</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.eq">Make.eq</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Equal">Make.Equal</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Exists">Make.Exists</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.For_all">Make.For_all</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.In">Make.In</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Make.Subset">Make.Subset</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.add">MSetIntervals.add</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.cardinal">MSetIntervals.cardinal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.choose">MSetIntervals.choose</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.compare">MSetIntervals.compare</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.diff">MSetIntervals.diff</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elements">MSetIntervals.elements</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.elt">MSetIntervals.elt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.empty">MSetIntervals.empty</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Empty">MSetIntervals.Empty</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq">MSetIntervals.eq</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.equal">MSetIntervals.equal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Equal">MSetIntervals.Equal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.eq_dec">MSetIntervals.eq_dec</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Exists">MSetIntervals.Exists</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.exists_">MSetIntervals.exists_</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.filter">MSetIntervals.filter</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.fold">MSetIntervals.fold</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.for_all">MSetIntervals.for_all</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.For_all">MSetIntervals.For_all</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.In">MSetIntervals.In</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.inter">MSetIntervals.inter</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.is_empty">MSetIntervals.is_empty</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.lt">MSetIntervals.lt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.max_elt">MSetIntervals.max_elt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.mem">MSetIntervals.mem</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.min_elt">MSetIntervals.min_elt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.partition">MSetIntervals.partition</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.remove">MSetIntervals.remove</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.singleton">MSetIntervals.singleton</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.subset">MSetIntervals.subset</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.Subset">MSetIntervals.Subset</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.t">MSetIntervals.t</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.union">MSetIntervals.union</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="definition_O"></a><h2>O </h2>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.add">Ops.add</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.add">Ops.add</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.addZ">Ops.addZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.addZ_aux">Ops.addZ_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.add_list">Ops.add_list</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.cardinal">Ops.cardinal</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.cardinal">Ops.cardinal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.cardinalN">Ops.cardinalN</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.choose">Ops.choose</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.choose">Ops.choose</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.compare">Ops.compare</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.diff">Ops.diff</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.diff">Ops.diff</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.diff_aux2">Ops.diff_aux2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.diff_aux">Ops.diff_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.elements">Ops.elements</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.elements">Ops.elements</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.elementsZ">Ops.elementsZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.elements_dist">Ops.elements_dist</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.elt">Ops.elt</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.elt">Ops.elt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.empty">Ops.empty</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.empty">Ops.empty</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.equal">Ops.equal</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.equal">Ops.equal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.exists_">Ops.exists_</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.exists_">Ops.exists_</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.filter">Ops.filter</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filter">Ops.filter</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ">Ops.filterZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_aux">Ops.filterZ_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_single">Ops.filterZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_single_aux">Ops.filterZ_single_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.filterZ_fold_fun">Ops.filterZ_fold_fun</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.fold">Ops.fold</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold">Ops.fold</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ">Ops.fold_elementsZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_single">Ops.fold_elementsZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.fold_elementsZ_aux">Ops.fold_elementsZ_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.for_all">Ops.for_all</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.for_all">Ops.for_all</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.from_elements">Ops.from_elements</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.insert_intervalZ_guarded">Ops.insert_intervalZ_guarded</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.insert_interval_begin">Ops.insert_interval_begin</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.inter">Ops.inter</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.inter">Ops.inter</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.interval_1_compare">Ops.interval_1_compare</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.interval_compare">Ops.interval_compare</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.inter_aux2">Ops.inter_aux2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.inter_aux">Ops.inter_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.is_empty">Ops.is_empty</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