diff --git a/Archive/Imo/Imo2019Q4.lean b/Archive/Imo/Imo2019Q4.lean
index a34672f2393f90..2f016b94e97dea 100644
--- a/Archive/Imo/Imo2019Q4.lean
+++ b/Archive/Imo/Imo2019Q4.lean
@@ -34,7 +34,6 @@ open Finset
 
 namespace Imo2019Q4
 
-set_option backward.isDefEq.respectTransparency false in
 theorem upper_bound {k n : ℕ} (hk : k > 0)
     (h : (k ! : ℤ) = ∏ i ∈ range n, ((2 : ℤ) ^ n - (2 : ℤ) ^ i)) : n < 6 := by
   have h2 : ∑ i ∈ range n, i < k := by
diff --git a/Archive/Imo/Imo2024Q6.lean b/Archive/Imo/Imo2024Q6.lean
index 8c315b62297251..8e26eb7b9494f6 100644
--- a/Archive/Imo/Imo2024Q6.lean
+++ b/Archive/Imo/Imo2024Q6.lean
@@ -225,7 +225,6 @@ lemma floor_fExample (x : ℚ) :
     rw [Int.floor_eq_iff]
     simp [(Int.fract_nonneg x).lt_of_ne' h, (Int.fract_lt_one x).le]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma card_range_fExample : #(Set.range (fun x ↦ fExample x + fExample (-x))) = 2 := by
   have h : Set.range (fun x ↦ fExample x + fExample (-x)) = {0, -2} := by
     ext x
diff --git a/Mathlib/Algebra/Category/ContinuousCohomology/Basic.lean b/Mathlib/Algebra/Category/ContinuousCohomology/Basic.lean
index da5f2f53763393..9308090b730c82 100644
--- a/Mathlib/Algebra/Category/ContinuousCohomology/Basic.lean
+++ b/Mathlib/Algebra/Category/ContinuousCohomology/Basic.lean
@@ -136,7 +136,6 @@ def complex : CochainComplex (Action (TopModuleCat R) G ⥤ Action (TopModuleCat
 
 end MultiInd
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The functor taking an `R`-linear `G`-representation to its `G`-invariant submodule. -/
 def invariants : Action (TopModuleCat R) G ⥤ TopModuleCat R where
   obj M := .of R
diff --git a/Mathlib/Algebra/Category/ModuleCat/Abelian.lean b/Mathlib/Algebra/Category/ModuleCat/Abelian.lean
index aec3ef7a88a85f..783a23e67141ce 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Abelian.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Abelian.lean
@@ -52,7 +52,6 @@ def normalMono (hf : Mono f) : NormalMono f where
               (LinearMap.quotKerEquivRange f.hom ≪≫ₗ
               LinearEquiv.ofEq _ _ (Submodule.ker_mkQ _).symm))) <| by ext; rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- In the category of modules, every epimorphism is normal. -/
 def normalEpi (hf : Epi f) : NormalEpi f where
   W := of R (LinearMap.ker f.hom)
diff --git a/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean b/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
index 490979e953274c..9e0360a27a0de6 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
@@ -252,7 +252,6 @@ namespace Free
 
 section
 
-set_option backward.whnf.reducibleClassField false in
 instance : Preadditive (Free R C) where
   homGroup _ _ := Finsupp.instAddCommGroup
   add_comp X Y Z f f' g := by
@@ -264,7 +263,6 @@ instance : Preadditive (Free R C) where
     congr; ext r h
     rw [Finsupp.sum_add_index'] <;> · simp [mul_add]
 
-set_option backward.whnf.reducibleClassField false in
 instance : Linear R (Free R C) where
   homModule _ _ := Finsupp.module _ R
   smul_comp X Y Z r f g := by
@@ -276,7 +274,6 @@ instance : Linear R (Free R C) where
     congr; ext h s
     rw [Finsupp.sum_smul_index] <;> simp [mul_left_comm]
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 theorem single_comp_single {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (r s : R) :
     (single f r ≫ single g s : Free.of R X ⟶ Free.of R Z) = single (f ≫ g) (r * s) := by
@@ -303,7 +300,6 @@ variable {C} {D : Type u} [Category.{v} D] [Preadditive D] [Linear R D]
 
 open Preadditive Linear
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 /-- A functor to an `R`-linear category lifts to a functor from its `R`-linear completion.
 -/
diff --git a/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean b/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
index 4bf5078ce2a29f..8d4a56263b4c75 100644
--- a/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
@@ -28,7 +28,6 @@ namespace ModuleCat
 variable {R : Type u} [Ring R] {X Y : ModuleCat.{v} R} (f : X ⟶ Y)
 variable {M : Type v} [AddCommGroup M] [Module R M]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ker_eq_bot_of_mono [Mono f] : LinearMap.ker f.hom = ⊥ :=
   LinearMap.ker_eq_bot_of_cancel fun u v h => ModuleCat.hom_ext_iff.mp <|
     (@cancel_mono _ _ _ _ _ f _ (↟u) (↟v)).1 <| ModuleCat.hom_ext_iff.mpr h
@@ -55,7 +54,6 @@ theorem epi_iff_surjective : Epi f ↔ Function.Surjective f := by
 def uniqueOfEpiZero (X) [h : Epi (0 : X ⟶ of R M)] : Unique M :=
   uniqueOfSurjectiveZero X ((ModuleCat.epi_iff_surjective _).mp h)
 
-set_option backward.isDefEq.respectTransparency false in
 instance mono_as_hom'_subtype (U : Submodule R X) : Mono (ModuleCat.ofHom U.subtype) :=
   (mono_iff_ker_eq_bot _).mpr (Submodule.ker_subtype U)
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/ExteriorPower.lean b/Mathlib/Algebra/Category/ModuleCat/ExteriorPower.lean
index 3f1d9de2029dfd..c1bf6b14d57304 100644
--- a/Mathlib/Algebra/Category/ModuleCat/ExteriorPower.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/ExteriorPower.lean
@@ -26,7 +26,6 @@ namespace ModuleCat
 
 variable {R : Type u} [CommRing R]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The exterior power of an object in `ModuleCat R`. -/
 def exteriorPower (M : ModuleCat.{v} R) (n : ℕ) : ModuleCat.{max u v} R :=
   ModuleCat.of R (⋀[R]^n M)
@@ -76,7 +75,6 @@ lemma hom_ext {M : ModuleCat.{v} R} {N : ModuleCat.{max u v} R} {n : ℕ}
   ext : 1
   exact exteriorPower.linearMap_ext h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The morphism `M.exteriorPower n ⟶ N` induced by an alternating map. -/
 noncomputable def desc {M : ModuleCat.{v} R} {n : ℕ} {N : ModuleCat.{max u v} R}
     (φ : M.AlternatingMap N n) : M.exteriorPower n ⟶ N :=
@@ -88,7 +86,6 @@ lemma desc_mk {M : ModuleCat.{v} R} {n : ℕ} {N : ModuleCat.{max u v} R}
     desc φ (mk x) = φ x := by
   apply exteriorPower.alternatingMapLinearEquiv_apply_ιMulti
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The morphism `M.exteriorPower n ⟶ N.exteriorPower n` induced by a morphism `M ⟶ N`
 in `ModuleCat R`. -/
 noncomputable def map {M N : ModuleCat.{v} R} (f : M ⟶ N) (n : ℕ) :
diff --git a/Mathlib/Algebra/Category/ModuleCat/Images.lean b/Mathlib/Algebra/Category/ModuleCat/Images.lean
index c61a130b47f8f4..4f0f8454e0c352 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Images.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Images.lean
@@ -33,13 +33,11 @@ attribute [local ext] Subtype.ext
 
 section
 
-set_option backward.isDefEq.respectTransparency false in
 -- implementation details of `HasImage` for ModuleCat; use the API, not these
 /-- The image of a morphism in `ModuleCat R` is just the bundling of `LinearMap.range f` -/
 def image : ModuleCat R :=
   ModuleCat.of R (LinearMap.range f.hom)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The inclusion of `image f` into the target -/
 def image.ι : image f ⟶ H :=
   ofHom (LinearMap.range f.hom).subtype
@@ -47,7 +45,6 @@ def image.ι : image f ⟶ H :=
 instance : Mono (image.ι f) :=
   ConcreteCategory.mono_of_injective (image.ι f) Subtype.val_injective
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The corestriction map to the image -/
 def factorThruImage : G ⟶ image f :=
   ofHom f.hom.rangeRestrict
@@ -59,7 +56,6 @@ attribute [local simp] image.fac
 
 variable {f}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The universal property for the image factorisation -/
 noncomputable def image.lift (F' : MonoFactorisation f) : image f ⟶ F'.I :=
   ofHom
@@ -99,19 +95,16 @@ noncomputable def isImage : IsImage (monoFactorisation f) where
   lift := image.lift
   lift_fac := image.lift_fac
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The categorical image of a morphism in `ModuleCat R` agrees with the linear algebraic range. -/
 noncomputable def imageIsoRange {G H : ModuleCat.{v} R} (f : G ⟶ H) :
     Limits.image f ≅ ModuleCat.of R (LinearMap.range f.hom) :=
   IsImage.isoExt (Image.isImage f) (isImage f)
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp, reassoc, elementwise]
 theorem imageIsoRange_inv_image_ι {G H : ModuleCat.{v} R} (f : G ⟶ H) :
     (imageIsoRange f).inv ≫ Limits.image.ι f = ModuleCat.ofHom (LinearMap.range f.hom).subtype :=
   IsImage.isoExt_inv_m _ _
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp, reassoc, elementwise]
 theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
     (imageIsoRange f).hom ≫ ModuleCat.ofHom (LinearMap.range f.hom).subtype = Limits.image.ι f := by
diff --git a/Mathlib/Algebra/Category/ModuleCat/Kernels.lean b/Mathlib/Algebra/Category/ModuleCat/Kernels.lean
index ff5cd9c7ea8762..30fa3921a7d585 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Kernels.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Kernels.lean
@@ -29,7 +29,6 @@ section
 
 variable {M N P : ModuleCat.{v} R} (f : M ⟶ N)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The kernel cone induced by the concrete kernel. -/
 def kernelCone : KernelFork f :=
   KernelFork.ofι (ofHom (LinearMap.ker f.hom).subtype) <| by aesop
@@ -104,7 +103,6 @@ attribute [local instance] hasCokernels_moduleCat
 
 variable {G H : ModuleCat.{v} R} (f : G ⟶ H)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The categorical kernel of a morphism in `ModuleCat`
 agrees with the usual module-theoretical kernel.
 -/
@@ -113,7 +111,6 @@ noncomputable def kernelIsoKer {G H : ModuleCat.{v} R} (f : G ⟶ H) :
     kernel f ≅ ModuleCat.of R (LinearMap.ker f.hom) :=
   limit.isoLimitCone ⟨_, kernelIsLimit f⟩
 
-set_option backward.isDefEq.respectTransparency false in
 -- We now show this isomorphism commutes with the inclusion of the kernel into the source.
 @[simp, elementwise]
     -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11036): broken dot notation
@@ -121,7 +118,6 @@ theorem kernelIsoKer_inv_kernel_ι : (kernelIsoKer f).inv ≫ kernel.ι f =
     ofHom (LinearMap.ker f.hom).subtype :=
   limit.isoLimitCone_inv_π _ _
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp, elementwise]
 theorem kernelIsoKer_hom_ker_subtype :
     -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11036): broken dot notation
diff --git a/Mathlib/Algebra/Category/ModuleCat/Subobject.lean b/Mathlib/Algebra/Category/ModuleCat/Subobject.lean
index 6166dd8de5b658..8c555f4aeb015b 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Subobject.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Subobject.lean
@@ -35,7 +35,6 @@ namespace ModuleCat
 
 variable {R : Type u} [Ring R] (M : ModuleCat.{v} R)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The categorical subobjects of a module `M` are in one-to-one correspondence with its
 submodules. -/
 noncomputable def subobjectModule : Subobject M ≃o Submodule R M :=
@@ -81,7 +80,6 @@ noncomputable def toKernelSubobject {M N : ModuleCat.{v} R} {f : M ⟶ N} :
     LinearMap.ker f.hom →ₗ[R] kernelSubobject f :=
   (kernelSubobjectIso f ≪≫ ModuleCat.kernelIsoKer f).inv.hom
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem toKernelSubobject_arrow {M N : ModuleCat R} {f : M ⟶ N} (x : LinearMap.ker f.hom) :
     (kernelSubobject f).arrow (toKernelSubobject x) = x.1 := by
diff --git a/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean b/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean
index ed357a7bd341df..a280a5ef7987a9 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean
@@ -34,17 +34,14 @@ variable {M N : TopModuleCat.{v} R} (φ : M ⟶ N)
 
 section kernel
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Kernel in `TopModuleCat R` is the kernel of the linear map with the subspace topology. -/
 abbrev ker : TopModuleCat R := .of R φ.hom.ker
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The inclusion map from the kernel in `TopModuleCat R`. -/
 def kerι : ker φ ⟶ M := ofHom ⟨Submodule.subtype _, continuous_subtype_val⟩
 
 instance : Mono (kerι φ) := ConcreteCategory.mono_of_injective (kerι φ) <| Subtype.val_injective
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma kerι_comp : kerι φ ≫ φ = 0 := by ext ⟨_, hm⟩; exact hm
 
 @[simp] lemma kerι_apply (x) : kerι φ x = x.1 := rfl
diff --git a/Mathlib/Algebra/DirectSum/LinearMap.lean b/Mathlib/Algebra/DirectSum/LinearMap.lean
index 975a384b3b6791..4cb9d438df2c70 100644
--- a/Mathlib/Algebra/DirectSum/LinearMap.lean
+++ b/Mathlib/Algebra/DirectSum/LinearMap.lean
@@ -97,7 +97,6 @@ lemma trace_eq_zero_of_mapsTo_ne (h : IsInternal N) [IsNoetherian R M]
     diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne h b σ hσ hf (by simp [s]),
     Pi.zero_apply, Finset.sum_const_zero]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `f` and `g` are commuting endomorphisms of a finite, free `R`-module `M`, such that `f`
 is triangularizable, then to prove that the trace of `g ∘ f` vanishes, it is sufficient to prove
 that the trace of `g` vanishes on each generalized eigenspace of `f`. -/
@@ -138,7 +137,6 @@ lemma mapsTo_biSup_of_mapsTo {ι : Type*} {N : ι → Submodule R M}
 
 end IsInternal
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The trace of an endomorphism of a direct sum is the sum of the traces on each component.
 
 Note that it is important the statement gives the user definitional control over `p` since the
diff --git a/Mathlib/Algebra/DirectSum/Module.lean b/Mathlib/Algebra/DirectSum/Module.lean
index 597e91f32d47f7..04926c31e07be6 100644
--- a/Mathlib/Algebra/DirectSum/Module.lean
+++ b/Mathlib/Algebra/DirectSum/Module.lean
@@ -496,7 +496,6 @@ theorem isInternal_ne_bot_iff {A : ι → Submodule R M} :
     IsInternal (fun i : {i // A i ≠ ⊥} ↦ A i) ↔ IsInternal A := by
   simp [isInternal_submodule_iff_iSupIndep_and_iSup_eq_top]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isInternal_biSup_submodule_of_iSupIndep {A : ι → Submodule R M} (s : Set ι)
     (h : iSupIndep <| fun i : s ↦ A i) :
     IsInternal <| fun (i : s) ↦ (A i).comap (⨆ i ∈ s, A i).subtype := by
diff --git a/Mathlib/Algebra/DualQuaternion.lean b/Mathlib/Algebra/DualQuaternion.lean
index 315fe2d902afa4..c04f0414c7495e 100644
--- a/Mathlib/Algebra/DualQuaternion.lean
+++ b/Mathlib/Algebra/DualQuaternion.lean
@@ -96,7 +96,6 @@ theorem imK_snd_dualNumberEquiv (q : Quaternion (DualNumber R)) :
     (dualNumberEquiv q).snd.imK = q.imK.snd :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem fst_re_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
     (dualNumberEquiv.symm d).re.fst = d.fst.re :=
diff --git a/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean b/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
index f27de7cacd1705..45034617f760ea 100644
--- a/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
@@ -282,7 +282,6 @@ lemma leftUnshift_add {n' a : ℤ} (γ₁ γ₂ : Cochain (K⟦a⟧) L n') (n :
   change (leftShiftAddEquiv K L n a n' hn).symm (γ₁ + γ₂) = _
   apply map_add
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma rightShift_smul (a n' : ℤ) (hn' : n' + a = n) (x : R) :
     (x • γ).rightShift a n' hn' = x • γ.rightShift a n' hn' := by
diff --git a/Mathlib/Algebra/Homology/HomotopyCofiber.lean b/Mathlib/Algebra/Homology/HomotopyCofiber.lean
index 58db5c89b1be1b..a1be484621a962 100644
--- a/Mathlib/Algebra/Homology/HomotopyCofiber.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCofiber.lean
@@ -458,7 +458,6 @@ noncomputable def nullHomotopicMap : K.cylinder ⟶ K.cylinder :=
 noncomputable def nullHomotopy : Homotopy (nullHomotopicMap K) 0 :=
   Homotopy.nullHomotopy' _
 
-set_option backward.isDefEq.respectTransparency false in
 lemma inlX_nullHomotopy_f (i j : ι) (hij : c.Rel j i) :
     inlX K i j hij ≫ (nullHomotopicMap K).f j =
       inlX K i j hij ≫ (π K ≫ ι₀ K - 𝟙 _).f j := by
diff --git a/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean b/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
index 2bc12348acf112..2fcf6ec474cc7d 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
@@ -206,14 +206,12 @@ variable {M : Type v} [AddCommGroup M] [Module R M] {N : Type v} [AddCommGroup N
 
 open CategoryTheory
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a linear map `f : M → N`, we can obtain a short complex `0 → ker(f) → M → N`. -/
 abbrev LinearMap.shortComplexKer (f : M →ₗ[R] N) : ShortComplex (ModuleCat.{v} R) where
   f := ModuleCat.ofHom.{v} (LinearMap.ker f).subtype
   g := ModuleCat.ofHom.{v} f
   zero := by ext; simp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem LinearMap.shortExact_shortComplexKer {f : M →ₗ[R] N} (h : Function.Surjective f) :
     f.shortComplexKer.ShortExact where
   exact := (ShortComplex.ShortExact.moduleCat_exact_iff_function_exact _).mpr
diff --git a/Mathlib/Algebra/Lie/TraceForm.lean b/Mathlib/Algebra/Lie/TraceForm.lean
index ed87d4941099a0..6a8e14dfb61959 100644
--- a/Mathlib/Algebra/Lie/TraceForm.lean
+++ b/Mathlib/Algebra/Lie/TraceForm.lean
@@ -217,7 +217,6 @@ open TensorProduct
 
 variable [LieRing.IsNilpotent L] [IsDomain R]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma traceForm_eq_sum_genWeightSpaceOf [IsPrincipalIdealRing R]
     [Module.IsTorsionFree R M] [IsNoetherian R M] [IsTriangularizable R L M] (z : L) :
     traceForm R L M =
@@ -408,7 +407,6 @@ namespace LieModule
 variable [Field K] [LieAlgebra K L] [Module K M] [LieModule K L M] [FiniteDimensional K M]
 variable [LieRing.IsNilpotent L] [LinearWeights K L M] [IsTriangularizable K L M]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma traceForm_eq_sum_finrank_nsmul_mul (x y : L) :
     traceForm K L M x y = ∑ χ : Weight K L M, finrank K (genWeightSpace M χ) • (χ x * χ y) := by
   have hxy : ∀ χ : Weight K L M, MapsTo (toEnd K L M x ∘ₗ toEnd K L M y)
diff --git a/Mathlib/Algebra/Lie/Weights/Chain.lean b/Mathlib/Algebra/Lie/Weights/Chain.lean
index 4b44e6dbc2408a..c252b53290b060 100644
--- a/Mathlib/Algebra/Lie/Weights/Chain.lean
+++ b/Mathlib/Algebra/Lie/Weights/Chain.lean
@@ -203,7 +203,6 @@ lemma trace_toEnd_genWeightSpaceChain_eq_zero
   | add => simp_all
   | smul => simp_all
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a (potential) root `α` relative to a Cartan subalgebra `H`, if we restrict to the ideal
 `I = corootSpace α` of `H` (informally, `I = ⁅H(α), H(-α)⁆`), we may find an
 integral linear combination between `α` and any weight `χ` of a representation.
diff --git a/Mathlib/Algebra/Lie/Weights/Killing.lean b/Mathlib/Algebra/Lie/Weights/Killing.lean
index 953138293f877c..b98e7034e4381f 100644
--- a/Mathlib/Algebra/Lie/Weights/Killing.lean
+++ b/Mathlib/Algebra/Lie/Weights/Killing.lean
@@ -100,7 +100,6 @@ variable [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra]
 section
 variable [IsTriangularizable K H L]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- For any `α` and `β`, the corresponding root spaces are orthogonal with respect to the Killing
 form, provided `α + β ≠ 0`. -/
 lemma killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero {α β : H → K} {x y : L}
diff --git a/Mathlib/Algebra/Module/FinitePresentation.lean b/Mathlib/Algebra/Module/FinitePresentation.lean
index 9ea47051e5f46a..6d3d0331d80d68 100644
--- a/Mathlib/Algebra/Module/FinitePresentation.lean
+++ b/Mathlib/Algebra/Module/FinitePresentation.lean
@@ -210,7 +210,6 @@ lemma Module.FinitePresentation.fg_ker_iff [Module.FinitePresentation R M]
     Submodule.FG (LinearMap.ker l) ↔ Module.FinitePresentation R N :=
   ⟨finitePresentation_of_surjective l hl, fun _ ↦ fg_ker l hl⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Module.finitePresentation_of_ker [Module.FinitePresentation R N]
     (l : M →ₗ[R] N) (hl : Function.Surjective l) [Module.FinitePresentation R (LinearMap.ker l)] :
     Module.FinitePresentation R M := by
@@ -285,7 +284,6 @@ instance (priority := 900) of_subsingleton [Subsingleton M] :
     Module.FinitePresentation R M :=
   .of_equiv (default : (Fin 0 → R) ≃ₗ[R] M)
 
-set_option backward.isDefEq.respectTransparency false in
 variable (M N) in
 instance prod [Module.FinitePresentation R M] [Module.FinitePresentation R N] :
     Module.FinitePresentation R (M × N) := by
@@ -330,7 +328,6 @@ lemma Module.FinitePresentation.trans (S : Type*) [CommRing S] [Algebra R S]
     simp only [f, LinearMap.ker_restrictScalars, ← Module.Finite.iff_fg]
     exact Module.Finite.trans S _
 
-set_option backward.isDefEq.respectTransparency false in
 open TensorProduct in
 instance {A} [CommRing A] [Algebra R A] [Module.FinitePresentation R M] :
     Module.FinitePresentation A (A ⊗[R] M) := by
diff --git a/Mathlib/Algebra/Module/Injective.lean b/Mathlib/Algebra/Module/Injective.lean
index 040cef53c5730f..7e157df954a9e4 100644
--- a/Mathlib/Algebra/Module/Injective.lean
+++ b/Mathlib/Algebra/Module/Injective.lean
@@ -402,7 +402,6 @@ protected theorem injective (h : Module.Baer R Q) : Module.Injective R Q where
     obtain ⟨h, H⟩ := Module.Baer.extension_property h i hi f
     exact ⟨h, DFunLike.congr_fun H⟩
 
-set_option backward.isDefEq.respectTransparency false in
 protected theorem of_injective [Small.{v} R] (inj : Module.Injective R Q) : Module.Baer R Q := by
   intro I g
   let eI := Shrink.linearEquiv R I
diff --git a/Mathlib/Algebra/Module/Lattice.lean b/Mathlib/Algebra/Module/Lattice.lean
index 75baa3ed570c40..21b18d9d9f0800 100644
--- a/Mathlib/Algebra/Module/Lattice.lean
+++ b/Mathlib/Algebra/Module/Lattice.lean
@@ -134,7 +134,6 @@ lemma _root_.Submodule.span_range_eq_top_of_injective_of_rank_le {M N : Type u}
 
 variable (K) {V : Type*} [AddCommGroup V] [Module K V] [Module R V] [IsScalarTower R K V]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Any basis of an `R`-lattice in `V` defines a `K`-basis of `V`. -/
 noncomputable def _root_.Module.Basis.extendOfIsLattice [IsFractionRing R K] {κ : Type*}
     {M : Submodule R V} [IsLattice K M] (b : Basis κ R M) :
@@ -157,7 +156,6 @@ lemma _root_.Module.Basis.extendOfIsLattice_apply [IsFractionRing R K] {κ : Typ
 
 variable [IsDomain R]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A finitely-generated `R`-submodule of `V` of rank at least the `K`-rank of `V`
 is a lattice. -/
 lemma of_rank_le [Module.Finite K V] [IsFractionRing R K] {M : Submodule R V}
@@ -168,7 +166,6 @@ lemma of_rank_le [Module.Finite K V] [IsFractionRing R K] {M : Submodule R V}
 
 variable [IsPrincipalIdealRing R]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Any lattice over a PID is a free `R`-module.
 Note that under our conditions, `Module.IsTorsionFree R K` simply says that `algebraMap R K` is
 injective. -/
@@ -194,7 +191,6 @@ lemma finrank_of_pi {ι : Type*} [Fintype ι] [IsFractionRing R K] (M : Submodul
     [IsLattice K M] : Module.finrank R M = Fintype.card ι :=
   Module.finrank_eq_of_rank_eq (IsLattice.rank_of_pi K M)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The intersection of two lattices is a lattice. -/
 instance inf [Module.Finite K V] [IsFractionRing R K] (M N : Submodule R V)
     [IsLattice K M] [IsLattice K N] : IsLattice K (M ⊓ N) where
diff --git a/Mathlib/Algebra/Module/PID.lean b/Mathlib/Algebra/Module/PID.lean
index 2efd59d5930f49..9a965d65d15378 100644
--- a/Mathlib/Algebra/Module/PID.lean
+++ b/Mathlib/Algebra/Module/PID.lean
@@ -64,7 +64,6 @@ open Submodule
 
 open UniqueFactorizationMonoid
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Submodule.isSemisimple_torsionBy_of_irreducible {a : R} (h : Irreducible a) :
     IsSemisimpleModule R (torsionBy R M a) :=
   haveI := PrincipalIdealRing.isMaximal_of_irreducible h
@@ -167,7 +166,6 @@ theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^
 
 open Finset Multiset
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 omit dec in
 /-- A finitely generated `p ^ ∞`-torsion module over a PID is isomorphic to a direct sum of some
@@ -232,7 +230,6 @@ theorem torsion_by_prime_power_decomposition (hM : Module.IsTorsion' M (Submonoi
 
 end PTorsion
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A finitely generated torsion module over a PID is isomorphic to a direct sum of some
   `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers. -/
 theorem equiv_directSum_of_isTorsion [h' : Module.Finite R M] (hM : Module.IsTorsion R M) :
@@ -258,7 +255,6 @@ theorem equiv_directSum_of_isTorsion [h' : Module.Finite R M] (hM : Module.IsTor
 
 variable (R M)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Structure theorem of finitely generated modules over a PID** : A finitely generated
   module over a PID is isomorphic to the product of a free module and a direct sum of some
   `R ⧸ R ∙ (p i ^ e i)` where the `p i ^ e i` are prime powers. -/
diff --git a/Mathlib/Algebra/Module/SpanRank.lean b/Mathlib/Algebra/Module/SpanRank.lean
index ccce93b0ee7b12..fe57567d2feac1 100644
--- a/Mathlib/Algebra/Module/SpanRank.lean
+++ b/Mathlib/Algebra/Module/SpanRank.lean
@@ -167,7 +167,6 @@ theorem exists_span_set_card_eq_spanRank (p : Submodule R M) :
     Set.range (fun (s : {s : Set M // span R s = p}) ↦ #s) := csInf_mem ⟨#p, ⟨⟨p, by simp⟩, rfl⟩⟩
   exact ⟨s.1, ⟨hs, s.2⟩⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Constructs a generating set with cardinality equal to the `spanFinrank` of the submodule when
   the submodule is finitely generated. -/
 theorem FG.exists_span_set_encard_eq_spanFinrank {p : Submodule R M} (h : p.FG) :
diff --git a/Mathlib/Algebra/Module/Submodule/Defs.lean b/Mathlib/Algebra/Module/Submodule/Defs.lean
index 8348fcab601a94..8d9c319b7e96f4 100644
--- a/Mathlib/Algebra/Module/Submodule/Defs.lean
+++ b/Mathlib/Algebra/Module/Submodule/Defs.lean
@@ -322,6 +322,7 @@ protected theorem neg_mem (hx : x ∈ p) : -x ∈ p :=
   neg_mem hx
 
 /-- Reinterpret a submodule as an additive subgroup. -/
+@[reducible]
 def toAddSubgroup : AddSubgroup M :=
   { p.toAddSubmonoid with neg_mem' := fun {_} => p.neg_mem }
 
@@ -329,14 +330,12 @@ def toAddSubgroup : AddSubgroup M :=
 theorem coe_toAddSubgroup : (p.toAddSubgroup : Set M) = p :=
   rfl
 
-@[simp]
 theorem mem_toAddSubgroup : x ∈ p.toAddSubgroup ↔ x ∈ p :=
   Iff.rfl
 
 theorem toAddSubgroup_injective : Injective (toAddSubgroup : Submodule R M → AddSubgroup M)
   | _, _, h => SetLike.ext (SetLike.ext_iff.1 h :)
 
-@[simp]
 theorem toAddSubgroup_inj : p.toAddSubgroup = p'.toAddSubgroup ↔ p = p' :=
   toAddSubgroup_injective.eq_iff
 
diff --git a/Mathlib/Algebra/Module/Submodule/Ker.lean b/Mathlib/Algebra/Module/Submodule/Ker.lean
index cf684c7304b534..a9819a07f082b6 100644
--- a/Mathlib/Algebra/Module/Submodule/Ker.lean
+++ b/Mathlib/Algebra/Module/Submodule/Ker.lean
@@ -195,7 +195,6 @@ theorem ker_eq_bot {f : M →ₛₗ[τ₁₂] M₂} : ker f = ⊥ ↔ Injective
 @[deprecated (since := "2025-12-23")]
 alias _root_.LinearMapClass.ker_eq_bot := ker_eq_bot
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma injective_domRestrict_iff {f : M →ₛₗ[τ₁₂] M₂} {S : Submodule R M} :
     Injective (f.domRestrict S) ↔ S ⊓ LinearMap.ker f = ⊥ := by
   rw [← LinearMap.ker_eq_bot]
@@ -209,7 +208,6 @@ set_option backward.isDefEq.respectTransparency false in
     rw [h] at this
     simpa [mk_eq_zero] using this
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] theorem injective_restrict_iff_disjoint {p : Submodule R M} {f : M →ₗ[R] M}
     (hf : ∀ x ∈ p, f x ∈ p) :
     Injective (f.restrict hf) ↔ Disjoint p (ker f) := by
diff --git a/Mathlib/Algebra/Module/Submodule/LinearMap.lean b/Mathlib/Algebra/Module/Submodule/LinearMap.lean
index 26229b800aef89..68d86fc001983e 100644
--- a/Mathlib/Algebra/Module/Submodule/LinearMap.lean
+++ b/Mathlib/Algebra/Module/Submodule/LinearMap.lean
@@ -213,7 +213,6 @@ theorem restrict_apply {f : M →ₗ[R] M₁} {p : Submodule R M} {q : Submodule
     (hf : ∀ x ∈ p, f x ∈ q) (x : p) : f.restrict hf x = ⟨f x, hf x.1 x.2⟩ :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 lemma restrict_sub {R M M₁ : Type*}
     [Ring R] [AddCommGroup M] [AddCommGroup M₁] [Module R M] [Module R M₁]
     {p : Submodule R M} {q : Submodule R M₁} {f g : M →ₗ[R] M₁}
diff --git a/Mathlib/Algebra/Module/Submodule/Union.lean b/Mathlib/Algebra/Module/Submodule/Union.lean
index 2b3c1fca5ef70a..56916404bc4581 100644
--- a/Mathlib/Algebra/Module/Submodule/Union.lean
+++ b/Mathlib/Algebra/Module/Submodule/Union.lean
@@ -102,7 +102,6 @@ lemma Module.Dual.exists_forall_ne_zero_of_forall_exists
   obtain ⟨x, hx⟩ := Submodule.exists_forall_notMem_of_forall_ne_top p h
   exact ⟨x, by simpa [p] using hx⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A convenience variation of `Module.Dual.exists_forall_ne_zero_of_forall_exists` where we are
 concerned only about behaviour on a fixed submodule. -/
 lemma Module.Dual.exists_forall_mem_ne_zero_of_forall_exists (p : Submodule K M)
diff --git a/Mathlib/Algebra/Module/Torsion/Basic.lean b/Mathlib/Algebra/Module/Torsion/Basic.lean
index 5dc4bbb8513b82..eb7c0f00b43ddb 100644
--- a/Mathlib/Algebra/Module/Torsion/Basic.lean
+++ b/Mathlib/Algebra/Module/Torsion/Basic.lean
@@ -624,7 +624,6 @@ namespace Submodule
 
 variable [CommRing R] [AddCommGroup M] [Module R M]
 
-set_option backward.isDefEq.respectTransparency false in
 instance (I : Ideal R) : Module (R ⧸ I) (torsionBySet R M I) :=
   -- Porting note: times out without the (R := R)
   Module.IsTorsionBySet.module <| torsionBySet_isTorsionBySet (R := R) I
@@ -638,7 +637,6 @@ instance (I : Ideal R) {S : Type*} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : IsScalarTower S (R ⧸ I) (torsionBySet R M I) :=
   inferInstance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `a`-torsion submodule as an `(R ⧸ R ∙ a)`-module. -/
 instance instModuleQuotientTorsionBy (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
   Module.IsTorsionBySet.module <|
diff --git a/Mathlib/Algebra/Module/ZLattice/Basic.lean b/Mathlib/Algebra/Module/ZLattice/Basic.lean
index 0caf0441241bf3..8a355443e4c310 100644
--- a/Mathlib/Algebra/Module/ZLattice/Basic.lean
+++ b/Mathlib/Algebra/Module/ZLattice/Basic.lean
@@ -348,7 +348,6 @@ theorem fundamentalDomain_measurableSet [MeasurableSpace E] [OpensMeasurableSpac
     simp only [D, fundamentalDomain, Set.mem_Ico, Set.mem_setOf_eq, LinearEquiv.coe_coe,
       Set.mem_preimage, Basis.equivFun_apply, Set.mem_pi, Set.mem_univ, forall_true_left]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- For a ℤ-lattice `Submodule.span ℤ (Set.range b)`, proves that the set defined
 by `ZSpan.fundamentalDomain` is a fundamental domain. -/
 protected theorem isAddFundamentalDomain [Finite ι] [MeasurableSpace E] [OpensMeasurableSpace E]
@@ -364,7 +363,6 @@ protected theorem isAddFundamentalDomain' [Finite ι] [MeasurableSpace E] [Opens
     IsAddFundamentalDomain (span ℤ (Set.range b)).toAddSubgroup (fundamentalDomain b) μ :=
   ZSpan.isAddFundamentalDomain b μ
 
-set_option backward.isDefEq.respectTransparency false in
 theorem measure_fundamentalDomain_ne_zero [Finite ι] [MeasurableSpace E] [BorelSpace E]
     {μ : Measure E} [Measure.IsAddHaarMeasure μ] :
     μ (fundamentalDomain b) ≠ 0 := by
@@ -514,14 +512,12 @@ instance instModuleFinite_of_discrete_submodule {E : Type*} [NormedAddCommGroup
       Submodule.map_top, range_subtype, h_img]⟩
   exact ZLattice.module_finite ℝ L₀
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ZLattice.module_free [IsZLattice K L] : Module.Free ℤ L := by
   have : Module.Finite ℤ L := module_finite K L
   have : Module ℚ E := Module.compHom E (algebraMap ℚ K)
   have : IsAddTorsionFree E := .of_module_rat _
   infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 instance instModuleFree_of_discrete_submodule {E : Type*} [NormedAddCommGroup E]
     [NormedSpace ℝ E] [FiniteDimensional ℝ E] (L : Submodule ℤ E) [DiscreteTopology L] :
     Module.Free ℤ L := by
@@ -529,7 +525,6 @@ instance instModuleFree_of_discrete_submodule {E : Type*} [NormedAddCommGroup E]
   have : IsAddTorsionFree E := .of_module_rat _
   infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ZLattice.rank [hs : IsZLattice K L] : finrank ℤ L = finrank K E := by
   classical
   have : Module.Finite ℤ L := module_finite K L
@@ -616,7 +611,6 @@ variable {ι : Type*} [hs : IsZLattice K L] (b : Basis ι ℤ L)
 
 namespace Module.Basis
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Any `ℤ`-basis of `L` is also a `K`-basis of `E`. -/
 def ofZLatticeBasis : Basis ι K E := by
   have : Module.Finite ℤ L := ZLattice.module_finite K L
@@ -628,12 +622,10 @@ def ofZLatticeBasis : Basis ι K E := by
       range_subtype, top_le_iff, hs.span_top]
   · rw [← Fintype.card_congr e, ← finrank_eq_card_chooseBasisIndex, ZLattice.rank K L]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem ofZLatticeBasis_apply (i : ι) : b.ofZLatticeBasis K L i = b i := by
   simp [Basis.ofZLatticeBasis]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem ofZLatticeBasis_repr_apply (x : L) (i : ι) :
     (b.ofZLatticeBasis K L).repr x i = b.repr x i := by
@@ -645,7 +637,6 @@ theorem ofZLatticeBasis_repr_apply (x : L) (i : ι) :
     LinearEquiv.coe_coe, coe_subtype, ← b.ofZLatticeBasis_apply K, repr_self,
     Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ofZLatticeBasis_span : span ℤ (Set.range (b.ofZLatticeBasis K)) = L := by
   calc span ℤ (Set.range (b.ofZLatticeBasis K))
     _ = span ℤ (L.subtype '' Set.range b) := by congr; ext; simp
@@ -662,7 +653,6 @@ theorem ZLattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [Norm
   convert ZSpan.isAddFundamentalDomain (b.ofZLatticeBasis ℝ) μ
   all_goals exact (b.ofZLatticeBasis_span ℝ).symm
 
-set_option backward.isDefEq.respectTransparency false in
 instance instCountable_of_discrete_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [FiniteDimensional ℝ E] (L : Submodule ℤ E) [DiscreteTopology L] [IsZLattice ℝ L] :
     Countable L := by
@@ -694,7 +684,6 @@ section Basis
 
 variable {ι : Type*} [Fintype ι] (L : Submodule ℤ (ι → ℝ)) [DiscreteTopology L] [IsZLattice ℝ L]
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Return an arbitrary `ℤ`-basis of a lattice `L` of `ι → ℝ` indexed by `ι`.
 -/
@@ -794,7 +783,6 @@ variable {F : Type*} [NormedAddCommGroup F] [NormedSpace K F] [FiniteDimensional
   [ProperSpace F]
 variable (L : Submodule ℤ E) [DiscreteTopology L] [IsZLattice K L]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Module.Basis.ofZLatticeBasis_comap (e : F ≃L[K] E) {ι : Type*} (b : Basis ι ℤ L) :
     (b.ofZLatticeComap K L e.toLinearEquiv).ofZLatticeBasis K (ZLattice.comap K L e.toLinearMap) =
     (b.ofZLatticeBasis K L).map e.symm.toLinearEquiv := by
@@ -803,7 +791,6 @@ theorem Module.Basis.ofZLatticeBasis_comap (e : F ≃L[K] E) {ι : Type*} (b : B
 
 end NormedLinearOrderedField_comap
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `f` is periodic wrt a ℤ-lattice, then the range of `f` is compact. -/
 lemma IsZLattice.isCompact_range_of_periodic
     {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
diff --git a/Mathlib/Algebra/Module/ZLattice/Covolume.lean b/Mathlib/Algebra/Module/ZLattice/Covolume.lean
index 69430f7054a61a..6aa81b7721498f 100644
--- a/Mathlib/Algebra/Module/ZLattice/Covolume.lean
+++ b/Mathlib/Algebra/Module/ZLattice/Covolume.lean
@@ -88,7 +88,6 @@ theorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalD
   have : VAddInvariantMeasure L E μ := (inferInstance : VAddInvariantMeasure L.toAddSubgroup E μ)
   exact congr_arg ENNReal.toReal (h.covolume_eq_volume μ)
 
-set_option backward.isDefEq.respectTransparency false in
 set_option backward.privateInPublic true in
 theorem covolume_ne_zero : covolume L μ ≠ 0 := by
   rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain (Free.chooseBasis ℤ L) μ),
@@ -100,7 +99,6 @@ set_option backward.privateInPublic true in
 theorem covolume_pos : 0 < covolume L μ :=
   lt_of_le_of_ne ENNReal.toReal_nonneg (covolume_ne_zero L μ).symm
 
-set_option backward.isDefEq.respectTransparency false in
 set_option backward.privateInPublic true in
 theorem covolume_comap {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
     [MeasurableSpace F] [BorelSpace F] (ν : Measure F := by volume_tac) [Measure.IsAddHaarMeasure ν]
@@ -134,7 +132,6 @@ theorem covolume_eq_det {ι : Type*} [Fintype ι] [DecidableEq ι] (L : Submodul
   ext1
   exact b.ofZLatticeBasis_apply ℝ L _
 
-set_option backward.isDefEq.respectTransparency false in
 theorem covolume_eq_det_inv {ι : Type*} [Fintype ι] (L : Submodule ℤ (ι → ℝ))
     [DiscreteTopology L] [IsZLattice ℝ L] (b : Basis ι ℤ L) :
     covolume L = |(LinearEquiv.det (b.ofZLatticeBasis ℝ L).equivFun : ℝ)|⁻¹ := by
@@ -144,7 +141,6 @@ theorem covolume_eq_det_inv {ι : Type*} [Fintype ι] (L : Submodule ℤ (ι →
     IsUnit.unit_spec, ← Basis.det_basis, LinearEquiv.coe_det]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Let `L₁` be a sub-`ℤ`-lattice of `L₂`. Then the index of `L₁` inside `L₂` is equal to
 `covolume L₁ / covolume L₂`.
@@ -324,7 +320,6 @@ theorem tendsto_card_div_pow (b : Basis ι ℤ L) {s : Set (ι → ℝ)} (hs₁
       ENNReal.toReal_ofReal (covolume_pos L volume).le]
   · rw [frontier_equivFun, volume_image_eq_volume_div_covolume, hs₃, ENNReal.zero_div]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem tendsto_card_le_div {X : Set (ι → ℝ)} (hX : ∀ ⦃x⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X)
     {F : (ι → ℝ) → ℝ} (h₁ : ∀ x ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * (F x))
     (h₂ : IsBounded {x ∈ X | F x ≤ 1}) (h₃ : MeasurableSet {x ∈ X | F x ≤ 1})
@@ -353,7 +348,6 @@ variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDim
   [MeasurableSpace E] [BorelSpace E]
 variable (L : Submodule ℤ E) [DiscreteTopology L] [IsZLattice ℝ L]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A version of `ZLattice.covolume.tendsto_card_div_pow` for the `InnerProductSpace` case;
 see the `Naming convention` section in the introduction. -/
 theorem tendsto_card_div_pow' {s : Set E} (hs₁ : IsBounded s) (hs₂ : MeasurableSet s)
@@ -369,7 +363,6 @@ theorem tendsto_card_div_pow' {s : Set E} (hs₁ : IsBounded s) (hs₂ : Measura
   · rw [frontier_equivFun, volume_image_eq_volume_div_covolume', hs₃, ENNReal.zero_div]
     exact NullMeasurableSet.of_null hs₃
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A version of `ZLattice.covolume.tendsto_card_le_div` for the `InnerProductSpace` case;
 see the `Naming convention` section in the introduction. -/
 theorem tendsto_card_le_div' [Nontrivial E] {X : Set E} {F : E → ℝ}
diff --git a/Mathlib/Algebra/Order/Group/PartialSups.lean b/Mathlib/Algebra/Order/Group/PartialSups.lean
index b4afcb06ce2c2c..4fc60bf0c5403f 100644
--- a/Mathlib/Algebra/Order/Group/PartialSups.lean
+++ b/Mathlib/Algebra/Order/Group/PartialSups.lean
@@ -18,7 +18,6 @@ variable {α ι : Type*}
 
 variable [SemilatticeSup α] [Group α] [Preorder ι] [LocallyFiniteOrderBot ι]
 
-set_option backward.isDefEq.respectTransparency false in
 @[to_additive]
 lemma partialSups_const_mul [MulLeftMono α] (f : ι → α) (c : α) (i : ι) :
     partialSups (c * f ·) i = c * partialSups f i := map_partialSups (OrderIso.mulLeft _) ..
diff --git a/Mathlib/Algebra/Order/Interval/Basic.lean b/Mathlib/Algebra/Order/Interval/Basic.lean
index a2ac88d45a9575..37f12f4894bb9a 100644
--- a/Mathlib/Algebra/Order/Interval/Basic.lean
+++ b/Mathlib/Algebra/Order/Interval/Basic.lean
@@ -641,7 +641,6 @@ theorem length_neg : ∀ s : Interval α, (-s).length = s.length
   | ⊥ => rfl
   | (s : NonemptyInterval α) => s.length_neg
 
-set_option backward.isDefEq.respectTransparency false in
 theorem length_add_le : ∀ s t : Interval α, (s + t).length ≤ s.length + t.length
   | ⊥, _ => by simp
   | _, ⊥ => by simp
diff --git a/Mathlib/Algebra/Order/Module/HahnEmbedding.lean b/Mathlib/Algebra/Order/Module/HahnEmbedding.lean
index 647ec54b933fe0..efdec4b67e255e 100644
--- a/Mathlib/Algebra/Order/Module/HahnEmbedding.lean
+++ b/Mathlib/Algebra/Order/Module/HahnEmbedding.lean
@@ -772,7 +772,6 @@ theorem eval_lt [IsOrderedAddMonoid R] [Archimedean R] {x : M} (hx : x ∉ f.val
       exact (f.coeff_eq_of_mem y.val (by simp) hzy (by simp)).le
   exact hieq ▸ hi
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Extend `f` to a larger partial linear map by adding a new `x`. -/
 noncomputable
 def extendFun [IsOrderedAddMonoid R] [Archimedean R] {x : M} (hx : x ∉ f.val.domain) :
diff --git a/Mathlib/Algebra/Order/Ring/StandardPart.lean b/Mathlib/Algebra/Order/Ring/StandardPart.lean
index 8cc23926055062..c556eda3d24a9a 100644
--- a/Mathlib/Algebra/Order/Ring/StandardPart.lean
+++ b/Mathlib/Algebra/Order/Ring/StandardPart.lean
@@ -58,7 +58,6 @@ namespace FiniteElement
 noncomputable instance : CommRing (FiniteElement K) := by
   unfold FiniteElement; infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 instance : IsDomain (FiniteElement K) := by
   unfold FiniteElement; infer_instance
 
diff --git a/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean b/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
index a70bc170e19337..86c2963278f7dd 100644
--- a/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
+++ b/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
@@ -94,7 +94,6 @@ theorem trailingDegree_eq_natTrailingDegree (hp : p ≠ 0) :
     trailingDegree p = (natTrailingDegree p : ℕ∞) :=
   .symm <| ENat.coe_toNat <| mt trailingDegree_eq_top.1 hp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) :
     p.trailingDegree = n ↔ p.natTrailingDegree = n := by
   rw [trailingDegree_eq_natTrailingDegree hp, Nat.cast_inj]
@@ -159,7 +158,6 @@ theorem trailingDegree_ne_of_natTrailingDegree_ne {n : ℕ} :
     p.natTrailingDegree ≠ n → trailingDegree p ≠ n :=
   mt fun h => by rw [natTrailingDegree, h, ENat.toNat_coe]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
     (H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p := by
   rwa [trailingDegree_eq_natTrailingDegree hp, Nat.cast_le] at H
diff --git a/Mathlib/Algebra/Star/CentroidHom.lean b/Mathlib/Algebra/Star/CentroidHom.lean
index 7d80ae36ebdbc8..f9426a4854b033 100644
--- a/Mathlib/Algebra/Star/CentroidHom.lean
+++ b/Mathlib/Algebra/Star/CentroidHom.lean
@@ -61,7 +61,6 @@ instance instStarAddMonoidCenter : StarAddMonoid (Subsemiring.center (CentroidHo
   star_involutive f := SetCoe.ext (star_involutive f.val)
   star_add f g := SetCoe.ext (star_add f.val g.val)
 
-set_option backward.isDefEq.respectTransparency false in
 instance : StarRing (Subsemiring.center (CentroidHom α)) where
   __ := instStarAddMonoidCenter
   star_mul f g := by
diff --git a/Mathlib/AlgebraicGeometry/Fiber.lean b/Mathlib/AlgebraicGeometry/Fiber.lean
index 2f021ec360d2d4..ea80f2171db8aa 100644
--- a/Mathlib/AlgebraicGeometry/Fiber.lean
+++ b/Mathlib/AlgebraicGeometry/Fiber.lean
@@ -58,7 +58,6 @@ lemma Scheme.Hom.fiberToSpecResidueField_apply (f : X ⟶ Y) (y : Y) (x : f.fibe
   Subsingleton.elim (α := PrimeSpectrum _) _ _
 
 set_option backward.whnf.reducibleClassField false in
-set_option backward.isDefEq.respectTransparency false in
 lemma isPullback_fiberToSpecResidueField_of_isPullback {P X Y Z : Scheme.{u}} {fst : P ⟶ X}
     {snd : P ⟶ Y} {f : X ⟶ Z} {g : Y ⟶ Z} (h : IsPullback fst snd f g) (y : Y) :
     IsPullback (pullback.map _ _ _ _ fst (Spec.map (g.residueFieldMap y)) g h.w.symm (by simp))
@@ -70,7 +69,6 @@ lemma isPullback_fiberToSpecResidueField_of_isPullback {P X Y Z : Scheme.{u}} {f
   · simpa using (IsPullback.of_hasPullback _ _).paste_horiz h
   · simp [Scheme.Hom.fiberToSpecResidueField]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The morphism from the fiber of `Spec S ⟶ Spec R` at some prime `p` to `Spec κ(p)`
 is isomorphic to the map induced by `κ(p) ⟶ κ(p) ⊗[R] S`. -/
 noncomputable def Spec.fiberToSpecResidueFieldIso (R S : Type u) [CommRing R] [CommRing S]
diff --git a/Mathlib/AlgebraicGeometry/IdealSheaf/Subscheme.lean b/Mathlib/AlgebraicGeometry/IdealSheaf/Subscheme.lean
index fe242476886e1a..41badf6aa6f078 100644
--- a/Mathlib/AlgebraicGeometry/IdealSheaf/Subscheme.lean
+++ b/Mathlib/AlgebraicGeometry/IdealSheaf/Subscheme.lean
@@ -659,7 +659,6 @@ abbrev Hom.image : Scheme.{u} := f.ker.subscheme
 /-- The embedding from the scheme-theoretic image to the codomain. -/
 abbrev Hom.imageι : f.image ⟶ Y := f.ker.subschemeι
 
-set_option backward.isDefEq.respectTransparency false in
 lemma ideal_ker_le_ker_ΓSpecIso_inv_comp :
     f.ker.ideal U ≤ RingHom.ker ((ΓSpecIso Γ(Y, ↑U)).inv ≫
       (pullback.snd f U.1.ι ≫ U.1.toSpecΓ).appTop).hom := by
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean b/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean
index daa38a510bca6b..2ef75b5ce55e2b 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean
@@ -86,7 +86,6 @@ instance {X : Scheme} (r : Γ(X, ⊤)) :
 lemma isRetrocompact_basicOpen (s : Γ(X, ⊤)) : IsRetrocompact (X := X) (X.basicOpen s) :=
   IsRetrocompact_iff_isSpectralMap_subtypeVal.mpr (X.basicOpen s).ι.isSpectralMap
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Superseded by `isAffine_of_isAffineOpen_basicOpen`. -/
 private lemma isAffine_of_isAffineOpen_basicOpen_aux (s : Set Γ(X, ⊤))
     (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)) :
@@ -106,7 +105,6 @@ private lemma isAffine_of_isAffineOpen_basicOpen_aux (s : Set Γ(X, ⊤))
   · rw [← Opens.coe_inf, ← X.basicOpen_res _ (homOfLE le_top).op]
     exact (V.2.basicOpen _).isCompact
 
-set_option backward.isDefEq.respectTransparency false in
 @[stacks 01QF]
 lemma isAffine_of_isAffineOpen_basicOpen (s : Set Γ(X, ⊤))
     (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)) :
@@ -132,7 +130,6 @@ lemma isAffine_of_isAffineOpen_basicOpen (s : Set Γ(X, ⊤))
     convert isIso_ΓSpec_adjunction_unit_app_basicOpen i.1 using 0
     exact congr(IsIso ((ΓSpec.adjunction.unit.app X).app $(by simp)))
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 If `s` is a spanning set of `Γ(X, U)`, such that each `X.basicOpen i` is affine, then `U` is also
 affine.
@@ -185,7 +182,6 @@ instance (priority := 100) isAffineHom_of_isAffine [IsAffine X] [IsAffine Y] : I
 lemma isAffine_of_isAffineHom [IsAffineHom f] [IsAffine Y] : IsAffine X :=
   (HasAffineProperty.iff_of_isAffine (P := @IsAffineHom) (f := f)).mp inferInstance
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isAffineHom_of_forall_exists_isAffineOpen
     (H : ∀ x : Y, ∃ U : Y.Opens, x ∈ U ∧ IsAffineOpen U ∧ IsAffineOpen (f ⁻¹ᵁ U)) :
     IsAffineHom f := by
@@ -204,7 +200,6 @@ instance {X Y S : Scheme} (f : X ⟶ S) (g : Y ⟶ S) [IsAffineHom g] [IsAffine
   letI : IsAffineHom (pullback.fst f g) := MorphismProperty.pullback_fst _ _ ‹_›
   isAffine_of_isAffineHom (pullback.fst f g)
 
-set_option backward.isDefEq.respectTransparency false in
 instance {U V X : Scheme.{u}} (f : U ⟶ X) (g : V ⟶ X) [IsAffineHom f] [IsAffineHom g] :
     IsAffineHom (coprod.desc f g) := by
   refine ⟨fun W hW ↦ ?_⟩
@@ -268,7 +263,6 @@ lemma IsAffineOpen.isCompact_pullback_inf {X Y Z : Scheme.{u}} {f : X ⟶ Z} {g
   convert isCompact_range p.continuous
   simp [p, Scheme.Pullback.range_map]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isIso_morphismRestrict_iff_isIso_app [IsAffineHom f] {U : Y.Opens} (hU : IsAffineOpen U) :
     IsIso (f ∣_ U) ↔ IsIso (f.app U) := by
   have : IsAffine U := hU
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/SmoothFiber.lean b/Mathlib/AlgebraicGeometry/Morphisms/SmoothFiber.lean
index 1c1cd73be70a45..246eabc7614c1e 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/SmoothFiber.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/SmoothFiber.lean
@@ -28,7 +28,6 @@ namespace AlgebraicGeometry
 
 variable {X Y : Scheme.{u}} (f : X ⟶ Y)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `f : X ⟶ Y` is locally of finite presentation, flat and has smooth fibers, then `f` is
 smooth. -/
 lemma Smooth.of_smooth_fiberToSpecResidueField [LocallyOfFinitePresentation f] [Flat f]
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean
index a2eca72432a325..f1565891c0d8bf 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean
@@ -407,7 +407,6 @@ def CStarAlgebra.spectralOrder : PartialOrder A where
       quasispectrumRestricts_iff_spectrumRestricts_inr' ℂ] at hxy hyz ⊢
     exact ⟨by simpa using hyz.1.add hxy.1, by simpa using hyz.2.nnreal_add hyz.1 hxy.1 hxy.2⟩
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 variable (A) in
 /-- The `CStarAlgebra.spectralOrder` on a C⋆-algebra is a `StarOrderedRing`. -/
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
index 0add8ceb705083..ff56f4cf8b1f0d 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
@@ -61,7 +61,6 @@ theorem cfc_tsub {A : Type*} [TopologicalSpace A] [Ring A] [PartialOrder A] [Sta
     exact continuous_subtype_val.comp_continuousOn <|
       ContinuousOn.comp ‹_› continuous_real_toNNReal.continuousOn <| ha'.image ▸ Set.mapsTo_image ..
 
