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+---
+title: Contextual Semantics
+layout: page
+---
+
+```
+module Untyped.ContextualSemantics where
+```
+
+## Imports
+
+```
+open import Untyped
+open import Data.Nat using (ℕ; zero; suc; _+_; _∸_; _<_; _≤_; _≤ᵇ_)
+open import Data.List using (List; _++_)
+import Data.List as List
+open import Data.List.Relation.Unary.All using (All)
+open import Builtin
+open import Builtin.Signature using (_/_⊢⋆; args♯; Sig; fv)
+open import Untyped.RenamingSubstitution using (_[_])
+open import Data.Vec using (Vec)
+open import Data.Maybe using (Maybe; just; nothing)
+open import Data.Bool using (if_then_else_)
+import Data.List.NonEmpty as List⁺
+import Data.List.NonEmpty as NE
+import Data.Vec as Vec
+import Data.Fin as Fin
+open import Relation.Binary.PropositionalEquality using (_≡_)
+
+```
+
+## TODO
+
+```
+variable
+  n : ℕ
+
+data B : n ⊢ → Set 
+data Value : n ⊢ → Set
+β : {X : n ⊢} → B X → Builtin
+-- the signature of the builtin underlying the spine 'b'
+sigOf : {X : n ⊢} → B X → Sig
+-- the type of a term argument expected by the 'b' builtin
+ArgTy : {X : n ⊢} → B X → Set
+data A : {X : n ⊢} → (b : B X) → ℕ → NE.List⁺ (ArgTy b) → Set
+
+data B where
+  builtinB
+    : (b : Builtin)
+    → B {n} (builtin b)
+  appB
+    : {t₁ t₂ : n ⊢}
+    → B t₁
+    → Value t₂
+    → B (t₁ · t₂)
+  forceB
+    : {t : n ⊢}
+    → B t
+    → B (force t)
+
+β (builtinB b) = b
+β (appB b v) = β b
+β (forceB b) = β b
+
+sigOf b = signature (β b)
+
+ArgTy b = Sig.fv⋆ (sigOf b) / Sig.fv♯ (sigOf b) ⊢⋆
+
+||_|| : {X : n ⊢} → B X → ℕ
+||_|| (builtinB b) = 0 
+||_|| (appB b v) = 1 + || b ||
+||_|| (forceB b) = 1 + || b ||
+
+data A where
+  builtinA
+    : (b : Builtin)
+    → A (builtinB {n} b) (fv (signature b)) (Sig.args (signature b))
+  forceA
+    : {t : n ⊢} {b : B t} {fl : ℕ}
+      {al : NE.List⁺ (ArgTy b)}
+    -- a force is still expected: consume it, leaving the argument list suffix untouched
+    → A b (suc fl) al
+    → A (forceB b) fl al
+  appA
+    : {t₁ t₂ : n ⊢} {b : B t₁}
+      {τ : ArgTy b}
+      {as : NE.List⁺ (ArgTy b)}
+    -- all forces done: consume the head argument τ, non-empty suffix 'as' remains
+    → A b zero (τ NE.∷⁺ as)
+    → (v : Value t₂)
+    → A (appB b v) zero as
+
+βᴬ : {t : n ⊢} {b : B t} {fl : ℕ}
+     {al : NE.List⁺ (ArgTy b)}
+   → A b fl al → Builtin
+βᴬ {b = b} _ = β b
+
+-- number of forces performed so far: total type variables minus those still expected
+||_||ᶠᴬ
+  : {t : n ⊢} {b : B t} {fl : ℕ} {al : NE.List⁺ (ArgTy b)}
+  → A b fl al
+  → ℕ
+||_||ᶠᴬ {b = b} {fl = fl} _ = fv (sigOf b) ∸ fl
+
+-- number of forces + applications performed so far
+||_||ᴬ
+  : {t : n ⊢} {b : B t} {fl : ℕ} {al : NE.List⁺ (ArgTy b)}
+  → A b fl al
+  → ℕ
+||_||ᴬ {b = b} {fl = fl} {al = al} a =
+  || a ||ᶠᴬ + (args♯ (sigOf b) ∸ NE.length al)
+
+-- The type of the next argument the partial application expects:
+--   - nothing if the next expected argument is a type argument (a force)
+--   - just τ if the next expected argument is a term argument of type τ
+nextᴬ
+  : {t : n ⊢} {b : B t} {fl : ℕ} {al : NE.List⁺ (ArgTy b)}
+  → A b fl al
+  → Maybe (ArgTy b)
+nextᴬ {fl = suc _} _ = nothing
+nextᴬ {fl = zero} {al = al} _ = just (NE.head al)
+
+data Value where
+  conᵥ : (t : TmCon) → Value {n} (con t)
+  delayᵥ : (t : n ⊢) → Value (delay t)
+  ƛᵥ : (t : suc n ⊢) → Value (ƛ t)
+  constrᵥ : (i : ℕ) (ts : List (n ⊢)) → All Value ts → Value (constr i ts)
+  bAppᵥ
+    : {t : n ⊢} {b : B t} {fl : ℕ} {al : NE.List⁺ (ArgTy b)}
+    → A b fl al
+    → Value t
+
+data Frame : n ⊢ → Set where
+  □
+    : {t : n ⊢}
+    → Frame t
+  _ᶠ·_
+    : {t₁ : n ⊢}
+    → Frame t₁
+    → (t₂ : n ⊢)
+    → Frame (t₁ · t₂) 
+  _·ᶠ_
+    : {t₁ t₂ : n ⊢}
+    → Value t₁
+    → Frame t₂
+    → Frame (t₁ · t₂)
+  forceᶠ
+    : {t : n ⊢}
+    → Frame t
+    → Frame (force t)
+  constrᶠ
+    : {f : n ⊢} {vs : List (n ⊢)}
+    → (i : ℕ)
+    → All Value vs
+    → Frame f
+    → (ts : List (n ⊢))
+    → Frame (constr i (vs ++ List.[ f ] ++ ts))
+  caseᶠ
+    : {f : n ⊢}
+    → Frame f
+    → (ts : List (n ⊢))
+    → Frame (case f ts)
+
+-- TODO: implement
+postulate
+  multiApp : {t : n ⊢} → Value t → List (n ⊢) → n ⊢
+
+-- TODO: rename constructors
+data _⟶_ : n ⊢ → n ⊢ → Set where
+  lamapp
+    : {t₁ : suc n ⊢} {t₂ : n ⊢}
+    → Value t₂
+    → (ƛ t₁ · t₂) ⟶ (t₁ [ t₂ ])
+  forcedelay
+    : {t : n ⊢}
+    → (force (delay t)) ⟶ t
+  caseconstr
+    : {i l : ℕ} {vs ts : List (n ⊢)} {f : n ⊢} 
+    → All Value vs
+    -- TODO: lookup i in ts, we need to deal with empty ts's and index out of bounds
+    → (case (constr i vs) ts) ⟶ (multiApp {!   !} vs)
+
+
+```
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