.is_empty">Ops.is_empty</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.max_elt">Ops.max_elt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.max_eltZ">Ops.max_eltZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.mem">Ops.mem</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.mem">Ops.mem</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.memZ">Ops.memZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.merge_interval_size">Ops.merge_interval_size</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.min_elt">Ops.min_elt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.min_eltZ">Ops.min_eltZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.partition">Ops.partition</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partition">Ops.partition</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ">Ops.partitionZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_aux">Ops.partitionZ_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_single">Ops.partitionZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_single_aux">Ops.partitionZ_single_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_fun">Ops.partitionZ_fold_fun</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_skip">Ops.partitionZ_fold_skip</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.partitionZ_fold_insert">Ops.partitionZ_fold_insert</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.partition_aux">Ops.partition_aux</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.remove">Ops.remove</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.remove">Ops.remove</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.removeZ">Ops.removeZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.removeZ_aux">Ops.removeZ_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.remove_list">Ops.remove_list</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter">Ops.rev_filter</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.rev_filter_aux">Ops.rev_filter_aux</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.singleton">Ops.singleton</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.singleton">Ops.singleton</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.subset">Ops.subset</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.subset">Ops.subset</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.t">Ops.t</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.t">Ops.t</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#Ops.union">Ops.union</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.union">Ops.union</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Ops.union_aux">Ops.union_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="definition_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.choose_spec2">Raw.choose_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.choose_spec1">Raw.choose_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elementsZ_single">Raw.elementsZ_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.elements_single">Raw.elements_single</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Empty">Raw.Empty</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Equal">Raw.Equal</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Exists">Raw.Exists</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.fold_opt">Raw.fold_opt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.For_all">Raw.For_all</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.In">Raw.In</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_invariant">Raw.interval_list_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.interval_list_elements_greater">Raw.interval_list_elements_greater</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.InZ">Raw.InZ</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.IsOk">Raw.IsOk</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.is_encoded_elems_list">Raw.is_encoded_elems_list</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.lt">Raw.lt</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_elt_spec3">Raw.max_elt_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_elt_spec2">Raw.max_elt_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_elt_spec1">Raw.max_elt_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec3">Raw.max_eltZ_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.max_eltZ_spec1">Raw.max_eltZ_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec3">Raw.min_elt_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec2">Raw.min_elt_spec2</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_elt_spec1">Raw.min_elt_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec3">Raw.min_eltZ_spec3</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.min_eltZ_spec1">Raw.min_eltZ_spec1</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_fold_current">Raw.partitionZ_fold_current</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.partitionZ_aux_invariant">Raw.partitionZ_aux_invariant</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Subset">Raw.Subset</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_by_sortingA">RemoveDupsFromSorted.remove_dups_by_sortingA</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA">RemoveDupsFromSorted.remove_dups_from_sortedA</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.remove_dups_from_sortedA_aux">RemoveDupsFromSorted.remove_dups_from_sortedA_aux</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.le">RemoveDupsFromSorted.XTotalLeBool.le</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.leb">RemoveDupsFromSorted.XTotalLeBool.leb</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetListWithDups.html#RemoveDupsFromSorted.XTotalLeBool.t">RemoveDupsFromSorted.XTotalLeBool.t</a> [in <a href="MSetsExtra.MSetListWithDups.html">MSetsExtra.MSetListWithDups</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#rev_map">rev_map</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<a href="MSetsExtra.MSetIntervals.html#rev_map_aux">rev_map_aux</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="definition_W"></a><h2>W </h2>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Empty">WSetsOnWithDups.Empty</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.eq">WSetsOnWithDups.eq</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Equal">WSetsOnWithDups.Equal</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Exists">WSetsOnWithDups.Exists</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.For_all">WSetsOnWithDups.For_all</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOnWithDups.Subset">WSetsOnWithDups.Subset</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<a href="MSetsExtra.MSetWithDups.html#WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist">WSetsOn_TO_WSetsOnWithDupsExtra.elements_dist</a> [in <a href="MSetsExtra.MSetWithDups.html">MSetsExtra.MSetWithDups</a>]<br/>
+<br/><br/><hr/>
+<h1>Record Index</h1>
+<a id="record_H"></a><h2>H </h2>
+<a href="MSetsExtra.MSetFoldWithAbort.html#HasFoldWithAbortOps.FoldWithAbortSpecArg">HasFoldWithAbortOps.FoldWithAbortSpecArg</a> [in <a href="MSetsExtra.MSetFoldWithAbort.html">MSetsExtra.MSetFoldWithAbort</a>]<br/>