-set_option backward.isDefEq.respectTransparency false in
 theorem cfcₙ_tsub {A : Type*} [TopologicalSpace A] [NonUnitalRing A] [PartialOrder A] [StarRing A]
     [StarOrderedRing A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A]
     [IsTopologicalRing A] [T2Space A] [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
@@ -90,7 +89,6 @@ noncomputable instance instPartialOrder : PartialOrder A⁺¹ :=
 instance instStarOrderedRing : StarOrderedRing A⁺¹ :=
     CStarAlgebra.spectralOrderedRing _
 
-set_option backward.isDefEq.respectTransparency false in
 lemma inr_le_iff (a b : A) (ha : IsSelfAdjoint a := by cfc_tac)
     (hb : IsSelfAdjoint b := by cfc_tac) :
     (a : A⁺¹) ≤ (b : A⁺¹) ↔ a ≤ b := by
@@ -108,7 +106,6 @@ lemma inr_nonneg_iff {a : A} : 0 ≤ (a : A⁺¹) ↔ 0 ≤ a := by
     · exact isSelfAdjoint_inr (R := ℂ) |>.mp <| .of_nonneg h
     · exact .of_nonneg h
 
-set_option backward.isDefEq.respectTransparency false in
 lemma nnreal_cfcₙ_eq_cfc_inr (a : A) (f : ℝ≥0 → ℝ≥0)
     (hf₀ : f 0 = 0 := by cfc_zero_tac) : cfcₙ f a = cfc f (a : A⁺¹) :=
   cfcₙ_eq_cfc_inr inr_nonneg_iff ..
@@ -180,14 +177,12 @@ lemma CStarAlgebra.star_mul_le_algebraMap_norm_sq {a : A} :
 
 end StarOrderedRing
 
-set_option backward.isDefEq.respectTransparency false in
 lemma IsSelfAdjoint.toReal_spectralRadius_eq_norm {a : A} (ha : IsSelfAdjoint a) :
     (spectralRadius ℝ a).toReal = ‖a‖ := by
   simp [ha.spectrumRestricts.spectralRadius_eq, ha.spectralRadius_eq_nnnorm]
 
 namespace CStarAlgebra
 
-set_option backward.isDefEq.respectTransparency false in
 lemma norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
     (ha : IsSelfAdjoint a := by cfc_tac) : ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a := by
   have ha' : SpectrumRestricts a Complex.reCLM := ha.spectrumRestricts
@@ -196,7 +191,6 @@ lemma norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
 
 variable [PartialOrder A] [StarOrderedRing A]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma nnnorm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
     ‖a‖₊ ∈ spectrum ℝ≥0 a := by
   have : IsSelfAdjoint a := .of_nonneg ha
@@ -292,7 +286,6 @@ lemma CStarAlgebra.isUnit_of_le (a : A) {b : A} (hab : a ≤ b)
   peel h₀ with r hr _
   exact this.trans hab
 
-set_option backward.isDefEq.respectTransparency false in
 lemma le_iff_norm_sqrt_mul_rpow (a b : A) (ha : 0 ≤ a := by cfc_tac)
     (hb : IsStrictlyPositive b := by cfc_tac) :
     a ≤ b ↔ ‖sqrt a * (b : A) ^ (-(1 / 2) : ℝ)‖ ≤ 1 := by
@@ -412,14 +405,12 @@ variable {A : Type*} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing
 
 namespace CStarAlgebra
 
-set_option backward.isDefEq.respectTransparency false in
 open ComplexOrder in
 instance instNonnegSpectrumClassComplexNonUnital : NonnegSpectrumClass ℂ A where
   quasispectrum_nonneg_of_nonneg a ha x hx := by
     rw [Unitization.quasispectrum_eq_spectrum_inr' ℂ ℂ a] at hx
     exact spectrum_nonneg_of_nonneg (Unitization.inr_nonneg_iff.mpr ha) hx
 
-set_option backward.isDefEq.respectTransparency false in
 lemma norm_le_norm_of_nonneg_of_le {a b : A} (ha : 0 ≤ a := by cfc_tac) (hab : a ≤ b) :
     ‖a‖ ≤ ‖b‖ := by
   suffices ∀ a b : A⁺¹, 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖ by
@@ -445,7 +436,6 @@ theorem nnnorm_le_nnnorm_of_nonneg_of_le {a : A} {b : A} (ha : 0 ≤ a := by cfc
     ‖a‖₊ ≤ ‖b‖₊ :=
   norm_le_norm_of_nonneg_of_le ha hab
 
-set_option backward.isDefEq.respectTransparency false in
 lemma star_left_conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
     star a * b * a ≤ ‖b‖ • (star a * a) := by
   suffices ∀ a b : A⁺¹, IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a) by
@@ -467,7 +457,6 @@ lemma star_right_conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cf
 @[deprecated (since := "2025-10-20")] alias conjugate_le_norm_smul' :=
   star_right_conjugate_le_norm_smul
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The set of nonnegative elements in a C⋆-algebra is closed. -/
 lemma isClosed_nonneg : IsClosed {a : A | 0 ≤ a} := by
   suffices IsClosed {a : A⁺¹ | 0 ≤ a} by
@@ -489,7 +478,6 @@ section Icc
 
 open Unitization Set Metric
 
-set_option backward.isDefEq.respectTransparency false in
 lemma inr_mem_Icc_iff_norm_le {x : A} :
     (x : A⁺¹) ∈ Icc 0 1 ↔ 0 ≤ x ∧ ‖x‖ ≤ 1 := by
   simp only [mem_Icc, inr_nonneg_iff, and_congr_right_iff]
diff --git a/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean b/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
index 0f31ff4afcb676..615d3890440f8d 100644
--- a/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
+++ b/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
@@ -202,7 +202,6 @@ end PosRealAction
 
 section RealAddAction
 
-set_option backward.whnf.reducibleClassField false in
 instance : AddAction ℝ ℍ where
   vadd x z := mk (x + z) <| by simpa using z.im_pos
   zero_vadd _ := by simp [HVAdd.hVAdd]
diff --git a/Mathlib/Analysis/Convex/Intrinsic.lean b/Mathlib/Analysis/Convex/Intrinsic.lean
index 09b04d1d8795d6..14f5aa3e67192c 100644
--- a/Mathlib/Analysis/Convex/Intrinsic.lean
+++ b/Mathlib/Analysis/Convex/Intrinsic.lean
@@ -311,7 +311,6 @@ private theorem aux {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
 
 variable [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] {s : Set V}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The intrinsic interior of a nonempty convex set is nonempty. -/
 protected theorem Set.Nonempty.intrinsicInterior (hscv : Convex ℝ s) (hsne : s.Nonempty) :
     (intrinsicInterior ℝ s).Nonempty := by
diff --git a/Mathlib/Analysis/InnerProductSpace/Basic.lean b/Mathlib/Analysis/InnerProductSpace/Basic.lean
index 766902f83edf5e..a95d9fa0ba15c1 100644
--- a/Mathlib/Analysis/InnerProductSpace/Basic.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Basic.lean
@@ -916,7 +916,6 @@ local notation "⟪" x ", " y "⟫" => inner 𝕜 x y
 since `𝕜` does not appear in the return type `Inner ℝ E`. -/
 def Inner.rclikeToReal : Inner ℝ E where inner x y := re ⟪x, y⟫
 
-set_option backward.whnf.reducibleClassField false in
 /-- A general inner product space structure implies a real inner product structure.
 
 This is not registered as an instance since
diff --git a/Mathlib/Analysis/InnerProductSpace/PiL2.lean b/Mathlib/Analysis/InnerProductSpace/PiL2.lean
index c7d443374de325..f8d4d22b962e8a 100644
--- a/Mathlib/Analysis/InnerProductSpace/PiL2.lean
+++ b/Mathlib/Analysis/InnerProductSpace/PiL2.lean
@@ -1187,7 +1187,6 @@ noncomputable def LinearIsometry.extend (L : S →ₗᵢ[𝕜] V) : V →ₗᵢ[
     { toLinearMap := M
       norm_map' := M_norm_map }
 
-set_option backward.isDefEq.respectTransparency false in
 theorem LinearIsometry.extend_apply (L : S →ₗᵢ[𝕜] V) (s : S) : L.extend s = L s := by
   simp only [LinearIsometry.extend, ← LinearIsometry.coe_toLinearMap]
   simp only [add_eq_left, LinearIsometry.coe_toLinearMap,
diff --git a/Mathlib/Analysis/InnerProductSpace/ProdL2.lean b/Mathlib/Analysis/InnerProductSpace/ProdL2.lean
index 6151fe43713f85..ccb157e525112e 100644
--- a/Mathlib/Analysis/InnerProductSpace/ProdL2.lean
+++ b/Mathlib/Analysis/InnerProductSpace/ProdL2.lean
@@ -83,7 +83,6 @@ namespace Submodule
 
 variable (K : Submodule 𝕜 E) [K.HasOrthogonalProjection] (x : E)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If a subspace `K` of an inner product space `E` admits an orthogonal projection, then `E` is
 isometrically isomorphic to the `L²` product of `K` and `Kᗮ`. -/
 @[simps! symm_apply]
@@ -95,7 +94,6 @@ def orthogonalDecomposition : E ≃ₗᵢ[𝕜] WithLp 2 (K × Kᗮ) where
       K.norm_sq_eq_add_norm_sq_projection]
     simp [starProjection_apply_eq_isComplProjection]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem orthogonalDecomposition_apply :
     K.orthogonalDecomposition x =
diff --git a/Mathlib/Analysis/InnerProductSpace/Projection/Basic.lean b/Mathlib/Analysis/InnerProductSpace/Projection/Basic.lean
index b2870273babda1..ba54c4b4b01495 100644
--- a/Mathlib/Analysis/InnerProductSpace/Projection/Basic.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Projection/Basic.lean
@@ -139,7 +139,6 @@ theorem orthogonalProjectionFn_norm_sq (v : E) :
     orthogonalProjectionFn_inner_eq_zero _ _ (orthogonalProjectionFn_mem v)
   convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2 <;> simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The orthogonal projection onto a complete subspace. -/
 def orthogonalProjection : E →L[𝕜] K :=
   LinearMap.mkContinuous
diff --git a/Mathlib/Analysis/InnerProductSpace/Projection/FiniteDimensional.lean b/Mathlib/Analysis/InnerProductSpace/Projection/FiniteDimensional.lean
index d508b0c112a6ff..5bf40d464255a7 100644
--- a/Mathlib/Analysis/InnerProductSpace/Projection/FiniteDimensional.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Projection/FiniteDimensional.lean
@@ -57,7 +57,6 @@ variable [FiniteDimensional 𝕜 K]
 theorem topologicalClosure_eq_self : K.topologicalClosure = K :=
   K.closed_of_finiteDimensional.submodule_topologicalClosure_eq
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem det_reflection : LinearMap.det K.reflection.toLinearMap = (-1) ^ finrank 𝕜 Kᗮ := by
   by_cases hK : FiniteDimensional 𝕜 Kᗮ
diff --git a/Mathlib/Analysis/InnerProductSpace/Projection/Submodule.lean b/Mathlib/Analysis/InnerProductSpace/Projection/Submodule.lean
index 26181f4995d5f7..b1e411c8a0bdf2 100644
--- a/Mathlib/Analysis/InnerProductSpace/Projection/Submodule.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Projection/Submodule.lean
@@ -166,7 +166,6 @@ theorem topologicalClosure_eq_top_iff [CompleteSpace E] :
   · rw [← Submodule.triorthogonal_eq_orthogonal, h, Submodule.top_orthogonal_eq_bot]
   · rw [h, Submodule.bot_orthogonal_eq_top]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem orthogonalProjection_apply_eq_linearProjOfIsCompl [K.HasOrthogonalProjection] (x : E) :
     K.orthogonalProjection x =
       K.linearProjOfIsCompl _ Submodule.isCompl_orthogonal_of_hasOrthogonalProjection x := by
diff --git a/Mathlib/Analysis/InnerProductSpace/Semisimple.lean b/Mathlib/Analysis/InnerProductSpace/Semisimple.lean
index 645d0477bee7b5..2bc142dee93cee 100644
--- a/Mathlib/Analysis/InnerProductSpace/Semisimple.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Semisimple.lean
@@ -31,7 +31,6 @@ lemma orthogonalComplement_mem_invtSubmodule (hp : p ∈ T.invtSubmodule) :
     pᗮ ∈ T.invtSubmodule :=
   fun x hx y hy ↦ hT y x ▸ hx (T y) (hp hy)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Symmetric operators are semisimple on finite-dimensional subspaces. -/
 theorem isFinitelySemisimple :
     T.IsFinitelySemisimple := by
diff --git a/Mathlib/Analysis/InnerProductSpace/Subspace.lean b/Mathlib/Analysis/InnerProductSpace/Subspace.lean
index 59c63653b60b4b..3f2223e5325732 100644
--- a/Mathlib/Analysis/InnerProductSpace/Subspace.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Subspace.lean
@@ -250,7 +250,6 @@ local notation "⟪" x ", " y "⟫" => inner 𝕜 x y
 
 variable {ι : Type*} {G : ι → Type*}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- An orthogonal family forms an independent family of subspaces; that is, any collection of
 elements each from a different subspace in the family is linearly independent. In particular, the
 pairwise intersections of elements of the family are 0. -/
diff --git a/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean b/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean
index bb877bfad0ce27..fd6bf324e45160 100644
--- a/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean
+++ b/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean
@@ -422,7 +422,6 @@ set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma norm_assoc (x : E ⊗[𝕜] F ⊗[𝕜] G) :
     ‖TensorProduct.assoc 𝕜 E F G x‖ = ‖x‖ := assocIsometry 𝕜 E F G |>.norm_map x
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma nnnorm_assoc (x : E ⊗[𝕜] F ⊗[𝕜] G) :
     ‖TensorProduct.assoc 𝕜 E F G x‖₊ = ‖x‖₊ := assocIsometry 𝕜 E F G |>.nnnorm_map x
 
diff --git a/Mathlib/Analysis/InnerProductSpace/l2Space.lean b/Mathlib/Analysis/InnerProductSpace/l2Space.lean
index 8ec4ab17058f9d..a9ea606debc3ff 100644
--- a/Mathlib/Analysis/InnerProductSpace/l2Space.lean
+++ b/Mathlib/Analysis/InnerProductSpace/l2Space.lean
@@ -399,7 +399,6 @@ instance instFunLike : FunLike (HilbertBasis ι 𝕜 E) ι E where
     ext
     exact congr_fun h i
 
-set_option backward.whnf.reducibleClassField false in
 @[simp]
 protected theorem repr_symm_single [DecidableEq ι] (b : HilbertBasis ι 𝕜 E) (i : ι) :
     b.repr.symm (lp.single 2 i (1 : 𝕜)) = b i := by
diff --git a/Mathlib/Analysis/LocallyConvex/WeakDual.lean b/Mathlib/Analysis/LocallyConvex/WeakDual.lean
index 91a26136f644be..1e4786e7d1c3cf 100644
--- a/Mathlib/Analysis/LocallyConvex/WeakDual.lean
+++ b/Mathlib/Analysis/LocallyConvex/WeakDual.lean
@@ -180,7 +180,6 @@ theorem mem_span_iff_bound {f : ι → E →ₗ[𝕜] 𝕜} (φ : E →ₗ[𝕜]
 
 variable [AddCommGroup F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The Weak Representation Theorem: Every continuous functional on `E` endowed with
 the `σ(E, F; B)`-topology is of the form `x ↦ B(x, y)` for some `y : F`. -/
 theorem dualEmbedding_surjective : Function.Surjective (WeakBilin.eval B) := fun f ↦ by
diff --git a/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean b/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
index 034fa105d386a6..4f7173b9a9d296 100644
--- a/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
+++ b/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
@@ -983,7 +983,6 @@ theorem Topology.IsInducing.polynormableSpace [PolynormableSpace 𝕜₂ F]
     PolynormableSpace 𝕜 E :=
   hf.withSeminorms (PolynormableSpace.withSeminorms 𝕜₂ F) |>.toPolynormableSpace
 
-set_option backward.isDefEq.respectTransparency false in
 instance [PolynormableSpace 𝕜₂ F] {S : Submodule 𝕜₂ F} :
     PolynormableSpace 𝕜₂ S :=
   IsInducing.polynormableSpace (f := S.subtype) .subtypeVal
diff --git a/Mathlib/Analysis/Matrix/HermitianFunctionalCalculus.lean b/Mathlib/Analysis/Matrix/HermitianFunctionalCalculus.lean
index 56fa8f5c42a766..53c18c49d92b9f 100644
--- a/Mathlib/Analysis/Matrix/HermitianFunctionalCalculus.lean
+++ b/Mathlib/Analysis/Matrix/HermitianFunctionalCalculus.lean
@@ -135,12 +135,10 @@ protected noncomputable def cfc (f : ℝ → ℝ) : Matrix n n 𝕜 :=
   conjStarAlgAut 𝕜 _ hA.eigenvectorUnitary (diagonal (RCLike.ofReal ∘ f ∘ hA.eigenvalues))
 
 set_option backward.whnf.reducibleClassField false in
-set_option backward.isDefEq.respectTransparency false in
 lemma cfcHom_eq_cfcAux : cfcHom hA.isSelfAdjoint = hA.cfcAux :=
   cfcHom_eq_of_continuous_of_map_id hA hA.cfcAux
     hA.isClosedEmbedding_cfcAux.continuous hA.cfcAux_id
 
-set_option backward.isDefEq.respectTransparency false in
 instance instContinuousFunctionalCalculusIsClosedEmbedding :
     ClosedEmbeddingContinuousFunctionalCalculus ℝ (Matrix n n 𝕜) IsSelfAdjoint where
   isClosedEmbedding _ hA := cfcHom_eq_cfcAux hA ▸ hA.isHermitian.isClosedEmbedding_cfcAux
diff --git a/Mathlib/Analysis/Matrix/Normed.lean b/Mathlib/Analysis/Matrix/Normed.lean
index b276630e80576b..a2948c699380bf 100644
--- a/Mathlib/Analysis/Matrix/Normed.lean
+++ b/Mathlib/Analysis/Matrix/Normed.lean
@@ -325,7 +325,6 @@ section NonUnitalSeminormedRing
 
 variable [NonUnitalSeminormedRing α]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem linfty_opNNNorm_mul (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊ := by
   simp_rw [linfty_opNNNorm_def, Matrix.mul_apply]
   calc
diff --git a/Mathlib/Analysis/Matrix/Order.lean b/Mathlib/Analysis/Matrix/Order.lean
index 4bfd58faae5612..6dda18539fe821 100644
--- a/Mathlib/Analysis/Matrix/Order.lean
+++ b/Mathlib/Analysis/Matrix/Order.lean
@@ -174,7 +174,6 @@ set_option backward.isDefEq.respectTransparency false in
 @[deprecated CFC.sqrt_eq_zero_iff (since := "2025-09-22")]
 lemma sqrt_eq_zero_iff : CFC.sqrt A = 0 ↔ A = 0 := CFC.sqrt_eq_zero_iff A
 
-set_option backward.isDefEq.respectTransparency false in
 @[deprecated CFC.sqrt_eq_one_iff (since := "2025-09-23")]
 lemma sqrt_eq_one_iff : CFC.sqrt A = 1 ↔ A = 1 := CFC.sqrt_eq_one_iff A
 
diff --git a/Mathlib/Analysis/Normed/Affine/Isometry.lean b/Mathlib/Analysis/Normed/Affine/Isometry.lean
index f4d4485c4c32c4..ec549fa8c0989e 100644
--- a/Mathlib/Analysis/Normed/Affine/Isometry.lean
+++ b/Mathlib/Analysis/Normed/Affine/Isometry.lean
@@ -840,7 +840,6 @@ theorem isometryEquivMap.coe_apply (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : Affine
     (g : E) : ↑(E.isometryEquivMap φ g) = φ g :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem isometryEquivMap.toAffineMap_eq (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁')
     [Nonempty E] :
diff --git a/Mathlib/Analysis/Normed/Module/Basic.lean b/Mathlib/Analysis/Normed/Module/Basic.lean
index a9ee3448c575b5..7c5606bd3db3f1 100644
--- a/Mathlib/Analysis/Normed/Module/Basic.lean
+++ b/Mathlib/Analysis/Normed/Module/Basic.lean
@@ -168,7 +168,6 @@ instance SeparationQuotient.instNormedSpace : NormedSpace 𝕜 (SeparationQuotie
 instance MulOpposite.instNormedSpace : NormedSpace 𝕜 Eᵐᵒᵖ where
   norm_smul_le _ x := norm_smul_le _ x.unop
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A subspace of a normed space is also a normed space, with the restriction of the norm. -/
 instance Submodule.normedSpace {𝕜 R : Type*} [SMul 𝕜 R] [NormedField 𝕜] [Ring R] {E : Type*}
     [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [Module R E] [IsScalarTower 𝕜 R E]
diff --git a/Mathlib/Analysis/Normed/Module/Complemented.lean b/Mathlib/Analysis/Normed/Module/Complemented.lean
index 81dab8b60651de..5d5b9d5351d6cf 100644
--- a/Mathlib/Analysis/Normed/Module/Complemented.lean
+++ b/Mathlib/Analysis/Normed/Module/Complemented.lean
@@ -38,7 +38,6 @@ section
 
 variable [CompleteSpace 𝕜]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ker_closedComplemented_of_finiteDimensional_range (f : E →L[𝕜] F)
     [FiniteDimensional 𝕜 f.range] : f.ker.ClosedComplemented := by
   set f' : E →L[𝕜] f.range := f.codRestrict _ (LinearMap.mem_range_self (f : E →ₗ[𝕜] F))
diff --git a/Mathlib/Analysis/Normed/Module/Connected.lean b/Mathlib/Analysis/Normed/Module/Connected.lean
index 33ea545b239bfd..bdc874ff7c15c6 100644
--- a/Mathlib/Analysis/Normed/Module/Connected.lean
+++ b/Mathlib/Analysis/Normed/Module/Connected.lean
@@ -255,7 +255,6 @@ section
 variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F]
   [IsTopologicalAddGroup F] [ContinuousSMul ℝ F]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `E` be a linear subspace in a real vector space.
 If `E` has codimension at least two, its complement is path-connected. -/
 theorem isPathConnected_compl_of_one_lt_codim {E : Submodule ℝ F}
diff --git a/Mathlib/Analysis/Normed/Module/HahnBanach.lean b/Mathlib/Analysis/Normed/Module/HahnBanach.lean
index 9c0e3d9ef33527..1d9d31eaec6787 100644
--- a/Mathlib/Analysis/Normed/Module/HahnBanach.lean
+++ b/Mathlib/Analysis/Normed/Module/HahnBanach.lean
@@ -126,7 +126,6 @@ lemma ContinuousLinearMap.exist_extension_of_finiteDimensional_range {p : Submod
   ext x
   simp [fi, e, hgf]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A finite-dimensional submodule over `ℝ` or `ℂ` is `Submodule.ClosedComplemented`. -/
 lemma Submodule.ClosedComplemented.of_finiteDimensional (p : Submodule 𝕜 F)
     [FiniteDimensional 𝕜 p] : p.ClosedComplemented :=
diff --git a/Mathlib/Analysis/Normed/Module/Span.lean b/Mathlib/Analysis/Normed/Module/Span.lean
index d2ec11305eb102..f0bb564181ed2d 100644
--- a/Mathlib/Analysis/Normed/Module/Span.lean
+++ b/Mathlib/Analysis/Normed/Module/Span.lean
@@ -101,7 +101,6 @@ namespace LinearIsometryEquiv
 
 variable [NormedDivisionRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [NormSMulClass 𝕜 E]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a unit element `x` of a normed space `E` over a field `𝕜`, the natural
 linear isometry equivalence from `𝕜` to the span of `x`. -/
 noncomputable def toSpanUnitSingleton (x : E) (hx : ‖x‖ = 1) :
diff --git a/Mathlib/Analysis/Normed/Operator/Extend.lean b/Mathlib/Analysis/Normed/Operator/Extend.lean
index 49c365b2facdf4..d531d90d03b091 100644
--- a/Mathlib/Analysis/Normed/Operator/Extend.lean
+++ b/Mathlib/Analysis/Normed/Operator/Extend.lean
@@ -149,7 +149,6 @@ variable [DivisionRing 𝕜] [DivisionRing 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜
 
 variable (f : E →ₛₗ[σ₁₂] F) (g : E →ₗ[𝕜] Eₗ)
 
-set_option backward.isDefEq.respectTransparency false in
 open scoped Classical in
 /-- Composition of a semilinear map `f` with the left inverse of a linear map `g` as a continuous
 linear map provided that the norm estimate `‖f x‖ ≤ C * ‖g x‖` holds for all `x : E`. -/
diff --git a/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean b/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
index acc127fb8ec14e..7da83abfe03fff 100644
--- a/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
+++ b/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
@@ -378,7 +378,6 @@ variable {R R₁ R₂ M M₁ : Type*}
 variable [Ring R] [SeminormedAddCommGroup M] [SeminormedAddCommGroup M₁]
 variable [Module R M] [Module R M₁]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A linear isometry between two modules restricts to a linear isometry
 from any submodule `p` of the domain onto the image of that submodule.
 
@@ -403,7 +402,6 @@ namespace Submodule
 
 variable {R' : Type*} [Ring R'] [Module R' E] (p : Submodule R' E)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- `Submodule.subtype` as a `LinearIsometry`. -/
 def subtypeₗᵢ : p →ₗᵢ[R'] E :=
   ⟨p.subtype, fun _ => rfl⟩
@@ -412,12 +410,10 @@ def subtypeₗᵢ : p →ₗᵢ[R'] E :=
 theorem coe_subtypeₗᵢ : ⇑p.subtypeₗᵢ = p.subtype :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem subtypeₗᵢ_toLinearMap : p.subtypeₗᵢ.toLinearMap = p.subtype :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem subtypeₗᵢ_toContinuousLinearMap : p.subtypeₗᵢ.toContinuousLinearMap = p.subtypeL :=
   rfl
@@ -1027,7 +1023,6 @@ theorem coe_prodAssoc_symm [Module R E₂] [Module R E₃] :
     ((prodAssoc R E E₂ E₃).symm : E × E₂ × E₃ → (E × E₂) × E₃) = (Equiv.prodAssoc E E₂ E₃).symm :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `p` is a submodule that is equal to `⊤`, then `LinearIsometryEquiv.ofTop p hp` is the
 "identity" equivalence between `p` and `E`. -/
 @[simps! toLinearEquiv apply symm_apply_coe]
@@ -1037,7 +1032,6 @@ def ofTop {R : Type*} [Ring R] [Module R E] (p : Submodule R E) (hp : p = ⊤) :
 variable {R E E₂ E₃} {R' : Type*} [Ring R']
 variable [Module R' E] (p q : Submodule R' E)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- `LinearEquiv.ofEq` as a `LinearIsometryEquiv`. -/
 def ofEq (hpq : p = q) : p ≃ₗᵢ[R'] q :=
   { LinearEquiv.ofEq p q hpq with norm_map' := fun _ => rfl }
@@ -1048,12 +1042,10 @@ variable {p q}
 theorem coe_ofEq_apply (h : p = q) (x : p) : (ofEq p q h x : E) = x :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem ofEq_symm (h : p = q) : (ofEq p q h).symm = ofEq q p h.symm :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem ofEq_rfl : ofEq p p rfl = LinearIsometryEquiv.refl R' p := rfl
 
@@ -1064,7 +1056,6 @@ variable [Ring R] [Ring R₂] [SeminormedAddCommGroup M] [SeminormedAddCommGroup
 variable [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R}
 variable {re₁₂ : RingHomInvPair σ₁₂ σ₂₁} {re₂₁ : RingHomInvPair σ₂₁ σ₁₂}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A linear isometry equivalence between two modules restricts to a
 linear isometry equivalence from any submodule `p` of the domain onto
 the image of that submodule.
@@ -1089,7 +1080,6 @@ theorem Module.Basis.ext_linearIsometryEquiv {ι : Type*} (b : Basis ι R E) {f
     (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ :=
   LinearIsometryEquiv.toLinearEquiv_injective <| b.ext' h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Reinterpret a `LinearIsometry` as a `LinearIsometryEquiv` to the range. -/
 @[simps! apply_coe]
 noncomputable def LinearIsometry.equivRange {R S : Type*} [Semiring R] [Ring S] [Module S E]
diff --git a/Mathlib/Analysis/ODE/PicardLindelof.lean b/Mathlib/Analysis/ODE/PicardLindelof.lean
index 1c7c34993a9b6a..f6b62ce57e9463 100644
--- a/Mathlib/Analysis/ODE/PicardLindelof.lean
+++ b/Mathlib/Analysis/ODE/PicardLindelof.lean
@@ -656,7 +656,6 @@ lemma weaken_lipschitz (hf : IsPicardLindelof f t₀ x₀ a r L K) {K' : ℝ≥0
   norm_le := hf.norm_le
   mul_max_le := hf.mul_max_le
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given `IsPicardLindelof` on a symmetric interval `[t₀ - ε, t₀ + ε]`, if we shrink the radius
 from `a` to `a'` with `a' ≤ a`, and choose any `r' < a'`, then there exists `ε' > 0` such that
 `IsPicardLindelof` holds on `[t₀ - ε', t₀ + ε']` with the new parameters. -/
diff --git a/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean b/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean
index 887731b608d81f..d42dd109fcb6be 100644
--- a/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean
@@ -156,7 +156,6 @@ def latticeBasis : Basis (Fin 2) ℤ L.lattice :=
 def latticeEquivProd : L.lattice ≃ₗ[ℤ] ℤ × ℤ :=
   L.latticeBasis.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite _ _ _ ≪≫ₗ .finTwoArrow ℤ ℤ
 
-set_option backward.isDefEq.respectTransparency false in
 lemma latticeEquiv_symm_apply (x : ℤ × ℤ) :
     (L.latticeEquivProd.symm x).1 = x.1 * L.ω₁ + x.2 * L.ω₂ := by
   simp [latticeEquivProd, Finsupp.linearCombination]
@@ -248,7 +247,6 @@ lemma differentiableOn_weierstrassPExcept (l₀ : ℂ) :
   · simp
   · exact .sub (.div (by fun_prop) (by fun_prop) (by aesop (add simp sub_eq_zero))) (by fun_prop)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma weierstrassPExcept_neg (l₀ : ℂ) (z : ℂ) :
     ℘[L - l₀] (-z) = ℘[L - -l₀] z := by
   simp only [weierstrassPExcept]
@@ -307,7 +305,6 @@ lemma differentiableOn_weierstrassP :
   convert L.differentiableOn_weierstrassPExcept _
   simp [L.ω₁_div_two_notMem_lattice]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma weierstrassP_neg (z : ℂ) : ℘[L] (-z) = ℘[L] z := by
   simp only [weierstrassP]
@@ -434,7 +431,6 @@ end derivWeierstrassPExcept
 
 section Periodicity
 
-set_option backward.isDefEq.respectTransparency false in
 lemma derivWeierstrassPExcept_add_coe (l₀ : ℂ) (z : ℂ) (l : L.lattice) :
     ℘'[L - l₀] (z + l) = ℘'[L - (l₀ - l)] z := by
   simp only [derivWeierstrassPExcept]
@@ -563,7 +559,6 @@ lemma differentiableOn_derivWeierstrassP :
   convert L.differentiableOn_derivWeierstrassPExcept _
   simp [L.ω₁_div_two_notMem_lattice]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma derivWeierstrassP_neg (z : ℂ) : ℘'[L] (-z) = - ℘'[L] z := by
   simp only [derivWeierstrassP]
@@ -573,7 +568,6 @@ lemma derivWeierstrassP_neg (z : ℂ) : ℘'[L] (-z) = - ℘'[L] z := by
   congr! with l
   ring
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma derivWeierstrassP_add_coe (z : ℂ) (l : L.lattice) :
     ℘'[L] (z + l) = ℘'[L] z := by
@@ -955,7 +949,6 @@ def G (n : ℕ) : ℂ := ∑' l : L.lattice, (l ^ n)⁻¹
 lemma sumInvPow_zero : L.sumInvPow 0 = L.G := by
   ext; simp [sumInvPow, G]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma G_eq_zero_of_odd (n : ℕ) (hn : Odd n) : L.G n = 0 := by
   rw [← CharZero.eq_neg_self_iff, G, ← tsum_neg, ← (Equiv.neg _).tsum_eq]
   congr with l
diff --git a/Mathlib/Analysis/SpecialFunctions/Exp.lean b/Mathlib/Analysis/SpecialFunctions/Exp.lean
index 8f0ba9147bdd03..2c930f7e77874f 100644
--- a/Mathlib/Analysis/SpecialFunctions/Exp.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Exp.lean
@@ -121,7 +121,6 @@ theorem ContinuousOn.cexp (h : ContinuousOn f s) : ContinuousOn (fun y => exp (f
 theorem Continuous.cexp (h : Continuous f) : Continuous fun y => exp (f y) :=
   continuous_iff_continuousAt.2 fun _ => h.continuousAt.cexp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The complex exponential function is uniformly continuous on left half planes. -/
 lemma UniformContinuousOn.cexp (a : ℝ) : UniformContinuousOn exp {x : ℂ | x.re ≤ a} := by
   have : Continuous (cexp - 1) := Continuous.sub (by fun_prop) continuous_one
diff --git a/Mathlib/CategoryTheory/Bicategory/Functor/StrictlyUnitary.lean b/Mathlib/CategoryTheory/Bicategory/Functor/StrictlyUnitary.lean
index 68db64fef309e6..725c25e7a34094 100644
--- a/Mathlib/CategoryTheory/Bicategory/Functor/StrictlyUnitary.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Functor/StrictlyUnitary.lean
@@ -171,7 +171,6 @@ def comp (F : StrictlyUnitaryLaxFunctor B C)
 section
 attribute [local ext] StrictlyUnitaryLaxFunctor
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Composition of `StrictlyUnitaryLaxFunctor` is strictly right unitary -/
 lemma comp_id (F : StrictlyUnitaryLaxFunctor B C) :
     F.comp (.id C) = F := by
diff --git a/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean b/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean
index e437f5c386a029..9d8c630450f7aa 100644
--- a/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean
@@ -374,7 +374,6 @@ lemma map_id_eq : map (𝟙 F) = 𝟭 (∫ᶜ F) :=
 
 end
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The natural isomorphism witnessing the pseudo-functoriality of `CoGrothendieck.map`. -/
 def mapCompIso (α : F ⟶ G) (β : G ⟶ H) : map (α ≫ β) ≅ map α ⋙ map β :=
   NatIso.ofComponents (fun _ ↦ eqToIso (by cat_disch)) (fun f ↦ by
diff --git a/Mathlib/CategoryTheory/Bicategory/Modification/Pseudo.lean b/Mathlib/CategoryTheory/Bicategory/Modification/Pseudo.lean
index 92bfb391a3da94..acc58bde5d0dff 100644
--- a/Mathlib/CategoryTheory/Bicategory/Modification/Pseudo.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Modification/Pseudo.lean
@@ -64,7 +64,6 @@ namespace Modification
 
 variable (Γ : Modification η θ)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The modification between the corresponding strong transformation of the underlying oplax
 functors. -/
 @[simps]
diff --git a/Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean b/Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean
index 245c5d38840f7d..8a763683680f46 100644
--- a/Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean
+++ b/Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean
@@ -227,7 +227,6 @@ def TransportEnrichment.ofOrdinaryEnrichedCategoryEquiv : TransportEnrichment F
 
 open EnrichedCategory
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 /-- If for a lax monoidal functor `F : V ⥤ W` the canonical function
 `(𝟙_ V ⟶ v) → (𝟙_ W ⟶ F.obj v)` is bijective, and `C` is an enriched ordinary category on `V`,
diff --git a/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean b/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean
index 95ce828c874d32..729e23b0280e8f 100644
--- a/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean
+++ b/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean
@@ -135,7 +135,6 @@ noncomputable def Fib.homMk {S : 𝒮} {a b : Fib p S} (φ : (ι S).obj a ⟶ (
     [IsHomLift p (𝟙 S) φ] : a ⟶ b :=
   (inducedFunctor _ S).preimage (Fiber.homMk p S φ)
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma Fib.map_homMk {S : 𝒮} {a b : Fib p S} (φ : (ι S).obj a ⟶ (ι S).obj b)
     [IsHomLift p (𝟙 S) φ] : (ι S).map (homMk φ) = φ := by
diff --git a/Mathlib/CategoryTheory/Join/Sum.lean b/Mathlib/CategoryTheory/Join/Sum.lean
index b5bf9e693e58b4..8ac7ac99bcacd4 100644
--- a/Mathlib/CategoryTheory/Join/Sum.lean
+++ b/Mathlib/CategoryTheory/Join/Sum.lean
@@ -53,7 +53,6 @@ instance : (fromSum C D).EssSurj where
     | left c => Functor.obj_mem_essImage _ (Sum.inl c)
     | right d => Functor.obj_mem_essImage _ (Sum.inr d)
 
-set_option backward.isDefEq.respectTransparency false in
 instance : (fromSum C D).Faithful where
   map_injective {x y} h h' heq := by
     cases h <;> cases h'
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Bifunctor.lean b/Mathlib/CategoryTheory/Limits/Preserves/Bifunctor.lean
index a685c31e2a294f..1d47d8d0519418 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Bifunctor.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Bifunctor.lean
@@ -30,7 +30,6 @@ open Category Limits Functor
 variable {J₁ J₂ : Type*} [Category* J₁] [Category* J₂]
   {C₁ C₂ C : Type*} [Category* C₁] [Category* C₂] [Category* C]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a bifunctor `G : C₁ ⥤ C₂ ⥤ C`, diagrams `K₁ : J₁ ⥤ C₁` and `K₂ : J₂ ⥤ C₂`, and cocones
 over these diagrams, `G.mapCocone₂ c₁ c₂` is the cocone over the diagram `J₁ × J₂ ⥤ C` obtained
 by applying `G` to both `c₁` and `c₂`. -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/Basic.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/Basic.lean
index 4c3d79c52dca34..82f7fa6364c135 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/Basic.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Categorical/Basic.lean
@@ -322,7 +322,6 @@ def toCatCommSqOver : (X ⥤ F ⊡ G) ⥤ CatCommSqOver F G X where
   map_id := by intros; ext <;> simp
   map_comp := by intros; ext <;> simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Interpret a `CatCommSqOver` as a functor to the categorical pullback. -/
 @[simps!]
 def CatCommSqOver.toFunctorToCategoricalPullback :
@@ -420,7 +419,6 @@ lemma toCatCommSqOver_mapIso_mkNatIso_eq_mkIso
       (by simpa [functorEquiv, toCatCommSqOver] using coh) := by
   ext <;> simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Comparing mkNatIso with the corresponding construction one can deduce from
 `functorEquiv`. -/
 lemma mkNatIso_eq
@@ -457,7 +455,6 @@ variable {A₁ : Type u₄} {B₁ : Type u₅} {C₁ : Type u₆}
   [Category.{v₄} A₁] [Category.{v₅} B₁] [Category.{v₆} C₁]
   {F₁ : A₁ ⥤ B₁} {G₁ : C₁ ⥤ B₁}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Functorially transform a `CatCommSqOver F G X` by whiskering it with a
 `CatCospanTransform`. -/
 @[simps!]
@@ -503,7 +500,6 @@ variable {A₂ : Type u₇} {B₂ : Type u₈} {C₂ : Type u₉}
   [Category.{v₇} A₂] [Category.{v₈} B₂] [Category.{v₉} C₂]
   {F₂ : A₂ ⥤ B₂} {G₂ : C₂ ⥤ B₂}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The construction `CatCommSqOver.transform` respects vertical composition
 of `CatCospanTransform`s. -/
 @[simps!]
@@ -516,7 +512,6 @@ def transformObjComp (X : Type u₁₀) [Category.{v₁₀} X]
       (Functor.associator _ _ _).symm)
     (fun {x y} f ↦ by ext <;> simp)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The construction `CatCommSqOver.transform` respects the identity
 `CatCospanTransform`s. -/
 @[simps!]
@@ -592,7 +587,6 @@ variable
     {X : Type u₄} {Y : Type u₅} {Z : Type u₆}
     [Category.{v₄} X] [Category.{v₅} Y] [Category.{v₆} Z]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A functor `U : X ⥤ Y` (functorially) induces a functor
 `CatCommSqOver F G Y ⥤ CatCommSqOver F G X` by whiskering left the underlying
 categorical commutative square by U. -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean b/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
index 9cd751f7699a6f..5e35a2aa24760e 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
@@ -165,7 +165,6 @@ theorem eq_zero_of_src {X Y : C} (o : IsZero X) (f : X ⟶ Y) : f = 0 :=
 theorem eq_zero_of_tgt {X Y : C} (o : IsZero Y) (f : X ⟶ Y) : f = 0 :=
   o.eq_of_tgt _ _
 
-set_option backward.whnf.reducibleClassField false in
 theorem iff_id_eq_zero (X : C) : IsZero X ↔ 𝟙 X = 0 :=
   ⟨fun h => h.eq_of_src _ _, fun h =>
     ⟨fun Y => ⟨⟨⟨0⟩, fun f => by
diff --git a/Mathlib/CategoryTheory/Localization/Monoidal/Basic.lean b/Mathlib/CategoryTheory/Localization/Monoidal/Basic.lean
index 00a1c9ebf884c0..d6250f253b9f38 100644
--- a/Mathlib/CategoryTheory/Localization/Monoidal/Basic.lean
+++ b/Mathlib/CategoryTheory/Localization/Monoidal/Basic.lean
@@ -108,7 +108,6 @@ local notation "L'" => toMonoidalCategory L W ε
 
 instance : (L').IsLocalization W := inferInstanceAs (L.IsLocalization W)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isInvertedBy₂ :
     MorphismProperty.IsInvertedBy₂ W W
       (curriedTensor C ⋙ (Functor.whiskeringRight C C D).obj L') := by
diff --git a/Mathlib/CategoryTheory/LocallyCartesianClosed/ExponentiableMorphism.lean b/Mathlib/CategoryTheory/LocallyCartesianClosed/ExponentiableMorphism.lean
index 42981de6d5ea3f..bba1179297f966 100644
--- a/Mathlib/CategoryTheory/LocallyCartesianClosed/ExponentiableMorphism.lean
+++ b/Mathlib/CategoryTheory/LocallyCartesianClosed/ExponentiableMorphism.lean
@@ -152,7 +152,6 @@ section
 def id (I : C) [ChosenPullbacksAlong (𝟙 I)] : ExponentiableMorphism (𝟙 I) :=
   ⟨𝟭 _, ofNatIsoLeft (F := 𝟭 _) Adjunction.id (pullbackId I).symm⟩
 
-set_option backward.whnf.reducibleClassField false in
 theorem id_pushforward (I : C) [ChosenPullbacksAlong (𝟙 I)] :
     (id I).pushforward = 𝟭 (Over I) := by
   dsimp +instances only [id]
@@ -186,7 +185,6 @@ def comp {I J K : C} (f : I ⟶ J) (g : J ⟶ K)
     ofNatIsoLeft (pullbackPushforwardAdj g |>.comp <| pullbackPushforwardAdj f)
     (pullbackComp f g).symm⟩
 
-set_option backward.whnf.reducibleClassField false in
 theorem comp_pushforward {I J K : C} (f : I ⟶ J) (g : J ⟶ K)
     [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)]
     [ExponentiableMorphism f] [ExponentiableMorphism g] :
diff --git a/Mathlib/CategoryTheory/Monoidal/Bimod.lean b/Mathlib/CategoryTheory/Monoidal/Bimod.lean
index 214857e25294e0..13a83ac8f49a60 100644
--- a/Mathlib/CategoryTheory/Monoidal/Bimod.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Bimod.lean
@@ -216,7 +216,6 @@ noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
           slice_lhs 3 4 => rw [whisker_exchange]
           simp))
 
-set_option backward.isDefEq.respectTransparency false in
 theorem whiskerLeft_π_actLeft :
     (R.X ◁ coequalizer.π _ _) ≫ actLeft P Q =
       (α_ _ _ _).inv ≫ (P.actLeft ▷ Q.X) ≫ coequalizer.π _ _ := by
diff --git a/Mathlib/CategoryTheory/Monoidal/Cartesian/Comon_.lean b/Mathlib/CategoryTheory/Monoidal/Cartesian/Comon_.lean
index 6cfef3f013ab5f..8fdbd61387a90b 100644
--- a/Mathlib/CategoryTheory/Monoidal/Cartesian/Comon_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Cartesian/Comon_.lean
@@ -28,7 +28,6 @@ variable (C : Type u) [Category.{v} C] [CartesianMonoidalCategory C]
 
 attribute [local simp] leftUnitor_hom rightUnitor_hom
 
-set_option backward.whnf.reducibleClassField false in
 /--
 The functor from a Cartesian monoidal category to comonoids in that category,
 equipping every object with the diagonal map as a comultiplication.
diff --git a/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean b/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean
index 3865284cb2e5fb..412cddbdf552a7 100644
--- a/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean
+++ b/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean
@@ -1163,7 +1163,6 @@ lemma tensorHom_id {x x' : D} (f : x ⟶ x') (y : D) :
     f ⊗ₘ (𝟙 y) = f ▷ y :=
   rfl
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 lemma ι_map_tensorHom_eq {d₁ d₁' d₂ d₂' : D} (f : d₁ ⟶ d₂) (f' : d₁' ⟶ d₂') :
     letI := mkMonoidalCategoryStruct C V D
diff --git a/Mathlib/CategoryTheory/Monoidal/Grp_.lean b/Mathlib/CategoryTheory/Monoidal/Grp_.lean
index 128c50f943e185..75506fbaa2bbf9 100644
--- a/Mathlib/CategoryTheory/Monoidal/Grp_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Grp_.lean
@@ -186,7 +186,6 @@ theorem inv_comp_inv (A : C) [GrpObj A] : ι ≫ ι = 𝟙 A := by
   apply lift_left_mul_ext ι[A]
   rw [right_inv, ← comp_toUnit_assoc ι, ← left_inv, comp_lift_assoc, Category.comp_id]
 
-set_option backward.whnf.reducibleClassField false in
 /-- Transfer `GrpObj` along an isomorphism. -/
 -- Note: The simps lemmas are not tagged simp because their `#discr_tree_simp_key` are too generic.
 @[simps! -isSimp]
diff --git a/Mathlib/CategoryTheory/Monoidal/Internal/Types/Basic.lean b/Mathlib/CategoryTheory/Monoidal/Internal/Types/Basic.lean
index ea94ad40dd7329..4af2b8a22e647f 100644
--- a/Mathlib/CategoryTheory/Monoidal/Internal/Types/Basic.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Internal/Types/Basic.lean
@@ -44,7 +44,6 @@ noncomputable def functor : Mon (Type u) ⥤ MonCat.{u} where
       map_one' := congr_fun (IsMonHom.one_hom f.hom) PUnit.unit
       map_mul' x y := congr_fun (IsMonHom.mul_hom f.hom) (x, y) }
 
-set_option backward.whnf.reducibleClassField false in
 /-- Converting a bundled monoid to a monoid object in `Type`.
 -/
 noncomputable def inverse : MonCat.{u} ⥤ Mon (Type u) where
diff --git a/Mathlib/CategoryTheory/Shift/Localization.lean b/Mathlib/CategoryTheory/Shift/Localization.lean
index 29bfc06775c410..7ac09cceef2265 100644
--- a/Mathlib/CategoryTheory/Shift/Localization.lean
+++ b/Mathlib/CategoryTheory/Shift/Localization.lean
@@ -279,7 +279,6 @@ noncomputable def commShift : G.CommShift M := by
   letI : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := ⟨e.symm⟩
   exact Functor.commShiftOfLocalization L₁ W₁ M (Φ.functor ⋙ L₂) G
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 @[reassoc]
 lemma commShift_iso_hom_app (m : M) (X : C₁) :
@@ -291,7 +290,6 @@ lemma commShift_iso_hom_app (m : M) (X : C₁) :
   simp [Functor.commShiftOfLocalization_iso_hom_app,
     Functor.commShiftIso_comp_hom_app]
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 @[reassoc]
 lemma commShift_iso_inv_app (m : M) (X : C₁) :
diff --git a/Mathlib/CategoryTheory/ShrinkYoneda.lean b/Mathlib/CategoryTheory/ShrinkYoneda.lean
index 452e593e54d18b..c5a87d33d2317c 100644
--- a/Mathlib/CategoryTheory/ShrinkYoneda.lean
+++ b/Mathlib/CategoryTheory/ShrinkYoneda.lean
@@ -71,7 +71,6 @@ noncomputable def shrinkYonedaObjObjEquiv {X : C} {Y : Cᵒᵖ} :
     ((shrinkYoneda.{w}.obj X).obj Y) ≃ (Y.unop ⟶ X) :=
   (equivShrink _).symm
 
-set_option backward.isDefEq.respectTransparency false in
 lemma shrinkYoneda_obj_map_shrinkYonedaObjObjEquiv_symm
     {X : C} {Y Y' : Cᵒᵖ} (g : Y ⟶ Y') (f : Y.unop ⟶ X) :
     (shrinkYoneda.obj _).map g (shrinkYonedaObjObjEquiv.symm f) =
@@ -83,7 +82,6 @@ lemma shrinkYonedaObjObjEquiv_symm_comp {X Y Y' : C} (g : Y' ⟶ Y) (f : Y ⟶ X
     (shrinkYoneda.obj _).map g.op (shrinkYonedaObjObjEquiv.symm f) :=
   (shrinkYoneda_obj_map_shrinkYonedaObjObjEquiv_symm g.op f).symm
 
-set_option backward.isDefEq.respectTransparency false in
 lemma shrinkYoneda_map_app_shrinkYonedaObjObjEquiv_symm
     {X X' : C} {Y : Cᵒᵖ} (f : Y.unop ⟶ X) (g : X ⟶ X') :
     (shrinkYoneda.map g).app _ (shrinkYonedaObjObjEquiv.symm f) =
diff --git a/Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean b/Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean
index 5f42c3bc5584d4..0127ff2da1c9f6 100644
--- a/Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean
+++ b/Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean
@@ -745,7 +745,6 @@ lemma sheafifyHomEquivOfIsEquivalence_naturality_right
 
 variable (A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Assuming that `(C, J)` is a dense subsite of `(D, K)` (via a functor `G : C ⥤ D`)
 and `sheafPushforwardContinuous G A J K` is an equivalence of categories, and
 that `HasWeakSheafify J A` holds, then this adjunction shows the existence
diff --git a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
index f442d82101ee3f..ef26b78edf5ff8 100644
--- a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
+++ b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
@@ -461,7 +461,6 @@ lemma ofArrows_mem_iff_isLocallySurjective_cofanIsColimitDesc_shrinkYoneda_map
     exact Presheaf.imageSieve_mem J (Cofan.IsColimit.desc hc (fun i ↦ shrinkYoneda.{w}.map (f i)))
       (shrinkYonedaObjObjEquiv.symm (𝟙 S))
 
-set_option backward.isDefEq.respectTransparency false in
 lemma ofArrows_mem_iff_isLocallySurjective_cofanIsColimitDesc_uliftYoneda_map
     {S : C} {ι : Type*} [Small.{max w v} ι] {X : ι → C}
     (f : ∀ i, X i ⟶ S)
diff --git a/Mathlib/CategoryTheory/Triangulated/Functor.lean b/Mathlib/CategoryTheory/Triangulated/Functor.lean
index 151e8763d9e80b..c00413f92f1f17 100644
--- a/Mathlib/CategoryTheory/Triangulated/Functor.lean
+++ b/Mathlib/CategoryTheory/Triangulated/Functor.lean
@@ -78,7 +78,6 @@ section Additive
 
 variable [Preadditive C] [Preadditive D] [F.Additive]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The functor `F.mapTriangle` commutes with the shift. -/
 noncomputable def mapTriangleCommShiftIso (n : ℤ) :
     Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n :=
diff --git a/Mathlib/Combinatorics/Additive/CauchyDavenport.lean b/Mathlib/Combinatorics/Additive/CauchyDavenport.lean
index c31549aa2de418..ef89454ed7ee78 100644
--- a/Mathlib/Combinatorics/Additive/CauchyDavenport.lean
+++ b/Mathlib/Combinatorics/Additive/CauchyDavenport.lean
@@ -107,7 +107,6 @@ private lemma wellFoundedOn_devosMulRel :
     add_le_add ((card_le_card_mul_right hx.1.2).trans_eq hx.2) <|
       (card_le_card_mul_left hx.1.1).trans_eq hx.2
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
 lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
 subgroup. -/
@@ -196,7 +195,6 @@ lemma cauchy_davenport_of_isMulTorsionFree [DecidableEq G] [Group G] [IsMulTorsi
 
 /-! ### $ℤ/nℤ$ -/
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in `ℤ/pℤ`, then the size of
 `s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
 lemma ZMod.cauchy_davenport {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
diff --git a/Mathlib/Combinatorics/Additive/DoublingConst.lean b/Mathlib/Combinatorics/Additive/DoublingConst.lean
index 5d53bf1d31c0fa..e10a6b1d546b1e 100644
--- a/Mathlib/Combinatorics/Additive/DoublingConst.lean
+++ b/Mathlib/Combinatorics/Additive/DoublingConst.lean
@@ -141,13 +141,11 @@ lemma divConst_le_card : δₘ[A, B] ≤ #B := by
 section Fintype
 variable [Fintype G]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Dense sets have small doubling. -/
 @[to_additive addConst_le_inv_dens /-- Dense sets have small doubling. -/]
 lemma mulConst_le_inv_dens : σₘ[A, B] ≤ A.dens⁻¹ := by
   rw [dens, inv_div, mulConst]; gcongr; exact card_le_univ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Dense sets have small difference constant. -/
 @[to_additive subConst_le_inv_dens /-- Dense sets have small difference constant. -/]
 lemma divConst_le_inv_dens : δₘ[A, B] ≤ A.dens⁻¹ := by
diff --git a/Mathlib/Combinatorics/Hall/Finite.lean b/Mathlib/Combinatorics/Hall/Finite.lean
index 9ca804a778207b..f22ab42c5c95e5 100644
--- a/Mathlib/Combinatorics/Hall/Finite.lean
+++ b/Mathlib/Combinatorics/Hall/Finite.lean
@@ -50,7 +50,6 @@ section Fintype
 
 variable [Fintype ι]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem hall_cond_of_erase {x : ι} (a : α)
     (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))
     (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by
diff --git a/Mathlib/Combinatorics/Matroid/IndepAxioms.lean b/Mathlib/Combinatorics/Matroid/IndepAxioms.lean
index 11d28e7e36d00d..8a09ca62ae8c32 100644
--- a/Mathlib/Combinatorics/Matroid/IndepAxioms.lean
+++ b/Mathlib/Combinatorics/Matroid/IndepAxioms.lean
@@ -383,7 +383,6 @@ instance ofBddAugment_rankFinite (E : Set α) Indep indep_empty indep_subset ind
   rw [IndepMatroid.ofBddAugment]
   infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `E` is finite, then any collection of subsets of `E` satisfying
   the usual independence axioms determines a matroid -/
 protected def ofFinite {E : Set α} (hE : E.Finite) (Indep : Set α → Prop)
diff --git a/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean b/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean
index a370ad8dadfdf2..b7ec099d904cc6 100644
--- a/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean
+++ b/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean
@@ -463,7 +463,6 @@ theorem Indep.augment (hI : M.Indep I) (hJ : M.Indep J) (hIJ : I.encard < J.enca
   rw [← hJ'.encard_eq_encard hb] at hIJ
   exact hIJ.not_ge (encard_mono hJJ')
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Indep.augment_finset {I J : Finset α} (hI : M.Indep I) (hJ : M.Indep J)
     (hIJ : I.card < J.card) : ∃ e ∈ J, e ∉ I ∧ M.Indep (insert e I) := by
   obtain ⟨x, hx, hxI⟩ := hI.augment hJ (by simpa [encard_eq_coe_toFinset_card])
diff --git a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
index 69f1b8e57e4eb0..d6ed1e4c632184 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
@@ -200,7 +200,6 @@ This is `⊤` (infinity) iff `G` isn't colorable with finitely many colors.
 If `G` is colorable, then `ENat.toNat G.chromaticNumber` is the `ℕ`-valued chromatic number. -/
 noncomputable def chromaticNumber : ℕ∞ := ⨅ n ∈ setOf G.Colorable, (n : ℕ∞)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma le_chromaticNumber_iff_colorable : n ≤ G.chromaticNumber ↔ ∀ m, G.Colorable m → n ≤ m := by
   simp [chromaticNumber]
 
@@ -314,7 +313,6 @@ theorem colorable_set_nonempty_of_colorable {n : ℕ} (hc : G.Colorable n) :
 theorem chromaticNumber_bddBelow : BddBelow { n : ℕ | G.Colorable n } :=
   ⟨0, fun _ _ => zero_le _⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Colorable.chromaticNumber_le {n : ℕ} (hc : G.Colorable n) : G.chromaticNumber ≤ n := by
   rw [hc.chromaticNumber_eq_sInf]
   norm_cast
@@ -339,7 +337,6 @@ theorem chromaticNumber_le_iff_colorable {n : ℕ} : G.chromaticNumber ≤ n ↔
   rw [Set.mem_setOf_eq] at this
   exact this.mono h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the chromatic number of `G` is `n + 1`, then `G` is colorable in no fewer than `n + 1`
 colors. -/
 theorem chromaticNumber_eq_iff_colorable_not_colorable :
@@ -359,7 +356,6 @@ theorem colorable_chromaticNumber_of_fintype (G : SimpleGraph V) [Finite V] :
   cases nonempty_fintype V
   exact colorable_chromaticNumber G.colorable_of_fintype
 
-set_option backward.isDefEq.respectTransparency false in
 theorem chromaticNumber_le_one_of_subsingleton (G : SimpleGraph V) [Subsingleton V] :
     G.chromaticNumber ≤ 1 := by
   rw [← Nat.cast_one, chromaticNumber_le_iff_colorable]
@@ -368,7 +364,6 @@ theorem chromaticNumber_le_one_of_subsingleton (G : SimpleGraph V) [Subsingleton
   cases Subsingleton.elim v w
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem chromaticNumber_pos [Nonempty V] {n : ℕ} (hc : G.Colorable n) : 0 < G.chromaticNumber := by
   rw [hc.chromaticNumber_eq_sInf, Nat.cast_pos]
   apply le_csInf (colorable_set_nonempty_of_colorable hc)
@@ -385,7 +380,6 @@ theorem colorable_of_chromaticNumber_ne_top (h : G.chromaticNumber ≠ ⊤) :
   obtain ⟨n, hn⟩ := h
   exact colorable_chromaticNumber hn
 
-set_option backward.isDefEq.respectTransparency false in
 theorem chromaticNumber_eq_zero_of_isEmpty [IsEmpty V] : G.chromaticNumber = 0 := by
   rw [← nonpos_iff_eq_zero, ← Nat.cast_zero, chromaticNumber_le_iff_colorable]; exact .of_isEmpty _
 
@@ -424,7 +418,6 @@ theorem chromaticNumber_mono_of_embedding {V' : Type*} {G' : SimpleGraph V'}
     (f : G ↪g G') : G.chromaticNumber ≤ G'.chromaticNumber :=
   chromaticNumber_mono_of_hom f
 
-set_option backward.isDefEq.respectTransparency false in
 lemma card_le_chromaticNumber_iff_forall_surjective [Fintype α] :
     card α ≤ G.chromaticNumber ↔ ∀ C : G.Coloring α, Surjective C := by
   refine ⟨fun h C ↦ ?_, fun h ↦ ?_⟩
@@ -475,7 +468,6 @@ theorem chromaticNumber_top_eq_top_of_infinite (V : Type*) [Infinite V] :
   obtain ⟨n, ⟨hn⟩⟩ := hc
   exact not_injective_infinite_finite _ hn.injective_of_top_hom
 
-set_option backward.isDefEq.respectTransparency false in
 theorem eq_top_of_chromaticNumber_eq_card [Fintype V]
     (h : G.chromaticNumber = Fintype.card V) : G = ⊤ := by
   classical
@@ -503,7 +495,6 @@ theorem two_le_chromaticNumber_of_adj {u v : V} (hadj : G.Adj u v) : 2 ≤ G.chr
   obtain ⟨c⟩ := chromaticNumber_le_iff_colorable.mp (Order.le_of_lt_add_one h)
   exact c.valid hadj (Subsingleton.elim (c u) (c v))
 
-set_option backward.isDefEq.respectTransparency false in
 theorem chromaticNumber_eq_one_iff : G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V := by
   refine ⟨fun h ↦ ⟨?_, ?_⟩, fun ⟨h₁, _⟩ ↦ h₁ ▸ chromaticNumber_bot⟩
   · contrapose! h
@@ -571,7 +562,6 @@ protected theorem Colorable.cliqueFree {n m : ℕ} (hc : G.Colorable n) (hm : n
   obtain ⟨s, h, rfl⟩ := h
   exact Nat.lt_le_asymm hm (h.card_le_of_colorable hc)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem cliqueFree_of_chromaticNumber_lt {n : ℕ} (hc : G.chromaticNumber < n) :
     G.CliqueFree n := by
   have hne : G.chromaticNumber ≠ ⊤ := hc.ne_top
@@ -582,7 +572,6 @@ theorem cliqueFree_of_chromaticNumber_lt {n : ℕ} (hc : G.chromaticNumber < n)
   rw [← hne] at hc
   simpa using hc
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Given a colouring `α` of `G`, and a clique of size at least the number of colours, the clique
 contains a vertex of each colour.
diff --git a/Mathlib/Combinatorics/SimpleGraph/Connectivity/EdgeConnectivity.lean b/Mathlib/Combinatorics/SimpleGraph/Connectivity/EdgeConnectivity.lean
index 8c8fea89bee1ea..21d32c54c68137 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Connectivity/EdgeConnectivity.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Connectivity/EdgeConnectivity.lean
@@ -56,18 +56,15 @@ lemma IsEdgeReachable.trans (h1 : G.IsEdgeReachable k u v) (h2 : G.IsEdgeReachab
 lemma IsEdgeReachable.mono (hGH : G ≤ H) (h : G.IsEdgeReachable k u v) : H.IsEdgeReachable k u v :=
   fun _ hk ↦ h hk |>.mono <| deleteEdges_mono hGH
 
-set_option backward.isDefEq.respectTransparency false in
 @[gcongr]
 lemma IsEdgeReachable.anti (hkl : k ≤ l) (h : G.IsEdgeReachable l u v) : G.IsEdgeReachable k u v :=
   fun _ hk ↦ h <| by grw [← hkl]; exact hk
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 protected lemma IsEdgeReachable.zero : G.IsEdgeReachable 0 u v := by simp [IsEdgeReachable]
 
 @[simp] protected lemma IsEdgeConnected.zero : G.IsEdgeConnected 0 := fun _ _ ↦ .zero
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma isEdgeReachable_one : G.IsEdgeReachable 1 u v ↔ G.Reachable u v := by
   simp [IsEdgeReachable, ENat.lt_one_iff_eq_zero]
diff --git a/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean b/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
index 16a5879a0ddfeb..b895b8fadb9bd8 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
@@ -241,7 +241,6 @@ theorem toSubgraph_reverse (p : G.Walk u v) : p.reverse.toSubgraph = p.toSubgrap
     congr
     ext <;> simp [-Set.bot_eq_empty]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem toSubgraph_rotate [DecidableEq V] (c : G.Walk v v) (h : u ∈ c.support) :
     (c.rotate h).toSubgraph = c.toSubgraph := by
diff --git a/Mathlib/Combinatorics/SimpleGraph/Girth.lean b/Mathlib/Combinatorics/SimpleGraph/Girth.lean
index 2e859cfc1b8b69..3df2237596b69f 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Girth.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Girth.lean
@@ -61,7 +61,6 @@ lemma exists_egirth_eq_length :
       ← exists_prop, Subtype.exists', Sigma.exists', eq_comm] at h ⊢
     exact ciInf_mem _
 
-set_option backward.isDefEq.respectTransparency false in
 lemma three_le_egirth : 3 ≤ G.egirth := by
   simpa using fun _ _ a ↦ Walk.IsCycle.three_le_length a
 
diff --git a/Mathlib/Combinatorics/SimpleGraph/Metric.lean b/Mathlib/Combinatorics/SimpleGraph/Metric.lean
index daf824a3bd791d..c9e469edabbbff 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Metric.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Metric.lean
@@ -70,7 +70,6 @@ theorem edist_le (p : G.Walk u v) :
   sInf_le ⟨p, rfl⟩
 protected alias Walk.edist_le := edist_le
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem edist_eq_zero_iff :
     G.edist u v = 0 ↔ u = v := by
@@ -96,7 +95,6 @@ lemma exists_walk_of_edist_ne_top (h : G.edist u v ≠ ⊤) :
     ∃ p : G.Walk u v, p.length = G.edist u v :=
   (reachable_of_edist_ne_top h).exists_walk_length_eq_edist
 
-set_option backward.isDefEq.respectTransparency false in
 protected theorem edist_triangle : G.edist u w ≤ G.edist u v + G.edist v w := by
   cases eq_or_ne (G.edist u v) ⊤ with
   | inl huv => simp [huv]
@@ -245,7 +243,6 @@ protected theorem Connected.dist_triangle (hconn : G.Connected) :
   rw [← hp, ← hq, ← Walk.length_append]
   apply dist_le
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Reachable.dist_triangle_left (h : G.Reachable u v) (w) :
     G.dist u w ≤ G.dist u v + G.dist v w := by
   by_cases! h' : ¬G.Reachable u w
@@ -253,7 +250,6 @@ lemma Reachable.dist_triangle_left (h : G.Reachable u v) (w) :
   rw [← ENat.coe_le_coe, ENat.coe_add]
   grind [SimpleGraph.edist_triangle, Reachable.trans, Reachable.symm]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Reachable.dist_triangle_right (h : G.Reachable v w) (u) :
     G.dist u w ≤ G.dist u v + G.dist v w := by
   by_cases! h' : ¬G.Reachable u w
diff --git a/Mathlib/Combinatorics/SimpleGraph/VertexCover.lean b/Mathlib/Combinatorics/SimpleGraph/VertexCover.lean
index cd086362c7c7a1..be8d9555b09f9b 100644
--- a/Mathlib/Combinatorics/SimpleGraph/VertexCover.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/VertexCover.lean
@@ -127,7 +127,6 @@ theorem vertexCoverNum_eq_zero : vertexCoverNum G = 0 ↔ G = ⊥ := by
   refine ⟨fun h ↦ ?_, by simp_all⟩
   simpa [h] using vertexCoverNum_exists G
 
-set_option backward.isDefEq.respectTransparency false in
 theorem vertexCoverNum_le_card_sub_one : vertexCoverNum G ≤ ENat.card V - 1 := by
   nontriviality V
   obtain ⟨x⟩ := not_subsingleton_iff_nontrivial.mp (not_subsingleton V) |>.to_nonempty
@@ -146,7 +145,6 @@ theorem vertexCoverNum_lt_card [Nonempty V] [Finite V] : vertexCoverNum G < ENat
   enat_to_nat
   exact Nat.add_le_of_le_sub (Order.one_le_iff_pos.mpr Nat.card_pos) (le_refl _)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem vertexCoverNum_le_encard_edgeSet : vertexCoverNum G ≤ G.edgeSet.encard := by
   by_cases h' : G.edgeSet = ∅
   · simp [h', SimpleGraph.edgeSet_eq_empty.mp]
@@ -160,7 +158,6 @@ theorem vertexCoverNum_le_encard_edgeSet : vertexCoverNum G ≤ G.edgeSet.encard
 theorem vertexCoverNum_ne_top_of_finite_edgeSet (h : G.edgeSet.Finite) : vertexCoverNum G ≠ ⊤ :=
   ne_top_of_le_ne_top (Set.encard_ne_top_iff.mpr h) vertexCoverNum_le_encard_edgeSet
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem vertexCoverNum_top : vertexCoverNum (completeGraph V) = ENat.card V - 1 := by
   nontriviality V using tsub_eq_zero_of_le
diff --git a/Mathlib/Combinatorics/Tiling/Tile.lean b/Mathlib/Combinatorics/Tiling/Tile.lean
index 7a8279f5230242..073b8316ce1185 100644
--- a/Mathlib/Combinatorics/Tiling/Tile.lean
+++ b/Mathlib/Combinatorics/Tiling/Tile.lean
@@ -249,7 +249,6 @@ instance : SMul G (PlacedTile ps) where
   induction pt using PlacedTile.induction_on
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma coe_smul (g : G) (pt : PlacedTile ps) :
     (g • pt : PlacedTile ps) = g • (pt : Set X) := by
   induction pt using PlacedTile.induction_on
diff --git a/Mathlib/Computability/NFA.lean b/Mathlib/Computability/NFA.lean
index db5037a765f0fe..6e5a89afab2799 100644
--- a/Mathlib/Computability/NFA.lean
+++ b/Mathlib/Computability/NFA.lean
@@ -405,7 +405,6 @@ theorem disjoint_evalFrom_reverse_iff {x : List α} {S S' : Set σ} :
     Disjoint S (M.reverse.evalFrom S' x) ↔ Disjoint S' (M.evalFrom S x.reverse) :=
   ⟨disjoint_evalFrom_reverse, fun h ↦ List.reverse_reverse x ▸ disjoint_evalFrom_reverse h⟩
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem mem_accepts_reverse {x : List α} : x ∈ M.reverse.accepts ↔ x.reverse ∈ M.accepts := by
   simp [mem_accepts, ← Set.not_disjoint_iff, disjoint_evalFrom_reverse_iff]
diff --git a/Mathlib/Computability/Primrec/List.lean b/Mathlib/Computability/Primrec/List.lean
index fe38ce62dae448..ac0d82bf0e8cc7 100644
--- a/Mathlib/Computability/Primrec/List.lean
+++ b/Mathlib/Computability/Primrec/List.lean
@@ -585,7 +585,6 @@ theorem const {n} : ∀ m, @Primrec' n fun _ => m
 theorem head {n : ℕ} : @Primrec' n.succ head :=
   (get 0).of_eq fun v => by simp [get_zero]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem tail {n f} (hf : @Primrec' n f) : @Primrec' n.succ fun v => f v.tail :=
   (hf.comp _ fun i => @get _ i.succ).of_eq fun v => by
     rw [← ofFn_get v.tail]; congr; funext i; simp
diff --git a/Mathlib/Data/ENat/Pow.lean b/Mathlib/Data/ENat/Pow.lean
index c6c24d388e8dd7..b3794d14c0aa8e 100644
--- a/Mathlib/Data/ENat/Pow.lean
+++ b/Mathlib/Data/ENat/Pow.lean
@@ -110,7 +110,6 @@ lemma epow_eq_one_iff : x ^ y = 1 ↔ x = 1 ∨ y = 0 := by
     simp only [epow_one, h] at this
     exact (not_lt_of_ge this x_2).rec
 
-set_option backward.isDefEq.respectTransparency false in
 lemma epow_add : x ^ (y + z) = x ^ y * x ^ z := by
   rcases lt_trichotomy x 1 with x_0 | rfl | x_2
   · rw [lt_one_iff_eq_zero.1 x_0]
@@ -140,7 +139,6 @@ lemma mul_epow : (x * y) ^ z = x ^ z * y ^ z := by
         exact epow_top (one_lt_mul x_2.le y_2)
   · simp only [epow_natCast, mul_pow x y]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma epow_mul : x ^ (y * z) = (x ^ y) ^ z := by
   rcases eq_or_ne y 0 with y_0 | y_0
   · simp [y_0]
diff --git a/Mathlib/Data/Holor.lean b/Mathlib/Data/Holor.lean
index e157830b3922ff..a66431b1b9afe9 100644
--- a/Mathlib/Data/Holor.lean
+++ b/Mathlib/Data/Holor.lean
@@ -217,7 +217,6 @@ theorem slice_sum [AddCommMonoid α] {β : Type} (i : ℕ) (hid : i < d) (s : Fi
   · intro _ _ h_not_in ih
     rw [Finset.sum_insert h_not_in, ih, slice_add, Finset.sum_insert h_not_in]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The original holor can be recovered from its slices by multiplying with unit vectors and
 summing up. -/
 @[simp]
@@ -275,7 +274,6 @@ theorem cprankMax_add [Mul α] [AddMonoid α] :
     · assumption
     · exact cprankMax_add hx₂ hy
 
-set_option backward.isDefEq.respectTransparency false in
 theorem cprankMax_mul [NonUnitalNonAssocSemiring α] :
     ∀ (n : ℕ) (x : Holor α [d]) (y : Holor α ds), CPRankMax n y → CPRankMax n (x ⊗ y)
   | 0, x, _, CPRankMax.zero => by simp [mul_zero x, CPRankMax.zero]
diff --git a/Mathlib/Data/NNRat/Lemmas.lean b/Mathlib/Data/NNRat/Lemmas.lean
index 5a34e63b3a03f4..df6aac7d808201 100644
--- a/Mathlib/Data/NNRat/Lemmas.lean
+++ b/Mathlib/Data/NNRat/Lemmas.lean
@@ -48,7 +48,6 @@ lemma toNNRat_inv (q : ℚ) : toNNRat q⁻¹ = (toNNRat q)⁻¹ := by
 lemma toNNRat_div (hp : 0 ≤ p) : toNNRat (p / q) = toNNRat p / toNNRat q := by
   rw [div_eq_mul_inv, div_eq_mul_inv, ← toNNRat_inv, ← toNNRat_mul hp]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma toNNRat_div' (hq : 0 ≤ q) : toNNRat (p / q) = toNNRat p / toNNRat q := by
   rw [div_eq_inv_mul, div_eq_inv_mul, toNNRat_mul (inv_nonneg.2 hq), toNNRat_inv]
 
diff --git a/Mathlib/Data/Nat/Multiplicity.lean b/Mathlib/Data/Nat/Multiplicity.lean
index e0c7f2dfaf5436..5ed0419e2691e5 100644
--- a/Mathlib/Data/Nat/Multiplicity.lean
+++ b/Mathlib/Data/Nat/Multiplicity.lean
@@ -95,7 +95,6 @@ theorem emultiplicity_self {p : ℕ} (hp : p.Prime) : emultiplicity p p = 1 :=
 theorem emultiplicity_pow_self {p n : ℕ} (hp : p.Prime) : emultiplicity p (p ^ n) = n :=
   _root_.emultiplicity_pow_self hp.ne_zero hp.prime.not_unit n
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Legendre's Theorem**
 
 The multiplicity of a prime in `n!` is the sum of the quotients `n / p ^ i`. This sum is expressed
@@ -151,7 +150,6 @@ theorem emultiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
   rw [WithTop.add_left_inj h, sum_Ico_succ_top h2, hp.emultiplicity_mul, hp.emultiplicity_self,
     sum_congr rfl h4, sum_const_zero, zero_add, add_comm 1]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The multiplicity of `p` in `(p * n)!` is `n` more than that of `n!`. -/
 theorem emultiplicity_factorial_mul {n p : ℕ} (hp : p.Prime) :
     emultiplicity p (p * n)! = emultiplicity p n ! + n := by
@@ -163,7 +161,6 @@ theorem emultiplicity_factorial_mul {n p : ℕ} (hp : p.Prime) :
     congr 1
     rw [add_comm, add_assoc]
 
-set_option backward.isDefEq.respectTransparency false in
 /- The multiplicity of a prime `p` in `p ^ n` is the sum of `p ^ i`, where `i` ranges between `0`
   and `n - 1`. -/
 theorem multiplicity_factorial_pow {n p : ℕ} (hp : p.Prime) :
@@ -189,7 +186,6 @@ theorem emultiplicity_factorial_le_div_pred {p : ℕ} (hp : p.Prime) (n : ℕ) :
   apply WithTop.coe_mono
   exact Nat.geom_sum_Ico_le hp.two_le _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The multiplicity of `p` in `choose (n + k) k` is the number of carries when `k` and `n`
   are added in base `p`. The set is expressed by filtering `Ico 1 b` where `b`
   is any bound greater than `log p (n + k)`. -/
@@ -232,7 +228,6 @@ theorem emultiplicity_le_emultiplicity_choose_add {p : ℕ} (hp : p.Prime) :
 
 variable {p n k : ℕ}
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_choose_prime_pow_add_emultiplicity (hp : p.Prime) (hkn : k ≤ p ^ n)
     (hk0 : k ≠ 0) : emultiplicity p (choose (p ^ n) k) + emultiplicity p k = n :=
   le_antisymm
@@ -253,7 +248,6 @@ theorem emultiplicity_choose_prime_pow_add_emultiplicity (hp : p.Prime) (hkn : k
       rwa [card_Ico 1 n.succ] at filter_le_Ico)
     (by rw [← hp.emultiplicity_pow_self]; exact emultiplicity_le_emultiplicity_choose_add hp _ _)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_choose_prime_pow {p n k : ℕ} (hp : p.Prime) (hkn : k ≤ p ^ n) (hk0 : k ≠ 0) :
     emultiplicity p (choose (p ^ n) k) = ↑(n - multiplicity p k) := by
   push_cast
@@ -277,7 +271,6 @@ theorem dvd_choose_pow_iff (hp : Prime p) : p ∣ (p ^ n).choose k ↔ k ≠ 0 
 
 end Prime
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_two_factorial_lt : ∀ {n : ℕ} (_ : n ≠ 0), emultiplicity 2 n ! < n := by
   have h2 := prime_two.prime
   refine binaryRec ?_ ?_
diff --git a/Mathlib/Data/Nat/PartENat.lean b/Mathlib/Data/Nat/PartENat.lean
index 1a10b5bc3d3dcc..17ddc17cbd736b 100644
--- a/Mathlib/Data/Nat/PartENat.lean
+++ b/Mathlib/Data/Nat/PartENat.lean
@@ -551,7 +551,6 @@ theorem toWithTop_natCast' (n : ℕ) {_ : Decidable (n : PartENat).Dom} :
 theorem toWithTop_ofNat (n : ℕ) [n.AtLeastTwo] {_ : Decidable (OfNat.ofNat n : PartENat).Dom} :
     toWithTop (ofNat(n) : PartENat) = OfNat.ofNat n := toWithTop_natCast' n
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem toWithTop_le {x y : PartENat} [hx : Decidable x.Dom] [hy : Decidable y.Dom] :
     toWithTop x ≤ toWithTop y ↔ x ≤ y := by
@@ -617,7 +616,6 @@ theorem ofENat_lt {x y : ℕ∞} : ofENat x < ofENat y ↔ x < y := by
 
 section WithTopEquiv
 
-set_option backward.isDefEq.respectTransparency false in
 open scoped Classical in
 @[simp]
 theorem toWithTop_add {x y : PartENat} : toWithTop (x + y) = toWithTop x + toWithTop y := by
diff --git a/Mathlib/Data/PFunctor/Multivariate/M.lean b/Mathlib/Data/PFunctor/Multivariate/M.lean
index dcad336ffc2595..70431848031197 100644
--- a/Mathlib/Data/PFunctor/Multivariate/M.lean
+++ b/Mathlib/Data/PFunctor/Multivariate/M.lean
@@ -299,7 +299,6 @@ theorem M.dest_map {α β : TypeVec n} (g : α ⟹ β) (x : P.M α) :
     rw [M.dest, M.dest', map_eq, appendFun_comp_splitFun]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem M.map_dest {α β : TypeVec n} (g : (α ::: P.M α) ⟹ (β ::: P.M β)) (x : P.M α)
     (h : ∀ x : P.M α, lastFun g x = (dropFun g <$$> x : P.M β)) :
     g <$$> M.dest P x = M.dest P (dropFun g <$$> x) := by
diff --git a/Mathlib/Data/PFunctor/Univariate/M.lean b/Mathlib/Data/PFunctor/Univariate/M.lean
index 19578a51a5aabd..589051c33ce4e0 100644
--- a/Mathlib/Data/PFunctor/Univariate/M.lean
+++ b/Mathlib/Data/PFunctor/Univariate/M.lean
@@ -247,7 +247,6 @@ inductive Agree' : ℕ → M F → M F → Prop
 @[simp]
 theorem dest_mk (x : F (M F)) : dest (M.mk x) = x := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem mk_dest (x : M F) : M.mk (dest x) = x := by
   apply ext'
diff --git a/Mathlib/Data/PNat/Interval.lean b/Mathlib/Data/PNat/Interval.lean
index 1e432cc9c28b02..cb27965f890ed7 100644
--- a/Mathlib/Data/PNat/Interval.lean
+++ b/Mathlib/Data/PNat/Interval.lean
@@ -65,7 +65,6 @@ set_option backward.isDefEq.respectTransparency false in
 theorem card_Ico : #(Ico a b) = b - a := by
   rw [← Nat.card_Ico, ← map_subtype_embedding_Ico, card_map]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem card_Ioc : #(Ioc a b) = b - a := by
   rw [← Nat.card_Ioc, ← map_subtype_embedding_Ioc, card_map]
diff --git a/Mathlib/Data/Seq/Basic.lean b/Mathlib/Data/Seq/Basic.lean
index 42cf7a859bdc79..f4fb0af51a5541 100644
--- a/Mathlib/Data/Seq/Basic.lean
+++ b/Mathlib/Data/Seq/Basic.lean
@@ -41,7 +41,6 @@ set_option linter.flexible false in -- simp followed by exact rfl
 @[simp]
 theorem length_nil : length (nil : Seq α) terminates_nil = 0 := by simp [length]; exact rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem length'_nil : length' (nil : Seq α) = 0 := by
   simp -implicitDefEqProofs [length']
@@ -51,7 +50,6 @@ theorem length_cons {x : α} {s : Seq α} (h : s.Terminates) :
   apply Nat.find_comp_succ
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem length'_cons (x : α) (s : Seq α) :
     (cons x s).length' = s.length' + 1 := by
@@ -90,7 +88,6 @@ theorem length_le_iff {s : Seq α} {n : ℕ} {h : s.Terminates} :
     s.length h ≤ n ↔ s.TerminatedAt n := by
   rw [← length_le_iff']; simp [h]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem length'_le_iff {s : Seq α} {n : ℕ} :
     s.length' ≤ n ↔ s.TerminatedAt n := by
   by_cases h : s.Terminates
@@ -115,7 +112,6 @@ theorem lt_length_iff {s : Seq α} {n : ℕ} {h : s.Terminates} :
     n < s.length h ↔ ∃ a, a ∈ s.get? n := by
   rw [← lt_length_iff']; simp [h]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem lt_length'_iff {s : Seq α} {n : ℕ} :
     n < s.length' ↔ ∃ a, a ∈ s.get? n := by
   by_cases h : s.Terminates
@@ -530,7 +526,6 @@ theorem drop_nil {n : ℕ} : (@nil α).drop n = nil := by
   | zero => simp [drop]
   | succ m ih => simp [← dropn_tail, ih]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem drop_length' {n : ℕ} {s : Seq α} :
     (s.drop n).length' = s.length' - n := by
diff --git a/Mathlib/Data/Seq/Parallel.lean b/Mathlib/Data/Seq/Parallel.lean
index abcd64ed5ad5dc..1ebf9181b89e3a 100644
--- a/Mathlib/Data/Seq/Parallel.lean
+++ b/Mathlib/Data/Seq/Parallel.lean
@@ -116,7 +116,6 @@ theorem terminates_parallel.aux :
       have := H1 _ h
       rcases Seq.destruct S with (_ | ⟨_ | c, S'⟩) <;> apply IH <;> simp [this]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : Terminates c] :
     Terminates (parallel S) := by
   suffices
diff --git a/Mathlib/Data/Set/Card.lean b/Mathlib/Data/Set/Card.lean
index 3ba5ad2d1b00ea..abab08aede84a2 100644
--- a/Mathlib/Data/Set/Card.lean
+++ b/Mathlib/Data/Set/Card.lean
@@ -171,7 +171,6 @@ theorem finite_of_encard_le_coe {k : ℕ} (h : s.encard ≤ k) : s.Finite := by
 theorem finite_of_encard_eq_coe {k : ℕ} (h : s.encard = k) : s.Finite :=
   finite_of_encard_le_coe h.le
 
-set_option backward.isDefEq.respectTransparency false in
 theorem encard_le_coe_iff {k : ℕ} : s.encard ≤ k ↔ s.Finite ∧ ∃ (n₀ : ℕ), s.encard = n₀ ∧ n₀ ≤ k :=
   ⟨fun h ↦ ⟨finite_of_encard_le_coe h, by rwa [ENat.le_coe_iff] at h⟩,
     fun ⟨_,⟨n₀,hs, hle⟩⟩ ↦ by rwa [hs, Nat.cast_le]⟩
@@ -438,7 +437,6 @@ theorem Finite.eq_insert_of_subset_of_encard_eq_succ (hs : s.Finite) (h : s ⊆
     encard_eq_one] at hst
   obtain ⟨x, hx⟩ := hst; use x; rw [← diff_union_of_subset h, hx, singleton_union]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_subset_encard_eq {k : ℕ∞} (hk : k ≤ s.encard) : ∃ t, t ⊆ s ∧ t.encard = k := by
   induction k using ENat.nat_induction with
   | zero => exact ⟨∅, empty_subset _, by simp⟩
@@ -615,7 +613,6 @@ theorem encard_le_coe_iff_finite_ncard_le {k : ℕ} : s.encard ≤ k ↔ s.Finit
 theorem Infinite.ncard (hs : s.Infinite) : s.ncard = 0 := by
   rw [← _root_.Nat.card_coe_set_eq, @Nat.card_eq_zero_of_infinite _ hs.to_subtype]
 
-set_option backward.isDefEq.respectTransparency false in
 @[gcongr]
 theorem ncard_le_ncard (hst : s ⊆ t) (ht : t.Finite := by toFinite_tac) :
     s.ncard ≤ t.ncard := by
@@ -624,7 +621,6 @@ theorem ncard_le_ncard (hst : s ⊆ t) (ht : t.Finite := by toFinite_tac) :
 
 theorem ncard_mono [Finite α] : @Monotone (Set α) _ _ _ ncard := fun _ _ ↦ ncard_le_ncard
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] theorem ncard_eq_zero (hs : s.Finite := by toFinite_tac) :
     s.ncard = 0 ↔ s = ∅ := by
   rw [← Nat.cast_inj (R := ℕ∞), hs.cast_ncard_eq, Nat.cast_zero, encard_eq_zero]
@@ -660,7 +656,6 @@ theorem nonempty_of_ncard_ne_zero (hs : s.ncard ≠ 0) : s.Nonempty := by
 @[simp] theorem ncard_singleton (a : α) : ({a} : Set α).ncard = 1 := by
   simp [ncard]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_singleton_inter (a : α) (s : Set α) : ({a} ∩ s).ncard ≤ 1 := by
   rw [← Nat.cast_le (α := ℕ∞), (toFinite _).cast_ncard_eq, Nat.cast_one]
   apply encard_singleton_inter
@@ -678,7 +673,6 @@ theorem ncard_powerset (s : Set α) (hs : s.Finite := by toFinite_tac) :
 
 section InsertErase
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] theorem ncard_insert_of_notMem {a : α} (h : a ∉ s) (hs : s.Finite := by toFinite_tac) :
     (insert a s).ncard = s.ncard + 1 := by
   rw [← Nat.cast_inj (R := ℕ∞), (hs.insert a).cast_ncard_eq, Nat.cast_add, Nat.cast_one,
@@ -687,7 +681,6 @@ set_option backward.isDefEq.respectTransparency false in
 theorem ncard_insert_of_mem {a : α} (h : a ∈ s) : ncard (insert a s) = s.ncard := by
   rw [insert_eq_of_mem h]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_insert_le (a : α) (s : Set α) : (insert a s).ncard ≤ s.ncard + 1 := by
   obtain hs | hs := s.finite_or_infinite
   · to_encard_tac; rw [hs.cast_ncard_eq, (hs.insert _).cast_ncard_eq]; apply encard_insert_le
@@ -713,7 +706,6 @@ theorem ncard_le_ncard_insert (a : α) (s : Set α) : s.ncard ≤ (insert a s).n
 theorem ncard_pair {a b : α} (h : a ≠ b) : ({a, b} : Set α).ncard = 2 := by
   simp [h]
 
-set_option backward.isDefEq.respectTransparency false in
 -- removing `@[simp]` because the LHS is not in simp normal form
 theorem ncard_diff_singleton_add_one {a : α} (h : a ∈ s)
     (hs : s.Finite := by toFinite_tac) : (s \ {a}).ncard + 1 = s.ncard := by
@@ -763,7 +755,6 @@ end InsertErase
 
 variable {f : α → β}
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_image_le (hs : s.Finite := by toFinite_tac) : (f '' s).ncard ≤ s.ncard := by
   to_encard_tac; rw [hs.cast_ncard_eq, (hs.image _).cast_ncard_eq]; apply encard_image_le
 
@@ -772,7 +763,6 @@ theorem InjOn.ncard_image (H : Set.InjOn f s) : (f '' s).ncard = s.ncard :=
 
 @[deprecated (since := "2026-01-30")] alias ncard_image_of_injOn := InjOn.ncard_image
 
-set_option backward.isDefEq.respectTransparency false in
 theorem injOn_of_ncard_image_eq (h : (f '' s).ncard = s.ncard) (hs : s.Finite := by toFinite_tac) :
     Set.InjOn f s := by
   rw [← Nat.cast_inj (R := ℕ∞), hs.cast_ncard_eq, (hs.image _).cast_ncard_eq] at h
@@ -814,7 +804,6 @@ theorem ncard_inter_le_ncard_right (s t : Set α) (ht : t.Finite := by toFinite_
     (s ∩ t).ncard ≤ t.ncard :=
   ncard_le_ncard inter_subset_right ht
 
-set_option backward.isDefEq.respectTransparency false in
 theorem eq_of_subset_of_ncard_le (h : s ⊆ t) (h' : t.ncard ≤ s.ncard)
     (ht : t.Finite := by toFinite_tac) : s = t :=
   ht.eq_of_subset_of_encard_le' h
@@ -832,7 +821,6 @@ theorem sep_of_ncard_eq {a : α} {P : α → Prop} (h : { x ∈ s | P x }.ncard
     (hs : s.Finite := by toFinite_tac) : P a :=
   sep_eq_self_iff_mem_true.mp (eq_of_subset_of_ncard_le (by simp) h.symm.le hs) _ ha
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_lt_ncard (h : s ⊂ t) (ht : t.Finite := by toFinite_tac) :
     s.ncard < t.ncard := by
   rw [← Nat.cast_lt (α := ℕ∞), ht.cast_ncard_eq, (ht.subset h.subset).cast_ncard_eq]
@@ -879,7 +867,6 @@ theorem ncard_congr {t : Set β} (f : ∀ a ∈ s, β) (h₁ : ∀ a ha, f a ha
 theorem ncard_congr' {S : Set α} {T : Set β} (f : S ≃ T) : Set.ncard S = Set.ncard T :=
   Cardinal.toNat_congr f
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_le_ncard_of_injOn {t : Set β} (f : α → β) (hf : ∀ a ∈ s, f a ∈ t) (f_inj : InjOn f s)
     (ht : t.Finite := by toFinite_tac) :
     s.ncard ≤ t.ncard := by
@@ -956,7 +943,6 @@ theorem ncard_coe {α : Type*} (s : Set α) :
 
 section Lattice
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_union_add_ncard_inter (s t : Set α) (hs : s.Finite := by toFinite_tac)
     (ht : t.Finite := by toFinite_tac) : (s ∪ t).ncard + (s ∩ t).ncard = s.ncard + t.ncard := by
   to_encard_tac; rw [hs.cast_ncard_eq, ht.cast_ncard_eq, (hs.union ht).cast_ncard_eq,
@@ -966,7 +952,6 @@ theorem ncard_inter_add_ncard_union (s t : Set α) (hs : s.Finite := by toFinite
     (ht : t.Finite := by toFinite_tac) : (s ∩ t).ncard + (s ∪ t).ncard = s.ncard + t.ncard := by
   rw [add_comm, ncard_union_add_ncard_inter _ _ hs ht]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_union_le (s t : Set α) : (s ∪ t).ncard ≤ s.ncard + t.ncard := by
   obtain (h | h) := (s ∪ t).finite_or_infinite
   · to_encard_tac
@@ -976,7 +961,6 @@ theorem ncard_union_le (s t : Set α) : (s ∪ t).ncard ≤ s.ncard + t.ncard :=
   rw [h.ncard]
   apply zero_le
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_union_eq (h : Disjoint s t) (hs : s.Finite := by toFinite_tac)
     (ht : t.Finite := by toFinite_tac) : (s ∪ t).ncard = s.ncard + t.ncard := by
   to_encard_tac
@@ -993,7 +977,6 @@ theorem ncard_union_lt (hs : s.Finite := by toFinite_tac)
     (s ∪ t).ncard < s.ncard + t.ncard :=
   (ncard_union_le s t).lt_of_ne (mt (ncard_union_eq_iff hs ht).mp h)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_diff_add_ncard_of_subset (h : s ⊆ t) (ht : t.Finite := by toFinite_tac) :
     (t \ s).ncard + s.ncard = t.ncard := by
   to_encard_tac
@@ -1010,7 +993,6 @@ lemma cast_ncard_sdiff {R : Type*} [AddGroupWithOne R] (hst : s ⊆ t) (ht : t.F
     ((t \ s).ncard : R) = t.ncard - s.ncard := by
   rw [ncard_diff hst (ht.subset hst), Nat.cast_sub (ncard_le_ncard hst ht)]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ncard_le_ncard_diff_add_ncard (s t : Set α) (ht : t.Finite := by toFinite_tac) :
     s.ncard ≤ (s \ t).ncard + t.ncard := by
   rcases s.finite_or_infinite with hs | hs
diff --git a/Mathlib/Data/ZMod/QuotientGroup.lean b/Mathlib/Data/ZMod/QuotientGroup.lean
index 8eeeaad65f3536..22d38b8fbc2941 100644
--- a/Mathlib/Data/ZMod/QuotientGroup.lean
+++ b/Mathlib/Data/ZMod/QuotientGroup.lean
@@ -197,7 +197,6 @@ lemma quotientEquivSigmaZMod_apply (q : orbitRel.Quotient (zpowers g) (G ⧸ H))
   rw [apply_eq_iff_eq_symm_apply, quotientEquivSigmaZMod_symm_apply, ZMod.coe_intCast,
     zpow_smul_mod_minimalPeriod]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The sum of minimal periods over all orbits equals the index `[G:H]`. -/
 lemma index_eq_sum_minimalPeriod (g : G) [Finite (G ⧸ H)]
     [Fintype (Quotient (MulAction.orbitRel (zpowers g) (G ⧸ H)))] :
diff --git a/Mathlib/FieldTheory/LinearDisjoint.lean b/Mathlib/FieldTheory/LinearDisjoint.lean
index a179797e7d0059..835cdc0aeec8e6 100644
--- a/Mathlib/FieldTheory/LinearDisjoint.lean
+++ b/Mathlib/FieldTheory/LinearDisjoint.lean
@@ -731,7 +731,6 @@ theorem isField_of_forall (A : Type v) [Field A] (B : Type w) [Field B]
   change Function.Injective (Ideal.Quotient.mk M) at H
   rwa [RingHom.injective_iff_ker_eq_bot, Ideal.mk_ker] at H
 
-set_option backward.isDefEq.respectTransparency false in
 variable (F E) in
 /-- If `E` and `K` are field extensions of `F`, one of them is algebraic, such that
 `E ⊗[F] K` is a domain, then `E ⊗[F] K` is also a field. It is a corollary of
diff --git a/Mathlib/FieldTheory/Separable.lean b/Mathlib/FieldTheory/Separable.lean
index 934c6c160ec635..74e2984cb52bd6 100644
--- a/Mathlib/FieldTheory/Separable.lean
+++ b/Mathlib/FieldTheory/Separable.lean
@@ -281,7 +281,6 @@ theorem separable_C_mul_X_pow_add_C_mul_X_add_C'
     (C a * X ^ n + C b * X + C c).Separable :=
   separable_C_mul_X_pow_add_C_mul_X_add_C a b c ((CharP.cast_eq_zero_iff R p n).2 hn) hb
 
-set_option backward.isDefEq.respectTransparency false in
 theorem rootMultiplicity_le_one_of_separable [Nontrivial R] {p : R[X]} (hsep : Separable p)
     (x : R) : rootMultiplicity x p ≤ 1 := by
   classical
diff --git a/Mathlib/Geometry/Euclidean/Angle/Sphere.lean b/Mathlib/Geometry/Euclidean/Angle/Sphere.lean
index 9dfddc20ec6fdb..34ba17744df3da 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Sphere.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Sphere.lean
@@ -425,7 +425,6 @@ theorem mem_circumsphere_of_two_zsmul_oangle_eq {t : Triangle ℝ P} {p : P} {i
 
 end Oriented
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The circumradius of a triangle may be expressed explicitly as half the length of a side
 divided by the sine of the angle at the third point (a version of the law of sines or sine rule). -/
 theorem dist_div_sin_angle_div_two_eq_circumradius (t : Triangle ℝ P) {i₁ i₂ i₃ : Fin 3}
diff --git a/Mathlib/Geometry/Euclidean/MongePoint.lean b/Mathlib/Geometry/Euclidean/MongePoint.lean
index 9221d1709f7bc8..c9fbe0ebbbafb9 100644
--- a/Mathlib/Geometry/Euclidean/MongePoint.lean
+++ b/Mathlib/Geometry/Euclidean/MongePoint.lean
@@ -123,7 +123,6 @@ theorem mongePoint_mem_affineSpan {n : ℕ} (s : Simplex ℝ P n) :
   smul_vsub_vadd_mem _ _ (centroid_mem_affineSpan_of_card_eq_add_one ℝ _ (card_fin (n + 1)))
     s.circumcenter_mem_affineSpan s.circumcenter_mem_affineSpan
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem mongePoint_restrict {n : ℕ} (s : Simplex ℝ P n) (S : AffineSubspace ℝ P)
     (hS : affineSpan ℝ (Set.range s.points) ≤ S) :
diff --git a/Mathlib/Geometry/Euclidean/NinePointCircle.lean b/Mathlib/Geometry/Euclidean/NinePointCircle.lean
index 53ea53cfb59532..f34245a633be17 100644
--- a/Mathlib/Geometry/Euclidean/NinePointCircle.lean
+++ b/Mathlib/Geometry/Euclidean/NinePointCircle.lean
@@ -83,7 +83,6 @@ theorem ninePointCircle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂] [Inner
   · simp [ninePointCircle_center, centroid_map]
   · simp [ninePointCircle_radius]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem ninePointCircle_restrict {n : ℕ} (s : Simplex ℝ P n) (S : AffineSubspace ℝ P)
     (hS : affineSpan ℝ (Set.range s.points) ≤ S) :
     haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
diff --git a/Mathlib/Geometry/Euclidean/Projection.lean b/Mathlib/Geometry/Euclidean/Projection.lean
index 785e37fcbc9293..6e1b28c0ffc256 100644
--- a/Mathlib/Geometry/Euclidean/Projection.lean
+++ b/Mathlib/Geometry/Euclidean/Projection.lean
@@ -61,7 +61,6 @@ theorem orthogonalProjection_apply' (s : AffineSubspace 𝕜 P) [Nonempty s]
       (Classical.arbitrary s : P) :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem orthogonalProjection_apply_mem (s : AffineSubspace 𝕜 P) [Nonempty s]
     [s.direction.HasOrthogonalProjection] {p x} (hx : x ∈ s) :
     orthogonalProjection s p = (s.direction.orthogonalProjection (p -ᵥ x) : V) +ᵥ x := by
@@ -84,7 +83,6 @@ theorem orthogonalProjection_congr {s₁ s₂ : AffineSubspace 𝕜 P} {p₁ p
   subst h hp
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The linear map corresponding to `orthogonalProjection`. -/
 @[simp]
 theorem orthogonalProjection_linear {s : AffineSubspace 𝕜 P} [Nonempty s]
@@ -551,7 +549,6 @@ variable [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
     LinearIsometry.inner_map_map, Submodule.inner_right_of_mem_orthogonal hv
       (vsub_orthogonalProjection_mem_direction_orthogonal _ _)]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma orthogonalProjection_subtype (s : AffineSubspace 𝕜 P) [Nonempty s] (s' : AffineSubspace 𝕜 s)
     [Nonempty s'] [s'.direction.HasOrthogonalProjection]
     [(s'.map s.subtype).direction.HasOrthogonalProjection] (p : s) :
@@ -568,7 +565,6 @@ lemma orthogonalProjection_subtype (s : AffineSubspace 𝕜 P) [Nonempty s] (s'
     reflection (s.map f.toAffineMap) (f p) = f (reflection s p) := by
   simp [reflection_apply']
 
-set_option backward.isDefEq.respectTransparency false in
 lemma reflection_subtype (s : AffineSubspace 𝕜 P) [Nonempty s] (s' : AffineSubspace 𝕜 s)
     [Nonempty s'] [s'.direction.HasOrthogonalProjection]
     [(s'.map s.subtype).direction.HasOrthogonalProjection] (p : s) :
diff --git a/Mathlib/Geometry/Euclidean/SignedDist.lean b/Mathlib/Geometry/Euclidean/SignedDist.lean
index 65c39dc45308a5..9dca0146f60d5c 100644
--- a/Mathlib/Geometry/Euclidean/SignedDist.lean
+++ b/Mathlib/Geometry/Euclidean/SignedDist.lean
@@ -271,7 +271,6 @@ variable {s p} in
   ext x
   simp [signedInfDist_def, orthogonalProjection_eq_self_iff.2 h]
 
-set_option backward.isDefEq.respectTransparency false in
 variable {s p} in
 lemma abs_signedInfDist_eq_dist_of_mem_affineSpan_insert {x : P}
     (h : x ∈ affineSpan ℝ (insert p s)) :
diff --git a/Mathlib/Geometry/Manifold/Instances/Sphere.lean b/Mathlib/Geometry/Manifold/Instances/Sphere.lean
index cf303f55cf0e70..a65c8862efb9ce 100644
--- a/Mathlib/Geometry/Manifold/Instances/Sphere.lean
+++ b/Mathlib/Geometry/Manifold/Instances/Sphere.lean
@@ -280,7 +280,6 @@ theorem stereographic_source (hv : ‖v‖ = 1) : (stereographic hv).source = {
 theorem stereographic_target (hv : ‖v‖ = 1) : (stereographic hv).target = Set.univ :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem stereographic_apply_neg (v : sphere (0 : E) 1) :
     stereographic (norm_eq_of_mem_sphere v) (-v) = 0 := by
diff --git a/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean b/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
index 487bed96525b03..b3c8434cb67c33 100644
--- a/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
+++ b/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
@@ -356,7 +356,6 @@ lemma isInvertible_mfderiv_extChartAt {y : M} (hy : y ∈ (extChartAt I x).sourc
   have : (extChartAt I x).symm ((extChartAt I x) y) = y := (extChartAt I x).left_inv hy
   rwa [this] at Z
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The trivialization of the tangent bundle at a point is the manifold derivative of the
 extended chart.
 Use with care as this abuses the defeq `TangentSpace 𝓘(𝕜, E) y = E` for `y : E`. -/
diff --git a/Mathlib/GroupTheory/Coprod/Basic.lean b/Mathlib/GroupTheory/Coprod/Basic.lean
index 0edee6c877c76d..b3ca0d710cf16e 100644
--- a/Mathlib/GroupTheory/Coprod/Basic.lean
+++ b/Mathlib/GroupTheory/Coprod/Basic.lean
@@ -665,7 +665,6 @@ end MulOneClass
 
 variable (M N P : Type*) [Monoid M] [Monoid N] [Monoid P]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A multiplicative equivalence between `(M ∗ N) ∗ P` and `M ∗ (N ∗ P)`. -/
 @[to_additive /-- An additive equivalence between `AddMonoid.Coprod (AddMonoid.Coprod M N) P` and
 `AddMonoid.Coprod M (AddMonoid.Coprod N P)`. -/]
diff --git a/Mathlib/GroupTheory/CoprodI.lean b/Mathlib/GroupTheory/CoprodI.lean
index 07c4a82f410fa0..d3a8f939490231 100644
--- a/Mathlib/GroupTheory/CoprodI.lean
+++ b/Mathlib/GroupTheory/CoprodI.lean
@@ -178,7 +178,6 @@ theorem of_leftInverse [DecidableEq ι] (i : ι) :
 theorem of_injective (i : ι) : Function.Injective (of : M i →* _) := by
   classical exact (of_leftInverse i).injective
 
-set_option backward.isDefEq.respectTransparency false in
 theorem mrange_eq_iSup {N} [Monoid N] (f : ∀ i, M i →* N) :
     MonoidHom.mrange (lift f) = ⨆ i, MonoidHom.mrange (f i) := by
   rw [lift, Equiv.coe_fn_mk, Con.lift_range, FreeMonoid.mrange_lift,
diff --git a/Mathlib/GroupTheory/FreeGroup/Orbit.lean b/Mathlib/GroupTheory/FreeGroup/Orbit.lean
index 072a91e70ad10d..516448f05a7c3b 100644
--- a/Mathlib/GroupTheory/FreeGroup/Orbit.lean
+++ b/Mathlib/GroupTheory/FreeGroup/Orbit.lean
@@ -34,7 +34,6 @@ The neutral element is not contained in one of the startsWith sets.
 theorem startsWith.ne_one {w : α × Bool} (g : FreeGroup α) (h : g ∈ FreeGroup.startsWith w) :
     g ≠ 1 := fun h1 ↦ by simp [h1, startsWith, FreeGroup.toWord_one] at h
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma startsWith.disjoint_iff_ne {w w' : α × Bool} :
     Disjoint (startsWith w) (startsWith w') ↔ w ≠ w' := by
diff --git a/Mathlib/GroupTheory/GroupAction/MultiplePrimitivity.lean b/Mathlib/GroupTheory/GroupAction/MultiplePrimitivity.lean
index b59aa6eaf7b942..f4362def6d7c7b 100644
--- a/Mathlib/GroupTheory/GroupAction/MultiplePrimitivity.lean
+++ b/Mathlib/GroupTheory/GroupAction/MultiplePrimitivity.lean
@@ -114,14 +114,12 @@ instance (n : ℕ) [IsMultiplyPreprimitive M α n] :
     IsMultiplyPretransitive M α n :=
   IsMultiplyPreprimitive.isMultiplyPretransitive M α n
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Any action is `0`-preprimitive. -/
 @[to_additive /-- Any action is `0`-preprimitive. -/]
 theorem is_zero_preprimitive : IsMultiplyPreprimitive M α 0 where
   isMultiplyPretransitive := MulAction.is_zero_pretransitive
   isPreprimitive_ofFixingSubgroup hs := by simp at hs
 
-set_option backward.isDefEq.respectTransparency false in
 /-- An action is preprimitive iff it is `1`-preprimitive. -/
 @[to_additive
 /-- An action is preprimitive iff it is `1`-preprimitive. -/]
@@ -137,7 +135,6 @@ theorem is_one_preprimitive_iff :
     · exact is_one_pretransitive_iff.mpr h.toIsPretransitive
     · simpa using isPreprimitive_of_fixingSubgroup_empty_iff.mpr h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The action of `stabilizer M a` is one-less preprimitive. -/
 @[to_additive /-- The action of `stabilizer M a` is one-less preprimitive. -/]
 theorem isMultiplyPreprimitive_ofStabilizer
@@ -159,7 +156,6 @@ theorem isMultiplyPreprimitive_ofStabilizer
       aesop
     exact IsPreprimitive.of_surjective ofFixingSubgroup_insert_map_bijective.surjective
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A pretransitive action is `n.succ-`preprimitive
 iff the action of stabilizers is `n`-preprimitive. -/
 @[to_additive /-- A pretransitive action is `n.succ-`preprimitive
@@ -209,7 +205,6 @@ theorem isMultiplyPreprimitive_succ_iff_ofStabilizer
       rw [Set.encard_insert_of_notMem, Subtype.coe_injective.encard_image, ENat.coe_one]
       exact notMem_val_image M t
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The fixator of a subset of cardinal `d` in an `n`-primitive action
 acts `n-d`-primitively on the remaining (`d ≤ n`). -/
 @[to_additive
diff --git a/Mathlib/GroupTheory/GroupAction/SubMulAction/OfFixingSubgroup.lean b/Mathlib/GroupTheory/GroupAction/SubMulAction/OfFixingSubgroup.lean
index cbbfbe018f3eea..34bc3031fad3ab 100644
--- a/Mathlib/GroupTheory/GroupAction/SubMulAction/OfFixingSubgroup.lean
+++ b/Mathlib/GroupTheory/GroupAction/SubMulAction/OfFixingSubgroup.lean
@@ -477,7 +477,6 @@ section TwoCriteria
 
 open MulAction
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A pretransitivity criterion. -/
 theorem IsPretransitive.isPretransitive_ofFixingSubgroup_inter
     (hs : IsPretransitive (fixingSubgroup M s) (ofFixingSubgroup M s))
diff --git a/Mathlib/GroupTheory/IsSubnormal.lean b/Mathlib/GroupTheory/IsSubnormal.lean
index 488b241f40bb60..00b94b901f6647 100644
--- a/Mathlib/GroupTheory/IsSubnormal.lean
+++ b/Mathlib/GroupTheory/IsSubnormal.lean
@@ -90,7 +90,6 @@ inductive _root_.AddSubgroup.IsSubnormal {G : Type*} [AddGroup G] : AddSubgroup
 
 attribute [simp] Subgroup.IsSubnormal.top
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A normal subgroup is subnormal. -/
 @[to_additive /-- A normal additive subgroup is subnormal. -/]
 lemma Normal.isSubnormal (hn : H.Normal) : IsSubnormal H :=
diff --git a/Mathlib/GroupTheory/NoncommCoprod.lean b/Mathlib/GroupTheory/NoncommCoprod.lean
index f4cbb143b63d9b..ed92127f85c0e1 100644
--- a/Mathlib/GroupTheory/NoncommCoprod.lean
+++ b/Mathlib/GroupTheory/NoncommCoprod.lean
@@ -140,7 +140,6 @@ lemma noncommCoprod_injective {M N P : Type*} [Group M] [Group N] [Group P]
     have key := hp ⟨⟨a⁻¹, by rwa [map_inv, inv_eq_iff_mul_eq_one]⟩, b, rfl⟩
     exact ⟨hf a (by rwa [key, mul_one] at h), hg b key⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma noncommCoprod_range {M N P : Type*} [Group M] [Group N] [Group P]
     (f : M →* P) (g : N →* P) (comm : ∀ (m : M) (n : N), Commute (f m) (g n)) :
     (noncommCoprod f g comm).range = f.range ⊔ g.range := by
diff --git a/Mathlib/GroupTheory/Perm/ClosureSwap.lean b/Mathlib/GroupTheory/Perm/ClosureSwap.lean
index f8d96dc6d8dddc..7cc762ffafd823 100644
--- a/Mathlib/GroupTheory/Perm/ClosureSwap.lean
+++ b/Mathlib/GroupTheory/Perm/ClosureSwap.lean
@@ -137,7 +137,6 @@ theorem closure_of_isSwap_of_isPretransitive [Finite α] {S : Set (Perm α)} (hS
     [MulAction.IsPretransitive (Subgroup.closure S) α] : Subgroup.closure S = ⊤ := by
   simp [eq_top_iff', mem_closure_isSwap hS, orbit_eq_univ, Set.toFinite]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A transitive permutation group generated by transpositions must be the whole symmetric group -/
 theorem surjective_of_isSwap_of_isPretransitive' [Finite α] (S : Set G)
     (hS1 : ∀ σ ∈ S, MulAction.toPermHom G α σ = 1 ∨ Perm.IsSwap (MulAction.toPermHom G α σ))
diff --git a/Mathlib/GroupTheory/SpecificGroups/Alternating.lean b/Mathlib/GroupTheory/SpecificGroups/Alternating.lean
index c77166d0bbc0d4..bcdc1c560ac230 100644
--- a/Mathlib/GroupTheory/SpecificGroups/Alternating.lean
+++ b/Mathlib/GroupTheory/SpecificGroups/Alternating.lean
@@ -460,7 +460,6 @@ namespace Equiv.Perm
 
 open Subgroup Group
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The alternating group is the only subgroup of index 2 of the permutation group. -/
 theorem eq_alternatingGroup_of_index_eq_two {G : Subgroup (Equiv.Perm α)} (hG : G.index = 2) :
     G = alternatingGroup α := by
diff --git a/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.lean b/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.lean
index 8f82307e27c0bf..b43f900fe4a553 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Basic.lean
@@ -90,14 +90,12 @@ theorem coe_vadd (s : AffineSubspace k P) [Nonempty s] (a : s.direction) (b : s)
     ↑(a +ᵥ b) = (a : V) +ᵥ (b : P) :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Embedding of an affine subspace to the ambient space, as an affine map. -/
 protected def subtype (s : AffineSubspace k P) [Nonempty s] : s →ᵃ[k] P where
   toFun := (↑)
   linear := s.direction.subtype
   map_vadd' _ _ := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem subtype_linear (s : AffineSubspace k P) [Nonempty s] :
     s.subtype.linear = s.direction.subtype := rfl
@@ -158,7 +156,6 @@ theorem preimage_coe_affineSpan_singleton (x : P) :
 
 variable (P)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The top affine subspace is linearly equivalent to the affine space.
 This is the affine version of `Submodule.topEquiv`. -/
 @[simps! linear apply symm_apply_coe]
@@ -615,7 +612,6 @@ lemma map_mk' (p : P₁) (direction : Submodule k V₁) :
 section inclusion
 variable {S₁ S₂ : AffineSubspace k P₁} [Nonempty S₁]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Affine map from a smaller to a larger subspace of the same space.
 