+<br/><br/><a id="record_M"></a><h2>M </h2>
+<a href="MSetsExtra.MSetIntervals.html#MSetIntervals.t_">MSetIntervals.t_</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><a id="record_R"></a><h2>R </h2>
+<a href="MSetsExtra.MSetIntervals.html#Raw.Ok">Raw.Ok</a> [in <a href="MSetsExtra.MSetIntervals.html">MSetsExtra.MSetIntervals</a>]<br/>
+<br/><br/><hr/><table>
+<tr>
+<td>Global Index</td>
+<td><a href="index.html#global_A">A</a></td>
+<td><a href="index.html#global_B">B</a></td>
+<td><a href="index.html#global_C">C</a></td>
+<td>D</td>
+<td><a href="index.html#global_E">E</a></td>
+<td><a href="index.html#global_F">F</a></td>
+<td>G</td>
+<td><a href="index.html#global_H">H</a></td>
+<td><a href="index.html#global_I">I</a></td>
+<td>J</td>
+<td>K</td>
+<td><a href="index.html#global_L">L</a></td>
+<td><a href="index.html#global_M">M</a></td>
+<td><a href="index.html#global_N">N</a></td>
+<td><a href="index.html#global_O">O</a></td>
+<td><a href="index.html#global_P">P</a></td>
+<td>Q</td>
+<td><a href="index.html#global_R">R</a></td>
+<td><a href="index.html#global_S">S</a></td>
+<td><a href="index.html#global_T">T</a></td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#global_W">W</a></td>
+<td><a href="index.html#global_X">X</a></td>
+<td><a href="index.html#global_Y">Y</a></td>
+<td><a href="index.html#global_Z">Z</a></td>
+<td>_</td>
+<td>other</td>
+<td>(2110 entries)</td>
+</tr>
+<tr>
+<td>Notation Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#notation_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#notation_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#notation_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(5 entries)</td>
+</tr>
+<tr>
+<td>Binder Index</td>
+<td><a href="index.html#binder_A">A</a></td>
+<td><a href="index.html#binder_B">B</a></td>
+<td><a href="index.html#binder_C">C</a></td>
+<td>D</td>
+<td><a href="index.html#binder_E">E</a></td>
+<td><a href="index.html#binder_F">F</a></td>
+<td>G</td>
+<td><a href="index.html#binder_H">H</a></td>
+<td><a href="index.html#binder_I">I</a></td>
+<td>J</td>
+<td>K</td>
+<td><a href="index.html#binder_L">L</a></td>
+<td>M</td>
+<td><a href="index.html#binder_N">N</a></td>
+<td>O</td>
+<td><a href="index.html#binder_P">P</a></td>
+<td>Q</td>
+<td><a href="index.html#binder_R">R</a></td>
+<td><a href="index.html#binder_S">S</a></td>
+<td><a href="index.html#binder_T">T</a></td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td><a href="index.html#binder_X">X</a></td>
+<td><a href="index.html#binder_Y">Y</a></td>
+<td><a href="index.html#binder_Z">Z</a></td>
+<td>_</td>
+<td>other</td>
+<td>(1515 entries)</td>
+</tr>
+<tr>
+<td>Module Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#module_E">E</a></td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#module_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#module_M">M</a></td>
+<td><a href="index.html#module_N">N</a></td>
+<td><a href="index.html#module_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#module_R">R</a></td>
+<td><a href="index.html#module_S">S</a></td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#module_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(43 entries)</td>
+</tr>
+<tr>
+<td>Variable Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#variable_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td>R</td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#variable_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(10 entries)</td>
+</tr>
+<tr>
+<td>Library Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#library_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td>R</td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(4 entries)</td>
+</tr>
+<tr>
+<td>Lemma Index</td>
+<td><a href="index.html#lemma_A">A</a></td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#lemma_E">E</a></td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#lemma_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#lemma_M">M</a></td>
+<td><a href="index.html#lemma_N">N</a></td>
+<td><a href="index.html#lemma_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#lemma_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#lemma_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td><a href="index.html#lemma_Z">Z</a></td>
+<td>_</td>
+<td>other</td>
+<td>(257 entries)</td>
+</tr>
+<tr>
+<td>Axiom Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#axiom_E">E</a></td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#axiom_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td>M</td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td>R</td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#axiom_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(32 entries)</td>
+</tr>
+<tr>
+<td>Constructor Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td>M</td>
+<td>N</td>
+<td><a href="index.html#constructor_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#constructor_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(10 entries)</td>
+</tr>
+<tr>
+<td>Projection Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#projection_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#projection_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#projection_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(7 entries)</td>
+</tr>
+<tr>
+<td>Inductive Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td>M</td>
+<td>N</td>
+<td><a href="index.html#inductive_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#inductive_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(2 entries)</td>
+</tr>
+<tr>
+<td>Section Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#section_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#section_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#section_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(3 entries)</td>
+</tr>
+<tr>
+<td>Instance Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#instance_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#instance_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#instance_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(17 entries)</td>
+</tr>
+<tr>
+<td>Abbreviation Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td>H</td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#abbreviation_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td>R</td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#abbreviation_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(3 entries)</td>
+</tr>
+<tr>
+<td>Definition Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td><a href="index.html#definition_E">E</a></td>
+<td><a href="index.html#definition_F">F</a></td>
+<td>G</td>
+<td><a href="index.html#definition_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#definition_M">M</a></td>
+<td>N</td>
+<td><a href="index.html#definition_O">O</a></td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#definition_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td><a href="index.html#definition_W">W</a></td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(199 entries)</td>
+</tr>
+<tr>
+<td>Record Index</td>
+<td>A</td>
+<td>B</td>
+<td>C</td>
+<td>D</td>
+<td>E</td>