 This is the affine version of `Submodule.inclusion`. -/
@@ -632,7 +628,6 @@ def inclusion (h : S₁ ≤ S₂) :
 theorem coe_inclusion_apply (h : S₁ ≤ S₂) (x : S₁) : (inclusion h x : P₁) = x :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem inclusion_rfl : inclusion (le_refl S₁) = AffineMap.id k S₁ := rfl
 
@@ -688,7 +683,6 @@ theorem ext_on {V₂ P₂ : Type*} [AddCommGroup V₂] [Module k V₂] [AddTorso
 section ofEq
 variable (S₁ S₂ : AffineSubspace k P₁) [Nonempty S₁] [Nonempty S₂]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Affine equivalence between two equal affine subspace.
 
 This is the affine version of `LinearEquiv.ofEq`. -/
@@ -702,13 +696,11 @@ def ofEq (h : S₁ = S₂) : S₁ ≃ᵃ[k] S₂ where
 theorem coe_ofEq_apply (h : S₁ = S₂) (x : S₁) : (ofEq S₁ S₂ h x : P₁) = x :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem ofEq_symm (h : S₁ = S₂) : (ofEq S₁ S₂ h).symm = ofEq S₂ S₁ h.symm := by
   ext
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem ofEq_rfl : ofEq S₁ S₁ rfl = AffineEquiv.refl k S₁ := rfl
 
diff --git a/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean b/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
index 49b8434fcc2016..9fce3a60e806d6 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
@@ -42,54 +42,46 @@ open AffineSubspace Module
 
 variable [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of a finite set is finite-dimensional. -/
 theorem finiteDimensional_vectorSpan_of_finite {s : Set P} (h : Set.Finite s) :
     FiniteDimensional k (vectorSpan k s) :=
   .span_of_finite k <| h.vsub h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The vector span of a singleton is finite-dimensional. -/
 instance finiteDimensional_vectorSpan_singleton (p : P) :
     FiniteDimensional k (vectorSpan k {p}) :=
   finiteDimensional_vectorSpan_of_finite _ (Set.finite_singleton p)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of a family indexed by a `Fintype` is
 finite-dimensional. -/
 instance finiteDimensional_vectorSpan_range [Finite ι] (p : ι → P) :
     FiniteDimensional k (vectorSpan k (Set.range p)) :=
   finiteDimensional_vectorSpan_of_finite k (Set.finite_range _)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of a subset of a family indexed by a `Fintype`
 is finite-dimensional. -/
 instance finiteDimensional_vectorSpan_image_of_finite [Finite ι] (p : ι → P) (s : Set ι) :
     FiniteDimensional k (vectorSpan k (p '' s)) :=
   finiteDimensional_vectorSpan_of_finite k (Set.toFinite _)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of a finite set is
 finite-dimensional. -/
 theorem finiteDimensional_direction_affineSpan_of_finite {s : Set P} (h : Set.Finite s) :
     FiniteDimensional k (affineSpan k s).direction :=
   (direction_affineSpan k s).symm ▸ finiteDimensional_vectorSpan_of_finite k h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of a singleton is finite-dimensional. -/
 instance finiteDimensional_direction_affineSpan_singleton (p : P) :
     FiniteDimensional k (affineSpan k {p}).direction := by
   rw [direction_affineSpan]
   infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of a family indexed by a
 `Fintype` is finite-dimensional. -/
 instance finiteDimensional_direction_affineSpan_range [Finite ι] (p : ι → P) :
     FiniteDimensional k (affineSpan k (Set.range p)).direction :=
   finiteDimensional_direction_affineSpan_of_finite k (Set.finite_range _)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of a subset of a family indexed
 by a `Fintype` is finite-dimensional. -/
 instance finiteDimensional_direction_affineSpan_image_of_finite [Finite ι] (p : ι → P) (s : Set ι) :
@@ -112,7 +104,6 @@ theorem finite_set_of_fin_dim_affineIndependent [FiniteDimensional k V] {s : Set
 
 variable {k}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The supremum of two finite-dimensional affine subspaces is finite-dimensional. -/
 instance AffineSubspace.finiteDimensional_sup (s₁ s₂ : AffineSubspace k P)
     [FiniteDimensional k s₁.direction] [FiniteDimensional k s₂.direction] :
@@ -124,7 +115,6 @@ instance AffineSubspace.finiteDimensional_sup (s₁ s₂ : AffineSubspace k P)
   rw [AffineSubspace.direction_sup hp₁ hp₂]
   infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The image of a finite-dimensional affine subspace under an affine map is finite-dimensional. -/
 instance finiteDimensional_direction_map {V₂ P₂ : Type*} [AddCommGroup V₂] [Module k V₂]
     [AffineSpace V₂ P₂] (s : AffineSubspace k P) [FiniteDimensional k s.direction]
@@ -257,7 +247,6 @@ lemma AffineIndependent.card_le_finrank_succ [Fintype ι] {p : ι → P} (hp : A
   exact (affineIndependent_iff_le_finrank_vectorSpan _ _
     (tsub_add_cancel_of_le <| Nat.one_le_iff_ne_zero.2 Fintype.card_ne_zero).symm).1 hp
 
-set_option backward.isDefEq.respectTransparency false in
 open Finset in
 /-- If an affine independent finset is contained in the affine span of another finset, then its
 cardinality is at most the cardinality of that finset. -/
@@ -299,7 +288,6 @@ lemma AffineIndependent.card_lt_card_of_affineSpan_lt_affineSpan {s t : Finset V
   erw [hs.finrank_vectorSpan_add_one] at finrank_lt
   simpa using finrank_lt.trans_le <| finrank_vectorSpan_range_add_one_le _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the `vectorSpan` of a finite subset of an affinely independent
 family lies in a submodule with dimension one less than its
 cardinality, it equals that submodule. -/
@@ -310,7 +298,6 @@ theorem AffineIndependent.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_ad
     vectorSpan k (s.image p : Set P) = sm :=
   Submodule.eq_of_le_of_finrank_eq hle <| hi.finrank_vectorSpan_image_finset hc
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the `vectorSpan` of a finite affinely independent
 family lies in a submodule with dimension one less than its
 cardinality, it equals that submodule. -/
@@ -320,7 +307,6 @@ theorem AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one [Fintyp
     vectorSpan k (Set.range p) = sm :=
   Submodule.eq_of_le_of_finrank_eq hle <| hi.finrank_vectorSpan hc
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the `affineSpan` of a finite subset of an affinely independent
 family lies in an affine subspace whose direction has dimension one
 less than its cardinality, it equals that subspace. -/
@@ -337,7 +323,6 @@ theorem AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_ad
   rw [direction_affineSpan] at hd ⊢
   exact hi.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one hd hc
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the `affineSpan` of a finite affinely independent family lies
 in an affine subspace whose direction has dimension one less than its
 cardinality, it equals that subspace. -/
@@ -372,7 +357,6 @@ theorem Affine.Simplex.span_eq_top [FiniteDimensional k V] {n : ℕ} (T : Affine
   rw [AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one T.independent,
     Fintype.card_fin, hrank]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of adding a point to a finite-dimensional subspace is finite-dimensional. -/
 instance finiteDimensional_vectorSpan_insert (s : AffineSubspace k P)
     [FiniteDimensional k s.direction] (p : P) :
@@ -385,7 +369,6 @@ instance finiteDimensional_vectorSpan_insert (s : AffineSubspace k P)
   · rw [affineSpan_coe, direction_affineSpan_insert hp₀]
     infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of adding a point to a finite-dimensional subspace is
 finite-dimensional. -/
 instance finiteDimensional_direction_affineSpan_insert (s : AffineSubspace k P)
@@ -395,7 +378,6 @@ instance finiteDimensional_direction_affineSpan_insert (s : AffineSubspace k P)
 
 variable (k)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of adding a point to a set with a finite-dimensional `vectorSpan` is
 finite-dimensional. -/
 instance finiteDimensional_vectorSpan_insert_set (s : Set P) [FiniteDimensional k (vectorSpan k s)]
@@ -405,7 +387,6 @@ instance finiteDimensional_vectorSpan_insert_set (s : Set P) [FiniteDimensional
   rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, direction_affineSpan]
   exact finiteDimensional_vectorSpan_insert (affineSpan k s) p
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of adding a point to a set with a set with finite-dimensional
 direction of the `affineSpan` is finite-dimensional. -/
 instance finiteDimensional_direction_affineSpan_insert_set (s : Set P)
@@ -426,7 +407,6 @@ theorem collinear_iff_rank_le_one (s : Set P) :
 
 variable {k}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A set of points, whose `vectorSpan` is finite-dimensional, is
 collinear if and only if their `vectorSpan` has dimension at most
 `1`. -/
@@ -442,14 +422,12 @@ alias ⟨Collinear.finrank_le_one, _⟩ := collinear_iff_finrank_le_one
 theorem Collinear.subset {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Collinear k s₂) : Collinear k s₁ :=
   (Submodule.rank_mono (vectorSpan_mono k hs)).trans h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of collinear points is finite-dimensional. -/
 theorem Collinear.finiteDimensional_vectorSpan {s : Set P} (h : Collinear k s) :
     FiniteDimensional k (vectorSpan k s) :=
   IsNoetherian.iff_fg.1
     (IsNoetherian.iff_rank_lt_aleph0.2 (lt_of_le_of_lt h Cardinal.one_lt_aleph0))
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of collinear points is finite-dimensional. -/
 theorem Collinear.finiteDimensional_direction_affineSpan {s : Set P} (h : Collinear k s) :
     FiniteDimensional k (affineSpan k s).direction :=
@@ -471,7 +449,6 @@ theorem collinear_singleton (p : P) : Collinear k ({p} : Set P) := by
 
 variable {k}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a point `p₀` in a set of points, that set is collinear if and
 only if the points can all be expressed as multiples of the same
 vector, added to `p₀`. -/
@@ -696,20 +673,17 @@ variable (k) in
 def Coplanar (s : Set P) : Prop :=
   Module.rank k (vectorSpan k s) ≤ 2
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `vectorSpan` of coplanar points is finite-dimensional. -/
 theorem Coplanar.finiteDimensional_vectorSpan {s : Set P} (h : Coplanar k s) :
     FiniteDimensional k (vectorSpan k s) := by
   refine IsNoetherian.iff_fg.1 (IsNoetherian.iff_rank_lt_aleph0.2 (lt_of_le_of_lt h ?_))
   exact Cardinal.lt_aleph0.2 ⟨2, rfl⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The direction of the affine span of coplanar points is finite-dimensional. -/
 theorem Coplanar.finiteDimensional_direction_affineSpan {s : Set P} (h : Coplanar k s) :
     FiniteDimensional k (affineSpan k s).direction :=
   (direction_affineSpan k s).symm ▸ h.finiteDimensional_vectorSpan
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A set of points, whose `vectorSpan` is finite-dimensional, is coplanar if and only if their
 `vectorSpan` has dimension at most `2`. -/
 theorem coplanar_iff_finrank_le_two {s : Set P} [FiniteDimensional k (vectorSpan k s)] :
@@ -761,7 +735,6 @@ open AffineSubspace Module Module
 
 variable [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Adding a point to a finite-dimensional subspace increases the dimension by at most one. -/
 theorem finrank_vectorSpan_insert_le (s : AffineSubspace k P) (p : P) :
     finrank k (vectorSpan k (insert p (s : Set P))) ≤ finrank k s.direction + 1 := by
@@ -802,7 +775,6 @@ theorem finrank_vectorSpan_insert_le_set (s : Set P) (p : P) :
     ← direction_affineSpan _ s]
   exact finrank_vectorSpan_insert_le ..
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Adding a point to a collinear set produces a coplanar set. -/
 theorem Collinear.coplanar_insert {s : Set P} (h : Collinear k s) (p : P) :
     Coplanar k (insert p s) := by
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Restrict.lean b/Mathlib/LinearAlgebra/AffineSpace/Restrict.lean
index b76a70f9a302b9..95f5a7340452e5 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Restrict.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Restrict.lean
@@ -36,7 +36,6 @@ instance AffineSubspace.nonempty_map {E : AffineSubspace k P₁} [Ene : Nonempty
   obtain ⟨x, hx⟩ := id Ene
   exact ⟨⟨φ x, AffineSubspace.mem_map.mpr ⟨x, hx, rfl⟩⟩⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restrict domain and codomain of an affine map to the given subspaces. -/
 def AffineMap.restrict (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁} {F : AffineSubspace k P₂}
     [Nonempty E] [Nonempty F] (hEF : E.map φ ≤ F) : E →ᵃ[k] F := by
@@ -59,7 +58,6 @@ theorem AffineMap.restrict.linear_aux {φ : P₁ →ᵃ[k] P₂} {E : AffineSubs
   rw [← Submodule.map_le_iff_le_comap, ← AffineSubspace.map_direction]
   exact AffineSubspace.direction_le hEF
 
-set_option backward.isDefEq.respectTransparency false in
 theorem AffineMap.restrict.linear (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁}
     {F : AffineSubspace k P₂} [Nonempty E] [Nonempty F] (hEF : E.map φ ≤ F) :
     (φ.restrict hEF).linear = φ.linear.restrict (AffineMap.restrict.linear_aux hEF) :=
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Simplex/Basic.lean b/Mathlib/LinearAlgebra/AffineSpace/Simplex/Basic.lean
index 05203a80cd5fef..33f0aa15954352 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Simplex/Basic.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Simplex/Basic.lean
@@ -291,7 +291,6 @@ lemma range_faceOpposite_reindex {m n : ℕ} [NeZero m] [NeZero n] (s : Simplex
 
 section restrict
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restrict an affine simplex to an affine subspace that contains it. -/
 @[simps]
 def restrict {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P)
@@ -302,7 +301,6 @@ def restrict {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P)
   { points i := ⟨s.points i, hS <| mem_affineSpan _ <| Set.mem_range_self _⟩
     independent := AffineIndependent.of_comp S.subtype s.independent }
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restricting to `S₁` then mapping to a larger `S₂` is the same as restricting to `S₂`. -/
 @[simp]
 theorem restrict_map_inclusion {n : ℕ} (s : Affine.Simplex k P n)
@@ -313,7 +311,6 @@ theorem restrict_map_inclusion {n : ℕ} (s : Affine.Simplex k P n)
       s.restrict S₂ (hS₁.trans hS₂) :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem map_subtype_restrict
     {n : ℕ} (S : AffineSubspace k P) [Nonempty S] (s : Affine.Simplex k S n) :
@@ -321,7 +318,6 @@ theorem map_subtype_restrict
       S (affineSpan_le.2 <| by rintro x ⟨y, rfl⟩; exact Subtype.prop _) = s := by
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restricting to `S₁` then mapping through the restriction of `f` to `S₁ →ᵃ[k] S₂` is the same
 as mapping through unrestricted `f`, then restricting to `S₂`. -/
 theorem restrict_map_restrict
@@ -336,7 +332,6 @@ theorem restrict_map_restrict
           (AffineSubspace.map_mono f hS₁) |>.trans hfS) := by
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restricting to `affineSpan k (Set.range s.points)` can be reversed by mapping through
 `AffineSubspace.subtype`. -/
 @[simp]
@@ -344,14 +339,12 @@ theorem restrict_map_subtype {n : ℕ} (s : Affine.Simplex k P n) :
     (s.restrict _ le_rfl).map (AffineSubspace.subtype _) Subtype.coe_injective = s :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 lemma restrict_reindex {m n : ℕ} (s : Affine.Simplex k P n) (e : Fin (n + 1) ≃ Fin (m + 1))
     {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) :
     letI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
     (s.reindex e).restrict S (s.reindex_range_points e ▸ hS) = (s.restrict S hS).reindex e :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 lemma face_restrict {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P}
     (hS : affineSpan k (Set.range s.points) ≤ S) {fs : Finset (Fin (n + 1))} {m : ℕ}
     (h : #fs = m + 1) :
@@ -363,7 +356,6 @@ lemma face_restrict {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P
   simp_rw [Affine.Simplex.face_points]
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 lemma faceOpposite_restrict {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) {S : AffineSubspace k P}
     (hS : affineSpan k (Set.range s.points) ≤ S) (i : Fin (n + 1)) :
     letI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
@@ -445,7 +437,6 @@ lemma setInterior_map (I : Set k) {n : ℕ} (s : Simplex k P n) {f : P →ᵃ[k]
       rw [s.map_points, Set.range_comp, ← AffineSubspace.map_span, AffineSubspace.mem_map]
       exact ⟨q, (Set.mem_of_mem_of_subset hq s.setInterior_subset_affineSpan), hqp⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma setInterior_restrict (I : Set k) {n : ℕ} (s : Simplex k P n) {S : AffineSubspace k P}
     (hS : affineSpan k (Set.range s.points) ≤ S) :
     letI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
@@ -608,14 +599,12 @@ lemma closedInterior_map {n : ℕ} (s : Simplex k P n) {f : P →ᵃ[k] P₂} (h
     (s.map f hf).closedInterior = f '' s.closedInterior :=
   s.setInterior_map _ hf
 
-set_option backward.isDefEq.respectTransparency false in
 lemma interior_restrict {n : ℕ} (s : Simplex k P n) {S : AffineSubspace k P}
     (hS : affineSpan k (Set.range s.points) ≤ S) :
     letI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
     (s.restrict S hS).interior = S.subtype ⁻¹' s.interior :=
   s.setInterior_restrict _ hS
 
-set_option backward.isDefEq.respectTransparency false in
 lemma closedInterior_restrict {n : ℕ} (s : Simplex k P n) {S : AffineSubspace k P}
     (hS : affineSpan k (Set.range s.points) ≤ S) :
     letI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Simplex/Centroid.lean b/Mathlib/LinearAlgebra/AffineSpace/Simplex/Centroid.lean
index 96592be9453dd3..7a5465fd0ebf70 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Simplex/Centroid.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Simplex/Centroid.lean
@@ -215,7 +215,6 @@ theorem centroid_reindex {m n : ℕ} (s : Simplex k P m)
   subst h_eq
   convert Finset.univ.affineCombination_map e.toEmbedding _ _ <;> simp [Function.comp_assoc]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem centroid_restrict [CharZero k] {n : ℕ} (s : Simplex k P n) (S : AffineSubspace k P)
     (hS : affineSpan k (Set.range s.points) ≤ S) :
     haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
@@ -399,7 +398,6 @@ theorem faceOppositeCentroid_eq_smul_vsub_vadd_point [CharZero k] (s : Simplex k
   rw [sum_centroidWeights_eq_one_of_card_ne_zero]
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] theorem faceOppositeCentroid_restrict [CharZero k] (s : Simplex k P n)
     (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) {i : Fin (n + 1)} :
     haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
@@ -439,7 +437,6 @@ theorem median_map [CharZero k] {V₂ P₂ : Type*} [AddCommGroup V₂] [Module
     (s.map f hf).median i = (s.median i).map f := by
   simp [median, map_span, Set.image_pair]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem median_restrict [CharZero k] (s : Simplex k P n) (i : Fin (n + 1)) (S : AffineSubspace k P)
     (hS : affineSpan k (Set.range s.points) ≤ S) :
     haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
@@ -585,7 +582,6 @@ theorem affineSpan_range_medial [CharZero k] (s : Simplex k P n) :
   congrm ∃ a b, ?_ = v
   simp [← smul_neg]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem medial_restrict [CharZero k] (s : Simplex k P n) (S : AffineSubspace k P)
     (hS : affineSpan k (Set.range s.points) ≤ S) :
     haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
diff --git a/Mathlib/LinearAlgebra/Basis/Exact.lean b/Mathlib/LinearAlgebra/Basis/Exact.lean
index 9673184a97237d..7fd1838c54644c 100644
--- a/Mathlib/LinearAlgebra/Basis/Exact.lean
+++ b/Mathlib/LinearAlgebra/Basis/Exact.lean
@@ -112,7 +112,6 @@ end
 section
 include hfg
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Submodule.linearProjOfIsCompl_comp_surjective_of_exact
     {p q : Submodule R M} (hpq : IsCompl p q)
     (hmap : Submodule.map g q = ⊤) :
@@ -124,7 +123,6 @@ lemma Submodule.linearProjOfIsCompl_comp_surjective_of_exact
   obtain ⟨a, haq, ha⟩ : g x.val ∈ q.map g := by rwa [hmap]
   exact ⟨x - a, by simp [← ha], by simpa⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Submodule.linearProjOfIsCompl_comp_bijective_of_exact
     (hf : Function.Injective f) {p q : Submodule R M} (hpq : IsCompl p q)
     (hker : Disjoint (LinearMap.ker g) q) (hmap : Submodule.map g q = ⊤) :
diff --git a/Mathlib/LinearAlgebra/Basis/Fin.lean b/Mathlib/LinearAlgebra/Basis/Fin.lean
index 9383619c631e23..2c92d83b9acf15 100644
--- a/Mathlib/LinearAlgebra/Basis/Fin.lean
+++ b/Mathlib/LinearAlgebra/Basis/Fin.lean
@@ -38,7 +38,6 @@ namespace Basis
 
 section Fin
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`
 and `y` and `N` together span the whole of `M`, then there is a basis for `M`
 whose basis vectors are given by `Fin.cons y b`. -/
@@ -64,7 +63,6 @@ theorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R
   unfold mkFinCons
   exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`
 and `y` and `N` together span the whole of `O`, then there is a basis for `O`
 whose basis vectors are given by `Fin.cons y b`. -/
@@ -75,7 +73,6 @@ noncomputable def mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y
     (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))
     fun z => hsp z z.2
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem coe_mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O) (b : Basis (Fin n) R N)
     (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)
diff --git a/Mathlib/LinearAlgebra/Basis/VectorSpace.lean b/Mathlib/LinearAlgebra/Basis/VectorSpace.lean
index a5bc3269df6a48..b9e1cea57f2518 100644
--- a/Mathlib/LinearAlgebra/Basis/VectorSpace.lean
+++ b/Mathlib/LinearAlgebra/Basis/VectorSpace.lean
@@ -276,7 +276,6 @@ theorem LinearMap.leftInverse_apply_of_inj {f : V →ₗ[K] V'} (h_inj : LinearM
     f.leftInverse (f x) = x :=
   LinearMap.ext_iff.mp (f.leftInverse_comp_of_inj h_inj) x
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Submodule.exists_isCompl (p : Submodule K V) : ∃ q : Submodule K V, IsCompl p q :=
   ⟨LinearMap.ker p.subtype.leftInverse,
     LinearMap.isCompl_of_proj <| LinearMap.leftInverse_apply_of_inj p.ker_subtype⟩
@@ -284,7 +283,6 @@ theorem Submodule.exists_isCompl (p : Submodule K V) : ∃ q : Submodule K V, Is
 instance Submodule.complementedLattice : ComplementedLattice (Submodule K V) :=
   ⟨Submodule.exists_isCompl⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Any linear map `f : p →ₗ[K] V'` defined on a subspace `p` can be extended to the whole
 space. -/
 theorem LinearMap.exists_extend {p : Submodule K V} (f : p →ₗ[K] V') :
@@ -343,7 +341,6 @@ variable {K : Type*} {V : Type*} [Field K] [AddCommGroup V] [Module K V]
 
 variable {f : V →ₗ[K] K} {v : V}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- In a vector space, given a nonzero linear form `f`,
 a nonzero vector `v` such that `f v ≠ 0`,
 there exists a basis `b` with an index `i`
@@ -386,7 +383,6 @@ theorem exists_basis_of_pairing_ne_zero
     apply Or.resolve_left (Set.mem_insert_iff.mpr j.prop)
     simp [← hi, b, Subtype.coe_inj, Ne.symm h]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- In a vector space, given a nonzero linear form `f`,
 a nonzero vector `v` such that `f v = 0`,
 there exists a basis `b` with two distinct indices `i`, `j`
diff --git a/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean b/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
index 3e15258dcc24bc..0f42be79c7aaf4 100644
--- a/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
+++ b/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
@@ -260,7 +260,6 @@ open Module Submodule
 
 variable {B : BilinForm K V}
 
-set_option backward.isDefEq.respectTransparency false in
 theorem finrank_add_finrank_orthogonal' (W : Submodule K V) :
     finrank K W + finrank K (B.orthogonal W) =
       finrank K V + finrank K (W ⊓ B.ker : Subspace K V) := by
diff --git a/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean b/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean
index 3b8c4931fbc6af..18d5455a583cfb 100644
--- a/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean
+++ b/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean
@@ -202,7 +202,6 @@ def aux (f : EvenHom Q A) : CliffordAlgebra.even Q →ₗ[R] A := by
   refine ?_ ∘ₗ (even Q).val.toLinearMap
   exact LinearMap.fst R _ _ ∘ₗ foldr Q (fFold f) (fFold_fFold f) (1, 0)
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem aux_one : aux f 1 = 1 :=
   congr_arg Prod.fst (foldr_one _ _ _ _)
@@ -215,7 +214,6 @@ theorem aux_ι (m₁ m₂ : M) : aux f ((even.ι Q).bilin m₁ m₂) = f.bilin m
       rw [foldr_ι, foldr_ι]
       exact mul_one _)
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem aux_algebraMap (r) :
     aux f (algebraMap R (even Q) r) = algebraMap R A r :=
diff --git a/Mathlib/LinearAlgebra/Dimension/Constructions.lean b/Mathlib/LinearAlgebra/Dimension/Constructions.lean
index 1266bd01a45645..fbe86213994e83 100644
--- a/Mathlib/LinearAlgebra/Dimension/Constructions.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Constructions.lean
@@ -48,7 +48,6 @@ section Quotient
 variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
 variable [Module R M]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem LinearIndependent.sumElim_of_quotient
     {M' : Submodule R M} {ι₁ ι₂} {f : ι₁ → M'} (hf : LinearIndependent R f) (g : ι₂ → M)
     (hg : LinearIndependent R (Submodule.Quotient.mk (p := M') ∘ g)) :
diff --git a/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean b/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
index 4b9a11da9b93c2..5d895922609122 100644
--- a/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
+++ b/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
@@ -53,7 +53,6 @@ theorem Module.Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0 (h : Module.ran
     (Basis.ofVectorSpaceIndex K V).Finite :=
   Set.finite_def.2 <| (Basis.ofVectorSpace K V).nonempty_fintype_index_of_rank_lt_aleph0 h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Also see `rank_quotient_add_rank`. -/
 theorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :
     Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V := by
@@ -76,7 +75,6 @@ variable [AddCommGroup V₃] [Module K V₃]
 
 open LinearMap
 
-set_option backward.isDefEq.respectTransparency false in
 /-- This is mostly an auxiliary lemma for `Submodule.rank_sup_add_rank_inf_eq`. -/
 theorem rank_add_rank_split (db : V₂ →ₗ[K] V) (eb : V₃ →ₗ[K] V) (cd : V₁ →ₗ[K] V₂)
     (ce : V₁ →ₗ[K] V₃) (hde : ⊤ ≤ LinearMap.range db ⊔ LinearMap.range eb) (hgd : ker cd = ⊥)
diff --git a/Mathlib/LinearAlgebra/Dimension/Finite.lean b/Mathlib/LinearAlgebra/Dimension/Finite.lean
index 4efecc1010ff97..b65336015f8d61 100644
--- a/Mathlib/LinearAlgebra/Dimension/Finite.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Finite.lean
@@ -434,7 +434,6 @@ lemma Submodule.bot_eq_top_of_rank_eq_zero (h : Module.rank R M = 0) : (⊥ : Su
   rw [rank_zero_iff] at h
   subsingleton
 
-set_option backward.isDefEq.respectTransparency false in
 /-- See `rank_subsingleton` for the reason that `Nontrivial R` is needed. -/
 @[simp]
 theorem Submodule.rank_eq_zero {S : Submodule R M} : Module.rank R S = 0 ↔ S = ⊥ :=
diff --git a/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean b/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
index 2f1333bdf3d307..6f248047572d50 100644
--- a/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
+++ b/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
@@ -118,7 +118,6 @@ theorem rank_eq_one_iff [Module.Free K V] :
     haveI := b.repr.toEquiv.subsingleton
     exact h (Subsingleton.elim _ _)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule has dimension at most `1` if and only if there is a
 single vector in the submodule such that the submodule is contained in
 its span. -/
@@ -127,7 +126,6 @@ theorem rank_submodule_le_one_iff (s : Submodule K V) [Module.Free K s] :
   simp_rw [rank_le_one_iff, le_span_singleton_iff]
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule has dimension `1` if and only if there is a
 single non-zero vector in the submodule such that the submodule is contained in
 its span. -/
@@ -143,7 +141,6 @@ theorem rank_submodule_eq_one_iff (s : Submodule K V) [Module.Free K s] :
   · obtain ⟨r, hr⟩ := h v hv
     exact ⟨r, by rwa [Subtype.ext_iff, coe_smul]⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule has dimension at most `1` if and only if there is a
 single vector, not necessarily in the submodule, such that the
 submodule is contained in its span. -/
@@ -159,7 +156,6 @@ theorem rank_submodule_le_one_iff' (s : Submodule K V) [Module.Free K s] :
     simpa [b.mk_eq_rank''] using b.linearIndependent.map' _ (ker_inclusion _ _ h)
       |>.cardinal_le_rank.trans (rank_span_le {v₀})
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Submodule.rank_le_one_iff_isPrincipal (W : Submodule K V) [Module.Free K W] :
     Module.rank K W ≤ 1 ↔ W.IsPrincipal := by
   simp only [rank_le_one_iff, Submodule.isPrincipal_iff, le_antisymm_iff, le_span_singleton_iff,
diff --git a/Mathlib/LinearAlgebra/Dimension/LinearMap.lean b/Mathlib/LinearAlgebra/Dimension/LinearMap.lean
index c263611fb4d84e..6a04777c2ea4d9 100644
--- a/Mathlib/LinearAlgebra/Dimension/LinearMap.lean
+++ b/Mathlib/LinearAlgebra/Dimension/LinearMap.lean
@@ -98,7 +98,6 @@ theorem rank_finset_sum_le {η} (s : Finset η) (f : η → V →ₗ[K] V') :
   @Finset.sum_hom_rel _ _ _ _ _ (fun a b => rank a ≤ b) f (fun d => rank (f d)) s
     (le_of_eq rank_zero) fun _ _ _ h => le_trans (rank_add_le _ _) (by gcongr)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem le_rank_iff_exists_linearIndependent {c : Cardinal} {f : V →ₗ[K] V'} :
     c ≤ rank f ↔ ∃ s : Set V,
     Cardinal.lift.{v'} #s = Cardinal.lift.{v} c ∧ LinearIndepOn K f s := by
diff --git a/Mathlib/LinearAlgebra/Dimension/Localization.lean b/Mathlib/LinearAlgebra/Dimension/Localization.lean
index 8200d9b4a06d68..dde2860c6aba15 100644
--- a/Mathlib/LinearAlgebra/Dimension/Localization.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Localization.lean
@@ -87,7 +87,6 @@ theorem exists_set_linearIndependent_of_isDomain [IsDomain R] :
     IsLocalization.rank_eq (FractionRing R) R⁰ le_rfl,
     IsLocalizedModule.lift_rank_eq R⁰ (LocalizedModule.mkLinearMap R⁰ M) le_rfl]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The **rank-nullity theorem** for commutative domains. Also see `rank_quotient_add_rank`. -/
 theorem rank_quotient_add_rank_of_isDomain [IsDomain R] (M' : Submodule R M) :
     Module.rank R (M ⧸ M') + Module.rank R M' = Module.rank R M := by
diff --git a/Mathlib/LinearAlgebra/Dimension/RankNullity.lean b/Mathlib/LinearAlgebra/Dimension/RankNullity.lean
index 82c7aeaeeb3ac5..a302292d05210b 100644
--- a/Mathlib/LinearAlgebra/Dimension/RankNullity.lean
+++ b/Mathlib/LinearAlgebra/Dimension/RankNullity.lean
@@ -163,7 +163,6 @@ theorem Submodule.exists_smul_notMem_of_rank_lt {N : Submodule R M}
 
 open Cardinal Basis Submodule Function Set LinearMap
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Submodule.rank_sup_add_rank_inf_eq (s t : Submodule R M) :
     Module.rank R (s ⊔ t : Submodule R M) + Module.rank R (s ⊓ t : Submodule R M) =
     Module.rank R s + Module.rank R t := by
@@ -220,7 +219,6 @@ lemma Submodule.finrank_quotient [Module.Finite R M] {S : Type*} [Ring S] [SMul
   rw [← (N.restrictScalars R).finrank_quotient_add_finrank]
   exact Nat.eq_sub_of_add_eq rfl
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Submodule.disjoint_ker_of_finrank_le [IsDomain R] [IsTorsionFree R M] {N : Type*}
     [AddCommGroup N] [Module R N] {L : Submodule R M} [Module.Finite R L] (f : M →ₗ[R] N)
     (h : finrank R L ≤ finrank R (L.map f)) :
diff --git a/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean b/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
index cd29f4d6e8ceeb..1189421e069d77 100644
--- a/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
+++ b/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
@@ -404,7 +404,6 @@ def Submodule.inductionOnRank {R M} [Ring R] [StrongRankCondition R] [AddCommGro
   Submodule.inductionOnRankAux b P ih (Fintype.card ι) N fun hs hli => by
     simpa using b.card_le_card_of_linearIndependent hli
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `S` a module-finite free `R`-algebra, then the `R`-rank of a nonzero `R`-free
 ideal `I` of `S` is the same as the rank of `S`. -/
 theorem Ideal.rank_eq {R S : Type*} [CommRing R] [StrongRankCondition R] [Ring S] [IsDomain S]
diff --git a/Mathlib/LinearAlgebra/Dual/Lemmas.lean b/Mathlib/LinearAlgebra/Dual/Lemmas.lean
index 8181cb6569d2dc..b669fbdaa6d7bb 100644
--- a/Mathlib/LinearAlgebra/Dual/Lemmas.lean
+++ b/Mathlib/LinearAlgebra/Dual/Lemmas.lean
@@ -437,7 +437,6 @@ theorem dualAnnihilator_inj {W W' : Subspace K V} :
     W.dualAnnihilator = W'.dualAnnihilator ↔ W = W' :=
   ⟨fun h ↦ (dualAnnihilatorGci K V).l_injective h, congr_arg _⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a subspace `W` of `V` and an element of its dual `φ`, `dualLift W φ` is
 an arbitrary extension of `φ` to an element of the dual of `V`.
 That is, `dualLift W φ` sends `w ∈ W` to `φ x` and `x` in a chosen complement of `W` to `0`. -/
@@ -446,7 +445,6 @@ noncomputable def dualLift (W : Subspace K V) : Module.Dual K W →ₗ[K] Module
 
 variable {W : Subspace K V}
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem dualLift_of_subtype {φ : Module.Dual K W} (w : W) : W.dualLift φ (w : V) = φ w :=
   congr_arg φ <| LinearMap.leftInverse_apply_of_inj W.ker_subtype _
@@ -529,7 +527,6 @@ noncomputable def quotDualEquivAnnihilator (W : Subspace K V) :
     (Module.Dual K V ⧸ LinearMap.range W.dualLift) ≃ₗ[K] W.dualAnnihilator :=
   LinearEquiv.quotEquivOfQuotEquiv <| LinearEquiv.trans W.quotAnnihilatorEquiv W.dualEquivDual
 
-set_option backward.isDefEq.respectTransparency false in
 open scoped Classical in
 /-- The quotient by a subspace is isomorphic to its dual annihilator. -/
 noncomputable def quotEquivAnnihilator (W : Subspace K V) : (V ⧸ W) ≃ₗ[K] W.dualAnnihilator :=
@@ -680,7 +677,6 @@ theorem flip_quotDualCoannihilatorToDual_injective (W : Submodule R (Dual R M))
     Function.Injective W.quotDualCoannihilatorToDual.flip :=
   fun _ _ he ↦ Subtype.ext <| LinearMap.ext fun m ↦ DFunLike.congr_fun he ⟦m⟧
 
-set_option backward.isDefEq.respectTransparency false in
 open LinearMap in
 theorem quotDualCoannihilatorToDual_nondegenerate (W : Submodule R (Dual R M)) :
     W.quotDualCoannihilatorToDual.Nondegenerate := by
@@ -699,7 +695,6 @@ theorem range_dualMap_eq_dualAnnihilator_ker_of_surjective (f : M →ₗ[R] M')
   ((f.quotKerEquivOfSurjective hf).dualMap.range_comp _).trans
     (LinearMap.ker f).range_dualMap_mkQ_eq
 
-set_option backward.isDefEq.respectTransparency false in
 -- Note, this can be specialized to the case where `R` is an injective `R`-module, or when
 -- `f.coker` is a projective `R`-module.
 theorem range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective (f : M →ₗ[R] M')
@@ -798,7 +793,6 @@ theorem dualMap_surjective_of_injective {f : V₁ →ₗ[K] V₂} (hf : Function
   have ⟨f', hf'⟩ := f.exists_leftInverse_of_injective (ker_eq_bot.mpr hf)
   ⟨φ.comp f', ext fun x ↦ congr(φ <| $hf' x)⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem range_dualMap_eq_dualAnnihilator_ker (f : V₁ →ₗ[K] V₂) :
     LinearMap.range f.dualMap = (LinearMap.ker f).dualAnnihilator :=
   range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective f <|
@@ -822,7 +816,6 @@ theorem dualPairing_eq (W : Subspace K V₁) :
   ext
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem dualPairing_nondegenerate (W : Subspace K V₁) : W.dualPairing.Nondegenerate := by
   constructor
   · rw [LinearMap.separatingLeft_iff_ker_eq_bot, dualPairing_eq]
@@ -901,7 +894,6 @@ open Module LinearMap
 
 namespace LinearMap
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem finrank_range_dualMap_eq_finrank_range (f : V₁ →ₗ[K] V₂) :
     finrank K (LinearMap.range f.dualMap) = finrank K (LinearMap.range f) := by
@@ -960,19 +952,16 @@ namespace Subspace
 
 variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotDualCoannihilatorToDual_bijective (W : Subspace K (Dual K V)) [FiniteDimensional K W] :
     Function.Bijective W.quotDualCoannihilatorToDual :=
   ⟨W.quotDualCoannihilatorToDual_injective, letI : AddCommGroup W := inferInstance
     flip_injective_iff₂.mp W.flip_quotDualCoannihilatorToDual_injective⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem flip_quotDualCoannihilatorToDual_bijective (W : Subspace K (Dual K V))
     [FiniteDimensional K W] : Function.Bijective W.quotDualCoannihilatorToDual.flip :=
   letI : AddCommGroup W := inferInstance
   flip_bijective_iff₂.mpr W.quotDualCoannihilatorToDual_bijective
 
-set_option backward.isDefEq.respectTransparency false in
 theorem dualCoannihilator_dualAnnihilator_eq {W : Subspace K (Dual K V)} [FiniteDimensional K W] :
     W.dualCoannihilator.dualAnnihilator = W :=
   let e := (LinearEquiv.ofBijective _ W.flip_quotDualCoannihilatorToDual_bijective).trans
@@ -981,13 +970,11 @@ theorem dualCoannihilator_dualAnnihilator_eq {W : Subspace K (Dual K V)} [Finite
   haveI : FiniteDimensional K W.dualCoannihilator.dualAnnihilator := LinearEquiv.finiteDimensional e
   (eq_of_le_of_finrank_eq W.le_dualCoannihilator_dualAnnihilator e.finrank_eq).symm
 
-set_option backward.isDefEq.respectTransparency false in
 theorem finiteDimensional_quot_dualCoannihilator_iff {W : Submodule K (Dual K V)} :
     FiniteDimensional K (V ⧸ W.dualCoannihilator) ↔ FiniteDimensional K W :=
   ⟨fun _ ↦ FiniteDimensional.of_injective _ W.flip_quotDualCoannihilatorToDual_injective,
     fun _ ↦ FiniteDimensional.of_injective _ W.quotDualCoannihilatorToDual_injective⟩
 
-set_option backward.isDefEq.respectTransparency false in
 open OrderDual in
 /-- For any vector space, `dualAnnihilator` and `dualCoannihilator` gives an antitone order
   isomorphism between the finite-codimensional subspaces in the vector space and the
@@ -1014,7 +1001,6 @@ def orderIsoFiniteDimensional [FiniteDimensional K V] :
   right_inv _ := dualCoannihilator_dualAnnihilator_eq
   map_rel_iff' := dualAnnihilator_le_dualAnnihilator_iff
 
-set_option backward.isDefEq.respectTransparency false in
 open Submodule in
 theorem dualAnnihilator_dualAnnihilator_eq_map (W : Subspace K V) [FiniteDimensional K W] :
     W.dualAnnihilator.dualAnnihilator = W.map (Dual.eval K V) := by
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Basic.lean b/Mathlib/LinearAlgebra/Eigenspace/Basic.lean
index 6c72ef181ae011..a48df9e5f8f438 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Basic.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Basic.lean
@@ -82,13 +82,11 @@ lemma mem_genEigenspace {f : End R M} {μ : R} {k : ℕ∞} {x : M} :
   simp_rw [genEigenspace, OrderHom.coe_mk, LinearMap.mem_ker, iSup_subtype',
     Submodule.mem_iSup_of_directed _ this, LinearMap.mem_ker, Subtype.exists, exists_prop]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma genEigenspace_directed {f : End R M} {μ : R} {k : ℕ∞} :
     Directed (· ≤ ·) (fun l : {l : ℕ // l ≤ k} ↦ f.genEigenspace μ l) := by
   have aux : Monotone ((↑) : {l : ℕ // l ≤ k} → ℕ∞) := fun x y h ↦ by simpa using h
   exact ((genEigenspace f μ).monotone.comp aux).directed_le
 
-set_option backward.isDefEq.respectTransparency false in
 lemma mem_genEigenspace_nat {f : End R M} {μ : R} {k : ℕ} {x : M} :
     x ∈ f.genEigenspace μ k ↔ x ∈ LinearMap.ker ((f - μ • 1) ^ k) := by
   rw [mem_genEigenspace]
@@ -116,7 +114,6 @@ lemma genEigenspace_top (f : End R M) (μ : R) :
   rw [genEigenspace_eq_iSup_genEigenspace_nat, iSup_subtype]
   simp only [le_top, iSup_pos]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma genEigenspace_one {f : End R M} {μ : R} :
     f.genEigenspace μ 1 = LinearMap.ker (f - μ • 1) := by
   rw [← Nat.cast_one, genEigenspace_nat, pow_one]
@@ -127,7 +124,6 @@ lemma mem_genEigenspace_one {f : End R M} {μ : R} {x : M} :
   rw [genEigenspace_one, LinearMap.mem_ker, LinearMap.sub_apply,
     sub_eq_zero, LinearMap.smul_apply, Module.End.one_apply]
 
-set_option backward.isDefEq.respectTransparency false in
 -- `simp` can prove this using `genEigenspace_zero`
 lemma mem_genEigenspace_zero {f : End R M} {μ : R} {x : M} :
     x ∈ f.genEigenspace μ 0 ↔ x = 0 := by
@@ -254,7 +250,6 @@ or the infimum of these ranges if `k = ∞`. -/
 def genEigenrange (f : End R M) (μ : R) (k : ℕ∞) : Submodule R M :=
   ⨅ l : ℕ, ⨅ (_ : l ≤ k), LinearMap.range ((f - μ • 1) ^ l)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma genEigenrange_nat {f : End R M} {μ : R} {k : ℕ} :
     f.genEigenrange μ k = LinearMap.range ((f - μ • 1) ^ k) := by
   ext x
@@ -267,7 +262,6 @@ lemma genEigenrange_nat {f : End R M} {μ : R} {k : ℕ} :
     rw [this, pow_add]
     exact ⟨_, rfl⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The exponent of a generalized eigenvalue is never 0. -/
 lemma HasUnifEigenvalue.exp_ne_zero {f : End R M} {μ : R} {k : ℕ}
     (h : f.HasUnifEigenvalue μ k) : k ≠ 0 := by
@@ -299,7 +293,6 @@ lemma genEigenspace_le_genEigenspace_maxUnifEigenspaceIndex [IsNoetherian R M] (
   rw [← genEigenspace_top_eq_maxUnifEigenspaceIndex]
   exact (f.genEigenspace μ).monotone le_top
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Generalized eigenspaces for exponents at least `finrank K V` are equal to each other. -/
 theorem genEigenspace_eq_genEigenspace_maxUnifEigenspaceIndex_of_le [IsNoetherian R M]
     (f : End R M) (μ : R) {k : ℕ} (hk : maxUnifEigenspaceIndex f μ ≤ k) :
@@ -350,7 +343,6 @@ lemma maxUnifEigenspaceIndex_le_finrank [FiniteDimensional K V] (f : End K V) (
     rw [genEigenspace_nat, genEigenspace_nat]
     apply ker_pow_le_ker_pow_finrank
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Every generalized eigenvector is a generalized eigenvector for exponent `finrank K V`.
 (Lemma 8.20 of [axler2024]) -/
 lemma genEigenspace_le_genEigenspace_finrank [FiniteDimensional K V] (f : End K V)
@@ -361,7 +353,6 @@ lemma genEigenspace_le_genEigenspace_finrank [FiniteDimensional K V] (f : End K
       rw [genEigenspace_top_eq_maxUnifEigenspaceIndex]
       exact (f.genEigenspace _).monotone <| by simpa using maxUnifEigenspaceIndex_le_finrank f μ
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Generalized eigenspaces for exponents at least `finrank K V` are equal to each other. -/
 theorem genEigenspace_eq_genEigenspace_finrank_of_le [FiniteDimensional K V]
     (f : End K V) (μ : K) {k : ℕ} (hk : finrank K V ≤ k) :
@@ -414,7 +405,6 @@ lemma eigenspace_def {f : End R M} {μ : R} :
     f.eigenspace μ = LinearMap.ker (f - μ • 1) := by
   rw [eigenspace, genEigenspace_one]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by
   simp only [eigenspace, ← Nat.cast_one (R := ℕ∞), genEigenspace_zero_nat, pow_one]
@@ -552,7 +542,6 @@ theorem maxGenEigenspace_eq_maxGenEigenspace_zero (f : End R M) (μ : R) :
     maxGenEigenspace f μ = maxGenEigenspace (f - μ • 1) 0 := by
   ext; simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A generalized eigenvalue for some exponent `k` is also
 a generalized eigenvalue for exponents larger than `k`. -/
 theorem hasGenEigenvalue_of_hasGenEigenvalue_of_le {f : End R M} {μ : R} {k : ℕ}
@@ -560,7 +549,6 @@ theorem hasGenEigenvalue_of_hasGenEigenvalue_of_le {f : End R M} {μ : R} {k : 
     f.HasGenEigenvalue μ m :=
   hk.le <| by simpa using hm
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The eigenspace is a subspace of the generalized eigenspace. -/
 theorem eigenspace_le_genEigenspace {f : End R M} {μ : R} {k : ℕ} (hk : 0 < k) :
     f.eigenspace μ ≤ f.genEigenspace μ k :=
@@ -570,7 +558,6 @@ theorem eigenspace_le_maxGenEigenspace {f : End R M} {μ : R} :
     f.eigenspace μ ≤ f.maxGenEigenspace μ :=
   (f.genEigenspace _).monotone <| OrderTop.le_top _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- All eigenvalues are generalized eigenvalues. -/
 theorem hasGenEigenvalue_of_hasEigenvalue {f : End R M} {μ : R} {k : ℕ} (hk : 0 < k)
     (hμ : f.HasEigenvalue μ) : f.HasGenEigenvalue μ k :=
@@ -581,7 +568,6 @@ theorem hasEigenvalue_of_hasGenEigenvalue {f : End R M} {μ : R} {k : ℕ}
     (hμ : f.HasGenEigenvalue μ k) : f.HasEigenvalue μ :=
   hμ.lt zero_lt_one
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Generalized eigenvalues are actually just eigenvalues. -/
 theorem hasGenEigenvalue_iff_hasEigenvalue {f : End R M} {μ : R} {k : ℕ} (hk : 0 < k) :
     f.HasGenEigenvalue μ k ↔ f.HasEigenvalue μ := by
@@ -616,7 +602,6 @@ lemma isNilpotent_restrict_maxGenEigenspace_sub_algebraMap [IsNoetherian R M] (f
     _ (isNilpotent_restrict_genEigenspace_nat f μ (maxUnifEigenspaceIndex f μ))
   rw [maxGenEigenspace_eq]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma disjoint_genEigenspace [IsDomain R] [IsTorsionFree R M]
     (f : End R M) {μ₁ μ₂ : R} (hμ : μ₁ ≠ μ₂) (k l : ℕ∞) :
     Disjoint (f.genEigenspace μ₁ k) (f.genEigenspace μ₂ l) := by
@@ -728,7 +713,6 @@ theorem eigenvectors_linearIndependent [IsDomain R] [IsTorsionFree R M]
     (h_eigenvec : ∀ μ : μs, f.HasEigenvector μ (xs μ)) : LinearIndependent R xs :=
   f.eigenvectors_linearIndependent' (fun μ : μs ↦ μ) Subtype.coe_injective _ h_eigenvec
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `f` maps a subspace `p` into itself, then the generalized eigenspace of the restriction
 of `f` to `p` is the part of the generalized eigenspace of `f` that lies in `p`. -/
 theorem genEigenspace_restrict (f : End R M) (p : Submodule R M) (k : ℕ∞) (μ : R)
@@ -750,14 +734,12 @@ theorem genEigenspace_restrict (f : End R M) (p : Submodule R M) (k : ℕ∞) (
     erw [pow_succ, pow_succ, LinearMap.ker_comp, LinearMap.ker_comp, ih, ← LinearMap.ker_comp,
       LinearMap.comp_assoc]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma _root_.Submodule.inf_genEigenspace (f : End R M) (p : Submodule R M) {k : ℕ∞} {μ : R}
     (hfp : ∀ x : M, x ∈ p → f x ∈ p) :
     p ⊓ f.genEigenspace μ k =
       (genEigenspace (LinearMap.restrict f hfp) μ k).map p.subtype := by
   rw [f.genEigenspace_restrict _ _ _ hfp, Submodule.map_comap_eq, Submodule.range_subtype]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo
     {p : Submodule R M} (f g : End R M) (hf : MapsTo f p p) (hg : MapsTo g p p) {μ₁ μ₂ : R}
     (h : MapsTo f (g.maxGenEigenspace μ₁) (g.maxGenEigenspace μ₂)) :
@@ -770,7 +752,6 @@ lemma mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo
     Submodule.mk_eq_zero, ← mem_maxGenEigenspace] at hx ⊢
   exact h hx
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `p` is an invariant submodule of an endomorphism `f`, then the `μ`-eigenspace of the
 restriction of `f` to `p` is a submodule of the `μ`-eigenspace of `f`. -/
 theorem eigenspace_restrict_le_eigenspace (f : End R M) {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p)
@@ -798,7 +779,6 @@ theorem generalized_eigenvec_disjoint_range_ker [FiniteDimensional K V] (f : End
     Submodule.map_inf_eq_map_inf_comap, top_inf_eq, h, genEigenspace_nat]
   apply Submodule.map_comap_le
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If an invariant subspace `p` of an endomorphism `f` is disjoint from the `μ`-eigenspace of `f`,
 then the restriction of `f` to `p` has trivial `μ`-eigenspace. -/
 theorem eigenspace_restrict_eq_bot {f : End R M} {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p)
@@ -807,7 +787,6 @@ theorem eigenspace_restrict_eq_bot {f : End R M} {p : Submodule R M} (hfp : ∀
   intro x hx
   simpa using hμp.le_bot ⟨eigenspace_restrict_le_eigenspace f hfp μ ⟨x, hx, rfl⟩, x.prop⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The generalized eigenspace of an eigenvalue has positive dimension for positive exponents. -/
 theorem pos_finrank_genEigenspace_of_hasEigenvalue [FiniteDimensional K V] {f : End K V}
     {k : ℕ} {μ : K} (hx : f.HasEigenvalue μ) (hk : 0 < k) :
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Pi.lean b/Mathlib/LinearAlgebra/Eigenspace/Pi.lean
index 1037e5a5eceef9..72c964cab3c592 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Pi.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Pi.lean
@@ -39,7 +39,6 @@ theorem mem_iInf_maxGenEigenspace_iff (χ : ι → R) (m : M) :
     m ∈ ⨅ i, (f i).maxGenEigenspace (χ i) ↔ ∀ j, ∃ k : ℕ, ((f j - χ j • ↑1) ^ k) m = 0 := by
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a family of endomorphisms `i ↦ f i`, a family of candidate eigenvalues `i ↦ μ i`, and a
 submodule `p` which is invariant w.r.t. every `f i`, the intersection of `p` with the simultaneous
 maximal generalised eigenspace (taken over all `i`), is the same as the simultaneous maximal
@@ -52,7 +51,6 @@ lemma _root_.Submodule.inf_iInf_maxGenEigenspace_of_forall_mapsTo {μ : ι → R
   · simp [iInf_of_isEmpty]
   · simp_rw [inf_iInf, p.inf_genEigenspace _ (hfp _), Submodule.map_iInf _ p.injective_subtype]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a family of endomorphisms `i ↦ f i`, a family of candidate eigenvalues `i ↦ μ i`, and a
 distinguished index `i` whose maximal generalised `μ i`-eigenspace is invariant w.r.t. every `f j`,
 taking simultaneous maximal generalised eigenspaces is unaffected by first restricting to the
@@ -144,7 +142,6 @@ lemma independent_iInf_maxGenEigenspace_of_forall_mapsTo
     simp only [Submodule.coe_iInf]
     exact h l χ
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a family of endomorphisms `i ↦ f i` which are compatible in the sense that every maximal
 generalised eigenspace of `f i` is invariant w.r.t. `f j`, if each `f i` is triangularizable, the
 family is simultaneously triangularizable. -/
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean b/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean
index 85ba8c43d786ad..c1e6cb804cee15 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean
@@ -29,7 +29,6 @@ namespace Module.End
 
 variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] {f g : End R M}
 
-set_option backward.isDefEq.respectTransparency false in
 lemma apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNil
     {μ : R} {k : ℕ∞} {m : M} (hm : m ∈ f.genEigenspace μ k)
     (hfg : Commute f g) (hss : g.IsFinitelySemisimple) (hnil : IsNilpotent (f - g)) :
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean b/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
index b103db2a4a4ce6..909bb28e8236b7 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
@@ -69,7 +69,6 @@ noncomputable instance [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f
     Inhabited f.Eigenvalues :=
   ⟨⟨f.exists_eigenvalue.choose, f.exists_eigenvalue.choose_spec⟩⟩
 
-set_option backward.isDefEq.respectTransparency false in
 -- Lemma 8.22(c) of [axler2024]
 /-- In finite dimensions, over an algebraically closed field, the generalized eigenspaces of any
 linear endomorphism span the whole space. -/
@@ -143,7 +142,6 @@ namespace Submodule
 
 variable {p : Submodule K V} {f : Module.End K V}
 
-set_option backward.isDefEq.respectTransparency false in
 theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈ p) (k : ℕ∞) :
     p ⊓ ⨆ μ, f.genEigenspace μ k = ⨆ μ, p ⊓ f.genEigenspace μ k := by
   refine le_antisymm (fun m hm ↦ ?_)
@@ -232,7 +230,6 @@ theorem eq_iSup_inf_genEigenspace [FiniteDimensional K V] (k : ℕ∞)
 
 end Submodule
 
-set_option backward.isDefEq.respectTransparency false in
 /-- In finite dimensions, if the generalized eigenspaces of a linear endomorphism span the whole
 space then the same is true of its restriction to any invariant submodule. -/
 theorem Module.End.genEigenspace_restrict_eq_top
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Zero.lean b/Mathlib/LinearAlgebra/Eigenspace/Zero.lean
index c93f11b256397e..bb2c8df2fe52a9 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Zero.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Zero.lean
@@ -137,7 +137,6 @@ lemma not_hasEigenvalue_zero_tfae (φ : Module.End K M) :
   have aux₂ : ker φ = ⊥ ↔ ¬ ⊥ < ker φ := by rw [bot_lt_iff_ne_bot, not_not]
   simpa only [aux₁, aux₂] using this
 
-set_option backward.isDefEq.respectTransparency false in
 open Module.Free in
 lemma finrank_maxGenEigenspace_zero_eq (φ : Module.End K M) :
     finrank K (φ.maxGenEigenspace 0) = natTrailingDegree (φ.charpoly) := by
diff --git a/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean b/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean
index 054dd8ed39df28..a15cc2f17da5c2 100644
--- a/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean
+++ b/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean
@@ -105,7 +105,6 @@ lemma ιMulti_span_fixedDegree_of_span_eq_top {s : Set M} (hs : span R s = ⊤)
       obtain ⟨m, -, hm⟩ := (Set.mem_image _ _ _).mp (f i).2
       rw [Function.comp_apply, Function.comp_apply, ← hm, ExteriorAlgebra.ι_leftInverse]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The image of `exteriorPower.ιMulti` spans `⋀[R]^n M`. -/
 lemma ιMulti_span :
     Submodule.span R (Set.range (ιMulti R n)) = (⊤ : Submodule R (⋀[R]^n M)) := by
@@ -115,7 +114,6 @@ lemma ιMulti_span :
     Submodule.range_subtype]
   exact ExteriorAlgebra.ιMulti_span_fixedDegree R n
 
-set_option backward.isDefEq.respectTransparency false in
 open Set Submodule in
 /-- A version of `ιMulti_span_fixedDegree_of_span` that works in the exterior power. -/
 lemma ιMulti_span_of_span {s : Set M} (hs : span R s = ⊤) :
@@ -181,7 +179,6 @@ noncomputable def relationsSolutionEquiv {ι : Type*} [DecidableEq ι] {M : Type
         · simp
         · simpa using f.map_eq_zero_of_eq v hm hij }
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The universal property of the exterior power. -/
 noncomputable def isPresentationCore :
     (relationsSolutionEquiv.symm (ιMulti R n (M := M))).IsPresentationCore where
@@ -195,7 +192,6 @@ noncomputable def isPresentationCore :
 
 end presentation
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The standard presentation of the `R`-module `⋀[R]^n M`. -/
 @[simps! G R relation var]
 noncomputable def presentation : Module.Presentation R (⋀[R]^n M) :=
@@ -203,7 +199,6 @@ noncomputable def presentation : Module.Presentation R (⋀[R]^n M) :=
 
 variable {R M n}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Two linear maps on `⋀[R]^n M` that agree on the image of `exteriorPower.ιMulti`
 are equal. -/
 @[ext]
@@ -211,7 +206,6 @@ lemma linearMap_ext {f : ⋀[R]^n M →ₗ[R] N} {g : ⋀[R]^n M →ₗ[R] N}
     (heq : f.compAlternatingMap (ιMulti R n) = g.compAlternatingMap (ιMulti R n)) : f = g :=
   (presentation R n M).postcomp_injective (by ext f; apply DFunLike.congr_fun heq)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The linear equivalence between `n`-fold alternating maps from `M` to `N` and linear maps from
 `⋀[R]^n M` to `N`: this is the universal property of the `n`th exterior power of `M`. -/
 noncomputable def alternatingMapLinearEquiv : (M [⋀^Fin n]→ₗ[R] N) ≃ₗ[R] ⋀[R]^n M →ₗ[R] N :=
@@ -240,7 +234,6 @@ lemma alternatingMapLinearEquiv_symm_apply (F : ⋀[R]^n M →ₗ[R] N) (m : Fin
   obtain ⟨f, rfl⟩ := alternatingMapLinearEquiv.surjective F
   simp only [LinearEquiv.symm_apply_apply, alternatingMapLinearEquiv_comp_ιMulti]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma alternatingMapLinearEquiv_ιMulti :
     alternatingMapLinearEquiv (ιMulti R n (M := M)) = LinearMap.id := by
@@ -261,24 +254,20 @@ lemma alternatingMapLinearEquiv_comp (g : N →ₗ[R] N') (f : M [⋀^Fin n]→
 
 /-! Functoriality of the exterior powers. -/
 
-set_option backward.isDefEq.respectTransparency false in
 variable (n) in
 /-- The linear map between `n`th exterior powers induced by a linear map between the modules. -/
 noncomputable def map (f : M →ₗ[R] N) : ⋀[R]^n M →ₗ[R] ⋀[R]^n N :=
   alternatingMapLinearEquiv ((ιMulti R n).compLinearMap f)
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma alternatingMapLinearEquiv_symm_map (f : M →ₗ[R] N) :
     alternatingMapLinearEquiv.symm (map n f) = (ιMulti R n).compLinearMap f := by
   simp only [map, LinearEquiv.symm_apply_apply]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem map_comp_ιMulti (f : M →ₗ[R] N) :
     (map n f).compAlternatingMap (ιMulti R n) = (ιMulti R n).compLinearMap f := by
   simp only [map, alternatingMapLinearEquiv_comp_ιMulti]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem map_apply_ιMulti (f : M →ₗ[R] N) (m : Fin n → M) :
     map n f (ιMulti R n m) = ιMulti R n (f ∘ m) := by
@@ -292,7 +281,6 @@ lemma map_comp_ιMulti_family {I : Type*} [LinearOrder I] (v : I → M) (f : M 
   simp only [ιMulti_family, Function.comp_apply, map_apply_ιMulti]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma map_apply_ιMulti_family {I : Type*} [LinearOrder I] (v : I → M) (f : M →ₗ[R] N)
     (s : powersetCard I n) :
@@ -300,13 +288,11 @@ lemma map_apply_ιMulti_family {I : Type*} [LinearOrder I] (v : I → M) (f : M
   simp only [ιMulti_family, map, alternatingMapLinearEquiv_apply_ιMulti]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem map_id :
     map n (LinearMap.id (R := R) (M := M)) = LinearMap.id := by
   aesop
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem map_comp (f : M →ₗ[R] N) (g : N →ₗ[R] N') :
     map n (g ∘ₗ f) = map n g ∘ₗ map n f := by
@@ -390,7 +376,6 @@ lemma ιMulti_family_span_fixedDegree_of_span {I : Type*} [LinearOrder I] {v : I
     obtain ⟨α, rfl⟩ := Set.range_subset_range_iff_exists_comp.mp f_range
     exact ιMulti_family_span_fixedDegree_aux R v α
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If a family of vectors spans `M`, then the family of its `n`-fold exterior products spans
 `⋀[R]^n M`. This is a variant of `ιMulti_family_span_fixedDegree_of_span` where we
 work in the exterior power and not the exterior algebra. -/
@@ -401,7 +386,6 @@ lemma ιMulti_family_span_of_span {I : Type*} [LinearOrder I]
   rw [LinearMap.map_span, ← Set.image_univ, Set.image_image]
   simpa using ιMulti_family_span_fixedDegree_of_span R hv
 
-set_option backward.isDefEq.respectTransparency false in
 open Set Submodule in
 /-- If `v` is a family of vectors of `M` indexed by a linearly ordered type, then the span of the
 range of `exteriorPower.ιMulti_family R n v`, i.e., of the family of `n`-fold exterior products
@@ -423,7 +407,6 @@ end ιMulti_family
 
 /-! Linear equivalences in degrees 0 and 1. -/
 
-set_option backward.isDefEq.respectTransparency false in
 variable (R M) in
 /-- The linear equivalence ` ⋀[R]^0 M ≃ₗ[R] R`. -/
 @[simps! -isSimp symm_apply]
@@ -443,7 +426,6 @@ lemma zeroEquiv_ιMulti (f : Fin 0 → M) :
 lemma zeroEquiv_naturality (f : M →ₗ[R] N) :
     (zeroEquiv R N).comp (map 0 f) = zeroEquiv R M := by aesop
 
-set_option backward.isDefEq.respectTransparency false in
 variable (R M) in
 /-- The linear equivalence `M ≃ₗ[R] ⋀[R]^1 M`. -/
 @[simps! -isSimp symm_apply]
diff --git a/Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean b/Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean
index 4c243c6e8ca43e..0169aea3796ead 100644
--- a/Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean
+++ b/Mathlib/LinearAlgebra/FiniteDimensional/Basic.lean
@@ -57,7 +57,6 @@ theorem _root_.LinearIndependent.lt_aleph0_of_finiteDimensional {ι : Type w} [F
     {v : ι → V} (h : LinearIndependent K v) : #ι < ℵ₀ :=
   h.lt_aleph0_of_finite
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If a submodule has maximal dimension in a finite-dimensional space, then it is equal to the
 whole space. -/
 theorem _root_.Submodule.eq_top_of_finrank_eq [FiniteDimensional K V] {S : Submodule K V}
@@ -154,7 +153,6 @@ theorem FiniteDimensional.of_rank_eq_one (h : Module.rank K V = 1) : FiniteDimen
 
 variable (K V)
 
-set_option backward.isDefEq.respectTransparency false in
 instance finiteDimensional_bot : FiniteDimensional K (⊥ : Submodule K V) :=
   .of_rank_eq_zero <| by simp
 
@@ -168,35 +166,30 @@ section DivisionRing
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule contained in a finite-dimensional submodule is
 finite-dimensional. -/
 theorem finiteDimensional_of_le {S₁ S₂ : Submodule K V} [FiniteDimensional K S₂] (h : S₁ ≤ S₂) :
     FiniteDimensional K S₁ :=
   (isNoetherian_of_le h).finite
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The inf of two submodules, the first finite-dimensional, is
 finite-dimensional. -/
 instance finiteDimensional_inf_left (S₁ S₂ : Submodule K V) [FiniteDimensional K S₁] :
     FiniteDimensional K (S₁ ⊓ S₂ : Submodule K V) :=
   finiteDimensional_of_le inf_le_left
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The inf of two submodules, the second finite-dimensional, is
 finite-dimensional. -/
 instance finiteDimensional_inf_right (S₁ S₂ : Submodule K V) [FiniteDimensional K S₂] :
     FiniteDimensional K (S₁ ⊓ S₂ : Submodule K V) :=
   finiteDimensional_of_le inf_le_right
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The sup of two finite-dimensional submodules is
 finite-dimensional. -/
 instance finiteDimensional_sup (S₁ S₂ : Submodule K V) [h₁ : FiniteDimensional K S₁]
     [h₂ : FiniteDimensional K S₂] : FiniteDimensional K (S₁ ⊔ S₂ : Submodule K V) :=
   finite_sup _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The submodule generated by a finite supremum of finite-dimensional submodules is
 finite-dimensional.
 
@@ -206,7 +199,6 @@ instance finiteDimensional_finset_sup {ι : Type*} (s : Finset ι) (S : ι → S
     [∀ i, FiniteDimensional K (S i)] : FiniteDimensional K (s.sup S : Submodule K V) :=
   Submodule.finite_finset_sup _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The submodule generated by a supremum of finite-dimensional submodules, indexed by a finite
 sort is finite-dimensional. -/
 instance finiteDimensional_iSup {ι : Sort*} [Finite ι] (S : ι → Submodule K V)
@@ -230,7 +222,6 @@ end
 namespace Submodule
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If a submodule is contained in a finite-dimensional
 submodule with the same or smaller dimension, they are equal. -/
 theorem eq_of_le_of_finrank_le {S₁ S₂ : Submodule K V} [FiniteDimensional K S₂] (hle : S₁ ≤ S₂)
@@ -239,7 +230,6 @@ theorem eq_of_le_of_finrank_le {S₁ S₂ : Submodule K V} [FiniteDimensional K
   exact le_antisymm hle (Submodule.comap_subtype_eq_top.1
     (eq_top_of_finrank_eq (le_antisymm (comap (Submodule.subtype S₂) S₁).finrank_le hd)))
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If a submodule is contained in a finite-dimensional
 submodule with the same dimension, they are equal. -/
 theorem eq_of_le_of_finrank_eq {S₁ S₂ : Submodule K V} [FiniteDimensional K S₂] (hle : S₁ ≤ S₂)
@@ -289,7 +279,6 @@ theorem finiteDimensional_of_surjective [FiniteDimensional K V] (f : V →ₗ[K]
     (hf : LinearMap.range f = ⊤) : FiniteDimensional K V₂ :=
   Module.Finite.of_surjective f <| range_eq_top.1 hf
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The range of a linear map defined on a finite-dimensional space is also finite-dimensional. -/
 instance finiteDimensional_range [FiniteDimensional K V] (f : V →ₗ[K] V₂) :
     FiniteDimensional K (LinearMap.range f) :=
@@ -304,7 +293,6 @@ theorem injective_iff_surjective [FiniteDimensional K V] {f : V →ₗ[K] V} :
     (leftInverse_of_surjective_of_rightInverse (surjective_of_injective this.injective)
         this).injective⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma injOn_iff_surjOn {p : Submodule K V} [FiniteDimensional K p]
     {f : V →ₗ[K] V} (h : ∀ x ∈ p, f x ∈ p) :
     Set.InjOn f p ↔ Set.SurjOn f p p := by
@@ -361,7 +349,6 @@ they are inverse to each other on the other side. -/
 theorem mul_eq_one_comm [FiniteDimensional K V] {f g : V →ₗ[K] V} : f * g = 1 ↔ g * f = 1 :=
   mul_eq_one_comm
 
-set_option backward.isDefEq.respectTransparency false in
 theorem comap_eq_sup_ker_of_disjoint {p : Submodule K V} [FiniteDimensional K p] {f : V →ₗ[K] V}
     (h : ∀ x ∈ p, f x ∈ p) (h' : Disjoint p (ker f)) :
     p.comap f = p ⊔ ker f := by
@@ -528,7 +515,6 @@ section Span
 
 open Submodule
 
-set_option backward.isDefEq.respectTransparency false in
 theorem finrank_span_singleton {v : V} (hv : v ≠ 0) : finrank K (K ∙ v) = 1 := by
   apply le_antisymm
   · exact finrank_span_le_card ({v} : Set V)
diff --git a/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean b/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
index 2534fef930f788..55f86eec9f9c02 100644
--- a/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
+++ b/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
@@ -127,7 +127,6 @@ theorem of_finite_basis {ι : Type w} {s : Set ι} (h : Basis s K V) (hs : Set.F
   haveI := hs.fintype
   h.finiteDimensional_of_finite
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A subspace of a finite-dimensional space is also finite-dimensional.
 
 This is a shortcut instance to simplify inference in the presence of `[FiniteDimensional K V]`.
@@ -199,22 +198,18 @@ variable (K)
 
 instance finiteDimensional_self : FiniteDimensional K K := inferInstance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The submodule generated by a finite set is finite-dimensional. -/
 theorem span_of_finite {A : Set V} (hA : Set.Finite A) : FiniteDimensional K (Submodule.span K A) :=
   Module.Finite.span_of_finite K hA
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The submodule generated by a single element is finite-dimensional. -/
 instance span_singleton (x : V) : FiniteDimensional K (K ∙ x) :=
   Module.Finite.span_singleton K x
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The submodule generated by a finset is finite-dimensional. -/
 instance span_finset (s : Finset V) : FiniteDimensional K (span K (s : Set V)) :=
   Module.Finite.span_finset K s
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Pushforwards of finite-dimensional submodules are finite-dimensional. -/
 instance (f : V →ₗ[K] V₂) (p : Submodule K V) [FiniteDimensional K p] :
     FiniteDimensional K (p.map f) :=
@@ -240,7 +235,6 @@ section DivisionRing
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule is finitely generated if and only if it is finite-dimensional -/
 theorem fg_iff_finiteDimensional (s : Submodule K V) : s.FG ↔ FiniteDimensional K s :=
   Module.Finite.iff_fg.symm
diff --git a/Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean b/Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean
index bf6df711dec700..218606ff59b7e7 100644
--- a/Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean
+++ b/Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean
@@ -49,7 +49,6 @@ theorem finrank_lt [FiniteDimensional K V] {s : Submodule K V} (h : s ≠ ⊤) :
   rw [← Quotient.nontrivial_iff] at h
   exact Nat.lt_add_of_pos_right finrank_pos
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The sum of the dimensions of s + t and s ∩ t is the sum of the dimensions of s and t -/
 theorem finrank_sup_add_finrank_inf_eq (s t : Submodule K V) [FiniteDimensional K s]
     [FiniteDimensional K t] :
@@ -59,7 +58,6 @@ theorem finrank_sup_add_finrank_inf_eq (s t : Submodule K V) [FiniteDimensional
   repeat rw [← finrank_eq_rank] at key
   norm_cast at key
 
-set_option backward.isDefEq.respectTransparency false in
 theorem finrank_add_le_finrank_add_finrank (s t : Submodule K V) [FiniteDimensional K s]
     [FiniteDimensional K t] : finrank K (s ⊔ t : Submodule K V) ≤ finrank K s + finrank K t := by
   rw [← finrank_sup_add_finrank_inf_eq]
@@ -111,7 +109,6 @@ noncomputable def LinearEquiv.quotEquivOfEquiv {p : Subspace K V} {q : Subspace
         LinearEquiv.finrank_eq f₁, Submodule.finrank_quotient_add_finrank,
         LinearEquiv.finrank_eq f₂])
 
-set_option backward.isDefEq.respectTransparency false in
 -- TODO: generalize to the case where one of `p` and `q` is finite-dimensional.
 /-- Given the subspaces `p q`, if `p.quotient ≃ₗ[K] q`, then `q.quotient ≃ₗ[K] p` -/
 noncomputable def LinearEquiv.quotEquivOfQuotEquiv {p q : Subspace K V} (f : (V ⧸ p) ≃ₗ[K] q) :
@@ -197,7 +194,6 @@ section DivisionRing
 variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
   [Module K V₂]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem finrank_lt_finrank_of_lt {s t : Submodule K V} [FiniteDimensional K t] (hst : s < t) :
     finrank K s < finrank K t :=
   (comapSubtypeEquivOfLe hst.le).finrank_eq.symm.trans_lt <|
diff --git a/Mathlib/LinearAlgebra/FreeModule/Finite/CardQuotient.lean b/Mathlib/LinearAlgebra/FreeModule/Finite/CardQuotient.lean
index b4cc35558ae83a..49c112d32c4a6e 100644
--- a/Mathlib/LinearAlgebra/FreeModule/Finite/CardQuotient.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/Finite/CardQuotient.lean
@@ -28,7 +28,6 @@ section Submodule
 
 variable {M : Type*} [AddCommGroup M] [Module.Free ℤ M] [Module.Finite ℤ M]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `e : M ≃ N` be an additive isomorphism (therefore a `ℤ`-linear equiv).
 Then an alternative way to compute the cardinality of the quotient `M ⧸ N` is given by taking
 the determinant of `e`.
@@ -80,7 +79,6 @@ theorem Submodule.natAbs_det_equiv (N : Submodule ℤ M) {E : Type*} [EquivLike
   -- which maps `(M ⧸ N)` to `Π i, ZMod (a i).nat_abs`.
   simp_rw [Nat.card_congr (quotientEquivPiZMod N b h).toEquiv, Nat.card_pi, Nat.card_zmod, a]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `b` be a basis for `M` over `ℤ` and `bN` a basis for `N` over `ℤ` of the same dimension.
 Then an alternative way to compute the cardinality of `M ⧸ N` is given by taking the determinant
 of `bN` over `b`. -/
@@ -112,7 +110,6 @@ theorem AddSubgroup.relIndex_eq_natAbs_det {E : Type*} [AddCommGroup E]
   rw [relIndex, index_eq_natAbs_det b₂ _ (b₁.map (addSubgroupOfEquivOfLe H).toIntLinearEquiv.symm)]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem AddSubgroup.relIndex_eq_abs_det {E : Type*} [AddCommGroup E] [Module ℚ E]
     (L₁ L₂ : AddSubgroup E) (H : L₁ ≤ L₂) {ι : Type*} [DecidableEq ι] [Fintype ι]
     (b₁ b₂ : Basis ι ℚ E) (h₁ : L₁ = .closure (Set.range b₁)) (h₂ : L₂ = .closure (Set.range b₂)) :
diff --git a/Mathlib/LinearAlgebra/FreeModule/PID.lean b/Mathlib/LinearAlgebra/FreeModule/PID.lean
index 002d687b9df65f..d5084ac95fa998 100644
--- a/Mathlib/LinearAlgebra/FreeModule/PID.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/PID.lean
@@ -138,7 +138,6 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 
 variable [IsDomain R]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.
 
 Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
@@ -308,7 +307,6 @@ theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
   obtain rfl : n = 0 := by simpa using Fintype.card_eq.mpr ⟨e⟩
   exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule inside a free `R`-submodule of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -390,7 +388,6 @@ instance Module.free_of_finite_type_torsion_free' [Module.Finite R M] [IsTorsion
   obtain ⟨n, b⟩ : Σ n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
   exact Module.Free.of_basis b
 
-set_option backward.isDefEq.respectTransparency false in
 instance {S : Type*} [CommRing S] [Algebra R S] {I : Ideal S} [hI₁ : Module.Finite R I]
     [hI₂ : IsTorsionFree R I] : Free R I := by
   have : Module.Finite R (restrictScalars R I) := hI₁
@@ -505,7 +502,6 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
   obtain ⟨as, has⟩ := h'' as' has'
   exact ⟨_, _, hmn, bM, bN, as, has⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -525,7 +521,6 @@ noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,
     Submodule.coe_smul_of_tower, Fin.castLEEmb_apply]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
diff --git a/Mathlib/LinearAlgebra/Goursat.lean b/Mathlib/LinearAlgebra/Goursat.lean
index 11f64be2dec4c9..4090de1ada933a 100644
--- a/Mathlib/LinearAlgebra/Goursat.lean
+++ b/Mathlib/LinearAlgebra/Goursat.lean
@@ -51,12 +51,12 @@ def goursatSnd : Submodule R N :=
 lemma goursatFst_toAddSubgroup :
     (goursatFst L).toAddSubgroup = L.toAddSubgroup.goursatFst := by
   ext x
-  simp [mem_toAddSubgroup, goursatFst, AddSubgroup.mem_goursatFst]
+  simp [goursatFst, AddSubgroup.mem_goursatFst]
 
 lemma goursatSnd_toAddSubgroup :
     (goursatSnd L).toAddSubgroup = L.toAddSubgroup.goursatSnd := by
   ext x
-  simp [mem_toAddSubgroup, goursatSnd, AddSubgroup.mem_goursatSnd]
+  simp [goursatSnd, AddSubgroup.mem_goursatSnd]
 
 variable (L) in
 lemma goursatFst_prod_goursatSnd_le : L.goursatFst.prod L.goursatSnd ≤ L := by
@@ -95,7 +95,6 @@ lemma goursat_surjective : ∃ e : (M ⧸ L.goursatFst) ≃ₗ[R] N ⧸ L.goursa
   rw [mem_toAddSubgroup, mem_graph_iff, Eq.comm]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Goursat's lemma** for an arbitrary submodule of a product.
 
 If `L` is a submodule of `M × N`, then there exist submodules `M'' ≤ M' ≤ M` and `N'' ≤ N' ≤ N` such
diff --git a/Mathlib/LinearAlgebra/Isomorphisms.lean b/Mathlib/LinearAlgebra/Isomorphisms.lean
index 9a81a5f9d4993c..578dbff5593052 100644
--- a/Mathlib/LinearAlgebra/Isomorphisms.lean
+++ b/Mathlib/LinearAlgebra/Isomorphisms.lean
@@ -66,19 +66,16 @@ theorem quotKerEquivOfSurjective_symm_apply (hf : Function.Surjective f) (x : M)
     (f.quotKerEquivOfSurjective hf).symm (f x) = Submodule.Quotient.mk x := by
   simp [LinearEquiv.symm_apply_eq]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Linear map from `p` to `p+p'/p'` where `p p'` are submodules of `R` -/
 abbrev subToSupQuotient (p p' : Submodule R M) :
     { x // x ∈ p } →ₗ[R] { x // x ∈ p ⊔ p' } ⧸ comap (Submodule.subtype (p ⊔ p')) p' :=
   (comap (p ⊔ p').subtype p').mkQ.comp (Submodule.inclusion le_sup_left)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem comap_leq_ker_subToSupQuotient (p p' : Submodule R M) :
     comap (Submodule.subtype p) (p ⊓ p') ≤ ker (subToSupQuotient p p') := by
   rw [LinearMap.ker_comp, Submodule.inclusion, comap_codRestrict, ker_mkQ, map_comap_subtype]
   exact comap_mono (inf_le_inf_right _ le_sup_left)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Canonical linear map from the quotient `p/(p ∩ p')` to `(p+p')/p'`, mapping `x + (p ∩ p')`
 to `x + p'`, where `p` and `p'` are submodules of an ambient module.
 
@@ -107,7 +104,6 @@ theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
   use ⟨y, hy⟩; apply (Submodule.Quotient.eq _).2
   simp only [mem_comap, map_sub, coe_subtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Second Isomorphism Law : the canonical map from `p/(p ∩ p')` to `(p+p')/p'` as a linear isomorphism.
 
@@ -126,14 +122,12 @@ theorem coe_quotientInfToSupQuotient (p p' : Submodule R M) :
     ⇑(quotientInfToSupQuotient p p') = quotientInfEquivSupQuotient p p' :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientInfEquivSupQuotient_apply_mk (p p' : Submodule R M) (x : p) :
     let map := inclusion (le_sup_left : p ≤ p ⊔ p')
     quotientInfEquivSupQuotient p p' (Submodule.Quotient.mk x) =
       @Submodule.Quotient.mk R (p ⊔ p' : Submodule R M) _ _ _ (comap (p ⊔ p').subtype p') (map x) :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientInfEquivSupQuotient_symm_apply_left (p p' : Submodule R M) (x : ↥(p ⊔ p'))
     (hx : (x : M) ∈ p) :
     (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) =
@@ -142,12 +136,10 @@ theorem quotientInfEquivSupQuotient_symm_apply_left (p p' : Submodule R M) (x :
     rw [quotientInfEquivSupQuotient_apply_mk, inclusion_apply]
 
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientInfEquivSupQuotient_symm_apply_eq_zero_iff {p p' : Submodule R M} {x : ↥(p ⊔ p')} :
     (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0 ↔ (x : M) ∈ p' :=
   (LinearEquiv.symm_apply_eq _).trans <| by simp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientInfEquivSupQuotient_symm_apply_right (p p' : Submodule R M) {x : ↥(p ⊔ p')}
     (hx : (x : M) ∈ p') : (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x)
     = 0 :=
diff --git a/Mathlib/LinearAlgebra/LinearDisjoint.lean b/Mathlib/LinearAlgebra/LinearDisjoint.lean
index fb86215511ca01..a478007cac3be6 100644
--- a/Mathlib/LinearAlgebra/LinearDisjoint.lean
+++ b/Mathlib/LinearAlgebra/LinearDisjoint.lean
@@ -486,7 +486,6 @@ section
 
 variable [Nontrivial R]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `M` and `N` are linearly disjoint, if `M` is flat, then any two commutative
 elements of `↥(M ⊓ N)` are not `R`-linearly independent (namely, their span is not `R ^ 2`). -/
 theorem not_linearIndependent_pair_of_commute_of_flat_left [Module.Flat R M]
@@ -503,7 +502,6 @@ theorem not_linearIndependent_pair_of_commute_of_flat_left [Module.Flat R M]
   repeat rw [AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero] at hm
   exact h.ne_zero 0 hm.2
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `M` and `N` are linearly disjoint, if `N` is flat, then any two commutative
 elements of `↥(M ⊓ N)` are not `R`-linearly independent (namely, their span is not `R ^ 2`). -/
 theorem not_linearIndependent_pair_of_commute_of_flat_right [Module.Flat R N]
diff --git a/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean b/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean
index 648a2d62ac8a03..501b3f16950a31 100644
--- a/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean
+++ b/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean
@@ -805,7 +805,6 @@ theorem linearIndepOn_iff_disjoint : LinearIndepOn R v s ↔
       Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.linearCombination R v) := by
   rw [linearIndepOn_iff, LinearMap.disjoint_ker]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem linearIndepOn_iff_linearCombinationOn :
     LinearIndepOn R v s ↔ (LinearMap.ker <| Finsupp.linearCombinationOn ι M R v s) = ⊥ :=
   linearIndepOn_iff_linearCombinationOnₛ.trans <|
diff --git a/Mathlib/LinearAlgebra/LinearPMap.lean b/Mathlib/LinearAlgebra/LinearPMap.lean
index a80ac18dc3c884..2ab477b8410a9c 100644
--- a/Mathlib/LinearAlgebra/LinearPMap.lean
+++ b/Mathlib/LinearAlgebra/LinearPMap.lean
@@ -656,13 +656,11 @@ end LinearMap
 
 namespace LinearPMap
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restrict codomain of a `LinearPMap` -/
 def codRestrict (f : E →ₗ.[R] F) (p : Submodule R F) (H : ∀ x, f x ∈ p) : E →ₗ.[R] p where
   domain := f.domain
   toFun := f.toFun.codRestrict p H
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Compose two `LinearPMap`s -/
 def comp (g : F →ₗ.[R] G) (f : E →ₗ.[R] F) (H : ∀ x : f.domain, f x ∈ g.domain) : E →ₗ.[R] G :=
   g.toFun.compPMap <| f.codRestrict _ H
diff --git a/Mathlib/LinearAlgebra/Matrix/Ideal.lean b/Mathlib/LinearAlgebra/Matrix/Ideal.lean
index 65878094af3eab..d4f761ab71964e 100644
--- a/Mathlib/LinearAlgebra/Matrix/Ideal.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Ideal.lean
@@ -160,12 +160,10 @@ theorem matrix_strictMono_of_nonempty [Nonempty n] :
     StrictMono (matrix (R := R) n) :=
   matrix_monotone n |>.strictMono_of_injective <| matrix_injective _
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem matrix_bot : (⊥ : RingCon R).matrix n = ⊥ :=
   eq_bot_iff.2 fun _ _ h ↦ Matrix.ext h
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem matrix_top : (⊤ : RingCon R).matrix n = ⊤ :=
   eq_top_iff.2 fun _ _ _ _ _ ↦ by simp
diff --git a/Mathlib/LinearAlgebra/Multilinear/DFinsupp.lean b/Mathlib/LinearAlgebra/Multilinear/DFinsupp.lean
index 96be933487ffcc..d85ad2e86683bc 100644
--- a/Mathlib/LinearAlgebra/Multilinear/DFinsupp.lean
+++ b/Mathlib/LinearAlgebra/Multilinear/DFinsupp.lean
@@ -78,7 +78,6 @@ variable [DecidableEq ι] [Fintype ι] [Semiring R]
 variable [∀ i k, AddCommMonoid (M i k)] [∀ p, AddCommMonoid (N p)]
 variable [∀ i k, Module R (M i k)] [∀ p, Module R (N p)]
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Given a family of indices `κ` and a multilinear map `f p` for each way `p` to select one index from
 each family, `dfinsuppFamily f` maps a family of finitely-supported functions (one for each domain
diff --git a/Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean b/Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean
index c41a88d8c8fbf8..45a4ac8ce887f9 100644
--- a/Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean
+++ b/Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean
@@ -216,7 +216,6 @@ variable {M' N' : Type*}
   [AddCommGroup M'] [AddCommGroup N'] [Module K M'] [Module K N'] [IsScalarTower K L N]
   (i : M' →ₗ[K] M) (j : N' →ₗ[K] N) (hi : Injective i) (hj : Injective j)
 
-set_option backward.isDefEq.respectTransparency false in
 include hi hj in
 /-- An auxiliary definition used only to simplify the construction of the more general definition
 `PerfectPairing.restrictScalarsField`. -/
diff --git a/Mathlib/LinearAlgebra/PiTensorProduct.lean b/Mathlib/LinearAlgebra/PiTensorProduct.lean
index 956ad7230b0056..4a70416faba56b 100644
--- a/Mathlib/LinearAlgebra/PiTensorProduct.lean
+++ b/Mathlib/LinearAlgebra/PiTensorProduct.lean
@@ -311,7 +311,6 @@ lemma _root_.FreeAddMonoid.toPiTensorProduct (p : FreeAddMonoid (R × Π i, s i)
 def lifts (x : ⨂[R] i, s i) : Set (FreeAddMonoid (R × Π i, s i)) :=
   {p | AddCon.toQuotient (c := addConGen (PiTensorProduct.Eqv R s)) p = x}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- An element `p` of `FreeAddMonoid (R × Π i, s i)` lifts an element `x` of `⨂[R] i, s i`
 if and only if `x` is equal to the sum of `a • ⨂ₜ[R] i, m i` over all the entries
 `(a, m)` of `p`.
@@ -331,7 +330,6 @@ lemma nonempty_lifts (x : ⨂[R] i, s i) : Set.Nonempty (lifts x) := by
 lemma lifts_zero : 0 ∈ lifts (0 : ⨂[R] i, s i) := by
   rw [mem_lifts_iff, FreeAddMonoid.toList_zero, List.map_nil, List.sum_nil]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If elements `p,q` of `FreeAddMonoid (R × Π i, s i)` lift elements `x,y` of `⨂[R] i, s i`
 respectively, then `p + q` lifts `x + y`.
 -/
@@ -713,7 +711,6 @@ theorem lift_reindex_symm
     lift φ (reindex R s e |>.symm x) = lift (domDomCongrLinearEquiv' R R s _ e φ) x :=
   LinearMap.congr_fun (lift_comp_reindex_symm e φ) x
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem reindex_trans (e : ι ≃ ι₂) (e' : ι₂ ≃ ι₃) :
     (reindex R s e).trans (reindex R _ e') = reindex R s (e.trans e') := by
@@ -735,7 +732,6 @@ theorem reindex_symm (e : ι ≃ ι₂) :
   ext x
   simp [reindex]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem reindex_refl : reindex R s (Equiv.refl ι) = LinearEquiv.refl R _ := by
   ext
diff --git a/Mathlib/LinearAlgebra/Prod.lean b/Mathlib/LinearAlgebra/Prod.lean
index 64904085a1f3d3..6416808409864e 100644
--- a/Mathlib/LinearAlgebra/Prod.lean
+++ b/Mathlib/LinearAlgebra/Prod.lean
@@ -477,7 +477,6 @@ theorem ker_coprod_of_disjoint_range {M₂ : Type*} [AddCommGroup M₂] [Module
   rw [this] at h
   simpa [this] using h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a linear map `f : E →ₗ[R] F` and a complement `C` of its kernel, we get a linear
 equivalence between `C` and `range f`. -/
 @[simps!]
diff --git a/Mathlib/LinearAlgebra/Projection.lean b/Mathlib/LinearAlgebra/Projection.lean
index 4c3944a7819f92..ec916c60a6a827 100644
--- a/Mathlib/LinearAlgebra/Projection.lean
+++ b/Mathlib/LinearAlgebra/Projection.lean
@@ -52,7 +52,6 @@ theorem ker_id_sub_eq_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) :
 theorem range_eq_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) : range f = ⊤ :=
   range_eq_top.2 fun x => ⟨x, hf x⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isCompl_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) : IsCompl p (ker f) := by
   constructor
   · rw [disjoint_iff_inf_le]
@@ -71,7 +70,6 @@ namespace Submodule
 
 open LinearMap
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `q` is a complement of `p`, then `M/p ≃ q`. -/
 def quotientEquivOfIsCompl (h : IsCompl p q) : (E ⧸ p) ≃ₗ[R] q :=
   LinearEquiv.symm <|
@@ -94,7 +92,6 @@ theorem mk_quotientEquivOfIsCompl_apply (h : IsCompl p q) (x : E ⧸ p) :
     (Quotient.mk (quotientEquivOfIsCompl p q h x) : E ⧸ p) = x :=
   (quotientEquivOfIsCompl p q h).symm_apply_apply x
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `q` is a complement of `p`, then `p × q` is isomorphic to `E`. -/
 def prodEquivOfIsCompl (h : IsCompl p q) : (p × q) ≃ₗ[R] E := by
   apply LinearEquiv.ofBijective (p.subtype.coprod q.subtype)
diff --git a/Mathlib/LinearAlgebra/Projectivization/Basic.lean b/Mathlib/LinearAlgebra/Projectivization/Basic.lean
index 8722be4bb49df8..a70fb4efb0c8c6 100644
--- a/Mathlib/LinearAlgebra/Projectivization/Basic.lean
+++ b/Mathlib/LinearAlgebra/Projectivization/Basic.lean
@@ -144,7 +144,6 @@ theorem finrank_submodule (v : ℙ K V) : finrank K v.submodule = 1 := by
   rw [submodule_eq]
   exact finrank_span_singleton v.rep_nonzero
 
-set_option backward.isDefEq.respectTransparency false in
 instance (v : ℙ K V) : FiniteDimensional K v.submodule := by
   rw [← v.mk_rep]
   change FiniteDimensional K (K ∙ v.rep)
@@ -159,7 +158,6 @@ theorem submodule_injective :
 
 variable (K V)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The equivalence between the projectivization and the
 collection of subspaces of dimension 1. -/
 noncomputable def equivSubmodule : ℙ K V ≃ { H : Submodule K V // finrank K H = 1 } :=
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean b/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
index ef0f2ed92f93e3..b80e8cf1bc028a 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
@@ -1281,7 +1281,6 @@ open Module
 variable {V : Type u} {K : Type v} [Field K] [AddCommGroup V] [Module K V]
 variable [FiniteDimensional K V]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a symmetric bilinear form `B` on some vector space `V` over a field `K`
 in which `2` is invertible, there exists an orthogonal basis with respect to `B`. -/
 theorem exists_orthogonal_basis [hK : Invertible (2 : K)] {B : LinearMap.BilinForm K V}
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/Radical.lean b/Mathlib/LinearAlgebra/QuadraticForm/Radical.lean
index 8670cf2e1f5e2e..df01c340bda52d 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/Radical.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/Radical.lean
@@ -86,7 +86,6 @@ lemma lift_mk {N : Submodule R M} (hN : N ≤ Q.radical) (m : M) :
     Q.lift N hN (Submodule.Quotient.mk m) = Q m :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Universal property of the radical of a quadratic form:
 `Q.radical` is the largest subspace `N` such that
diff --git a/Mathlib/LinearAlgebra/Reflection.lean b/Mathlib/LinearAlgebra/Reflection.lean
index 08c7b7bfcab650..b22336e7424458 100644
--- a/Mathlib/LinearAlgebra/Reflection.lean
+++ b/Mathlib/LinearAlgebra/Reflection.lean
@@ -373,7 +373,6 @@ lemma Dual.eq_of_preReflection_mapsTo [CharZero R] [IsDomain R] [IsTorsionFree R
     simpa [hn₁, hn₀.ne', hx, sub_eq_zero] using hu n
   exact u.isOfFinOrder_of_finite_of_span_eq_top_of_mapsTo hΦ₁ hΦ₂ (hg₂.comp hf₂)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- This rather technical-looking lemma exists because it is exactly what is needed to establish a
 uniqueness result for root data. See the doc string of `Module.Dual.eq_of_preReflection_mapsTo` for
 further remarks. -/
diff --git a/Mathlib/LinearAlgebra/RootSystem/BaseChange.lean b/Mathlib/LinearAlgebra/RootSystem/BaseChange.lean
index 75275b9edf099f..b72ee38e87d3ff 100644
--- a/Mathlib/LinearAlgebra/RootSystem/BaseChange.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/BaseChange.lean
@@ -64,7 +64,6 @@ section SubfieldValued
 
 variable [P.IsValuedIn K]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restriction of scalars for a root pairing taking values in a subfield.
 
 See also `RootPairing.restrictScalars`. -/
@@ -88,7 +87,6 @@ def restrictScalars' :
   reflectionPerm_coroot i j := by
     ext; simpa [algebra_compatible_smul L] using P.reflectionPerm_coroot i j
 
-set_option backward.isDefEq.respectTransparency false in
 instance : (P.restrictScalars' K).IsRootSystem where
   span_root_eq_top := by
     rw [← span_setOf_mem_eq_top]
@@ -101,23 +99,19 @@ instance : (P.restrictScalars' K).IsRootSystem where
     ext ⟨x, hx⟩
     simp [restrictScalars']
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma restrictScalars_toLinearMap_apply_apply
     (x : span K (range P.root)) (y : span K (range P.coroot)) :
     algebraMap K L ((P.restrictScalars' K).toLinearMap x y) = P.toLinearMap x y := by
   simp [restrictScalars']
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma restrictScalars_coe_root (i : ι) :
     (P.restrictScalars' K).root i = P.root i :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma restrictScalars_coe_coroot (i : ι) :
     (P.restrictScalars' K).coroot i = P.coroot i :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma restrictScalars_pairing (i j : ι) :
     algebraMap K L ((P.restrictScalars' K).pairing i j) = P.pairing i j := by
   simp only [pairing, restrictScalars_toLinearMap_apply_apply, restrictScalars_coe_root,
@@ -125,7 +119,6 @@ set_option backward.isDefEq.respectTransparency false in
 
 end SubfieldValued
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Restriction of scalars for a crystallographic root pairing. -/
 abbrev restrictScalars [P.IsCrystallographic] :
     RootPairing ι K (span K (range P.root)) (span K (range P.coroot)) :=
diff --git a/Mathlib/LinearAlgebra/RootSystem/BaseExists.lean b/Mathlib/LinearAlgebra/RootSystem/BaseExists.lean
index aa256a3b00e382..1e81aab162db19 100644
--- a/Mathlib/LinearAlgebra/RootSystem/BaseExists.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/BaseExists.lean
@@ -59,7 +59,6 @@ lemma baseOf_pairwise_pairing_le_zero [CharZero R] [IsDomain R] [P.IsCrystallogr
   contrapose! this
   exact P.root_sub_root_mem_of_pairingIn_pos this hne
 
-set_option backward.isDefEq.respectTransparency false in
 /-- This lemma is usually established for root systems with coefficients `R` equal to `ℚ` or `ℝ`, in
 which case one may take `S = R`. However our statement allows for more general coefficients such as
 `R = ℂ` and `S = ℚ`.
@@ -114,7 +113,6 @@ section Field
 variable [Field R] [CharZero R] [Module R M] [Module R N] (P : RootPairing ι R M N)
   [P.IsRootSystem] [P.IsCrystallographic]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma linearIndepOn_root_baseOf (f : M →+ ℚ) (hf : ∀ i, f (P.root i) ≠ 0) :
     LinearIndepOn R P.root (baseOf P.root f) := by
   let _i : Module ℚ M := Module.compHom M (algebraMap ℚ R)
diff --git a/Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean b/Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean
index 6074e571edcc5a..a6fc707eedcc81 100644
--- a/Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/Finite/Lemmas.lean
@@ -46,7 +46,6 @@ variable (P : RootPairing ι R M N) [Finite ι]
 local notation "Φ" => range P.root
 local notation "α" => P.root
 
-set_option backward.isDefEq.respectTransparency false in
 /-- SGA3 XXI Prop. 2.3.1 -/
 lemma coxeterWeightIn_le_four (S : Type*)
     [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] [Algebra S R] [FaithfulSMul S R]
diff --git a/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean b/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean
index 334660187d5e33..ce49a7138e2d42 100644
--- a/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean
@@ -172,7 +172,6 @@ def rootLength (i : ι) : S :=
 lemma rootLength_pos (i : ι) : 0 < B.rootLength i := by
   simpa using B.zero_lt_posForm_apply_root i
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma rootLength_reflectionPerm_self (i : ι) :
     B.rootLength (P.reflectionPerm i i) = B.rootLength i := by
diff --git a/Mathlib/LinearAlgebra/Semisimple.lean b/Mathlib/LinearAlgebra/Semisimple.lean
index 7e82462a13356a..0310d476075bfb 100644
--- a/Mathlib/LinearAlgebra/Semisimple.lean
+++ b/Mathlib/LinearAlgebra/Semisimple.lean
@@ -60,7 +60,6 @@ structure on `M` is semisimple. This is equivalent to saying that every `f`-inva
 of `M` has an `f`-invariant complement: see `Module.End.isSemisimple_iff`. -/
 def IsSemisimple := IsSemisimpleModule R[X] (AEval' f)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A weaker version of semisimplicity that only prescribes behaviour on finitely-generated
 submodules. -/
 def IsFinitelySemisimple : Prop :=
@@ -95,7 +94,6 @@ lemma isSemisimple_restrict_iff (p) (hp : p ∈ invtSubmodule f) :
     exists_and_left, exists_and_right, invtSubmodule.mk_eq_bot_iff, exists_prop, and_assoc]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A linear endomorphism is finitely semisimple if it is semisimple on every finitely-generated
 invariant submodule.
 