+<td>F</td>
+<td>G</td>
+<td><a href="index.html#record_H">H</a></td>
+<td>I</td>
+<td>J</td>
+<td>K</td>
+<td>L</td>
+<td><a href="index.html#record_M">M</a></td>
+<td>N</td>
+<td>O</td>
+<td>P</td>
+<td>Q</td>
+<td><a href="index.html#record_R">R</a></td>
+<td>S</td>
+<td>T</td>
+<td>U</td>
+<td>V</td>
+<td>W</td>
+<td>X</td>
+<td>Y</td>
+<td>Z</td>
+<td>_</td>
+<td>other</td>
+<td>(3 entries)</td>
+</tr>
+</table>
+<hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
+"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<link href="coqdoc.css" rel="stylesheet" type="text/css" />
+<title>Table of contents</title>
+</head>
+
+<body>
+
+<div id="page">
+
+<div id="header">
+</div>
+
+<div id="main">
+
+<div id="toc">
+<h2><a href="MSetsExtra.MSetFoldWithAbort.html">Library MSetsExtra.MSetFoldWithAbort</a></h2>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab1">Fold with abort for sets</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab2">Fold With Abort Operations</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab3">Module Types</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab4">Derived operations</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab5">First lets define the other folding with abort variants</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab6">Specialisations for equality</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab7">FoldWithAbortSpecArgs</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab8">Filter with abort</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab9">Choose with abort</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab10">Exists and Forall with abort</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab11">Modules Types For Sets with Fold with Abort</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab12">Implementations</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab13">GenTree implementation</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab14">AVL implementation</a>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab15">RBT implementation</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab16">Sorted Lists Implementation</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetFoldWithAbort.html#lab17">Unsorted Lists without Dups Implementation</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="MSetsExtra.MSetIntervals.html">Library MSetsExtra.MSetIntervals</a></h2>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetIntervals.html#lab18">Weak sets implemented by interval lists</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetIntervals.html#lab19">Auxiliary stuff</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetIntervals.html#lab20">rev_map</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab21">Encoding Elements</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab22">Set Operations</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab23">Raw Module</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetIntervals.html#lab24">Defining invariant <span class="inlinecode"><span class="id" title="var">IsOk</span></span></a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab25">Defining notations</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab26">elements list properties</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab27">comparing intervals</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab28">Alternative definition of addZ</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab29">Auxiliary Lemmata about Invariant</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab30">Properties of In and InZ</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab31">Membership specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab32">insert if length not 0</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab33">Merging intervals</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab34">add specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab35">empty specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab36">is_empty specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab37">singleton specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab38">add_list specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab39">union specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab40">inter specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab41">diff specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab42">remove specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab43">remove_list specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab44">subset specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab45">elements and elementsZ specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab46">equal specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab47">compare</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab48">elements is sorted</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab49">choose specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab50">fold specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab51">cardinal specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab52">for_all specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab53">exists specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab54">filter specification</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab55">partition specification</a>
+
+</li>
+</ul>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab56">Main Module</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab57">Instantiations</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetIntervals.html#lab58">Z</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab59">N</a>
+
+</li>
+<li><a href="MSetsExtra.MSetIntervals.html#lab60">nat</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="MSetsExtra.MSetListWithDups.html">Library MSetsExtra.MSetListWithDups</a></h2>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetListWithDups.html#lab61">Weak sets implemented as lists with duplicates</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetListWithDups.html#lab62">Removing duplicates from sorted lists</a>
+
+</li>
+<li><a href="MSetsExtra.MSetListWithDups.html#lab63">Operations Module</a>
+
+</li>
+<li><a href="MSetsExtra.MSetListWithDups.html#lab64">Main Module</a>
+
+</li>
+<li><a href="MSetsExtra.MSetListWithDups.html#lab65">Proofs of set operation specifications.</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+<h2><a href="MSetsExtra.MSetWithDups.html">Library MSetsExtra.MSetWithDups</a></h2>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetWithDups.html#lab66">Signature for weak sets which may contain duplicates</a>
+<ul class="doclist">
+<li><a href="MSetsExtra.MSetWithDups.html#lab67">WSetsOnWithDups</a>
+
+</li>
+<li><a href="MSetsExtra.MSetWithDups.html#lab68">WSetsOnWithDupsExtra</a>
+
+</li>
+<li><a href="MSetsExtra.MSetWithDups.html#lab69">WSetOn to WSetsOnWithDupsExtra</a>
+
+</li>
+</ul>
+
+</li>
+</ul>
+</div>
+<hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
+</div>
+
+</div>
+
+</body>
+</html>
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