@@ -140,7 +138,6 @@ lemma eq_zero_of_isNilpotent_isSemisimple (hn : IsNilpotent f) (hs : f.IsSemisim
   rw [← RingHom.mem_ker, ← AEval.annihilator_eq_ker_aeval (M := M)] at h0 ⊢
   exact hs.annihilator_isRadical _ _ ⟨n, h0⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma eq_zero_of_isNilpotent_of_isFinitelySemisimple
     (hn : IsNilpotent f) (hs : IsFinitelySemisimple f) : f = 0 := by
   have (p) (hp₁ : p ∈ f.invtSubmodule) (hp₂ : Module.Finite R p) : f.restrict hp₁ = 0 := by
@@ -202,7 +199,6 @@ lemma isFinitelySemisimple_sub_algebraMap_iff {μ : R} :
   refine fun p ↦ ⟨fun h x hx ↦ ?_, fun h x hx ↦ p.sub_mem (h hx) (p.smul_mem μ hx)⟩
   simpa using p.add_mem (h hx) (p.smul_mem μ hx)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma IsFinitelySemisimple.restrict {p : Submodule R M} (hp : p ∈ f.invtSubmodule)
     (hf : f.IsFinitelySemisimple) :
     IsFinitelySemisimple (f.restrict hp) := by
diff --git a/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean b/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean
index 3263de18ec502a..b5627398ef5cae 100644
--- a/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean
+++ b/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean
@@ -802,7 +802,6 @@ lemma disjoint_ker_of_nondegenerate_restrict {B : M →ₗ[R] M →ₗ[R] M₁}
   rw [mem_ker] at hx'
   simp [x', hx']
 
-set_option backward.isDefEq.respectTransparency false in
 lemma IsSymm.nondegenerate_restrict_of_isCompl_ker {B : M →ₗ[R] M →ₗ[R] R} (hB : B.IsSymm)
     {W : Submodule R M} (hW : IsCompl W (LinearMap.ker B)) :
     (B.domRestrict₁₂ W W).Nondegenerate := by
@@ -820,7 +819,6 @@ lemma IsSymm.nondegenerate_restrict_of_isCompl_ker {B : M →ₗ[R] M →ₗ[R]
     exact hx' u hu
   simpa [hW.inf_eq_bot] using hx'
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The restriction of a reflexive bilinear map `B` onto a submodule `W` is
 nondegenerate if `W` has trivial intersection with its orthogonal complement,
 that is `Disjoint W (W.orthogonalBilin B)`. -/
@@ -990,7 +988,6 @@ lemma nondegenerate_iff' (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm) :
   contrapose!
   exact exists_congr fun x ↦ ⟨by aesop, fun ⟨h₀, h⟩ ↦ Or.inl ⟨le_antisymm h (hs x), h₀⟩⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma nondegenerate_restrict_iff_disjoint_ker (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm)
     {W : Submodule R M} :
     (B.domRestrict₁₂ W W).Nondegenerate ↔ Disjoint W (LinearMap.ker B) := by
diff --git a/Mathlib/LinearAlgebra/Span/Basic.lean b/Mathlib/LinearAlgebra/Span/Basic.lean
index c12110aadefe92..b4a063da3638d8 100644
--- a/Mathlib/LinearAlgebra/Span/Basic.lean
+++ b/Mathlib/LinearAlgebra/Span/Basic.lean
@@ -577,7 +577,6 @@ lemma biSup_comap_eq_top_of_surjective {ι : Type*} (s : Set ι) (hs : s.Nonempt
   rw [iSup_subtype'] at hp ⊢
   rw [← comap_map_eq, map_iSup_comap_of_surjective hf, hp, comap_top]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma biSup_comap_eq_top_of_range_eq_biSup
     {R R₂ : Type*} [Semiring R] [Ring R₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂]
     [Module R M] [Module R₂ M₂] {ι : Type*} (s : Set ι) (hs : s.Nonempty)
diff --git a/Mathlib/LinearAlgebra/Span/TensorProduct.lean b/Mathlib/LinearAlgebra/Span/TensorProduct.lean
index 2b51c38e6c93dd..b918f6da8d7809 100644
--- a/Mathlib/LinearAlgebra/Span/TensorProduct.lean
+++ b/Mathlib/LinearAlgebra/Span/TensorProduct.lean
@@ -108,7 +108,6 @@ variable [CommRing R] [CommRing A] [Nontrivial A]
   let b₂ : Basis ι A (span A (p : Set M)) := (b₁.baseChange A).map <| p.tensorEquivSpan A
   rw [finrank_eq_card_basis b₁, finrank_eq_card_basis b₂]
 
-set_option backward.isDefEq.respectTransparency false in
 variable (R) in
 lemma finrank_span_eq_finrank_span [IsPrincipalIdealRing R] [IsDomain R] [IsTorsionFree R M]
     (s : Set M) [Module.Finite R (span R s)] :
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean b/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean
index 8372c01950142e..fb1fcbd9f5d12f 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean
@@ -142,7 +142,6 @@ theorem auxEquiv_symm_one : (auxEquiv R 𝒜 ℬ).symm 1 = 1 :=
 
 variable [Module ι (Additive ℤˣ)]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Auxiliary construction used to build the `Mul` instance and get distributivity of `+` and
 `\smul`. -/
 noncomputable def mulHom : (𝒜 ᵍ⊗[R] ℬ) →ₗ[R] (𝒜 ᵍ⊗[R] ℬ) →ₗ[R] (𝒜 ᵍ⊗[R] ℬ) := by
@@ -151,7 +150,6 @@ noncomputable def mulHom : (𝒜 ᵍ⊗[R] ℬ) →ₗ[R] (𝒜 ᵍ⊗[R] ℬ) 
     fAB1.symm.toLinearMap
   exact this
 
-set_option backward.isDefEq.respectTransparency false in
 theorem mulHom_apply (x y : 𝒜 ᵍ⊗[R] ℬ) :
     mulHom 𝒜 ℬ x y
       = (auxEquiv R 𝒜 ℬ).symm (gradedMul R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) (auxEquiv R 𝒜 ℬ y)) :=
@@ -164,13 +162,11 @@ instance : Mul (𝒜 ᵍ⊗[R] ℬ) where mul x y := mulHom 𝒜 ℬ x y
 
 theorem mul_def (x y : 𝒜 ᵍ⊗[R] ℬ) : x * y = mulHom 𝒜 ℬ x y := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 -- Before https://github.com/leanprover-community/mathlib4/pull/8386 this was `@[simp]` but it times out when we try to apply it.
 theorem auxEquiv_mul (x y : 𝒜 ᵍ⊗[R] ℬ) :
     auxEquiv R 𝒜 ℬ (x * y) = gradedMul R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) (auxEquiv R 𝒜 ℬ y) :=
   LinearEquiv.eq_symm_apply _ |>.mp rfl
 
-set_option backward.isDefEq.respectTransparency false in
 instance instMonoid : Monoid (𝒜 ᵍ⊗[R] ℬ) where
   mul_one x := by
     rw [mul_def, mulHom_apply, auxEquiv_one, gradedMul_one, LinearEquiv.symm_apply_apply]
@@ -251,7 +247,6 @@ def includeLeftRingHom : A →+* 𝒜 ᵍ⊗[R] ℬ where
     rw [← SetLike.coe_gOne ℬ, tmul_coe_mul_coe_tmul, zero_mul, uzpow_zero, one_smul,
       SetLike.coe_gOne, one_mul]
 
-set_option backward.isDefEq.respectTransparency false in
 instance instAlgebra : Algebra R (𝒜 ᵍ⊗[R] ℬ) where
   algebraMap := (includeLeftRingHom 𝒜 ℬ).comp (algebraMap R A)
   commutes' r x := by
@@ -382,7 +377,6 @@ def comm : (𝒜 ᵍ⊗[R] ℬ) ≃ₐ[R] (ℬ ᵍ⊗[R] 𝒜) :=
       simp_rw [auxEquiv_mul, gradedComm_gradedMul, LinearEquiv.symm_apply_eq,
         ← gradedComm_gradedMul, auxEquiv_mul, LinearEquiv.apply_symm_apply, gradedComm_gradedMul])
 
-set_option backward.isDefEq.respectTransparency false in
 lemma auxEquiv_comm (x : 𝒜 ᵍ⊗[R] ℬ) :
     auxEquiv R ℬ 𝒜 (comm 𝒜 ℬ x) = gradedComm R (𝒜 ·) (ℬ ·) (auxEquiv R 𝒜 ℬ x) :=
   LinearEquiv.eq_symm_apply _ |>.mp rfl
diff --git a/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean b/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
index 7217c7fe22e25b..1080461a351f67 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
@@ -292,7 +292,6 @@ theorem lTensor_exact : Exact (lTensor Q f) (lTensor Q g) := by
   rw [LinearMap.ker_eq_bot]
   exact (lTensor.equiv Q hfg hg).symm.injective
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Right-exactness of tensor product -/
 lemma lTensor_mkQ (N : Submodule R M) :
     ker (lTensor Q N.mkQ) = range (lTensor Q N.subtype) := by
@@ -395,7 +394,6 @@ theorem rTensor_exact : Exact (rTensor Q f) (rTensor Q g) := by
   rw [rTensor_exact_iff_lTensor_exact]
   exact lTensor_exact Q hfg hg
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Right-exactness of tensor product (`rTensor`) -/
 lemma rTensor_mkQ (N : Submodule R M) :
     ker (rTensor Q N.mkQ) = range (rTensor Q N.subtype) := by
@@ -582,7 +580,6 @@ variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
   [IsScalarTower R S A] [IsScalarTower R S B]
   (f : A →ₐ[S] B) (g : C →ₐ[R] D)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `g` is surjective, then the kernel of `(id A) ⊗ g` is generated by the kernel of `g` -/
 lemma Algebra.TensorProduct.lTensor_ker (hg : Function.Surjective g) :
     RingHom.ker (map (AlgHom.id R A) g) =
@@ -594,7 +591,6 @@ lemma Algebra.TensorProduct.lTensor_ker (hg : Function.Surjective g) :
   rw [(lTensor_exact A g.toLinearMap.exact_subtype_ker_map hg).linearMap_ker_eq]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `f` is surjective, then the kernel of `f ⊗ (id B)` is generated by the kernel of `f` -/
 lemma Algebra.TensorProduct.rTensor_ker (hf : Function.Surjective f) :
     RingHom.ker (map f (AlgHom.id R C)) =
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean b/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
index bb9c19f02749f7..1e498515dd5226 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
@@ -182,7 +182,6 @@ theorem vanishesTrivially_iff_sum_tmul_eq_zero (hm : Submodule.span R (Set.range
     VanishesTrivially R m n ↔ ∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N) :=
   ⟨sum_tmul_eq_zero_of_vanishesTrivially R, vanishesTrivially_of_sum_tmul_eq_zero R hm⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Equational criterion for vanishing**
 [A. Altman and S. Kleiman, *A term of commutative algebra* (Lemma 8.16)][altman2021term],
 forward direction, generalization.
diff --git a/Mathlib/LinearAlgebra/Trace.lean b/Mathlib/LinearAlgebra/Trace.lean
index f86690b585c016..8a03d9f1cd8258 100644
--- a/Mathlib/LinearAlgebra/Trace.lean
+++ b/Mathlib/LinearAlgebra/Trace.lean
@@ -327,7 +327,6 @@ proof_wanted _root_.Matrix.trace_map' {K m F : Type*} [Field K] [Fintype m] [Dec
     [FunLike F (Matrix m m K) (Matrix m m K)] [AlgHomClass F K _ _] (f : F) (x : Matrix m m K) :
     (f x).trace = x.trace
 
-set_option backward.isDefEq.respectTransparency false in
 theorem IsProj.trace {p : Submodule R M} {f : M →ₗ[R] M} (h : IsProj p f) [Module.Free R p]
     [Module.Finite R p] [Module.Free R (ker f)] [Module.Finite R (ker f)] :
     trace R M f = (finrank R p : R) := by
diff --git a/Mathlib/LinearAlgebra/Transvection.lean b/Mathlib/LinearAlgebra/Transvection.lean
index f762226d3b6f8c..243139e421de91 100644
--- a/Mathlib/LinearAlgebra/Transvection.lean
+++ b/Mathlib/LinearAlgebra/Transvection.lean
@@ -284,7 +284,6 @@ theorem dilatransvections_pow_mono :
     Monotone (fun n : ℕ ↦ (dilatransvections R V) ^ n) :=
   Set.pow_right_monotone one_mem_dilatransvections
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Over a division ring, `dilatransvections` correspond to linear
 equivalences `e` such that the linear map `e - id` has rank at most 1.
 
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean
index 7d6fe8176312c8..49af52b180db3b 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean
@@ -359,7 +359,6 @@ section Induction
 
 variable {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 ≤ p)] [NormedSpace ℝ F]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Auxiliary lemma for `Lp.induction_stronglyMeasurable`. -/
 @[elab_as_elim]
 theorem Lp.induction_stronglyMeasurable_aux (hm : m ≤ m0) (hp_ne_top : p ≠ ∞) (P : Lp F p μ → Prop)
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
index 46496a378f6219..b80b443991c68c 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
@@ -410,7 +410,6 @@ theorem aestronglyMeasurable_condExpL1CLM (f : α →₁[μ] F') :
     refine IsClosed.preimage (condExpL1CLM F' hm μ).continuous ?_
     exact isClosed_aestronglyMeasurable hm
 
-set_option backward.isDefEq.respectTransparency false in
 theorem condExpL1CLM_lpMeas (f : lpMeas F' ℝ m 1 μ) :
     condExpL1CLM F' hm μ (f : α →₁[μ] F') = ↑f := by
   let g := lpMeasToLpTrimLie F' ℝ 1 μ hm f
diff --git a/Mathlib/MeasureTheory/Integral/IntervalIntegral/Periodic.lean b/Mathlib/MeasureTheory/Integral/IntervalIntegral/Periodic.lean
index c4d87d56cf9e2c..5b97f3a8563775 100644
--- a/Mathlib/MeasureTheory/Integral/IntervalIntegral/Periodic.lean
+++ b/Mathlib/MeasureTheory/Integral/IntervalIntegral/Periodic.lean
@@ -275,7 +275,6 @@ variable {E : Type*} [NormedAddCommGroup E]
 
 variable {f : ℝ → E} {T : ℝ}
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 A periodic function is interval integrable over every interval if it is interval integrable over one
 period.
@@ -342,7 +341,6 @@ theorem intervalIntegrable₀ (h₁f : Function.Periodic f T) (hT : T ≠ 0)
 
 variable [NormedSpace ℝ E]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `f` is a periodic function with period `T`, then its integral over `[t, t + T]` does not
 depend on `t`. -/
 theorem intervalIntegral_add_eq (hf : Periodic f T) (t s : ℝ) :
diff --git a/Mathlib/MeasureTheory/Measure/CharacteristicFunction.lean b/Mathlib/MeasureTheory/Measure/CharacteristicFunction.lean
index 9685b42f46e4b5..b167e4f1b8a479 100644
--- a/Mathlib/MeasureTheory/Measure/CharacteristicFunction.lean
+++ b/Mathlib/MeasureTheory/Measure/CharacteristicFunction.lean
@@ -248,7 +248,6 @@ theorem Measure.ext_of_charFun [CompleteSpace E]
   · exact fun v hv ↦ DFunLike.ne_iff.mpr ⟨v, inner_self_ne_zero.mpr hv⟩
   · exact continuous_inner
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The characteristic function of a convolution of measures
 is the product of the respective characteristic functions. -/
 lemma charFun_conv [IsFiniteMeasure μ] [IsFiniteMeasure ν] (t : E) :
diff --git a/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean b/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
index 9f7fc11d3d1c44..ff0b9cb2212e61 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
@@ -40,7 +40,6 @@ variable {𝕜 E F : Type*}
 variable [LocallyCompactSpace E]
 variable (L μ ν)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The image of an additive Haar measure under a surjective linear map is proportional to a given
 additive Haar measure. The proportionality factor will be infinite if the linear map has a
 nontrivial kernel. -/
diff --git a/Mathlib/MeasureTheory/SetSemiring.lean b/Mathlib/MeasureTheory/SetSemiring.lean
index df6fb2d5977a7e..16c5630003dd4b 100644
--- a/Mathlib/MeasureTheory/SetSemiring.lean
+++ b/Mathlib/MeasureTheory/SetSemiring.lean
@@ -183,7 +183,6 @@ lemma sUnion_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
   simp only [disjointOfDiff, coe_sdiff, coe_singleton]
   rw [sUnion_diff_singleton_empty]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma notMem_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
     t ∉ hC.disjointOfDiff hs ht := by
   intro hs_mem
diff --git a/Mathlib/ModelTheory/Algebra/Ring/Basic.lean b/Mathlib/ModelTheory/Algebra/Ring/Basic.lean
index 343f4032178407..0c8a8fe710e958 100644
--- a/Mathlib/ModelTheory/Algebra/Ring/Basic.lean
+++ b/Mathlib/ModelTheory/Algebra/Ring/Basic.lean
@@ -123,7 +123,6 @@ instance (α : Type*) : Neg (Language.ring.Term α) :=
 theorem neg_def (α : Type*) (t : Language.ring.Term α) :
     -t = negFunc.apply₁ t := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 instance : Fintype Language.ring.Symbols :=
   ⟨⟨Multiset.ofList
       [Sum.inl ⟨2, .add⟩,
diff --git a/Mathlib/ModelTheory/Definability.lean b/Mathlib/ModelTheory/Definability.lean
index 7a0742b047d2db..dce057782edc98 100644
--- a/Mathlib/ModelTheory/Definability.lean
+++ b/Mathlib/ModelTheory/Definability.lean
@@ -429,7 +429,6 @@ theorem TermDefinable.map_expansion (h : A.TermDefinable L f) (φ : L →ᴸ L')
   use (φ.addConstants A).onTerm ψ
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem termDefinable_empty_iff :
     (∅ : Set M).TermDefinable L f ↔ ∃ φ : L.Term α, f = φ.realize := by
   rw [TermDefinable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onTerm]
diff --git a/Mathlib/ModelTheory/Order.lean b/Mathlib/ModelTheory/Order.lean
index ee7845849e7f72..25ffab57fbe67d 100644
--- a/Mathlib/ModelTheory/Order.lean
+++ b/Mathlib/ModelTheory/Order.lean
@@ -385,7 +385,6 @@ def partialOrderOfModels [h : M ⊨ L.partialOrderTheory] : PartialOrder M where
   le_antisymm := (Relations.realize_antisymmetric.mp <|
     Theory.model_iff _ |>.mp h _ <| by simp [partialOrderTheory]).antisymm
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Any model of a theory of linear orders is a linear order. -/
 def linearOrderOfModels [h : M ⊨ L.linearOrderTheory]
     [DecidableRel (fun (a b : M) => Structure.RelMap (leSymb : L.Relations 2) ![a, b])] :
@@ -459,7 +458,6 @@ section Fraisse
 
 variable (M)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma dlo_isExtensionPair
     (M : Type w) [Language.order.Structure M] [M ⊨ Language.order.linearOrderTheory]
     (N : Type w') [Language.order.Structure N] [N ⊨ Language.order.dlo] [Nonempty N] :
diff --git a/Mathlib/ModelTheory/Types.lean b/Mathlib/ModelTheory/Types.lean
index e474b8f52d6621..8870155028160d 100644
--- a/Mathlib/ModelTheory/Types.lean
+++ b/Mathlib/ModelTheory/Types.lean
@@ -221,7 +221,6 @@ def realizedTypes (α : Type w) : Set (T.CompleteType α) :=
 
 section
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_modelType_is_realized_in (p : T.CompleteType α) :
     ∃ M : Theory.ModelType.{u, v, max u v w} T, p ∈ T.realizedTypes M α := by
   obtain ⟨M⟩ := p.isMaximal.1
diff --git a/Mathlib/NumberTheory/Cyclotomic/Basic.lean b/Mathlib/NumberTheory/Cyclotomic/Basic.lean
index 550c07788958ff..417091d6977f9a 100644
--- a/Mathlib/NumberTheory/Cyclotomic/Basic.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/Basic.lean
@@ -780,7 +780,6 @@ namespace CyclotomicRing
 instance : CommRing (CyclotomicRing n A K) := by
   delta CyclotomicRing; infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 instance : IsDomain (CyclotomicRing n A K) := by
   delta CyclotomicRing; infer_instance
 
diff --git a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
index 8548c681414a4c..80c75e9104170b 100644
--- a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
+++ b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
@@ -559,7 +559,6 @@ theorem exists_primitive_element_lt_of_isReal {w₀ : InfinitePlace K} (hw₀ :
       (fun w h_ne ↦ by convert (if_neg h_ne) ▸ h_le w) (Or.inl hw₀)
   · split_ifs <;> simp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_primitive_element_lt_of_isComplex {w₀ : InfinitePlace K} (hw₀ : IsComplex w₀)
     {B : ℝ≥0} (hB : minkowskiBound K ↑1 < convexBodyLT'Factor K * B) :
     ∃ a : 𝓞 K, ℚ⟮(a : K)⟯ = ⊤ ∧
diff --git a/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean b/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean
index e48ac87580ab33..ee34fba4c51200 100644
--- a/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean
+++ b/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean
@@ -50,7 +50,6 @@ instance (I : FractionalIdeal (𝓞 K)⁰ K) : Module.Finite ℤ I := by
     (LinearEquiv.restrictScalars ℤ (I.equivNum ?_)).symm.toLinearMap (LinearEquiv.surjective _)
   exact nonZeroDivisors.coe_ne_zero I.den
 
-set_option backward.isDefEq.respectTransparency false in
 instance (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
     IsLocalizedModule ℤ⁰ ((Submodule.subtype (I : Submodule (𝓞 K) K)).restrictScalars ℤ) where
   map_units x := by
@@ -75,19 +74,16 @@ instance (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
   exists_of_eq h :=
     ⟨1, by rwa [one_smul, one_smul, ← (Submodule.injective_subtype I.1.coeToSubmodule).eq_iff]⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A `ℤ`-basis of a fractional ideal. -/
 noncomputable def fractionalIdealBasis (I : FractionalIdeal (𝓞 K)⁰ K) :
     Basis (Free.ChooseBasisIndex ℤ I) ℤ I := Free.chooseBasis ℤ I
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A `ℚ`-basis of `K` that spans `I` over `ℤ`, see `mem_span_basisOfFractionalIdeal` below. -/
 noncomputable def basisOfFractionalIdeal (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
     Basis (Free.ChooseBasisIndex ℤ I) ℚ K :=
   (fractionalIdealBasis K I.1).ofIsLocalizedModule ℚ ℤ⁰
     ((Submodule.subtype (I : Submodule (𝓞 K) K)).restrictScalars ℤ)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem basisOfFractionalIdeal_apply (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
     (i : Free.ChooseBasisIndex ℤ I) :
     basisOfFractionalIdeal K I i = fractionalIdealBasis K I.1 i :=
@@ -98,7 +94,6 @@ theorem mem_span_basisOfFractionalIdeal {I : (FractionalIdeal (𝓞 K)⁰ K)ˣ}
   rw [basisOfFractionalIdeal, (fractionalIdealBasis K I.1).ofIsLocalizedModule_span ℚ ℤ⁰ _]
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 open Module in
 theorem fractionalIdeal_rank (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
     finrank ℤ I = finrank ℤ (𝓞 K) := by
@@ -111,7 +106,6 @@ section Norm
 
 open Module
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The absolute value of the determinant of the base change from `integralBasis` to
 `basisOfFractionalIdeal I` is equal to the norm of `I`. -/
 theorem det_basisOfFractionalIdeal_eq_absNorm (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
diff --git a/Mathlib/NumberTheory/NumberField/Ideal/KummerDedekind.lean b/Mathlib/NumberTheory/NumberField/Ideal/KummerDedekind.lean
index f1064423717c56..8cebbecf904a8a 100644
--- a/Mathlib/NumberTheory/NumberField/Ideal/KummerDedekind.lean
+++ b/Mathlib/NumberTheory/NumberField/Ideal/KummerDedekind.lean
@@ -80,7 +80,6 @@ theorem exponent_eq_sInf : exponent θ = sInf {d : ℕ | 0 < d ∧ (d : 𝓞 K)
 
 variable [NumberField K] {θ : 𝓞 K} {p : ℕ} [Fact p.Prime]
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 /--
 If `p` doesn't divide the exponent of `θ`, then `(ℤ / pℤ)[X] / (minpoly θ) ≃+* 𝓞 K / p(𝓞 K)`.
@@ -94,7 +93,6 @@ def ZModXQuotSpanEquivQuotSpan (hp : ¬ p ∣ exponent θ) :
         (quotientEquivAlgOfEq ℤ (by simp [map_span])).toRingEquiv))
 
 set_option backward.isDefEq.respectTransparency false in
-set_option backward.whnf.reducibleClassField false in
 theorem ZModXQuotSpanEquivQuotSpan_mk_apply (hp : ¬ p ∣ exponent θ) (Q : ℤ[X]) :
     (ZModXQuotSpanEquivQuotSpan hp)
       (Ideal.Quotient.mk (span {map (Int.castRingHom (ZMod p)) (minpoly ℤ θ)})
diff --git a/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean b/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean
index 12ec6f153c57ee..b25d1ed2713a19 100644
--- a/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean
+++ b/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean
@@ -140,7 +140,6 @@ theorem le_iff_le (x : K) (r : ℝ) : (∀ w : InfinitePlace K, w x ≤ r) ↔ 
 
 theorem pos_iff {w : InfinitePlace K} {x : K} : 0 < w x ↔ x ≠ 0 := AbsoluteValue.pos_iff w.1
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
   constructor
diff --git a/Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean b/Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean
index b0e33681065c62..bfbe19e74c968e 100644
--- a/Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean
+++ b/Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean
@@ -181,7 +181,6 @@ instance isTotallyReal_sup {E F : Subfield K} [hE : IsTotallyReal E] [hF : IsTot
     ← isTotallyReal_iff_le_maximalRealSubfield, ← isTotallyReal_iff_le_maximalRealSubfield]
   exact ⟨hE, hF⟩
 
-set_option backward.isDefEq.respectTransparency false in
 instance isTotallyReal_iSup {ι : Type*} {k : ι → Subfield K} [∀ i, IsTotallyReal (k i)] :
     IsTotallyReal (⨆ i, k i : Subfield K) := by
   obtain hι | ⟨⟨i⟩⟩ := isEmpty_or_nonempty ι
diff --git a/Mathlib/NumberTheory/Padics/PadicVal/Basic.lean b/Mathlib/NumberTheory/Padics/PadicVal/Basic.lean
index 6bc0fbc0e6e51c..04666004744e5a 100644
--- a/Mathlib/NumberTheory/Padics/PadicVal/Basic.lean
+++ b/Mathlib/NumberTheory/Padics/PadicVal/Basic.lean
@@ -91,7 +91,6 @@ theorem maxPowDiv_eq_emultiplicity {p n : ℕ} (hp : 1 < p) (hn : n ≠ 0) :
   apply Nat.not_lt.mpr <| le_of_dvd hp hn h
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem maxPowDiv_eq_multiplicity {p n : ℕ} (hp : 1 < p) (hn : n ≠ 0) (h : FiniteMultiplicity p n) :
     p.maxPowDiv n = multiplicity p n := by
   exact_mod_cast h.emultiplicity_eq_multiplicity ▸ maxPowDiv_eq_emultiplicity hp hn
@@ -450,7 +449,6 @@ theorem pow_padicValNat_dvd {n : ℕ} : p ^ padicValNat p n ∣ n := by
   apply pow_dvd_of_le_multiplicity
   rw [padicValNat_def'] <;> assumption
 
-set_option backward.isDefEq.respectTransparency false in
 theorem padicValNat_dvd_iff_le [hp : Fact p.Prime] {a n : ℕ} (ha : a ≠ 0) :
     p ^ n ∣ a ↔ n ≤ padicValNat p a := by
   rw [pow_dvd_iff_le_emultiplicity, ← padicValNat_eq_emultiplicity ha,
@@ -556,7 +554,6 @@ theorem range_pow_padicValNat_subset_divisors' {n : ℕ} [hp : Fact p.Prime] :
   refine ⟨?_, (pow_dvd_pow p <| succ_le_iff.2 hk).trans pow_padicValNat_dvd, hn⟩
   exact (Nat.one_lt_pow k.succ_ne_zero hp.out.one_lt).ne'
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The `p`-adic valuation of `(p * n)!` is `n` more than that of `n!`. -/
 theorem padicValNat_factorial_mul (n : ℕ) [hp : Fact p.Prime] :
     padicValNat p (p * n)! = padicValNat p n ! + n := by
@@ -590,7 +587,6 @@ largest multiple of `p` below `n`, i.e. `(p * ⌊n / p⌋)!`. -/
   nth_rw 2 [← div_add_mod n p]
   exact (padicValNat_factorial_mul_add (n / p) <| mod_lt n hp.out.pos).symm
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Legendre's Theorem**
 
 The `p`-adic valuation of `n!` is the sum of the quotients `n / p ^ i`. This sum is expressed
@@ -635,7 +631,6 @@ theorem padicValNat_factorial_le [hp : Fact p.Prime] (n : ℕ) : padicValNat p n
 
 variable {p}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Kummer's Theorem**
 
 The `p`-adic valuation of `n.choose k` is the number of carries when `k` and `n - k` are added
@@ -646,7 +641,6 @@ theorem padicValNat_choose {n k b : ℕ} [hp : Fact p.Prime] (hkn : k ≤ n) (hn
   exact_mod_cast (padicValNat_eq_emultiplicity (p := p) <| (choose_ne_zero hkn)) ▸
     Prime.emultiplicity_choose hp.out hkn hnb
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Kummer's Theorem**
 
 The `p`-adic valuation of `(n + k).choose k` is the number of carries when `k` and `n` are added
diff --git a/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean b/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean
index bca1345f570efc..3847e2e193572b 100644
--- a/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean
+++ b/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean
@@ -50,7 +50,6 @@ theorem padicValNat_def [hp : Fact p.Prime] {n : ℕ} (hn : n ≠ 0) :
     padicValNat p n = multiplicity p n :=
   padicValNat_def' hp.out.ne_one hn
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A simplification of `padicValNat` when one input is prime, by analogy with
 `padicValRat_def`. -/
 theorem padicValNat_eq_emultiplicity [hp : Fact p.Prime] {n : ℕ} (hn : n ≠ 0) :
@@ -86,7 +85,6 @@ theorem le_emultiplicity_iff_replicate_subperm_primeFactorsList {a b : ℕ} {n :
   (replicate_subperm_primeFactorsList_iff ha hb).trans
     pow_dvd_iff_le_emultiplicity |>.symm
 
-set_option backward.isDefEq.respectTransparency false in
 theorem le_padicValNat_iff_replicate_subperm_primeFactorsList {a b : ℕ} {n : ℕ} (ha : a.Prime)
     (hb : b ≠ 0) :
     n ≤ padicValNat a b ↔ replicate n a <+~ b.primeFactorsList := by
diff --git a/Mathlib/NumberTheory/RamificationInertia/Basic.lean b/Mathlib/NumberTheory/RamificationInertia/Basic.lean
index 851a1f695b5e85..ed6c2f268870c6 100644
--- a/Mathlib/NumberTheory/RamificationInertia/Basic.lean
+++ b/Mathlib/NumberTheory/RamificationInertia/Basic.lean
@@ -619,7 +619,6 @@ noncomputable def powQuotSuccInclusion (i : ℕ) :
   map_add' _ _ := rfl
   map_smul' _ _ := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem powQuotSuccInclusion_injective (i : ℕ) :
     Function.Injective (powQuotSuccInclusion p P i) := by
   rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']
@@ -627,7 +626,6 @@ theorem powQuotSuccInclusion_injective (i : ℕ) :
   rw [Subtype.ext_iff] at hx0 ⊢
   rwa [powQuotSuccInclusion_apply_coe] at hx0
 
-set_option backward.isDefEq.respectTransparency false in
 /-- `S ⧸ P` embeds into the quotient by `P^(i+1) ⧸ P^e` as a subspace of `P^i ⧸ P^e`.
 See `quotientToQuotientRangePowQuotSucc` for this as a linear map,
 and `quotientRangePowQuotSuccInclusionEquiv` for this as a linear equivalence.
@@ -644,7 +642,6 @@ noncomputable def quotientToQuotientRangePowQuotSuccAux {i : ℕ} {a : S} (a_mem
     rw [powQuotSuccInclusion_apply_coe, Subtype.coe_mk, Submodule.coe_sub, Subtype.coe_mk,
       Subtype.coe_mk, map_mul, map_sub, mul_sub]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientToQuotientRangePowQuotSuccAux_mk {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) :
     quotientToQuotientRangePowQuotSuccAux p P a_mem (Submodule.Quotient.mk x) =
       Submodule.Quotient.mk ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩ := by
@@ -653,7 +650,6 @@ theorem quotientToQuotientRangePowQuotSuccAux_mk {i : ℕ} {a : S} (a_mem : a 
 section
 variable [hfp : NeZero (ramificationIdx (algebraMap R S) p P)]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- `S ⧸ P` embeds into the quotient by `P^(i+1) ⧸ P^e` as a subspace of `P^i ⧸ P^e`. -/
 noncomputable def quotientToQuotientRangePowQuotSucc
     {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) :
@@ -674,13 +670,11 @@ noncomputable def quotientToQuotientRangePowQuotSucc
       Algebra.smul_def, Quotient.algebraMap_quotient_pow_ramificationIdx]
     ring
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientToQuotientRangePowQuotSucc_mk {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) :
     quotientToQuotientRangePowQuotSucc p P a_mem (Submodule.Quotient.mk x) =
       Submodule.Quotient.mk ⟨_, Ideal.mem_map_of_mem _ (Ideal.mul_mem_right x _ a_mem)⟩ :=
   quotientToQuotientRangePowQuotSuccAux_mk p P a_mem x
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientToQuotientRangePowQuotSucc_injective [IsDedekindDomain S] [P.IsPrime]
     {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i) (a_notMem : a ∉ P ^ (i + 1)) :
     Function.Injective (quotientToQuotientRangePowQuotSucc p P a_mem) := fun x =>
@@ -700,7 +694,6 @@ theorem quotientToQuotientRangePowQuotSucc_injective [IsDedekindDomain S] [P.IsP
               ((Submodule.sub_mem_iff_right _ hz).mp (Pe_le_Pi1 h))).resolve_left
           a_notMem
 
-set_option backward.isDefEq.respectTransparency false in
 theorem quotientToQuotientRangePowQuotSucc_surjective [IsDedekindDomain S]
     (hP0 : P ≠ ⊥) [hP : P.IsPrime] {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i)
     (a_notMem : a ∉ P ^ (i + 1)) :
@@ -730,7 +723,6 @@ theorem quotientToQuotientRangePowQuotSucc_surjective [IsDedekindDomain S]
     have := (P ^ (i + 1)).zero_mem
     contradiction
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Quotienting `P^i / P^e` by its subspace `P^(i+1) ⧸ P^e` is
 `R ⧸ p`-linearly isomorphic to `S ⧸ P`. -/
 noncomputable def quotientRangePowQuotSuccInclusionEquiv [IsDedekindDomain S]
@@ -745,7 +737,6 @@ noncomputable def quotientRangePowQuotSuccInclusionEquiv [IsDedekindDomain S]
   · exact quotientToQuotientRangePowQuotSucc_injective p P hi a_mem a_notMem
   · exact quotientToQuotientRangePowQuotSucc_surjective p P hP hi a_mem a_notMem
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Since the inclusion `(P^(i + 1) / P^e) ⊂ (P^i / P^e)` has a kernel isomorphic to `P / S`,
 `[P^i / P^e : R / p] = [P^(i+1) / P^e : R / p] + [P / S : R / p]` -/
 theorem rank_pow_quot_aux [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
diff --git a/Mathlib/Order/Birkhoff.lean b/Mathlib/Order/Birkhoff.lean
index 072a11a521aded..1cf7a86da04a13 100644
--- a/Mathlib/Order/Birkhoff.lean
+++ b/Mathlib/Order/Birkhoff.lean
@@ -231,7 +231,6 @@ set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma birkhoffSet_sup (a b : α) : birkhoffSet (a ⊔ b) = birkhoffSet a ∪ birkhoffSet b := by
   unfold OrderEmbedding.birkhoffSet; split <;> simp [eq_iff_true_of_subsingleton]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma birkhoffSet_inf (a b : α) : birkhoffSet (a ⊓ b) = birkhoffSet a ∩ birkhoffSet b := by
   unfold OrderEmbedding.birkhoffSet; split <;> simp [eq_iff_true_of_subsingleton]
 
diff --git a/Mathlib/Order/Height.lean b/Mathlib/Order/Height.lean
index 445e5a93dd5c24..9d83a5f3ae56d9 100644
--- a/Mathlib/Order/Height.lean
+++ b/Mathlib/Order/Height.lean
@@ -52,7 +52,6 @@ theorem chainHeight_ne_top_of_finite (h : s.Finite) : s.chainHeight r ≠ ⊤ :=
   LT.lt.ne_top <| lt_of_le_of_lt (chainHeight_le_encard s r) <| lt_top_iff_ne_top.mpr <|
     encard_ne_top_iff.mpr h
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_isChain_of_le_chainHeight {r} {s : Set α} (n : ℕ) (h : n ≤ s.chainHeight r) :
     ∃ t ⊆ s, t.encard = n ∧ IsChain r t := by
   by_cases h' : n = 0
@@ -145,7 +144,6 @@ section Rel
 
 variable {r : α → α → Prop} {r' : β → β → Prop} (s : Set α)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem chainHeight_eq_of_relEmbedding (e : r ↪r r') :
     (e '' s).chainHeight r' = s.chainHeight r := by
   refine eq_of_forall_natCast_le_iff fun n ↦ ⟨fun hn ↦ ?_, fun hn ↦ ?_⟩
diff --git a/Mathlib/Order/KrullDimension.lean b/Mathlib/Order/KrullDimension.lean
index 2b5f7cf2a75d99..7db2e4a5bc2574 100644
--- a/Mathlib/Order/KrullDimension.lean
+++ b/Mathlib/Order/KrullDimension.lean
@@ -137,7 +137,6 @@ lemma coheight_le_iff {a : α} {n : ℕ∞} :
     coheight a ≤ n ↔ ∀ ⦃p : LTSeries α⦄, a ≤ p.head → p.length ≤ n := by
   rw [coheight_eq, iSup₂_le_iff]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma height_le {a : α} {n : ℕ∞} (h : ∀ (p : LTSeries α), p.last = a → p.length ≤ n) :
     height a ≤ n := by
   apply height_le_iff.mpr
@@ -193,7 +192,6 @@ lemma coheight_le {a : α} {n : ℕ∞} (h : ∀ (p : LTSeries α), p.head = a 
     coheight a ≤ n :=
   coheight_le_iff'.mpr h
 
-set_option backward.isDefEq.respectTransparency false in
 lemma length_le_height {p : LTSeries α} {x : α} (hlast : p.last ≤ x) :
     p.length ≤ height x := by
   by_cases hlen0 : p.length ≠ 0
@@ -242,7 +240,6 @@ The coheight of an element in a series is larger or equal to its reverse index i
 lemma rev_index_le_coheight (p : LTSeries α) (i : Fin (p.length + 1)) : i.rev ≤ coheight (p i) := by
   simpa using index_le_height (α := αᵒᵈ) p.reverse i.rev
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 In a maximally long series, i.e one as long as the height of the last element, the height of each
 element is its index in the series.
@@ -344,7 +341,6 @@ private lemma exists_eq_iSup_of_iSup_eq_coe {α : Type*} [Nonempty α] {f : α 
   use x
   simpa [hx] using h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- There exists a series ending in an element for any length up to the element’s height. -/
 lemma exists_series_of_le_height (a : α) {n : ℕ} (h : n ≤ height a) :
     ∃ p : LTSeries α, p.last = a ∧ p.length = n := by
@@ -383,7 +379,6 @@ lemma exists_series_of_coheight_eq_coe (a : α) {n : ℕ} (h : coheight a = n) :
     ∃ p : LTSeries α, p.head = a ∧ p.length = n :=
   exists_series_of_le_coheight a (le_of_eq h.symm)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Another characterization of height, based on the supremum of the heights of elements below. -/
 lemma height_eq_iSup_lt_height (x : α) : height x = ⨆ y < x, height y + 1 := by
   apply le_antisymm
@@ -407,7 +402,6 @@ Another characterization of coheight, based on the supremum of the coheights of
 lemma coheight_eq_iSup_gt_coheight (x : α) : coheight x = ⨆ y > x, coheight y + 1 :=
   height_eq_iSup_lt_height (α := αᵒᵈ) x
 
-set_option backward.isDefEq.respectTransparency false in
 lemma height_le_coe_iff {x : α} {n : ℕ} : height x ≤ n ↔ ∀ y < x, height y < n := by
   conv_lhs => rw [height_eq_iSup_lt_height, iSup₂_le_iff]
   congr! 2 with y _
@@ -444,7 +438,6 @@ lemma coheight_eq_top_iff {x : α} :
   convert height_eq_top_iff (α := αᵒᵈ) (x := x) using 2 with n
   constructor <;> (intro ⟨p, hp, hl⟩; use p.reverse; constructor <;> simpa)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The elements of height zero are the minimal elements. -/
 @[simp] lemma height_eq_zero {x : α} : height x = 0 ↔ IsMin x := by
   simpa [isMin_iff_forall_not_lt] using height_le_coe_iff (x := x) (n := 0)
@@ -479,7 +472,6 @@ lemma coheight_pos_of_lt_top {x : α} [OrderTop α] (h : x < ⊤) : 0 < coheight
   rw [coheight_pos]
   grind [not_isMax_iff]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma coe_lt_height_iff {x : α} {n : ℕ} (hfin : height x < ⊤) :
     n < height x ↔ ∃ y < x, height y = n where
   mp h := by
@@ -499,7 +491,6 @@ lemma coe_lt_coheight_iff {x : α} {n : ℕ} (hfin : coheight x < ⊤) :
     n < coheight x ↔ ∃ y > x, coheight y = n :=
   coe_lt_height_iff (α := αᵒᵈ) hfin
 
-set_option backward.isDefEq.respectTransparency false in
 lemma height_eq_coe_add_one_iff {x : α} {n : ℕ} :
     height x = n + 1 ↔ height x < ⊤ ∧ (∃ y < x, height y = n) ∧ (∀ y < x, height y ≤ n) := by
   wlog hfin : height x < ⊤
@@ -517,7 +508,6 @@ lemma coheight_eq_coe_add_one_iff {x : α} {n : ℕ} :
       coheight x < ⊤ ∧ (∃ y > x, coheight y = n) ∧ (∀ y > x, coheight y ≤ n) :=
   height_eq_coe_add_one_iff (α := αᵒᵈ)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma height_eq_coe_iff {x : α} {n : ℕ} :
     height x = n ↔
       height x < ⊤ ∧ (n = 0 ∨ ∃ y < x, height y = n - 1) ∧ (∀ y < x, height y < n) := by
@@ -539,7 +529,6 @@ lemma coheight_eq_coe_iff {x : α} {n : ℕ} :
       coheight x < ⊤ ∧ (n = 0 ∨ ∃ y > x, coheight y = n - 1) ∧ (∀ y > x, coheight y < n) :=
   height_eq_coe_iff (α := αᵒᵈ)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The elements of finite height `n` are the minimal elements among those of height `≥ n`. -/
 lemma height_eq_coe_iff_minimal_le_height {a : α} {n : ℕ} :
     height a = n ↔ Minimal (fun y => n ≤ height y) a := by
@@ -728,13 +717,11 @@ lemma le_krullDim_iff {n : ℕ} : n ≤ krullDim α ↔ ∃ l : LTSeries α, l.l
     · exact fun ⟨l, hl⟩ ↦ hl ▸ l.longestOf_is_longest
   · simpa [krullDim_eq_top] using SetRel.InfiniteDimensional.exists_relSeries_with_length n
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A definition of krullDim for nonempty `α` that avoids `WithBot` -/
 lemma krullDim_eq_iSup_length [Nonempty α] :
     krullDim α = ⨆ (p : LTSeries α), (p.length : ℕ∞) := by
   simp [krullDim, WithBot.coe_iSup (OrderTop.bddAbove _), WithBot.coe_natCast]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma krullDim_lt_coe_iff {n : ℕ} : krullDim α < n ↔ ∀ l : LTSeries α, l.length < n := by
   rw [krullDim, ← WithBot.coe_natCast]
   rcases n with - | n
@@ -767,7 +754,6 @@ lemma height_le_krullDim (a : α) : height a ≤ krullDim α := by
 lemma coheight_le_krullDim (a : α) : coheight a ≤ krullDim α := by
   simpa using height_le_krullDim (α := αᵒᵈ) a
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma _root_.LTSeries.height_last_longestOf [FiniteDimensionalOrder α] :
     height (LTSeries.longestOf α).last = krullDim α := by
@@ -808,7 +794,6 @@ lemma krullDim_eq_iSup_coheight_of_nonempty [Nonempty α] :
     krullDim α = ↑(⨆ (a : α), coheight a) := by
   simpa using krullDim_eq_iSup_height_of_nonempty (α := αᵒᵈ)
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 The Krull dimension is the supremum of the elements' height plus coheight.
 -/
@@ -956,7 +941,6 @@ proof_wanted coheight_of_linearOrder {α : Type*} [LinearOrder α] (a : α) :
   coheight a = (Set.Ioi a).encard
 -/
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma height_nat (n : ℕ) : height n = n := by
   induction n using Nat.strongRecOn with | ind n ih =>
   apply le_antisymm
@@ -999,7 +983,6 @@ lemma coheight_int (n : ℤ) : coheight n = ⊤ := coheight_of_noMaxOrder ..
 
 lemma krullDim_int : krullDim ℤ = ⊤ := krullDim_of_noMaxOrder ..
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma height_coe_withBot (x : α) : height (x : WithBot α) = height x + 1 := by
   apply le_antisymm
   · apply height_le
@@ -1097,7 +1080,6 @@ section orderHom
 variable {α β : Type*} [Preorder α] [PartialOrder β]
 variable {m : ℕ} (f : α →o β) (h : ∀ (x : β), Order.krullDim (f ⁻¹' {x}) ≤ m)
 
-set_option backward.isDefEq.respectTransparency false in
 include h in
 lemma height_le_of_krullDim_preimage_le (x : α) :
     Order.height x ≤ (m + 1) * Order.height (f x) + m := by
@@ -1135,7 +1117,6 @@ lemma coheight_le_of_krullDim_preimage_le (x : α) :
   apply height_le_of_krullDim_preimage_le (f := f.dual)
   exact fun x ↦ le_of_eq_of_le (krullDim_orderDual (α := f ⁻¹' {x})) (h x)
 
-set_option backward.isDefEq.respectTransparency false in
 include f h in
 lemma krullDim_le_of_krullDim_preimage_le :
     Order.krullDim α ≤ (m + 1) * Order.krullDim β + m := by
diff --git a/Mathlib/Order/PrimeSeparator.lean b/Mathlib/Order/PrimeSeparator.lean
index bf7ca2b2f13d4b..e40fdfde9c1c76 100644
--- a/Mathlib/Order/PrimeSeparator.lean
+++ b/Mathlib/Order/PrimeSeparator.lean
@@ -45,7 +45,6 @@ lemma mem_ideal_sup_principal (a b : α) (J : Ideal α) : b ∈ J ⊔ principal
   ⟨fun ⟨j, ⟨jJ, _, ha', bja'⟩⟩ => ⟨j, jJ, le_trans bja' (sup_le_sup_left ha' j)⟩,
     fun ⟨j, hj, hbja⟩ => ⟨j, hj, a, le_refl a, hbja⟩⟩
 
-set_option backward.isDefEq.respectTransparency false in
 set_option linter.style.whitespace false in -- manual alignment is not recognised
 theorem prime_ideal_of_disjoint_filter_ideal (hFI : Disjoint (F : Set α) (I : Set α)) :
     ∃ J : Ideal α, (IsPrime J) ∧ I ≤ J ∧ Disjoint (F : Set α) J := by
diff --git a/Mathlib/Order/SuccPred/Tree.lean b/Mathlib/Order/SuccPred/Tree.lean
index dcf58343c78c01..cb6dccb7be2ab5 100644
--- a/Mathlib/Order/SuccPred/Tree.lean
+++ b/Mathlib/Order/SuccPred/Tree.lean
@@ -205,7 +205,6 @@ lemma RootedTree.mem_subtrees_disjoint_iff {t₁ t₂ : SubRootedTree t}
     simpa only [ht₂.le_iff_eq ht₁.1, ht₁.le_iff_eq ht₂.1, eq_comm, or_self] using
       le_total_of_directed oh.2 h₂
 
-set_option backward.isDefEq.respectTransparency false in
 lemma RootedTree.subtrees_disjoint : t.subtrees.PairwiseDisjoint ((↑) : _ → Set t) := by
   intro t₁ ht₁ t₂ ht₂ h
   rw [Function.onFun_apply, Set.disjoint_left]
diff --git a/Mathlib/Probability/Process/Stopping.lean b/Mathlib/Probability/Process/Stopping.lean
index c0e5ccfbf66b57..7f03e30d007b91 100644
--- a/Mathlib/Probability/Process/Stopping.lean
+++ b/Mathlib/Probability/Process/Stopping.lean
@@ -1078,7 +1078,6 @@ theorem integrable_stoppedValue_of_mem_finset (hτ : IsStoppingTime ℱ τ)
 
 variable (ι)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem integrable_stoppedValue [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
     (hu : ∀ n, Integrable (u n) μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) :
     Integrable (stoppedValue u τ) μ := by
diff --git a/Mathlib/Probability/StrongLaw.lean b/Mathlib/Probability/StrongLaw.lean
index 86066d7d853f34..9b38a0e29bef04 100644
--- a/Mathlib/Probability/StrongLaw.lean
+++ b/Mathlib/Probability/StrongLaw.lean
@@ -591,7 +591,6 @@ theorem strong_law_aux7 :
 
 end StrongLawNonneg
 
-set_option backward.whnf.reducibleClassField false in
 set_option backward.isDefEq.respectTransparency false in
 /-- **Strong law of large numbers**, almost sure version: if `X n` is a sequence of independent
 identically distributed integrable real-valued random variables, then `∑ i ∈ range n, X i / n`
diff --git a/Mathlib/RepresentationTheory/Character.lean b/Mathlib/RepresentationTheory/Character.lean
index e227b7f52fa806..c17a088739c7db 100644
--- a/Mathlib/RepresentationTheory/Character.lean
+++ b/Mathlib/RepresentationTheory/Character.lean
@@ -92,7 +92,6 @@ theorem char_linHom (V W : FDRep k G) (g : G) :
 
 variable [Fintype G] [Invertible (Fintype.card G : k)]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem average_char_eq_finrank_invariants (V : FDRep k G) :
     ⅟(Fintype.card G : k) • ∑ g : G, V.character g = finrank k (invariants V.ρ) := by
   rw [← (isProj_averageMap V.ρ).trace]
diff --git a/Mathlib/RepresentationTheory/Coinduced.lean b/Mathlib/RepresentationTheory/Coinduced.lean
index ca6219adad0b43..8b2eb90c01be5e 100644
--- a/Mathlib/RepresentationTheory/Coinduced.lean
+++ b/Mathlib/RepresentationTheory/Coinduced.lean
@@ -87,7 +87,6 @@ variable {k G H : Type u} [CommRing k] [Monoid G] [Monoid H] (φ : G →* H) (A
 
 section Coind
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 If `φ : G →* H` and  `A : Rep k G` then `coind φ A` is the coinduction of `A` along `φ`,
 defined by letting `H` act on the `G`-equivariant functions `H → A` by `(h • f) h₁ := f (h₁ * h)`.
diff --git a/Mathlib/RepresentationTheory/Coinvariants.lean b/Mathlib/RepresentationTheory/Coinvariants.lean
index c233512fe991b0..77c25c5000d937 100644
--- a/Mathlib/RepresentationTheory/Coinvariants.lean
+++ b/Mathlib/RepresentationTheory/Coinvariants.lean
@@ -299,7 +299,6 @@ noncomputable section
 
 variable [Group G] (A : Rep k G) (S : Subgroup G) [S.Normal]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a normal subgroup `S ≤ G`, a `G`-representation `A` restricts to a `G`-representation on
 the kernel of the quotient map to the `S`-coinvariants `A_S`. -/
 abbrev toCoinvariantsKer : Rep k G := Rep.of (A.ρ.toCoinvariantsKer S)
@@ -319,7 +318,6 @@ lemma toCoinvariantsMkQ_hom :
 the coinvariants of `ρ|_S`. -/
 abbrev quotientToCoinvariants : Rep k (G ⧸ S) := ofQuotient (toCoinvariants A S) S
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a normal subgroup `S ≤ G`, a `G`-representation `A` induces a short exact sequence of
 `G`-representations `0 ⟶ Ker(mk) ⟶ A ⟶ A_S ⟶ 0` where `mk` is the quotient map to the
 `S`-coinvariants `A_S`. -/
diff --git a/Mathlib/RepresentationTheory/FiniteIndex.lean b/Mathlib/RepresentationTheory/FiniteIndex.lean
index 7ed8b7c696d9e3..54c3f1956c7e40 100644
--- a/Mathlib/RepresentationTheory/FiniteIndex.lean
+++ b/Mathlib/RepresentationTheory/FiniteIndex.lean
@@ -99,7 +99,6 @@ lemma indToCoindAux_comm {A B : Rep k S} (f : A ⟶ B) (g₁ g₂ : G) (a : A) :
   · simp [S.1.smul_def, hom_comm_apply]
   · simp [indToCoindAux_of_not_rel (h := h)]
 
-set_option backward.isDefEq.respectTransparency false in
 variable (A) in
 /-- Let `S ≤ G` be a subgroup and `A` a `k`-linear `S`-representation. This is the `k`-linear map
 `Ind_S^G(A) →ₗ[k] Coind_S^G(A)` sending `(⟦g ⊗ₜ[k] a⟧, sg) ↦ ρ(s)(a)`. -/
@@ -144,7 +143,6 @@ lemma coindToInd_of_support_subset_orbit (g : G) (f : coind S.subtype A)
 
 variable (A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `S ≤ G` be a finite index subgroup, `g₁, ..., gₙ` a set of right coset representatives of
 `S`, and `A` a `k`-linear `S`-representation. This is an isomorphism `Ind_S^G(A) ≅ Coind_S^G(A)`.
 The forward map sends `(⟦g ⊗ₜ[k] a⟧, sg) ↦ ρ(s)(a)`, and the inverse sends `f : G → A` to
diff --git a/Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean b/Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean
index 838e8d65175bb7..914d56e9e2df08 100644
--- a/Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean
+++ b/Mathlib/RepresentationTheory/Homological/GroupCohomology/Functoriality.lean
@@ -105,7 +105,6 @@ lemma cochainsMap_f_map_epi (hf : Function.Injective f) [Epi φ] (i : ℕ) :
     ((Rep.epi_iff_surjective φ).1 inferInstance).comp_left.comp <|
     LinearMap.funLeft_surjective_of_injective k A _ hf.comp_left
 
-set_option backward.isDefEq.respectTransparency false in
 instance cochainsMap_id_f_map_epi {A B : Rep k G} (φ : A ⟶ B) [Epi φ] (i : ℕ) :
     Epi ((cochainsMap (MonoidHom.id G) φ).f i) :=
   cochainsMap_f_map_epi (MonoidHom.id G) φ (fun _ _ h => h) i
@@ -291,7 +290,6 @@ theorem mapShortComplexH1_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C
     mapShortComplexH1 (MonoidHom.id G) (φ ≫ ψ) =
       mapShortComplexH1 (MonoidHom.id G) φ ≫ mapShortComplexH1 (MonoidHom.id G) ψ := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
 this is induced map `Z¹(H, A) ⟶ Z¹(G, B)`. -/
 noncomputable abbrev mapCocycles₁ :
@@ -454,7 +452,6 @@ theorem mapShortComplexH2_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C
     mapShortComplexH2 (MonoidHom.id G) (φ ≫ ψ) =
       mapShortComplexH2 (MonoidHom.id G) φ ≫ mapShortComplexH2 (MonoidHom.id G) ψ := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a group homomorphism `f : G →* H` and a representation morphism `φ : Res(f)(A) ⟶ B`,
 this is induced map `Z²(H, A) ⟶ Z²(G, B)`. -/
 noncomputable abbrev mapCocycles₂ :
diff --git a/Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean b/Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean
index 10402006cf4b9e..28e3cc9a91eada 100644
--- a/Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean
+++ b/Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean
@@ -112,7 +112,6 @@ theorem d₀₁_ker_eq_invariants : LinearMap.ker (d₀₁ A).hom = invariants A
   rw [d₀₁_hom_apply, isTrivial_apply, sub_self]
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[reassoc (attr := simp), elementwise (attr := simp)]
 lemma subtype_comp_d₀₁ : ModuleCat.ofHom (A.ρ.invariants.subtype) ≫ d₀₁ A = 0 := by
   ext ⟨x, hx⟩ g
@@ -757,7 +756,6 @@ def dArrowIso₀₁ :
     Arrow.mk ((inhomogeneousCochains A).d 0 1) ≅ Arrow.mk (d₀₁ A) :=
   Arrow.isoMk (cochainsIso₀ A) (cochainsIso₁ A) (comp_d₀₁_eq A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 0-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
 `A.ρ.invariants`, which is a simpler type. -/
 def cocyclesIso₀ : cocycles A 0 ≅ ModuleCat.of k A.ρ.invariants :=
@@ -799,7 +797,6 @@ def isoShortComplexH1 : (inhomogeneousCochains A).sc 1 ≅ shortComplexH1 A :=
     isoMk (cochainsIso₀ A) (cochainsIso₁ A) (cochainsIso₂ A)
       (comp_d₀₁_eq A) (comp_d₁₂_eq A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 1-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
 `cocycles₁ A`, which is a simpler type. -/
 def isoCocycles₁ : cocycles A 1 ≅ ModuleCat.of k (cocycles₁ A) :=
@@ -849,7 +846,6 @@ def isoShortComplexH2 :
     isoMk (cochainsIso₁ A) (cochainsIso₂ A) (cochainsIso₃ A)
       (comp_d₁₂_eq A) (comp_d₂₃_eq A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 2-cocycles of the complex of inhomogeneous cochains of `A` are isomorphic to
 `cocycles₂ A`, which is a simpler type. -/
 def isoCocycles₂ : cocycles A 2 ≅ ModuleCat.of k (cocycles₂ A) :=
@@ -897,7 +893,6 @@ section H0
 defined as the 0th cohomology of the complex of inhomogeneous cochains of `A`. -/
 abbrev H0 := groupCohomology A 0
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 0th group cohomology of `A`, defined as the 0th cohomology of the complex of inhomogeneous
 cochains, is isomorphic to the invariants of the representation on `A`. -/
 def H0Iso : H0 A ≅ ModuleCat.of k A.ρ.invariants :=
@@ -945,7 +940,6 @@ section H1
 defined as the 1st cohomology of the complex of inhomogeneous cochains of `A`. -/
 abbrev H1 := groupCohomology A 1
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The quotient map from the 1-cocycles of `A`, as a submodule of `G → A`, to `H¹(G, A)`. -/
 def H1π : ModuleCat.of k (cocycles₁ A) ⟶ H1 A :=
   (isoCocycles₁ A).inv ≫ π A 1
@@ -966,7 +960,6 @@ lemma H1π_eq_zero_iff (x : cocycles₁ A) : H1π A x = 0 ↔ ⇑x ∈ coboundar
     ((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
   simp [LinearMap.range_codRestrict, coboundaries₁, shortComplexH1, cocycles₁]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma H1π_eq_iff (x y : cocycles₁ A) :
     H1π A x = H1π A y ↔ ⇑x - ⇑y ∈ coboundaries₁ A := by
   rw [← sub_eq_zero, ← map_sub, H1π_eq_zero_iff]
@@ -1028,7 +1021,6 @@ section H2
 defined as the 2nd cohomology of the complex of inhomogeneous cochains of `A`. -/
 abbrev H2 := groupCohomology A 2
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The quotient map from the 2-cocycles of `A`, as a submodule of `G × G → A`, to `H²(G, A)`. -/
 def H2π : ModuleCat.of k (cocycles₂ A) ⟶ H2 A :=
   (isoCocycles₂ A).inv ≫ π A 2
@@ -1049,7 +1041,6 @@ lemma H2π_eq_zero_iff (x : cocycles₂ A) : H2π A x = 0 ↔ ⇑x ∈ coboundar
     ((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
   simp [LinearMap.range_codRestrict, coboundaries₂, shortComplexH2, cocycles₂]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma H2π_eq_iff (x y : cocycles₂ A) :
     H2π A x = H2π A y ↔ ⇑x - ⇑y ∈ coboundaries₂ A := by
   rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
diff --git a/Mathlib/RepresentationTheory/Homological/GroupHomology/FiniteCyclic.lean b/Mathlib/RepresentationTheory/Homological/GroupHomology/FiniteCyclic.lean
index e5a15912394ddf..6b070566dfb9bb 100644
--- a/Mathlib/RepresentationTheory/Homological/GroupHomology/FiniteCyclic.lean
+++ b/Mathlib/RepresentationTheory/Homological/GroupHomology/FiniteCyclic.lean
@@ -90,7 +90,6 @@ noncomputable def groupHomologyIsoEven
     (by induction i generalizing h₀ with | zero => exact (NeZero.ne 0 rfl).elim | succ n _ => simp)
     hi
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a finite cyclic group `G` generated by `g` and `A : Rep k G`, this is the quotient map
 `Ker(N) ⟶ Ker(N)/Im(ρ(g) - Id(A)) ≅ Hᵢ(G, A)` for any nonzero even `i`. -/
 noncomputable abbrev groupHomologyπEven
@@ -107,7 +106,6 @@ lemma groupHomologyπEven_eq_zero_iff (hg : ∀ x, x ∈ Subgroup.zpowers g)
   simp [groupHomologyπEven, map_eq_zero_iff _ ((ModuleCat.mono_iff_injective _).1 inferInstance),
     ShortComplex.moduleCatToCycles, -LinearMap.mem_range, LinearMap.range_codRestrict]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma groupHomologyπEven_eq_iff (hg : ∀ x, x ∈ Subgroup.zpowers g)
     (i : ℕ) [NeZero i] (hi : Even i) (x y : LinearMap.ker A.ρ.norm) :
     groupHomologyπEven A g hg i hi x = groupHomologyπEven A g hg i hi y ↔
diff --git a/Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean b/Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean
index f3e6c38770cb99..8c2bff9fb52fc0 100644
--- a/Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean
+++ b/Mathlib/RepresentationTheory/Homological/GroupHomology/Functoriality.lean
@@ -327,7 +327,6 @@ theorem mapShortComplexH1_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C
       mapShortComplexH1 (MonoidHom.id G) φ ≫ mapShortComplexH1 (MonoidHom.id G) ψ :=
   mapShortComplexH1_comp (MonoidHom.id G) (MonoidHom.id G) _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
 this is the induced map `Z₁(G, A) ⟶ Z₁(H, B)`. -/
 noncomputable abbrev mapCycles₁ :
@@ -759,7 +758,6 @@ theorem mapShortComplexH2_id_comp {A B C : Rep k G} (φ : A ⟶ B) (ψ : B ⟶ C
       mapShortComplexH2 (MonoidHom.id G) φ ≫ mapShortComplexH2 (MonoidHom.id G) ψ :=
   mapShortComplexH2_comp (MonoidHom.id G) (MonoidHom.id G) _ _
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a group homomorphism `f : G →* H` and a representation morphism `φ : A ⟶ Res(f)(B)`,
 this is the induced map `Z₂(G, A) ⟶ Z₂(H, B)`. -/
 noncomputable abbrev mapCycles₂ :
diff --git a/Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean b/Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean
index c6ec0d0514cad6..97603f7a271ecb 100644
--- a/Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean
+++ b/Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean
@@ -125,7 +125,6 @@ lemma d₁₀_comp_coinvariantsMk : d₁₀ A ≫ (coinvariantsMk k G).app A = 0
   ext
   simp [d₁₀]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 0th differential in the complex of inhomogeneous chains of a `G`-representation `A` as a
 linear map into the `k`-submodule of `A` spanned by elements of the form
 `ρ(g)(x) - x, g ∈ G, x ∈ A`. -/
@@ -365,7 +364,6 @@ variable (A) in
 theorem cycles₁_eq_top_of_isTrivial [A.IsTrivial] : cycles₁ A = ⊤ := by
   rw [cycles₁, d₁₀_eq_zero_of_isTrivial, ModuleCat.hom_zero, LinearMap.ker_zero]
 
-set_option backward.isDefEq.respectTransparency false in
 variable (A) in
 /-- The natural inclusion `Z₁(G, A) ⟶ C₁(G, A)` is an isomorphism when the representation
 on `A` is trivial. -/
@@ -757,7 +755,6 @@ def isoShortComplexH1 : (inhomogeneousChains A).sc 1 ≅ shortComplexH1 A :=
   (inhomogeneousChains A).isoSc' 2 1 0 (by simp) (by simp) ≪≫
     isoMk (chainsIso₂ A) (chainsIso₁ A) (chainsIso₀ A) (comp_d₂₁_eq A) (comp_d₁₀_eq A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 1-cycles of the complex of inhomogeneous chains of `A` are isomorphic to
 `cycles₁ A`, which is a simpler type. -/
 def isoCycles₁ : cycles A 1 ≅ ModuleCat.of k (cycles₁ A) :=
@@ -804,7 +801,6 @@ def isoShortComplexH2 : (inhomogeneousChains A).sc 2 ≅ shortComplexH2 A :=
   (inhomogeneousChains A).isoSc' 3 2 1 (by simp) (by simp) ≪≫
     isoMk (chainsIso₃ A) (chainsIso₂ A) (chainsIso₁ A) (comp_d₃₂_eq A) (comp_d₂₁_eq A)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The 2-cycles of the complex of inhomogeneous chains of `A` are isomorphic to
 `cycles₂ A`, which is a simpler type. -/
 def isoCycles₂ : cycles A 2 ≅ ModuleCat.of k (cycles₂ A) :=
@@ -919,7 +915,6 @@ section H1
 defined as the 1st homology of the complex of inhomogeneous chains of `A`. -/
 abbrev H1 := groupHomology A 1
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The quotient map from the 1-cycles of `A`, as a submodule of `G →₀ A`, to `H₁(G, A)`. -/
 def H1π : ModuleCat.of k (cycles₁ A) ⟶ H1 A :=
   (isoCycles₁ A).inv ≫ π A 1
@@ -940,7 +935,6 @@ lemma H1π_eq_zero_iff (x : cycles₁ A) : H1π A x = 0 ↔ x.1 ∈ boundaries
     ((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
   simp [LinearMap.range_codRestrict, boundaries₁, shortComplexH1, cycles₁]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma H1π_eq_iff (x y : cycles₁ A) :
     H1π A x = H1π A y ↔ x.1 - y.1 ∈ boundaries₁ A := by
   rw [← sub_eq_zero, ← map_sub, H1π_eq_zero_iff]
@@ -1065,7 +1059,6 @@ section H2
 defined as the 2nd homology of the complex of inhomogeneous chains of `A`. -/
 abbrev H2 := groupHomology A 2
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The quotient map from the 2-cycles of `A`, as a submodule of `G × G →₀ A`, to `H₂(G, A)`. -/
 def H2π : ModuleCat.of k (cycles₂ A) ⟶ H2 A :=
   (isoCycles₂ A).inv ≫ π A 2
@@ -1086,7 +1079,6 @@ lemma H2π_eq_zero_iff (x : cycles₂ A) : H2π A x = 0 ↔ x.1 ∈ boundaries
     ((ModuleCat.mono_iff_injective <| _).1 inferInstance)]
   simp [LinearMap.range_codRestrict, boundaries₂, shortComplexH2, cycles₂]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma H2π_eq_iff (x y : cycles₂ A) :
     H2π A x = H2π A y ↔ x.1 - y.1 ∈ boundaries₂ A := by
   rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
diff --git a/Mathlib/RepresentationTheory/Invariants.lean b/Mathlib/RepresentationTheory/Invariants.lean
index e75be2ca285901..bdfdb506842568 100644
--- a/Mathlib/RepresentationTheory/Invariants.lean
+++ b/Mathlib/RepresentationTheory/Invariants.lean
@@ -204,12 +204,10 @@ open CategoryTheory
 
 variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (S : Subgroup G) [S.Normal]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` restricts to a `G`-representation on
 the invariants of `ρ|_S`. -/
 abbrev toInvariants : Rep k G := Rep.of <| A.ρ.toInvariants S
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a normal subgroup `S ≤ G`, a `G`-representation `ρ` induces a `G ⧸ S`-representation on
 the invariants of `ρ|_S`. -/
 abbrev quotientToInvariants : Rep k (G ⧸ S) := Rep.of (A.ρ.quotientToInvariants S)
diff --git a/Mathlib/RepresentationTheory/Maschke.lean b/Mathlib/RepresentationTheory/Maschke.lean
index ea73fe903ab43a..f3598431481640 100644
--- a/Mathlib/RepresentationTheory/Maschke.lean
+++ b/Mathlib/RepresentationTheory/Maschke.lean
@@ -158,7 +158,6 @@ theorem exists_leftInverse_of_injective (f : V →ₗ[k[G]] W) (hf : LinearMap.k
 
 namespace Submodule
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_isCompl (p : Submodule k[G] V) : ∃ q : Submodule k[G] V, IsCompl p q := by
   rcases MonoidAlgebra.exists_leftInverse_of_injective p.subtype p.ker_subtype with ⟨f, hf⟩
   exact ⟨LinearMap.ker f, LinearMap.isCompl_of_proj <| DFunLike.congr_fun hf⟩
diff --git a/Mathlib/RepresentationTheory/Rep.lean b/Mathlib/RepresentationTheory/Rep.lean
index 78f543a5968250..536788b0bbc188 100644
--- a/Mathlib/RepresentationTheory/Rep.lean
+++ b/Mathlib/RepresentationTheory/Rep.lean
@@ -173,14 +173,12 @@ end
 
 variable (A : Rep k G)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given a `k`-linear `G`-representation `(V, ρ)`, this is the representation defined by
 restricting `ρ` to a `G`-invariant `k`-submodule of `V`. -/
 abbrev subrepresentation (W : Submodule k A) (le_comap : ∀ g, W ≤ W.comap (A.ρ g)) :
     Rep k G :=
   Rep.of (A.ρ.subrepresentation W le_comap)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The natural inclusion of a subrepresentation into the ambient representation. -/
 @[simps]
 def subtype (W : Submodule k A) (le_comap : ∀ g, W ≤ W.comap (A.ρ g)) :
diff --git a/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean b/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean
index 20cc334e60ff96..3f7cd2cb71edd9 100644
--- a/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean
+++ b/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean
@@ -232,14 +232,12 @@ variable (hf : Function.Surjective f)
 
 include hf
 
-set_option backward.isDefEq.respectTransparency false in
 private lemma tens_exact : Function.Exact (lTensorKerIncl I M f) (lTensorf I M f) :=
   lTensor_exact (AdicCompletion I R) (f.exact_subtype_ker_map) hf
 
 private lemma tens_surj : Function.Surjective (lTensorf I M f) :=
   LinearMap.lTensor_surjective (AdicCompletion I R) hf
 
-set_option backward.isDefEq.respectTransparency false in
 private lemma adic_exact [IsNoetherianRing R] [Finite ι] :
     Function.Exact (map I (LinearMap.ker f).subtype) (map I f) :=
   map_exact (Submodule.injective_subtype _) (f.exact_subtype_ker_map) hf
@@ -247,7 +245,6 @@ private lemma adic_exact [IsNoetherianRing R] [Finite ι] :
 private lemma adic_surj : Function.Surjective (map I f) :=
   map_surjective I hf
 
-set_option backward.isDefEq.respectTransparency false in
 private
 lemma ofTensorProduct_bijective_of_map_from_fin [Finite ι] [IsNoetherianRing R] :
     Function.Bijective (ofTensorProduct I M) :=
@@ -345,7 +342,6 @@ lemma tensor_map_id_left_injective_of_injective (hf : Function.Injective f) :
 
 end
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Adic completion of a Noetherian ring `R` is flat over `R`. -/
 instance flat_of_isNoetherian [IsNoetherianRing R] : Module.Flat R (AdicCompletion I R) :=
   Module.Flat.iff_lTensor_injective'.mpr fun J ↦
diff --git a/Mathlib/RingTheory/AdicCompletion/Basic.lean b/Mathlib/RingTheory/AdicCompletion/Basic.lean
index 2cf93c82680364..5ce835a93b2eed 100644
--- a/Mathlib/RingTheory/AdicCompletion/Basic.lean
+++ b/Mathlib/RingTheory/AdicCompletion/Basic.lean
@@ -86,8 +86,7 @@ theorem IsHausdorff.of_map [CommRing S] [Module S M] {J : Ideal S} [Algebra R S]
   apply SModEq.of_toAddSubgroup_le
       (U := (I ^ n • ⊤ : Submodule R M)) (V := (J ^ n • ⊤ : Submodule S M))
   · rw [← AddSubgroup.toAddSubmonoid_le]
-    simp only [Submodule.toAddSubgroup_toAddSubmonoid, Submodule.smul_toAddSubmonoid,
-      Submodule.top_toAddSubmonoid]
+    simp only [Submodule.smul_toAddSubmonoid, Submodule.top_toAddSubmonoid]
     rw [AddSubmonoid.smul_le]
     intro r hr m hm
     rw [← algebraMap_smul S r m]
diff --git a/Mathlib/RingTheory/Artinian/Module.lean b/Mathlib/RingTheory/Artinian/Module.lean
index 79f412474fb64e..948ee4aa82e9a8 100644
--- a/Mathlib/RingTheory/Artinian/Module.lean
+++ b/Mathlib/RingTheory/Artinian/Module.lean
@@ -247,7 +247,6 @@ instance isArtinian_of_quotient_of_artinian
     (N : Submodule R M) [IsArtinian R M] : IsArtinian R (M ⧸ N) :=
   isArtinian_of_surjective M (Submodule.mkQ N) (Submodule.Quotient.mk_surjective N)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isArtinian_of_range_eq_ker [IsArtinian R M] [IsArtinian R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P)
     (h : LinearMap.range f = LinearMap.ker g) : IsArtinian R N :=
   wellFounded_lt_exact_sequence (LinearMap.range f)
@@ -259,7 +258,6 @@ theorem isArtinian_of_range_eq_ker [IsArtinian R M] [IsArtinian R P] (f : M →
     (by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
     (by simp [Submodule.comap_map_eq, h])
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isArtinian_iff_submodule_quotient (S : Submodule R P) :
     IsArtinian R P ↔ IsArtinian R S ∧ IsArtinian R (P ⧸ S) := by
   refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
@@ -269,7 +267,6 @@ theorem isArtinian_iff_submodule_quotient (S : Submodule R P) :
 instance isArtinian_prod [IsArtinian R M] [IsArtinian R P] : IsArtinian R (M × P) :=
   isArtinian_of_range_eq_ker (LinearMap.inl R M P) (LinearMap.snd R M P) (LinearMap.range_inl R M P)
 
-set_option backward.isDefEq.respectTransparency false in
 instance isArtinian_sup (M₁ M₂ : Submodule R P) [IsArtinian R M₁] [IsArtinian R M₂] :
     IsArtinian R ↥(M₁ ⊔ M₂) := by
   have := isArtinian_range (M₁.subtype.coprod M₂.subtype)
@@ -436,7 +433,6 @@ instance isArtinian_of_fg_of_artinian' {R M} [Ring R] [AddCommGroup M] [Module R
   have ⟨_, _, h⟩ := Module.Finite.exists_fin' R M
   isArtinian_of_surjective _ _ h
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
     (N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by
   rw [← Module.Finite.iff_fg] at hN; infer_instance
diff --git a/Mathlib/RingTheory/Binomial.lean b/Mathlib/RingTheory/Binomial.lean
index 681dd95cfd4c9d..84cabf002fbabf 100644
--- a/Mathlib/RingTheory/Binomial.lean
+++ b/Mathlib/RingTheory/Binomial.lean
@@ -274,7 +274,6 @@ instance Int.instBinomialRing : BinomialRing ℤ where
 
 attribute [local instance] IsAddTorsionFree.of_module_nnrat
 
-set_option backward.isDefEq.respectTransparency false in
 noncomputable instance {R : Type*} [AddCommMonoid R] [Module ℚ≥0 R] [Pow R ℕ] : BinomialRing R where
   multichoose r n := (n.factorial : ℚ≥0)⁻¹ • Polynomial.smeval (ascPochhammer ℕ n) r
   factorial_nsmul_multichoose r n := by
diff --git a/Mathlib/RingTheory/DedekindDomain/Factorization.lean b/Mathlib/RingTheory/DedekindDomain/Factorization.lean
index e0eb2e55b9b92b..e357a0ae82c950 100644
--- a/Mathlib/RingTheory/DedekindDomain/Factorization.lean
+++ b/Mathlib/RingTheory/DedekindDomain/Factorization.lean
@@ -716,7 +716,6 @@ lemma divMod_zero_of_not_le {a b c : FractionalIdeal R⁰ K} (hac : ¬ a ≤ c)
     c.divMod b a = 0 := by
   simp [divMod, hac]
 
-set_option backward.isDefEq.respectTransparency false in
 set_option maxHeartbeats 212000 in
 -- changed for new compiler
 /-- Let `I J I' J'` be nonzero fractional ideals in a Dedekind domain with `J ≤ I` and `J' ≤ I'`.
diff --git a/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean b/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
index 595589750b39dc..21f66039595257 100644
--- a/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
+++ b/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
@@ -147,7 +147,6 @@ theorem FiniteDimensional.exists_is_basis_integral :
 
 variable [Algebra.IsSeparable K L]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
 integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
 Noetherian over `A`. -/
diff --git a/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean b/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
index bce61407ddfadd..85b7b2cfbfd78e 100644
--- a/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
+++ b/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
@@ -420,7 +420,6 @@ theorem addVal_mul {a b : R} :
 theorem addVal_pow (a : R) (n : ℕ) : addVal R (a ^ n) = n • addVal R a :=
   (addVal R).map_pow _ _
 
-set_option backward.isDefEq.respectTransparency false in
 nonrec theorem _root_.Irreducible.addVal_pow {ϖ : R} (h : Irreducible ϖ) (n : ℕ) :
     addVal R (ϖ ^ n) = n := by
   rw [addVal_pow, addVal_uniformizer h, nsmul_one]
@@ -455,7 +454,6 @@ theorem addVal_le_iff_dvd {a b : R} : addVal R a ≤ addVal R b ↔ a ∣ b := b
 theorem addVal_add {a b : R} : min (addVal R a) (addVal R b) ≤ addVal R (a + b) :=
   (addVal R).map_add _ _
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma addVal_eq_zero_of_unit (u : Rˣ) :
     addVal R u = 0 := by
diff --git a/Mathlib/RingTheory/Etale/Kaehler.lean b/Mathlib/RingTheory/Etale/Kaehler.lean
index 1f2a3b5babc3a0..5bfae9bb8389c6 100644
--- a/Mathlib/RingTheory/Etale/Kaehler.lean
+++ b/Mathlib/RingTheory/Etale/Kaehler.lean
@@ -136,7 +136,6 @@ def tensorCotangentSpace
       ext a
       simp; rfl }
 
-set_option backward.isDefEq.respectTransparency false in
 /-- (Implementation)
 If `J ≃ Q ⊗ₚ I` (e.g. when `T = Q ⊗ₚ S` and `P → Q` is flat), then `T ⊗ₛ I/I² ≃ J/J²`.
 This is the inverse. -/
diff --git a/Mathlib/RingTheory/Extension/Cotangent/Basic.lean b/Mathlib/RingTheory/Extension/Cotangent/Basic.lean
index 85340c457f12f2..119eeae3fb65a1 100644
--- a/Mathlib/RingTheory/Extension/Cotangent/Basic.lean
+++ b/Mathlib/RingTheory/Extension/Cotangent/Basic.lean
@@ -300,7 +300,6 @@ lemma CotangentSpace.map_sub_map (f g : Hom P P') :
         Function.comp_apply, Hom.sub_tmul, LinearMap.map_smul_of_tower, cotangentComplex_mk,
         Hom.subToKer_apply_coe, map_sub, ← algebraMap_eq_smul_one, tmul_sub, smul_sub]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Cotangent.map_sub_map (f g : Hom P P') :
     map f - map g = (f.sub g) ∘ₗ P.cotangentComplex := by
   ext x
diff --git a/Mathlib/RingTheory/FiniteLength.lean b/Mathlib/RingTheory/FiniteLength.lean
index 767d61b023d720..e978f7d821bbda 100644
--- a/Mathlib/RingTheory/FiniteLength.lean
+++ b/Mathlib/RingTheory/FiniteLength.lean
@@ -25,7 +25,6 @@ Finite length, Composition series
 
 variable (R : Type*) [Ring R]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A module of finite length is either trivial or a simple extension of a module known
 to be of finite length. -/
 inductive IsFiniteLength : ∀ (M : Type*) [AddCommGroup M] [Module R M], Prop
@@ -37,7 +36,6 @@ attribute [nontriviality] IsFiniteLength.of_subsingleton
 
 variable {R} {M N : Type*} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem LinearEquiv.isFiniteLength (e : M ≃ₗ[R] N)
     (h : IsFiniteLength R M) : IsFiniteLength R N := by
   induction h generalizing N with
@@ -54,7 +52,6 @@ theorem exists_compositionSeries_of_isNoetherian_isArtinian [IsNoetherian R M] [
   obtain ⟨f, f0, n, hn⟩ := exists_covBy_seq_of_wellFoundedLT_wellFoundedGT (Submodule R M)
   exact ⟨⟨n, fun i ↦ f i, fun i ↦ hn.2 i i.2⟩, f0.eq_bot, hn.1.eq_top⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isFiniteLength_of_exists_compositionSeries
     (h : ∃ s : CompositionSeries (Submodule R M), s.head = ⊥ ∧ s.last = ⊤) :
     IsFiniteLength R M :=
@@ -88,7 +85,6 @@ theorem isFiniteLength_iff_exists_compositionSeries :
     exists_compositionSeries_of_isNoetherian_isArtinian R M,
     isFiniteLength_of_exists_compositionSeries⟩
 
-set_option backward.isDefEq.respectTransparency false in
 open scoped IsSimpleOrder in
 theorem IsSemisimpleModule.finite_tfae [IsSemisimpleModule R M] :
     List.TFAE [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsFiniteLength R M,
diff --git a/Mathlib/RingTheory/FiniteType.lean b/Mathlib/RingTheory/FiniteType.lean
index 014e0e331069f2..c161e193e291c1 100644
--- a/Mathlib/RingTheory/FiniteType.lean
+++ b/Mathlib/RingTheory/FiniteType.lean
@@ -611,7 +611,6 @@ end MonoidAlgebra
 
 section Orzech
 
-set_option backward.isDefEq.respectTransparency false in
 open Submodule Module Module.Finite in
 /-- Any commutative ring `R` satisfies the `OrzechProperty`, that is, for any finitely generated
 `R`-module `M`, any surjective homomorphism `f : N →ₗ[R] M` from a submodule `N` of `M` to `M`
diff --git a/Mathlib/RingTheory/Finiteness/Descent.lean b/Mathlib/RingTheory/Finiteness/Descent.lean
index cbf8fb9477e5af..9efb7cb4d21149 100644
--- a/Mathlib/RingTheory/Finiteness/Descent.lean
+++ b/Mathlib/RingTheory/Finiteness/Descent.lean
@@ -46,7 +46,6 @@ lemma Module.Finite.of_finite_tensorProduct_of_faithfullyFlat {M : Type*} [AddCo
     simp [f₀, -Pi.basisFun_equivFun, -Pi.basisFun_apply, h i]
   rwa [← Module.FaithfullyFlat.lTensor_surjective_iff_surjective _ T]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Ideal.FG.of_FG_map_of_faithfullyFlat [Module.FaithfullyFlat R S] {I : Ideal R}
     (hI : (I.map (algebraMap R S)).FG) : I.FG := by
   change Submodule.FG I
diff --git a/Mathlib/RingTheory/Finiteness/Finsupp.lean b/Mathlib/RingTheory/Finiteness/Finsupp.lean
index f304004412c8b9..9dd33a3601aa36 100644
--- a/Mathlib/RingTheory/Finiteness/Finsupp.lean
+++ b/Mathlib/RingTheory/Finiteness/Finsupp.lean
@@ -127,7 +127,6 @@ lemma _root_.Module.Finite.of_exact {f : M →ₗ[R] N} {g : N →ₗ[R] P}
     exact Module.Finite.fg_top
   · simp [LinearMap.exact_iff.1 h_exact]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem _root_.Module.Finite.of_submodule_quotient (N : Submodule R M) [Module.Finite R N]
     [Module.Finite R (M ⧸ N)] : Module.Finite R M :=
   .of_exact (LinearMap.exact_subtype_mkQ N) (Quotient.mk_surjective _)
diff --git a/Mathlib/RingTheory/Finiteness/NilpotentKer.lean b/Mathlib/RingTheory/Finiteness/NilpotentKer.lean
index 58a0e5190f9653..07d454680f56ca 100644
--- a/Mathlib/RingTheory/Finiteness/NilpotentKer.lean
+++ b/Mathlib/RingTheory/Finiteness/NilpotentKer.lean
@@ -17,7 +17,6 @@ public section
 
 open TensorProduct
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `I` is a finitely generated nilpotent ideal of an `R`-algebra `S`, and `T = S / I` is
 `R`-finite, then `S` is also `R`-finite. -/
 lemma Module.finite_of_surjective_of_ker_le_nilradical
diff --git a/Mathlib/RingTheory/Flat/Equalizer.lean b/Mathlib/RingTheory/Flat/Equalizer.lean
index 996ff4a5d5842e..c040345c47bf78 100644
--- a/Mathlib/RingTheory/Flat/Equalizer.lean
+++ b/Mathlib/RingTheory/Flat/Equalizer.lean
@@ -32,7 +32,6 @@ variable (M : Type*) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R
 variable {N P : Type*} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P]
   (f g : N →ₗ[R] P)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Module.Flat.ker_lTensor_eq [Module.Flat R M] :
     LinearMap.ker (AlgebraTensorModule.lTensor S M f) =
       LinearMap.range (AlgebraTensorModule.lTensor S M (LinearMap.ker f).subtype) := by
@@ -188,7 +187,6 @@ lemma LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm [Module.Flat R M
 
 variable {M}
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Given a short exact sequence `0 → M → N → P → 0` with `P` flat,
 then any `A ⊗ M → A ⊗ N` is injective.
@@ -228,7 +226,6 @@ to get `0 → A ⊗ K → A ⊗ M` exact.
   rw [Subsingleton.elim (SnakeLemma.δ' ..) 0] at this
   simpa using this
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Given surjection `f : N → P` with `P` flat, then `A ⊗ ker f ≃ ker (A ⊗ f)`.
 Also see `LinearMap.tensorKerEquiv` for the version with `A` flat instead. -/
 def LinearMap.kerLTensorEquivOfSurjective [Module.Flat R P]
@@ -321,7 +318,6 @@ lemma AlgHom.tensorEqualizerEquiv_apply [Module.Flat R T]
 
 #adaptation_note /-- After nightly-2026-02-23 this requires more heartbeats. -/
 set_option maxHeartbeats 400000 in -- see note
-set_option backward.isDefEq.respectTransparency false in
 variable (R A) in
 attribute [local instance] Algebra.TensorProduct.rightAlgebra in
 /--
diff --git a/Mathlib/RingTheory/Flat/EquationalCriterion.lean b/Mathlib/RingTheory/Flat/EquationalCriterion.lean
index 3b905c713cb8b5..a1c7b3220051c1 100644
--- a/Mathlib/RingTheory/Flat/EquationalCriterion.lean
+++ b/Mathlib/RingTheory/Flat/EquationalCriterion.lean
@@ -266,7 +266,6 @@ theorem exists_factorization_of_comp_eq_zero_of_free [Flat R M] {K N : Type*} [A
   ⟨k, a ∘ₗ e.symm, y, by rwa [← comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm], by
     rwa [comp_assoc]⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Every homomorphism from a finitely presented module to a flat module factors through a finite
 free module. -/
 @[stacks 058E "only if"]
diff --git a/Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.lean b/Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.lean
index da05b7ff50c9df..47924ebfdad1b2 100644
--- a/Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.lean
+++ b/Mathlib/RingTheory/Flat/FaithfullyFlat/Basic.lean
@@ -268,7 +268,6 @@ variable {N2 : Type*} [AddCommGroup N2] [Module R N2]
 variable {N3 : Type*} [AddCommGroup N3] [Module R N3]
 variable (l12 : N1 →ₗ[R] N2) (l23 : N2 →ₗ[R] N3)
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 If `M` is faithfully flat, then exactness of `N₁ ⊗ M -> N₂ ⊗ M -> N₃ ⊗ M` implies that the
 composition `N₁ -> N₂ -> N₃` is `0`.
@@ -317,7 +316,6 @@ lemma range_le_ker_of_exact_rTensor [fl : FaithfullyFlat R M]
   -- but `E ⊗ M = 0` implies `E = 0` because `M` is faithfully flat and this is a contradiction.
   exact not_subsingleton_iff_nontrivial.2 inferInstance <| fl.rTensor_reflects_triviality R M E
 
-set_option backward.isDefEq.respectTransparency false in
 lemma rTensor_reflects_exact [fl : FaithfullyFlat R M]
     (ex : Function.Exact (l12.rTensor M) (l23.rTensor M)) :
     Function.Exact l12 l23 := LinearMap.exact_iff.2 <| by
diff --git a/Mathlib/RingTheory/Flat/Tensor.lean b/Mathlib/RingTheory/Flat/Tensor.lean
index eac4caf96827b6..d9f90cd20ed276 100644
--- a/Mathlib/RingTheory/Flat/Tensor.lean
+++ b/Mathlib/RingTheory/Flat/Tensor.lean
@@ -61,7 +61,6 @@ theorem iff_characterModule_baer : Flat R M ↔ Baer R (CharacterModule M) := by
   rw [equiv_iff (N := ULift.{u} M) ULift.moduleEquiv.symm, iff_characterModule_injective,
     ← Baer.iff_injective, Baer.congr (CharacterModule.congr ULift.moduleEquiv)]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- An `R`-module `M` is flat iff for all ideals `I` of `R`, the tensor product of the
 inclusion `I → R` and the identity `M → M` is injective. See `iff_rTensor_injective` to
 restrict to finitely generated ideals `I`. -/
@@ -75,7 +74,6 @@ theorem iff_lTensor_injective' :
     Flat R M ↔ ∀ (I : Ideal R), Function.Injective (lTensor M I.subtype) := by
   simpa [← comm_comp_rTensor_comp_comm_eq] using iff_rTensor_injective'
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A module `M` over a ring `R` is flat iff for all finitely generated ideals `I` of `R`, the
 tensor product of the inclusion `I → R` and the identity `M → M` is injective. See
 `iff_rTensor_injective'` to extend to all ideals `I`. -/
diff --git a/Mathlib/RingTheory/HopkinsLevitzki.lean b/Mathlib/RingTheory/HopkinsLevitzki.lean
index e062cd04537961..4238581ff166c7 100644
--- a/Mathlib/RingTheory/HopkinsLevitzki.lean
+++ b/Mathlib/RingTheory/HopkinsLevitzki.lean
@@ -43,7 +43,6 @@ namespace IsSemiprimaryRing
 
 variable [IsSemiprimaryRing R]
 
-set_option backward.isDefEq.respectTransparency false in
 @[elab_as_elim] protected theorem induction
     {P : ∀ (M : Type u) [AddCommGroup M] [Module R₀ M] [Module R M], Prop}
     (h0 : ∀ (M) [AddCommGroup M] [Module R₀ M] [Module R M] [IsScalarTower R₀ R M]
@@ -78,7 +77,6 @@ section
 
 variable [IsScalarTower R₀ R R] [Module.Finite R₀ (R ⧸ Ring.jacobson R)]
 
-set_option backward.isDefEq.respectTransparency false in
 private theorem finite_of_isNoetherian_or_isArtinian :
     IsNoetherian R M ∨ IsArtinian R M → Module.Finite R₀ M := by
   refine IsSemiprimaryRing.induction R₀ R M (P := fun M ↦ IsNoetherian R M ∨ IsArtinian R M →
@@ -103,7 +101,6 @@ end
 
 variable {R M}
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isNoetherian_iff_isArtinian : IsNoetherian R M ↔ IsArtinian R M :=
   IsSemiprimaryRing.induction R R M (P := fun M ↦ IsNoetherian R M ↔ IsArtinian R M)
     (fun M _ _ _ _ _ _ ↦ IsSemisimpleModule.finite_tfae.out 1 2)
diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean
index feb5d9ade64495..75a03a417b0cc3 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean
@@ -46,7 +46,6 @@ universe u v
 
 variable {A : Type u} [CommRing A] {M : Type v} [AddCommGroup M] [Module A M]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A `Prop` asserting that two submodules `N₁, N₂` satisfy `N₁ ≤ N₂` and
 `N₂ / N₁` is isomorphic to `A / p` for some prime ideal `p` of `A`. -/
 def Submodule.IsQuotientEquivQuotientPrime (N₁ N₂ : Submodule A M) :=
@@ -105,7 +104,6 @@ theorem IsNoetherianRing.exists_relSeries_isQuotientEquivQuotientPrime :
   have := Submodule.isQuotientEquivQuotientPrime_iff.mpr ⟨x, hp, rfl⟩
   refine ⟨_, by simpa [hs₂], s.snoc _ (hs₂ ▸ this), by simpa, rfl⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If a property on finitely generated modules over a Noetherian ring satisfies that:
 
 - it holds for zero module (it's formalized as it holds for any module which is subsingleton),
@@ -205,7 +203,6 @@ theorem Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisors {I : Ideal A}
     simpa only [smul_eq_mul, mul_comm x, SetLike.le_def, Submodule.mem_colon_singleton] using hP,
       fun h : x = 0 ↦ prime.ne_top <| by simp [h]⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem Ideal.nonempty_inter_nonZeroDivisors_of_faithfulSMul {I : Ideal A} [FaithfulSMul A I] :
     ((I : Set A) ∩ nonZeroDivisors A).Nonempty := by
   by_contra!
diff --git a/Mathlib/RingTheory/Ideal/Cotangent.lean b/Mathlib/RingTheory/Ideal/Cotangent.lean
index 421592adb1beec..1ecdd7b9d8002f 100644
--- a/Mathlib/RingTheory/Ideal/Cotangent.lean
+++ b/Mathlib/RingTheory/Ideal/Cotangent.lean
@@ -37,19 +37,15 @@ universe u v w
 variable {R : Type u} {S : Type v} {S' : Type w} [CommRing R] [CommSemiring S] [Algebra S R]
 variable [CommSemiring S'] [Algebra S' R] [Algebra S S'] [IsScalarTower S S' R] (I : Ideal R)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- `I ⧸ I ^ 2` as a quotient of `I`. -/
 def Cotangent : Type _ := I ⧸ (I • ⊤ : Submodule R I)
 deriving Inhabited, AddCommGroup, Module (R ⧸ I)
 
-set_option backward.isDefEq.respectTransparency false in
 deriving instance Module S, IsScalarTower S S' for Cotangent I
 
-set_option backward.isDefEq.respectTransparency false in
 variable [IsNoetherian R I] in
 deriving instance IsNoetherian R for Cotangent I
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The quotient map from `I` to `I ⧸ I ^ 2`. -/
 @[simps! -isSimp apply]
 def toCotangent : I →ₗ[R] I.Cotangent := Submodule.mkQ _
@@ -69,10 +65,8 @@ theorem toCotangent_eq {x y : I} : I.toCotangent x = I.toCotangent y ↔ (x - y
 
 theorem toCotangent_eq_zero (x : I) : I.toCotangent x = 0 ↔ (x : R) ∈ I ^ 2 := I.mem_toCotangent_ker
 
-set_option backward.isDefEq.respectTransparency false in
 theorem toCotangent_surjective : Function.Surjective I.toCotangent := Submodule.mkQ_surjective _
 
-set_option backward.isDefEq.respectTransparency false in
 theorem toCotangent_range : LinearMap.range I.toCotangent = ⊤ := Submodule.range_mkQ _
 
 theorem cotangent_subsingleton_iff : Subsingleton I.Cotangent ↔ IsIdempotentElem I := by
@@ -85,7 +79,6 @@ theorem cotangent_subsingleton_iff : Subsingleton I.Cotangent ↔ IsIdempotentEl
         Quotient.inductionOn₂' x y fun x y =>
           I.toCotangent_eq.mpr <| ((pow_two I).trans e).symm ▸ I.sub_mem x.prop y.prop⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The inclusion map `I ⧸ I ^ 2` to `R ⧸ I ^ 2`. -/
 def cotangentToQuotientSquare : I.Cotangent →ₗ[R] R ⧸ I ^ 2 :=
   Submodule.mapQ (I • ⊤) (I ^ 2) I.subtype
@@ -202,7 +195,6 @@ def quotCotangent : (R ⧸ I ^ 2) ⧸ I.cotangentIdeal ≃+* R ⧸ I := by
   refine (DoubleQuot.quotQuotEquivQuotSup _ _).trans ?_
   exact Ideal.quotEquivOfEq (sup_eq_right.mpr <| Ideal.pow_le_self two_ne_zero)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The map `I/I² → J/J²` if `I ≤ f⁻¹(J)`. -/
 def mapCotangent (I₁ : Ideal A) (I₂ : Ideal B) (f : A →ₐ[R] B) (h : I₁ ≤ I₂.comap f) :
     I₁.Cotangent →ₗ[R] I₂.Cotangent := by
@@ -252,7 +244,6 @@ lemma subsingleton_cotangentSpace_iff [IsNoetherianRing R] :
     Ideal.isIdempotentElem_iff_eq_bot_or_top_of_isLocalRing]
   simp [(maximalIdeal.isMaximal R).ne_top]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma CotangentSpace.map_eq_top_iff [IsNoetherianRing R] {M : Submodule R (maximalIdeal R)} :
     M.map (maximalIdeal R).toCotangent = ⊤ ↔ M = ⊤ := by
   refine ⟨fun H ↦ eq_top_iff.mpr ?_, by rintro rfl; simp [Ideal.toCotangent_range]⟩
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index ca26456601451f..d9f8589a5254bf 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -360,7 +360,6 @@ lemma mem_minimalPrimes_of_primeHeight_eq_height {I J : Ideal R} [J.IsPrime] (e
   rw [← J.height_eq_primeHeight] at e'
   exact mem_minimalPrimes_of_height_eq e (e' ▸ le_refl _)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Ideal R) (r : ℕ)
     (hr : r ≤ I.height) : ∃ J ≤ I, J.spanRank ≤ r ∧ r ≤ J.height := by
   induction r with
diff --git a/Mathlib/RingTheory/Ideal/IsPrincipalPowQuotient.lean b/Mathlib/RingTheory/Ideal/IsPrincipalPowQuotient.lean
index faf834ad9430c1..5441a0febc29a8 100644
--- a/Mathlib/RingTheory/Ideal/IsPrincipalPowQuotient.lean
+++ b/Mathlib/RingTheory/Ideal/IsPrincipalPowQuotient.lean
@@ -77,7 +77,6 @@ def quotEquivPowQuotPowSucc (h : I.IsPrincipal) (h' : I ≠ ⊥) (n : ℕ) :
   refine hx.imp ?_
   simp [g, LinearMap.codRestrict, eq_comm, mul_comm]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- For a principal ideal `I`, `R ⧸ I ≃ I ^ n ⧸ I ^ (n + 1)`. Supplied as a plain equiv to bypass
 typeclass synthesis issues on complex `Module` goals.  To convert into a form
 that uses the ideal of `R ⧸ I ^ (n + 1)`, compose with
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index b6fcc9fdff954e..f142f02c124a14 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -163,7 +163,6 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
       ← comap_map_of_surjective f hf p]
     exact comap_mono hrp
 
-set_option backward.isDefEq.respectTransparency false in
 open IsLocalRing in
 /-- **Krull's height theorem** (also known as **Krullscher Höhensatz**) :
   In a commutative Noetherian ring `R`, any prime ideal that is minimal over an ideal generated
@@ -242,7 +241,6 @@ lemma Ideal.height_le_spanRank_toENat (I : Ideal R) (hI : I ≠ ⊤) :
   convert (I.height_le_spanRank_toENat_of_mem_minimal_primes J hJ)
   exact Eq.symm (@height_eq_primeHeight _ _ J hJ.1.1)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Ideal.height_le_spanFinrank (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanFinrank := by
   have : I.spanFinrank = I.spanRank.toENat := by
@@ -312,7 +310,6 @@ lemma Ideal.exists_finset_card_eq_height_of_isNoetherianRing (p : Ideal R) [p.Is
       simpa [Submodule.fg_iff_spanRank_eq_spanFinrank] using (IsNoetherian.noetherian I)
     · exact I.height_le_spanRank_toENat_of_mem_minimal_primes _ hI
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `I ≤ p` and `p` is prime, the height of `p` is bounded by the height of `p ⧸ I R` plus
 the span rank of `I`. -/
 lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (hrp : I ≤ p) :
@@ -343,7 +340,6 @@ lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (
   convert hps
   simp [Ideal.map_span, ← himgo]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma height_le_ringKrullDim_quotient_add_spanFinrank {p I : Ideal R} [p.IsPrime] (h : I ≤ p) :
     p.height ≤ ringKrullDim (R ⧸ I) + I.spanFinrank := by
   trans (p.map (Ideal.Quotient.mk I)).height + I.spanFinrank
@@ -371,7 +367,6 @@ lemma Ideal.height_le_height_add_encard_of_subset (s : Set R) {p : Ideal R} [p.I
   · gcongr
     exact Submodule.spanFinrank_span_le_encard _
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Ideal.height_le_ringKrullDim_quotient_add_encard {p : Ideal R} [p.IsPrime]
     (s : Set R) (hs : s ⊆ p) : p.height ≤ ringKrullDim (R ⧸ span s) + s.encard := by
   refine le_trans (height_le_ringKrullDim_quotient_add_spanFinrank (I := .span s) ?_) ?_
@@ -388,7 +383,6 @@ lemma Ideal.height_le_ringKrullDim_quotient_add_one {r : R} {p : Ideal R} [p.IsP
   convert Ideal.height_le_ringKrullDim_quotient_add_encard {r} (by simpa)
   simp
 
-set_option backward.isDefEq.respectTransparency false in
 lemma ringKrullDim_le_ringKrullDim_quotient_add_encard (s : Set R) (hs : s ⊆ Ring.jacobson R) :
     ringKrullDim R ≤ ringKrullDim (R ⧸ Ideal.span s) + s.encard := by
   refine le_trans (ringKrullDim_le_ringKrullDim_quotient_add_spanFinrank (Ideal.span s) ?_) ?_
@@ -405,7 +399,6 @@ section Algebra
 
 variable {S : Type*} [CommRing S] [Algebra R S]
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 If `P` lies over `p`, the height of `P` is bounded by the height of `p` plus
 the height of the image of `P` in `S ⧸ p S`.
@@ -444,7 +437,6 @@ lemma Ideal.height_le_height_add_of_liesOver [IsNoetherianRing S] (p : Ideal R)
   convert hP'
   simp [Ideal.map_span, ← himgo]
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 If `S` satisfies going-down as an `R`-algebra and `P` lies over `p`, the height of `P` is equal
 to the height of `p` plus the height of the image of `P` in `S ⧸ p S`
diff --git a/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean b/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean
index 815819eb779744..b972242031ac5c 100644
--- a/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean
+++ b/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean
@@ -269,7 +269,6 @@ theorem span_singleton_absNorm {I : Ideal S} (hI : (Ideal.absNorm I).Prime) :
 
 variable [Module.Finite ℤ S]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `e : S ≃ I` be an additive isomorphism (therefore a `ℤ`-linear equiv).
 Then an alternative way to compute the norm of `I` is given by taking the determinant of `e`.
 See `natAbs_det_basis_change` for a more familiar formulation of this result. -/
@@ -293,7 +292,6 @@ theorem natAbs_det_basis_change {ι : Type*} [Fintype ι] [DecidableEq ι] (b :
     (I : Ideal S) (bI : Basis ι ℤ I) : (b.det ((↑) ∘ bI)).natAbs = Ideal.absNorm I :=
   Submodule.natAbs_det_basis_change b (I.restrictScalars ℤ) bI
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem absNorm_span_singleton (r : S) :
     absNorm (span ({r} : Set S)) = (Algebra.norm ℤ r).natAbs := by
diff --git a/Mathlib/RingTheory/Ideal/Quotient/ChineseRemainder.lean b/Mathlib/RingTheory/Ideal/Quotient/ChineseRemainder.lean
index 5d67cfbaef54fa..d0647fb45356b0 100644
--- a/Mathlib/RingTheory/Ideal/Quotient/ChineseRemainder.lean
+++ b/Mathlib/RingTheory/Ideal/Quotient/ChineseRemainder.lean
@@ -42,7 +42,6 @@ theorem pi_tensorProductMk_quotient_surjective :
   classical rw [pi_mkQ_rTensor] at this
   simpa using this
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A form of Chinese remainder theorem for modules, part II: if ideals `Iᵢ` of `R` are pairwise
 coprime, then for any `R`-module `M`, the kernel of `M → Πᵢ (R ⧸ Iᵢ) ⊗[R] M` equals `(⋂ᵢ Iᵢ) • M`.
 -/
diff --git a/Mathlib/RingTheory/Ideal/Quotient/Index.lean b/Mathlib/RingTheory/Ideal/Quotient/Index.lean
index 35005712b1b575..ad8ffac0b28bd3 100644
--- a/Mathlib/RingTheory/Ideal/Quotient/Index.lean
+++ b/Mathlib/RingTheory/Ideal/Quotient/Index.lean
@@ -33,7 +33,6 @@ public section
 variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
 variable (I : Ideal R) {N : Submodule R M}
 
-set_option backward.isDefEq.respectTransparency false in
 open TensorProduct in
 /-- Let `N` be a finite index f.g. `R`-submodule, and `I` be a finite index ideal.
 Then `I • N` also has finite index. -/
@@ -56,7 +55,6 @@ lemma Submodule.finite_quotient_smul [Finite (R ⧸ I)] [Finite (M ⧸ N)] (hN :
   have : Finite ((R ⧸ I) ⊗[R] N) := Module.finite_of_finite (R ⧸ I)
   exact Nat.card_pos.ne'
 
-set_option backward.isDefEq.respectTransparency false in
 -- We have `hs` and `N` instead of using `span R s` in the goal to make it easier to use.
 -- Usually we would like to bound the index of some abstract `I • N`, and we may construct `s` while
 -- applying this lemma instead of having to provide it beforehand.
diff --git a/Mathlib/RingTheory/Ideal/Quotient/Operations.lean b/Mathlib/RingTheory/Ideal/Quotient/Operations.lean
index ccc148c9c90886..6e50892857b4e1 100644
--- a/Mathlib/RingTheory/Ideal/Quotient/Operations.lean
+++ b/Mathlib/RingTheory/Ideal/Quotient/Operations.lean
@@ -1174,7 +1174,6 @@ def powQuotPowSuccLinearEquivMapMkPowSuccPow :
     refine ⟨Submodule.Quotient.mk ⟨y, hy⟩, ?_⟩
     simp
 
-set_option backward.isDefEq.respectTransparency false in
 /-- `I ^ n ⧸ I ^ (n + 1)` can be viewed as a quotient module and as ideal of `R ⧸ I ^ (n + 1)`.
 This definition gives the equivalence between the two, instead of the `R`-linear equivalence,
 to bypass typeclass synthesis issues on complex `Module` goals. -/
diff --git a/Mathlib/RingTheory/Jacobson/Radical.lean b/Mathlib/RingTheory/Jacobson/Radical.lean
index a020ba3cbd1b3e..872b79eb91f821 100644
--- a/Mathlib/RingTheory/Jacobson/Radical.lean
+++ b/Mathlib/RingTheory/Jacobson/Radical.lean
@@ -193,7 +193,6 @@ theorem jacobson_smul_lt_top [Nontrivial M] [IsCoatomic (Submodule R M)] (N : Su
   ((smul_mono_right _ le_top).trans <| Ring.jacobson_smul_top_le R M).trans_lt
     (Module.jacobson_lt_top R M)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FG.jacobson_smul_lt {N : Submodule R M} (ne_bot : N ≠ ⊥) (fg : N.FG) :
     Ring.jacobson R • N < N := by
   rw [← Module.Finite.iff_fg] at fg
diff --git a/Mathlib/RingTheory/Jacobson/Semiprimary.lean b/Mathlib/RingTheory/Jacobson/Semiprimary.lean
index 90cc51212e5b51..bd19d851e082c6 100644
--- a/Mathlib/RingTheory/Jacobson/Semiprimary.lean
+++ b/Mathlib/RingTheory/Jacobson/Semiprimary.lean
@@ -26,7 +26,6 @@ variable {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂]
 theorem IsSimpleModule.jacobson_eq_bot [IsSimpleModule R M] : Module.jacobson R M = ⊥ :=
   le_bot_iff.mp <| sInf_le isCoatom_bot
 
-set_option backward.isDefEq.respectTransparency false in
 theorem IsSemisimpleModule.jacobson_eq_bot [IsSemisimpleModule R M] :
     Module.jacobson R M = ⊥ :=
   have ⟨s, e, simple⟩ := isSemisimpleModule_iff_exists_linearEquiv_dfinsupp.mp ‹_›
diff --git a/Mathlib/RingTheory/Kaehler/Basic.lean b/Mathlib/RingTheory/Kaehler/Basic.lean
index 7c606a1c272bbe..2838a8a3aab55a 100644
--- a/Mathlib/RingTheory/Kaehler/Basic.lean
+++ b/Mathlib/RingTheory/Kaehler/Basic.lean
@@ -157,20 +157,17 @@ deriving AddCommGroup, Module (S ⊗[R] S), IsScalarTower S (S ⊗[R] S), Inhabi
 @[inherit_doc KaehlerDifferential]
 notation "Ω[" S "⁄" R "]" => KaehlerDifferential R S
 
-set_option backward.isDefEq.respectTransparency false in
 instance KaehlerDifferential.module' {R' : Type*} [CommRing R'] [Algebra R' S]
     [SMulCommClass R R' S] :
     Module R' Ω[S⁄R] :=
   Submodule.Quotient.module' _
 
-set_option backward.isDefEq.respectTransparency false in
 instance KaehlerDifferential.isScalarTower_of_tower {R₁ R₂ : Type*} [CommRing R₁] [CommRing R₂]
     [Algebra R₁ S] [Algebra R₂ S] [SMul R₁ R₂]
     [SMulCommClass R R₁ S] [SMulCommClass R R₂ S] [IsScalarTower R₁ R₂ S] :
     IsScalarTower R₁ R₂ Ω[S⁄R] :=
   Submodule.Quotient.isScalarTower _ _
 
-set_option backward.isDefEq.respectTransparency false in
 instance KaehlerDifferential.isScalarTower' : IsScalarTower R (S ⊗[R] S) Ω[S⁄R] :=
   Submodule.Quotient.isScalarTower _ _
 
@@ -250,7 +247,6 @@ lemma KaehlerDifferential.subsingleton_of_surjective (h : Function.Surjective (a
 
 variable {R S}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The linear map from `Ω[S⁄R]`, associated with a derivation. -/
 def Derivation.liftKaehlerDifferential (D : Derivation R S M) : Ω[S⁄R] →ₗ[S] M := by
   refine LinearMap.comp ((((KaehlerDifferential.ideal R S) •
@@ -775,7 +771,6 @@ def KaehlerDifferential.kerToTensor :
     algebraMap_eq_smul_one, RingHom.mem_ker.mp x.prop, TensorProduct.zero_tmul, add_zero,
     RingHom.id_apply]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The map `I/I² → B ⊗[A] Ω[A⁄R]` where `I = ker(A → B)`. -/
 noncomputable
 def KaehlerDifferential.kerCotangentToTensor :
diff --git a/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean b/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean
index d4606de8045e74..eba090b661274e 100644
--- a/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean
+++ b/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean
@@ -33,7 +33,6 @@ lemma ringKrullDim_quotient (I : Ideal R) :
     ringKrullDim (R ⧸ I) = Order.krullDim (PrimeSpectrum.zeroLocus (R := R) I) := by
   rw [ringKrullDim, Order.krullDim_eq_of_orderIso I.primeSpectrumQuotientOrderIsoZeroLocus]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma ringKrullDim_quotient_succ_le_of_nonZeroDivisor
     {r : R} (hr : r ∈ R⁰) :
     ringKrullDim (R ⧸ Ideal.span {r}) + 1 ≤ ringKrullDim R := by
diff --git a/Mathlib/RingTheory/KrullDimension/PID.lean b/Mathlib/RingTheory/KrullDimension/PID.lean
index 3dcb5aa01665bb..f27f06f6ac2f49 100644
--- a/Mathlib/RingTheory/KrullDimension/PID.lean
+++ b/Mathlib/RingTheory/KrullDimension/PID.lean
@@ -43,7 +43,6 @@ theorem IsPrincipalIdealRing.ringKrullDim_eq_one (R : Type*) [CommRing R] [IsDom
     rw [← Nat.cast_zero, ← Ring.krullDimLE_iff] at h''
     exact Ring.KrullDimLE.isField_of_isDomain
 
-set_option backward.isDefEq.respectTransparency false in
 /-- In a PID that is not a field, every maximal ideal has height one. -/
 lemma IsPrincipalIdealRing.height_eq_one_of_isMaximal {R : Type*} [CommRing R] [IsDomain R]
     [IsPrincipalIdealRing R] (m : Ideal R) [m.IsMaximal] (h : ¬ IsField R) :
diff --git a/Mathlib/RingTheory/Length.lean b/Mathlib/RingTheory/Length.lean
index de8d7e875d7712..60b2ac3eaec844 100644
--- a/Mathlib/RingTheory/Length.lean
+++ b/Mathlib/RingTheory/Length.lean
@@ -65,7 +65,6 @@ lemma Module.length_pos_iff : 0 < Module.length R M ↔ Nontrivial M := by
 lemma Module.length_pos [Nontrivial M] : 0 < Module.length R M :=
   Module.length_pos_iff.mpr ‹_›
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Module.length_compositionSeries (s : CompositionSeries (Submodule R M)) (h₁ : s.head = ⊥)
     (h₂ : s.last = ⊤) : s.length = Module.length R M := by
   have H := isFiniteLength_of_exists_compositionSeries ⟨s, h₁, h₂⟩
@@ -108,7 +107,6 @@ lemma Module.length_ne_top [IsArtinian R M] [IsNoetherian R M] : Module.length R
   rw [length_ne_top_iff, isFiniteLength_iff_isNoetherian_isArtinian]
   exact ⟨‹_›, ‹_›⟩
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Module.length_submodule {N : Submodule R M} :
     Module.length R N = Order.height N := by
   apply WithBot.coe_injective
@@ -126,17 +124,14 @@ lemma LinearEquiv.length_eq {N : Type*} [AddCommGroup N] [Module R N] (e : M ≃
   rw [Module.coe_length, Module.coe_length,
     Order.krullDim_eq_of_orderIso (Submodule.orderIsoMapComap e)]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Module.length_bot :
     Module.length R (⊥ : Submodule R M) = 0 :=
   Module.length_eq_zero
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma Module.length_top :
     Module.length R (⊤ : Submodule R M) = Module.length R M := by
   rw [Module.length_submodule, Module.length_eq_height]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Submodule.height_lt_top [IsArtinian R M] [IsNoetherian R M] (N : Submodule R M) :
     Order.height N < ⊤ := by
   simpa only [← Module.length_submodule] using Module.length_ne_top.lt_top
@@ -145,7 +140,6 @@ lemma Submodule.height_strictMono [IsArtinian R M] [IsNoetherian R M] :
     StrictMono (Order.height : Submodule R M → ℕ∞) :=
   fun N _ h ↦ Order.height_strictMono h N.height_lt_top
 
-set_option backward.isDefEq.respectTransparency false in
 lemma Submodule.length_lt [IsArtinian R M] [IsNoetherian R M] {N : Submodule R M} (h : N ≠ ⊤) :
     Module.length R N < Module.length R M := by
   simpa [← Module.length_top (M := M), Module.length_submodule] using height_strictMono h.lt_top
@@ -189,7 +183,6 @@ lemma Module.length_le_of_injective : Module.length R N ≤ Module.length R M :=
     (Submodule.mkQ_surjective _) (LinearMap.exact_map_mkQ_range f)]
   exact le_self_add
 
-set_option backward.isDefEq.respectTransparency false in
 include hg in
 lemma Module.length_le_of_surjective : Module.length R P ≤ Module.length R M := by
   rw [Module.length_eq_add_of_exact (LinearMap.ker g).subtype g (Submodule.subtype_injective _) hg
@@ -235,7 +228,6 @@ lemma Module.length_finsupp {ι : Type*} :
   rw [this]
   exact ENat.self_le_mul_right _ length_pos.ne'
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma Module.length_pi {ι : Type*} :
     Module.length R (ι → M) = ENat.card ι * Module.length R M := by
diff --git a/Mathlib/RingTheory/LinearDisjoint.lean b/Mathlib/RingTheory/LinearDisjoint.lean
index 7fcae2dc22e407..34fe3f89371b5f 100644
--- a/Mathlib/RingTheory/LinearDisjoint.lean
+++ b/Mathlib/RingTheory/LinearDisjoint.lean
@@ -884,7 +884,6 @@ variable [Field R] [Ring S] [Algebra R S]
 
 variable {A B : Subalgebra R S}
 
-set_option backward.isDefEq.respectTransparency false in
 theorem inf_eq_bot_of_commute (H : A.LinearDisjoint B)
     (hc : ∀ (a b : ↥(A ⊓ B)), Commute a.1 b.1) : A ⊓ B = ⊥ :=
   eq_bot_of_rank_le_one (Submodule.LinearDisjoint.rank_inf_le_one_of_commute_of_flat_left H hc)
diff --git a/Mathlib/RingTheory/LocalRing/Module.lean b/Mathlib/RingTheory/LocalRing/Module.lean
index 323dbfad1e8dc0..bda4590c370ae6 100644
--- a/Mathlib/RingTheory/LocalRing/Module.lean
+++ b/Mathlib/RingTheory/LocalRing/Module.lean
@@ -312,7 +312,6 @@ theorem free_of_flat_of_isLocalRing [Module.Finite R P] [Flat R P] : Free R P :=
   .of_basis <| .mk (IsLocalRing.linearIndependent_of_flat _ (eq ▸ w.linearIndependent)) <| by
     exact (span_eq_top_of_tmul_eq_basis _ w <| congr_fun eq).ge
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 If `M → N → P → 0` is a presentation of `P` over a local ring `(R, 𝔪, k)` with
 `M` finite and `N` finite free, then injectivity of `k ⊗ M → k ⊗ N` implies that `P` is free.
@@ -332,7 +331,6 @@ theorem free_of_lTensor_residueField_injective (hg : Surjective g) (h : Exact f
 
 end Module
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Given a linear map `l : M → N` over a local ring `(R, 𝔪, k)`
 with `M` finite and `N` finite free,
diff --git a/Mathlib/RingTheory/Localization/Integral.lean b/Mathlib/RingTheory/Localization/Integral.lean
index 3ec7c39074886e..19ddee512c6e67 100644
--- a/Mathlib/RingTheory/Localization/Integral.lean
+++ b/Mathlib/RingTheory/Localization/Integral.lean
@@ -439,7 +439,6 @@ section
 variable {Rf Sf : Type*} [CommRing Rf] [CommRing Sf] [Algebra R Rf] [Algebra S Sf]
     [Algebra Rf Sf] [Algebra R Sf] [IsScalarTower R S Sf] [IsScalarTower R Rf Sf]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Taking integral closure commutes with localizations. -/
 -- We take in an arbitrary `Algebra (integralClosure R S) (integralClosure Rf Sf)` instance
 -- so that it applies more easily.
diff --git a/Mathlib/RingTheory/Morita/Matrix.lean b/Mathlib/RingTheory/Morita/Matrix.lean
index ec8f6d2b40df9e..fddb63710f5eb4 100644
--- a/Mathlib/RingTheory/Morita/Matrix.lean
+++ b/Mathlib/RingTheory/Morita/Matrix.lean
@@ -93,7 +93,6 @@ lemma MatrixModCat.isScalarTower_toModuleCat (M : ModuleCat (Matrix ι ι R)) :
   { smul_assoc r m x := show _ = Matrix.scalar ι r • (m • x) by
       rw [← SemigroupAction.mul_smul, Matrix.scalar_apply, Matrix.smul_eq_diagonal_mul] }
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The functor from the category of modules over `Mₙ(R)` to the category of modules over `R`
   induced by sending `M` to the image of `Eᵢᵢ • ·` where `Eᵢᵢ` is the elementary matrix. -/
 @[simps]
@@ -141,7 +140,6 @@ def MatrixModCat.unitIso (i : ι) :
     ext
     simp [fromModuleCatToModuleCatLinearEquivtoModuleCatObj]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The linear equiv induced by the equality `toMatrixModCat (toModuleCat M) = Mⁿ` -/
 def toModuleCatFromModuleCatLinearEquiv (M : ModuleCat (Matrix ι ι R)) (j : ι) :
     letI := Module.compHom M (Matrix.scalar (α := R) ι)
diff --git a/Mathlib/RingTheory/Multiplicity.lean b/Mathlib/RingTheory/Multiplicity.lean
index b017ee4630e5e1..6caa0b80562d5d 100644
--- a/Mathlib/RingTheory/Multiplicity.lean
+++ b/Mathlib/RingTheory/Multiplicity.lean
@@ -85,7 +85,6 @@ theorem FiniteMultiplicity.emultiplicity_eq_multiplicity (h : FiniteMultiplicity
   · simp [h] at hm
   rw [multiplicity_eq_of_emultiplicity_eq_some hm]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.emultiplicity_eq_iff_multiplicity_eq {n : ℕ}
     (h : FiniteMultiplicity a b) : emultiplicity a b = n ↔ multiplicity a b = n := by
   simp [h.emultiplicity_eq_multiplicity]
@@ -120,12 +119,10 @@ theorem multiplicity_eq_of_emultiplicity_eq {c d : β}
   unfold multiplicity
   rw [h]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem multiplicity_le_of_emultiplicity_le {n : ℕ} (h : emultiplicity a b ≤ n) :
     multiplicity a b ≤ n := by
   exact_mod_cast multiplicity_le_emultiplicity.trans h
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.emultiplicity_le_of_multiplicity_le (hfin : FiniteMultiplicity a b)
     {n : ℕ} (h : multiplicity a b ≤ n) : emultiplicity a b ≤ n := by
   rw [emultiplicity_eq_multiplicity hfin]
@@ -135,18 +132,15 @@ theorem le_emultiplicity_of_le_multiplicity {n : ℕ} (h : n ≤ multiplicity a
     n ≤ emultiplicity a b := by
   exact_mod_cast (WithTop.coe_mono h).trans multiplicity_le_emultiplicity
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.le_multiplicity_of_le_emultiplicity (hfin : FiniteMultiplicity a b)
     {n : ℕ} (h : n ≤ emultiplicity a b) : n ≤ multiplicity a b := by
   rw [emultiplicity_eq_multiplicity hfin] at h
   assumption_mod_cast
 
-set_option backward.isDefEq.respectTransparency false in
 theorem multiplicity_lt_of_emultiplicity_lt {n : ℕ} (h : emultiplicity a b < n) :
     multiplicity a b < n := by
   exact_mod_cast multiplicity_le_emultiplicity.trans_lt h
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.emultiplicity_lt_of_multiplicity_lt (hfin : FiniteMultiplicity a b)
     {n : ℕ} (h : multiplicity a b < n) : emultiplicity a b < n := by
   rw [emultiplicity_eq_multiplicity hfin]
@@ -156,7 +150,6 @@ theorem lt_emultiplicity_of_lt_multiplicity {n : ℕ} (h : n < multiplicity a b)
     n < emultiplicity a b := by
   exact_mod_cast (WithTop.coe_strictMono h).trans_le multiplicity_le_emultiplicity
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.lt_multiplicity_of_lt_emultiplicity (hfin : FiniteMultiplicity a b)
     {n : ℕ} (h : n < emultiplicity a b) : n < multiplicity a b := by
   rw [emultiplicity_eq_multiplicity hfin] at h
@@ -199,7 +192,6 @@ theorem FiniteMultiplicity.mul_left {c : α} :
     FiniteMultiplicity a (b * c) → FiniteMultiplicity a b := fun ⟨n, hn⟩ =>
   ⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem pow_dvd_of_le_emultiplicity {k : ℕ} (hk : k ≤ emultiplicity a b) :
     a ^ k ∣ b := by classical
   cases k
@@ -217,7 +209,6 @@ theorem pow_dvd_of_le_multiplicity {k : ℕ} (hk : k ≤ multiplicity a b) :
 theorem pow_multiplicity_dvd (a b : α) : a ^ (multiplicity a b) ∣ b :=
   pow_dvd_of_le_multiplicity le_rfl
 
-set_option backward.isDefEq.respectTransparency false in
 theorem not_pow_dvd_of_emultiplicity_lt {m : ℕ} (hm : emultiplicity a b < m) :
     ¬a ^ m ∣ b := fun nh => by
   unfold emultiplicity at hm
@@ -227,7 +218,6 @@ theorem not_pow_dvd_of_emultiplicity_lt {m : ℕ} (hm : emultiplicity a b < m) :
     exact hn2 ((pow_dvd_pow _ hn1).trans nh)
   · simp at hm
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt (hf : FiniteMultiplicity a b) {m : ℕ}
     (hm : multiplicity a b < m) : ¬a ^ m ∣ b := by
   apply not_pow_dvd_of_emultiplicity_lt
@@ -242,7 +232,6 @@ theorem multiplicity_pos_of_dvd (hdiv : a ∣ b) : 0 < multiplicity a b := by
 theorem emultiplicity_pos_of_dvd (hdiv : a ∣ b) : 0 < emultiplicity a b :=
   lt_emultiplicity_of_lt_multiplicity (multiplicity_pos_of_dvd hdiv)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_eq_of_dvd_of_not_dvd {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
     emultiplicity a b = k := by classical
   have : FiniteMultiplicity a b := ⟨k, hsucc⟩
@@ -266,7 +255,6 @@ theorem pow_dvd_iff_le_emultiplicity {k : ℕ} :
     a ^ k ∣ b ↔ k ≤ emultiplicity a b :=
   ⟨le_emultiplicity_of_pow_dvd, pow_dvd_of_le_emultiplicity⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.pow_dvd_iff_le_multiplicity (hf : FiniteMultiplicity a b) {k : ℕ} :
     a ^ k ∣ b ↔ k ≤ multiplicity a b := by
   exact_mod_cast hf.emultiplicity_eq_multiplicity ▸ pow_dvd_iff_le_emultiplicity
@@ -277,7 +265,6 @@ theorem emultiplicity_lt_iff_not_dvd {k : ℕ} :
 theorem FiniteMultiplicity.multiplicity_lt_iff_not_dvd {k : ℕ} (hf : FiniteMultiplicity a b) :
     multiplicity a b < k ↔ ¬a ^ k ∣ b := by rw [hf.pow_dvd_iff_le_multiplicity, not_le]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_eq_coe {n : ℕ} :
     emultiplicity a b = n ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
   constructor
@@ -292,7 +279,6 @@ theorem emultiplicity_eq_coe {n : ℕ} :
   · rw [and_imp]
     apply emultiplicity_eq_of_dvd_of_not_dvd
 
-set_option backward.isDefEq.respectTransparency false in
 theorem FiniteMultiplicity.multiplicity_eq_iff (hf : FiniteMultiplicity a b) {n : ℕ} :
     multiplicity a b = n ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
   simp [← emultiplicity_eq_coe, hf.emultiplicity_eq_multiplicity]
@@ -346,7 +332,6 @@ theorem FiniteMultiplicity.exists_eq_pow_mul_and_not_dvd (hfin : FiniteMultiplic
   have h₁ : a ^ (multiplicity a b + 1) ∣ b := ⟨k, hc⟩
   exact (hfin.multiplicity_eq_iff.1 (by simp)).2 h₁
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_le_emultiplicity_iff {c d : β} :
     emultiplicity a b ≤ emultiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d := by classical
   constructor
@@ -679,7 +664,6 @@ theorem multiplicity_self {a : α} : multiplicity a a = 1 := by
     · simpa using ha.ne_zero
   · simp [ha]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem FiniteMultiplicity.emultiplicity_self {a : α} (hfin : FiniteMultiplicity a a) :
     emultiplicity a a = 1 := by
@@ -719,14 +703,12 @@ theorem Finset.emultiplicity_prod {β : Type*} {p : α} (hp : Prime p) (s : Fins
     exact emultiplicity_of_one_right hp.not_unit
   | insert a s has ih => simpa [has, ← ih] using emultiplicity_mul hp
 
-set_option backward.isDefEq.respectTransparency false in
 theorem emultiplicity_pow {p a : α} (hp : Prime p) {k : ℕ} :
     emultiplicity p (a ^ k) = k * emultiplicity p a := by
   induction k with
   | zero => simp [emultiplicity_of_one_right hp.not_unit]
   | succ k hk => simp [pow_succ, emultiplicity_mul hp, hk, add_mul]
 
-set_option backward.isDefEq.respectTransparency false in
 protected theorem FiniteMultiplicity.multiplicity_pow {p a : α} (hp : Prime p)
     (ha : FiniteMultiplicity p a) {k : ℕ} : multiplicity p (a ^ k) = k * multiplicity p a := by
   exact_mod_cast (ha.pow hp).emultiplicity_eq_multiplicity ▸
diff --git a/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean b/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean
index d679f71680065a..488c54d425d4ef 100644
--- a/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean
+++ b/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean
@@ -147,7 +147,6 @@ instance : NoZeroDivisors (MvPowerSeries σ R) where
       exact IsWellOrder.toIsWellFounded
     simpa only [← lexOrder_eq_top_iff_eq_zero, lexOrder_mul, WithTop.add_eq_top] using h
 
-set_option backward.isDefEq.respectTransparency false in
 theorem weightedOrder_mul (w : σ → ℕ) (f g : MvPowerSeries σ R) :
     (f * g).weightedOrder w = f.weightedOrder w + g.weightedOrder w := by
   apply le_antisymm _ (le_weightedOrder_mul w)
diff --git a/Mathlib/RingTheory/MvPowerSeries/Order.lean b/Mathlib/RingTheory/MvPowerSeries/Order.lean
index 5b76ddb8267c27..a85341c6cc0e36 100644
--- a/Mathlib/RingTheory/MvPowerSeries/Order.lean
+++ b/Mathlib/RingTheory/MvPowerSeries/Order.lean
@@ -153,7 +153,6 @@ theorem weightedOrder_eq_top_iff :
     f.weightedOrder w = ⊤ ↔ f = 0 := by
   rw [← not_iff_not, ← ne_eq, ← ne_eq, ne_zero_iff_weightedOrder_finite w, coe_toNat_eq_self]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the order of a formal power series `f` is finite,
 then some coefficient of weight equal to the order of `f` is nonzero. -/
 theorem exists_coeff_ne_zero_and_weightedOrder
@@ -164,7 +163,6 @@ theorem exists_coeff_ne_zero_and_weightedOrder
   generalize_proofs h1
   exact Nat.find_spec h1
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the `d`th coefficient of a formal power series is nonzero,
 then the weighted order of the power series is less than or equal to `weight d w`. -/
 theorem weightedOrder_le {d : σ →₀ ℕ} (h : coeff d f ≠ 0) :
@@ -180,7 +178,6 @@ theorem coeff_eq_zero_of_lt_weightedOrder {d : σ →₀ ℕ} (h : (weight w d)
     coeff d f = 0 := by
   contrapose! h; exact weightedOrder_le w h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is at least `n` if
 the `d`th coefficient is `0` for all `d` such that `weight w d < n`. -/
 theorem nat_le_weightedOrder {n : ℕ} (h : ∀ d, weight w d < n → coeff d f = 0) :
@@ -192,7 +189,6 @@ theorem nat_le_weightedOrder {n : ℕ} (h : ∀ d, weight w d < n → coeff d f
   rw [← hd, Nat.cast_lt] at H
   exact hfd (h d H)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is at least `n` if
 the `d`th coefficient is `0` for all `d` such that `weight w d < n`. -/
 theorem le_weightedOrder {n : ℕ∞} (h : ∀ d : σ →₀ ℕ, weight w d < n → coeff d f = 0) :
@@ -203,7 +199,6 @@ theorem le_weightedOrder {n : ℕ∞} (h : ∀ d : σ →₀ ℕ, weight w d < n
   · apply nat_le_weightedOrder;
     simpa only [ENat.some_eq_coe, Nat.cast_lt] using h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is exactly `n` if and only if some coefficient of weight `n`
 is nonzero, and the `d`th coefficient is `0` for all `d` such that `weight w d < n`. -/
 theorem weightedOrder_eq_nat {n : ℕ} :
@@ -250,7 +245,6 @@ theorem min_weightedOrder_le_add :
     [coeff_eq_zero_of_lt_weightedOrder w, lt_min_iff, map_add, add_zero,
       imp_true_iff]
 
-set_option backward.isDefEq.respectTransparency false in
 private theorem weightedOrder_add_of_weightedOrder_lt.aux
     (H : f.weightedOrder w < g.weightedOrder w) :
     (f + g).weightedOrder w = f.weightedOrder w := by
@@ -280,7 +274,6 @@ theorem weightedOrder_add_of_weightedOrder_ne (h : f.weightedOrder w ≠ g.weigh
   simp only [le_inf_iff, weightedOrder_add_of_weightedOrder_lt.aux w H₁]
   exact ⟨le_rfl, le_of_lt H₁⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The weighted_order of the product of two formal power series
 is at least the sum of their orders. -/
 theorem le_weightedOrder_mul :
@@ -482,7 +475,6 @@ theorem le_order_prod {R : Type*} [CommSemiring R] {ι : Type*}
     (f : ι → MvPowerSeries σ R) (s : Finset ι) : ∑ i ∈ s, (f i).order ≤ (∏ i ∈ s, f i).order :=
   le_weightedOrder_prod _ _ _
 
-set_option backward.isDefEq.respectTransparency false in
 theorem one_le_order_iff_constCoeff_eq_zero :
     1 ≤ f.order ↔ f.constantCoeff = 0 := by
   constructor
@@ -622,7 +614,6 @@ theorem weightedHomogeneousComponent_of_lt_weightedOrder_eq_zero
     exact hf
   · rw [map_zero]
 
-set_option backward.isDefEq.respectTransparency false in
 variable {w} in
 theorem weightedHomogeneousComponent_of_weightedOrder
     {f : MvPowerSeries σ R} {p : ℕ} (hf : p = f.weightedOrder w) :
diff --git a/Mathlib/RingTheory/MvPowerSeries/Substitution.lean b/Mathlib/RingTheory/MvPowerSeries/Substitution.lean
index 43d442108f5206..8847a2e11e21d7 100644
--- a/Mathlib/RingTheory/MvPowerSeries/Substitution.lean
+++ b/Mathlib/RingTheory/MvPowerSeries/Substitution.lean
@@ -210,7 +210,6 @@ theorem coe_substAlgHom (ha : HasSubst a) :
   letI : UniformSpace S := ⊥
   rw [substAlgHom_eq_aeval, coe_aeval ha.hasEval, subst_eq_eval₂]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem subst_self : subst (MvPowerSeries.X : σ → MvPowerSeries σ R) = id := by
   rw [← coe_substAlgHom HasSubst.X]
   letI : UniformSpace R := ⊥
diff --git a/Mathlib/RingTheory/Noetherian/Basic.lean b/Mathlib/RingTheory/Noetherian/Basic.lean
index 6582dec1c1cd19..e80ace46b31d83 100644
--- a/Mathlib/RingTheory/Noetherian/Basic.lean
+++ b/Mathlib/RingTheory/Noetherian/Basic.lean
@@ -163,7 +163,6 @@ instance isNoetherian_prod [IsNoetherian R M] [IsNoetherian R N] : IsNoetherian
         fun x ⟨_, hx2⟩ => ⟨x.1, Prod.ext rfl <| Eq.symm <| LinearMap.mem_ker.1 hx2⟩
       Submodule.map_comap_eq_self this ▸ (noetherian _).map _⟩
 
-set_option backward.isDefEq.respectTransparency false in
 instance isNoetherian_sup (M₁ M₂ : Submodule R N) [IsNoetherian R M₁] [IsNoetherian R M₂] :
     IsNoetherian R ↥(M₁ ⊔ M₂) := by
   have := isNoetherian_range (M₁.subtype.coprod M₂.subtype)
@@ -192,7 +191,6 @@ instance isNoetherian_iSup :
   · intros; rw [iSup_of_empty]; infer_instance
   · intro _ _ ih _ _; rw [iSup_option]; infer_instance
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the first and final modules in an exact sequence are Noetherian,
   then the middle module is also Noetherian. -/
 theorem isNoetherian_of_range_eq_ker {P : Type*} [AddCommGroup P] [Module R P] [IsNoetherian R M]
@@ -209,7 +207,6 @@ theorem isNoetherian_of_range_eq_ker {P : Type*} [AddCommGroup P] [Module R P] [
       (by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
       (by simp [Submodule.comap_map_eq, h])
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isNoetherian_iff_submodule_quotient (S : Submodule R N) :
     IsNoetherian R N ↔ IsNoetherian R S ∧ IsNoetherian R (N ⧸ S) := by
   refine ⟨fun _ ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
@@ -326,7 +323,6 @@ instance isNoetherian_of_isNoetherianRing_of_finite (R M : Type*)
   have ⟨_, _, h⟩ := Module.Finite.exists_fin' R M
   isNoetherian_of_surjective _ (LinearMap.range_eq_top.mpr h)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module R M]
     (N : Submodule R M) [I : IsNoetherianRing R] (hN : N.FG) : IsNoetherian R N :=
   haveI : Module.Finite R N := .of_fg hN; inferInstance
@@ -377,7 +373,6 @@ variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M]
 theorem FG.of_le_of_isNoetherian {S T : Submodule R M} [IsNoetherian R T] (hST : S ≤ T) : S.FG :=
   isNoetherian_submodule.mp inferInstance _ hST
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A submodule contained in an FG submodule is FG over noetherian rings. -/
 lemma FG.of_le [IsNoetherianRing R] {S T : Submodule R M} (hT : T.FG) (hST : S ≤ T) : S.FG := by
   rw [← Module.Finite.iff_fg] at hT
diff --git a/Mathlib/RingTheory/Noetherian/Orzech.lean b/Mathlib/RingTheory/Noetherian/Orzech.lean
index 2e308120ddd72e..ed2d8f1243cb19 100644
--- a/Mathlib/RingTheory/Noetherian/Orzech.lean
+++ b/Mathlib/RingTheory/Noetherian/Orzech.lean
@@ -49,7 +49,6 @@ theorem IsNoetherian.injective_of_surjective_of_injective (i f : N →ₗ[R] M)
   exact LinearMap.ker_eq_bot.1 <| bot_unique <|
     f.ker_le_of_iterateMapComap_eq_succ i ⊥ n (H _ (Nat.le_succ _)) hf hi
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Orzech's theorem** for Noetherian modules: if `R` is a ring (not necessarily commutative),
 `M` is a Noetherian `R`-module, `N` is a submodule, `f : N →ₗ[R] M` is surjective, then `f` is also
 injective. -/
diff --git a/Mathlib/RingTheory/Polynomial/DegreeLT.lean b/Mathlib/RingTheory/Polynomial/DegreeLT.lean
index d7ac8ab9cd5896..eb85f8480eec66 100644
--- a/Mathlib/RingTheory/Polynomial/DegreeLT.lean
+++ b/Mathlib/RingTheory/Polynomial/DegreeLT.lean
@@ -174,7 +174,6 @@ noncomputable def taylorLinearEquiv (r : R) (n : ℕ) : R[X]_n ≃ₗ[R] R[X]_n
     (taylorLinearEquiv r n).symm = taylorLinearEquiv (-r) n :=
   LinearEquiv.ext <| fun _ ↦ rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] theorem det_taylorLinearEquiv_toLinearMap :
     (taylorLinearEquiv r n).toLinearMap.det = 1 := by
   nontriviality R
@@ -189,7 +188,6 @@ set_option backward.isDefEq.respectTransparency false in
     change (taylor r (degreeLT.basis R n j)).coeff i = 0
     rw [degreeLT.basis_val, coeff_eq_zero_of_degree_lt (by simpa [-taylor_X_pow, -taylor_pow])]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] theorem det_taylorLinearEquiv :
     (taylorLinearEquiv r n).det = 1 :=
   Units.ext <| by rw [LinearEquiv.coe_det, det_taylorLinearEquiv_toLinearMap, Units.val_one]
diff --git a/Mathlib/RingTheory/PowerSeries/Order.lean b/Mathlib/RingTheory/PowerSeries/Order.lean
index 5bc25eda91e9ab..f279617fef3484 100644
--- a/Mathlib/RingTheory/PowerSeries/Order.lean
+++ b/Mathlib/RingTheory/PowerSeries/Order.lean
@@ -80,7 +80,6 @@ theorem coeff_order (h : φ ≠ 0) : coeff φ.order.toNat φ ≠ 0 := by
   generalize_proofs h
   exact Nat.find_spec h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If the `n`th coefficient of a formal power series is nonzero,
 then the order of the power series is less than or equal to `n`. -/
 theorem order_le (n : ℕ) (h : coeff n φ ≠ 0) : order φ ≤ n := by
@@ -101,7 +100,6 @@ theorem coeff_of_lt_order_toNat (n : ℕ) (h : n < φ.order.toNat) : coeff n φ
   · refine coeff_of_lt_order _ ?_
     rwa [← coe_toNat_order h', ENat.coe_lt_coe]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is at least `n` if
 the `i`th coefficient is `0` for all `i < n`. -/
 theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff i φ = 0) : ↑n ≤ order φ := by
@@ -111,7 +109,6 @@ theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff i φ = 0) :
   · simp
   · simpa [Nat.le_find_iff]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is at least `n` if
 the `i`th coefficient is `0` for all `i < n`. -/
 theorem le_order (φ : R⟦X⟧) (n : ℕ∞) (h : ∀ i : ℕ, ↑i < n → coeff i φ = 0) :
@@ -122,7 +119,6 @@ theorem le_order (φ : R⟦X⟧) (n : ℕ∞) (h : ∀ i : ℕ, ↑i < n → coe
     convert nat_le_order φ n _
     simpa using h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
 and the `i`th coefficient is `0` for all `i < n`. -/
 theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} :
@@ -132,7 +128,6 @@ theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} :
   · simp
   simp [order, dif_neg hφ, Nat.find_eq_iff]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
 and the `i`th coefficient is `0` for all `i < n`. -/
 theorem order_eq {φ : R⟦X⟧} {n : ℕ∞} :
@@ -193,7 +188,6 @@ theorem le_order_smul {a : R} :
     φ.order ≤ (a • φ).order :=
   le_order _ φ.order fun i hi => by simp [coeff_of_lt_order i hi]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of the product of two formal power series
 is at least the sum of their orders. -/
 theorem le_order_mul (φ ψ : R⟦X⟧) : order φ + order ψ ≤ order (φ * ψ) := by
@@ -224,7 +218,6 @@ theorem le_order_prod {R : Type*} [CommSemiring R] {ι : Type*} (φ : ι → R
 
 alias order_mul_ge := le_order_mul
 
-set_option backward.isDefEq.respectTransparency false in
 theorem one_le_order_iff_constCoeff_eq_zero :
     1 ≤ φ.order ↔ φ.constantCoeff = 0 := by
   constructor
@@ -246,7 +239,6 @@ theorem le_order_pow_of_constantCoeff_eq_zero (n : ℕ) (hf : φ.constantCoeff =
   refine .trans ?_ (le_order_pow _ n)
   simpa using le_mul_of_one_le_right' (one_le_order_iff_constCoeff_eq_zero.mpr hf)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The order of the monomial `a*X^n` is infinite if `a = 0` and `n` otherwise. -/
 theorem order_monomial (n : ℕ) (a : R) [Decidable (a = 0)] :
     order (monomial n a) = if a = 0 then (⊤ : ℕ∞) else n := by
@@ -265,7 +257,6 @@ theorem order_monomial_of_ne_zero (n : ℕ) (a : R) (h : a ≠ 0) : order (monom
   classical
   rw [order_monomial, if_neg h]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `n` is strictly smaller than the order of `ψ`, then the `n`th coefficient of its product
 with any other power series is `0`. -/
 theorem coeff_mul_of_lt_order {φ ψ : R⟦X⟧} {n : ℕ} (h : ↑n < ψ.order) :
diff --git a/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean b/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean
index 811a31aed51d79..b09ba772520e46 100644
--- a/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean
+++ b/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean
@@ -259,7 +259,6 @@ theorem coeff_seq_mem (k : ℕ) {i : ℕ} (hi : i ≥ (g.map (Ideal.Quotient.mk
     rw [coeff_mk]
     exact hq (by simp)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem coeff_seq_succ_sub_seq_mem (k i : ℕ) :
     coeff i (H.seq f (k + 1) - H.seq f k) ∈ I ^ k := by
   rw [seq, add_sub_cancel_left]
diff --git a/Mathlib/RingTheory/Regular/IsSMulRegular.lean b/Mathlib/RingTheory/Regular/IsSMulRegular.lean
index d4f0fc2ef96395..f7560ab2f90cfa 100644
--- a/Mathlib/RingTheory/Regular/IsSMulRegular.lean
+++ b/Mathlib/RingTheory/Regular/IsSMulRegular.lean
@@ -74,7 +74,6 @@ variable [Ring R] [AddCommGroup M] [Module R M]
     [AddCommGroup M'] [Module R M'] [AddCommGroup M''] [Module R M'']
     (N : Submodule R M) (r : R)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isSMulRegular_submodule_iff_right_eq_zero_of_smul :
     IsSMulRegular N r ↔ ∀ x ∈ N, r • x = 0 → x = 0 :=
   isSMulRegular_iff_right_eq_zero_of_smul.trans <|
@@ -107,7 +106,6 @@ lemma isSMulRegular_of_range_eq_ker {f : M →ₗ[R] M'} {g : M' →ₗ[R] M''}
   refine (congrArg f (h1.right_eq_zero_of_smul ?_)).trans f.map_zero
   exact hf <| (f.map_smul r y).trans <| hx.trans f.map_zero.symm
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isSMulRegular_of_isSMulRegular_on_submodule_on_quotient
     (h1 : IsSMulRegular N r) (h2 : IsSMulRegular (M ⧸ N) r) : IsSMulRegular M r :=
   isSMulRegular_of_range_eq_ker N.injective_subtype
diff --git a/Mathlib/RingTheory/SimpleModule/Basic.lean b/Mathlib/RingTheory/SimpleModule/Basic.lean
index 01b9c26568ff90..f8170e3af50a83 100644
--- a/Mathlib/RingTheory/SimpleModule/Basic.lean
+++ b/Mathlib/RingTheory/SimpleModule/Basic.lean
@@ -106,7 +106,6 @@ theorem IsSimpleModule.congr (e : M ≃ₗ[R] N) [IsSimpleModule R N] : IsSimple
 theorem LinearEquiv.isSimpleModule_iff (e : M ≃ₗ[R] N) : IsSimpleModule R M ↔ IsSimpleModule R N :=
   ⟨(·.congr e.symm), (·.congr e)⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem isSimpleModule_iff_isAtom : IsSimpleModule R m ↔ IsAtom m := by
   rw [← Set.isSimpleOrder_Iic_iff_isAtom, isSimpleModule_iff]
   exact m.mapIic.isSimpleOrder_iff
@@ -115,7 +114,6 @@ theorem isSimpleModule_iff_isCoatom : IsSimpleModule R (M ⧸ m) ↔ IsCoatom m
   rw [← Set.isSimpleOrder_Ici_iff_isCoatom, isSimpleModule_iff]
   exact (Submodule.comapMkQRelIso m).isSimpleOrder_iff
 
-set_option backward.isDefEq.respectTransparency false in
 theorem covBy_iff_quot_is_simple {A B : Submodule R M} (hAB : A ≤ B) :
     A ⋖ B ↔ IsSimpleModule R (B ⧸ Submodule.comap B.subtype A) := by
   set f : Submodule R B ≃o Set.Iic B := B.mapIic with hf
@@ -124,7 +122,6 @@ theorem covBy_iff_quot_is_simple {A B : Submodule R M} (hAB : A ≤ B) :
 
 namespace IsSimpleModule
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem isAtom [IsSimpleModule R m] : IsAtom m :=
   isSimpleModule_iff_isAtom.1 ‹_›
@@ -195,7 +192,6 @@ theorem isSimpleModule_self_iff_isUnit :
     obtain ⟨z, hzy : z * y = 1⟩ := h y hy 1
     exact ⟨⟨x, y, left_inv_eq_right_inv hzy hyx ▸ hzy, hyx⟩, rfl⟩
 
-set_option backward.isDefEq.respectTransparency false in
 theorem IsSemisimpleModule.of_sSup_simples_eq_top
     (h : sSup { m : Submodule R M | IsSimpleModule R m } = ⊤) : IsSemisimpleModule R M where
   __ := complementedLattice_of_sSup_atoms_eq_top (by simp_rw [← h, isSimpleModule_iff_isAtom])
@@ -233,7 +229,6 @@ end Module.Finite
 
 namespace IsSemisimpleModule
 
-set_option backward.isDefEq.respectTransparency false in
 theorem eq_bot_or_exists_simple_le (N : Submodule R M) [IsSemisimpleModule R N] :
     N = ⊥ ∨ ∃ m ≤ N, IsSimpleModule R m := by
   rw [← N.subsingleton_iff_eq_bot, ← Submodule.subsingleton_iff R, ← subsingleton_iff_bot_eq_top]
@@ -267,36 +262,30 @@ theorem lifting_property {P} [AddCommGroup P] [Module R P] (f : M →ₗ[R] N)
   simp only [LinearMap.comp_apply, ← eq, LinearEquiv.coe_coe, e.symm_apply_apply]
   simp [e]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem sSup_simples_le (N : Submodule R M) :
     sSup { m : Submodule R M | IsSimpleModule R m ∧ m ≤ N } = N := by
   simpa only [isSimpleModule_iff_isAtom] using sSup_atoms_le_eq _
 
 variable (R M)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_simple_submodule [Nontrivial M] : ∃ m : Submodule R M, IsSimpleModule R m := by
   simpa only [isSimpleModule_iff_isAtom] using IsAtomic.exists_atom _
 
-set_option backward.isDefEq.respectTransparency false in
 theorem sSup_simples_eq_top : sSup { m : Submodule R M | IsSimpleModule R m } = ⊤ := by
   simpa only [isSimpleModule_iff_isAtom] using sSup_atoms_eq_top
 
-set_option backward.isDefEq.respectTransparency false in
 theorem exists_sSupIndep_sSup_simples_eq_top :
     ∃ s : Set (Submodule R M), sSupIndep s ∧ sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R m := by
   have := sSup_simples_eq_top R M
   simp_rw [isSimpleModule_iff_isAtom] at this ⊢
   exact exists_sSupIndep_of_sSup_atoms_eq_top this
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The annihilator of a semisimple module over a commutative ring is a radical ideal. -/
 theorem annihilator_isRadical (R) [CommRing R] [Module R M] [IsSemisimpleModule R M] :
     (Module.annihilator R M).IsRadical := by
   rw [← Submodule.annihilator_top, ← sSup_simples_eq_top, sSup_eq_iSup', Submodule.annihilator_iSup]
   exact Ideal.isRadical_iInf _ fun i ↦ (i.2.annihilator_isMaximal).isPrime.isRadical
 
-set_option backward.isDefEq.respectTransparency false in
 instance submodule {m : Submodule R M} : IsSemisimpleModule R m where
   __ := m.mapIic.complementedLattice_iff.2 IsModularLattice.complementedLattice_Iic
 
@@ -306,11 +295,9 @@ open LinearMap
 theorem congr (e : N ≃ₗ[R] M) : IsSemisimpleModule R N where
   __ := (Submodule.orderIsoMapComap e.symm).complementedLattice
 
-set_option backward.isDefEq.respectTransparency false in
 theorem of_injective (f : N →ₗ[R] M) (hf : Function.Injective f) : IsSemisimpleModule R N :=
   congr (Submodule.topEquiv.symm.trans <| Submodule.equivMapOfInjective f hf _)
 
-set_option backward.isDefEq.respectTransparency false in
 instance quotient : IsSemisimpleModule R (M ⧸ m) :=
   have ⟨_, ⟨e⟩⟩ := exists_submodule_linearEquiv_quotient m
   .congr e.symm
@@ -321,7 +308,6 @@ instance (priority := low) [Module.Finite R M] : IsNoetherian R M where
     letI := Module.Finite.equiv e.symm
     .of_finite
 
-set_option backward.isDefEq.respectTransparency false in
 -- does not work as an instance, not sure why
 protected theorem range (f : M →ₗ[R] N) : IsSemisimpleModule R (range f) :=
   congr (quotKerEquivRange _).symm
@@ -356,13 +342,11 @@ theorem LinearEquiv.isSemisimpleModule_iff (e : M ≃ₗ[R] N) :
     IsSemisimpleModule R M ↔ IsSemisimpleModule R N :=
   ⟨(·.congr e.symm), (·.congr e)⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A module is semisimple iff it is generated by its simple submodules. -/
 theorem sSup_simples_eq_top_iff_isSemisimpleModule :
     sSup { m : Submodule R M | IsSimpleModule R m } = ⊤ ↔ IsSemisimpleModule R M :=
   ⟨.of_sSup_simples_eq_top, fun _ ↦ IsSemisimpleModule.sSup_simples_eq_top _ _⟩
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A module generated by semisimple submodules is itself semisimple. -/
 lemma isSemisimpleModule_of_isSemisimpleModule_submodule {s : Set ι} {p : ι → Submodule R M}
     (hp : ∀ i ∈ s, IsSemisimpleModule R (p i)) (hp' : ⨆ i ∈ s, p i = ⊤) :
@@ -371,7 +355,6 @@ lemma isSemisimpleModule_of_isSemisimpleModule_submodule {s : Set ι} {p : ι 
   refine complementedLattice_of_complementedLattice_Iic (fun i hi ↦ ?_) hp'
   simpa only [← (p i).mapIic.complementedLattice_iff] using hp i hi
 
-set_option backward.isDefEq.respectTransparency false in
 open Submodule in
 lemma isSemisimpleModule_biSup_of_isSemisimpleModule_submodule {s : Set ι} {p : ι → Submodule R M}
     (hp : ∀ i ∈ s, IsSemisimpleModule R (p i)) :
@@ -381,7 +364,6 @@ lemma isSemisimpleModule_biSup_of_isSemisimpleModule_submodule {s : Set ι} {p :
     (biSup_comap_subtype_eq_top ..)
   simp_rw [range_subtype, le_biSup p ‹_›]
 
-set_option backward.isDefEq.respectTransparency false in
 lemma isSemisimpleModule_of_isSemisimpleModule_submodule' {p : ι → Submodule R M}
     (hp : ∀ i, IsSemisimpleModule R (p i)) (hp' : ⨆ i, p i = ⊤) :
     IsSemisimpleModule R M :=
@@ -393,7 +375,6 @@ instance {ι} (M : ι → Type*) [∀ i, AddCommGroup (M i)] [∀ i, Module R (M
   exact isSemisimpleModule_of_isSemisimpleModule_submodule'
     (fun _ ↦ .range _) DFinsupp.iSup_range_lsingle
 
-set_option backward.isDefEq.respectTransparency false in
 variable (R M) in
 theorem IsSemisimpleModule.exists_linearEquiv_dfinsupp [IsSemisimpleModule R M] :
     ∃ (s : Set (Submodule R M)) (_ : M ≃ₗ[R] Π₀ m : s, m.1),
@@ -412,7 +393,6 @@ theorem isSemisimpleModule_iff_exists_linearEquiv_dfinsupp : IsSemisimpleModule
   have ⟨s, e, h⟩ := IsSemisimpleModule.exists_linearEquiv_dfinsupp R M
   exact ⟨s, e, h.2⟩
 
-set_option backward.isDefEq.respectTransparency false in
 variable (R M) in
 theorem IsSemisimpleModule.exists_linearEquiv_fin_dfinsupp [IsSemisimpleModule R M]
     [Module.Finite R M] : ∃ (n : ℕ) (S : Fin n → Submodule R M)
@@ -429,7 +409,6 @@ instance {ι} [Finite ι] (M : ι → Type*) [∀ i, AddCommGroup (M i)] [∀ i,
   exact isSemisimpleModule_of_isSemisimpleModule_submodule' (p := (range <| single _ _ ·))
     (fun i ↦ .range _) (by simp_rw [range_eq_map, Submodule.iSup_map_single, Submodule.pi_top])
 
-set_option backward.isDefEq.respectTransparency false in
 theorem IsSemisimpleModule.sup {p q : Submodule R M}
     (_ : IsSemisimpleModule R p) (_ : IsSemisimpleModule R q) :
     IsSemisimpleModule R ↥(p ⊔ q) := by
@@ -445,7 +424,6 @@ theorem IsSemisimpleRing.exists_linearEquiv_ideal_of_isSimpleModule [IsSemisimpl
   have ⟨I, ⟨e'⟩⟩ := IsSemisimpleModule.exists_submodule_linearEquiv_quotient J
   ⟨I, ⟨e.trans e'.symm⟩⟩
 
-set_option backward.isDefEq.respectTransparency false in
 instance (ι) [IsSemisimpleModule R M] : IsSemisimpleModule R (ι →₀ M) :=
   isSemisimpleModule_of_isSemisimpleModule_submodule'
     (fun _ ↦ .congr (LinearMap.quotKerEquivRange _).symm) Finsupp.iSup_lsingle_range
@@ -536,7 +514,6 @@ theorem isCoatom_ker_of_surjective [IsSimpleModule R N] {f : M →ₗ[R] N}
   rw [← isSimpleModule_iff_isCoatom]
   exact IsSimpleModule.congr (f.quotKerEquivOfSurjective hf)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem linearEquiv_of_ne_zero [IsSemisimpleModule R M] [IsSimpleModule R N]
     {f : M →ₗ[R] N} (h : f ≠ 0) : ∃ S : Submodule R M, Nonempty (N ≃ₗ[R] S) :=
   have ⟨m, (_ : IsSimpleModule R m), ne⟩ :=
@@ -567,7 +544,6 @@ end LinearMap
 
 namespace JordanHolderModule
 
-set_option backward.isDefEq.respectTransparency false in
 instance instJordanHolderLattice : JordanHolderLattice (Submodule R M) where
   IsMaximal := (· ⋖ ·)
   lt_of_isMaximal := CovBy.lt
diff --git a/Mathlib/RingTheory/Smooth/IntegralClosure.lean b/Mathlib/RingTheory/Smooth/IntegralClosure.lean
index 19a0192da98b01..1b5a279dd820d6 100644
--- a/Mathlib/RingTheory/Smooth/IntegralClosure.lean
+++ b/Mathlib/RingTheory/Smooth/IntegralClosure.lean
@@ -49,7 +49,6 @@ def TensorProduct.toIntegralClosure
       (R := R) (A := S))).tower_top (A := S)).smul x
     simp [smul_tmul']
 
-set_option backward.isDefEq.respectTransparency false in
 lemma TensorProduct.toIntegralClosure_injective_of_flat [Module.Flat R S] :
     Function.Injective (toIntegralClosure R S B) := by
   refine Function.Injective.of_comp (f := (integralClosure _ _).val) ?_
diff --git a/Mathlib/RingTheory/Smooth/Kaehler.lean b/Mathlib/RingTheory/Smooth/Kaehler.lean
index f9b5bb05f0bc55..f8fc8ec690fe6c 100644
--- a/Mathlib/RingTheory/Smooth/Kaehler.lean
+++ b/Mathlib/RingTheory/Smooth/Kaehler.lean
@@ -99,7 +99,6 @@ lemma isScalarTower_of_section_of_ker_sqZero :
   refine Ideal.mul_mem_mul ?_ m.2
   simpa [RingHom.mem_ker, sub_eq_zero] using AlgHom.congr_fun hg (algebraMap P S p)
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Given a surjective algebra hom `f : P →ₐ[R] S` with square-zero kernel `I`,
 and a section `g : S →ₐ[R] P` (as algebra homs),
@@ -113,7 +112,6 @@ def retractionOfSectionOfKerSqZero : S ⊗[P] Ω[P⁄R] →ₗ[P] RingHom.ker (a
     (IsScalarTower.toAlgHom R P S) hf' g hg).liftKaehlerDifferential
   (f.liftBaseChange S).restrictScalars P
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma retractionOfSectionOfKerSqZero_tmul_D (s : S) (t : P) :
     retractionOfSectionOfKerSqZero g hf' hg (s ⊗ₜ .D _ _ t) =
@@ -142,7 +140,6 @@ include hl
 -- suppose we have a (set-theoretic) section
 variable (σ : S → P) (hσ : ∀ x, algebraMap P S (σ x) = x)
 
-set_option backward.isDefEq.respectTransparency false in
 lemma sectionOfRetractionKerToTensorAux_prop (x y) (h : algebraMap P S x = algebraMap P S y) :
     x - l (1 ⊗ₜ .D _ _ x) = y - l (1 ⊗ₜ .D _ _ y) := by
   rw [sub_eq_iff_eq_add, sub_add_comm, ← sub_eq_iff_eq_add, ← Submodule.coe_sub,
@@ -217,7 +214,6 @@ end ofRetraction
 variable [Algebra R S] [IsScalarTower R P S]
 variable (hf' : (RingHom.ker (algebraMap P S)) ^ 2 = ⊥) (hf : Surjective (algebraMap P S))
 
-set_option backward.isDefEq.respectTransparency false in
 /--
 Given a surjective algebra homomorphism `f : P →ₐ[R] S` with square-zero kernel `I`,
 there is a one-to-one correspondence between `P`-linear retractions of `I →ₗ[P] S ⊗[P] Ω[P/R]`
diff --git a/Mathlib/RingTheory/TensorProduct/MonoidAlgebra.lean b/Mathlib/RingTheory/TensorProduct/MonoidAlgebra.lean
index 25cd053dde70fa..e55d39b273c879 100644
--- a/Mathlib/RingTheory/TensorProduct/MonoidAlgebra.lean
+++ b/Mathlib/RingTheory/TensorProduct/MonoidAlgebra.lean
@@ -37,7 +37,6 @@ noncomputable def _root_.AddMonoidAlgebra.tensorEquiv.invFun [AddCommMonoid M] :
     (Algebra.TensorProduct.includeRight.toMonoidHom.comp <| AddMonoidAlgebra.of B M)
       fun _ _ ↦ .all ..
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Implementation detail. -/
 @[to_additive existing (dont_translate := R A B)]
 def tensorEquiv.invFun : (A ⊗[R] B)[M] →ₐ[A] A ⊗[R] B[M] :=
diff --git a/Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean b/Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean
index 868e718ad9d351..790fdd75e6c557 100644
--- a/Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean
+++ b/Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean
@@ -318,7 +318,6 @@ variable [CommMonoidWithZero α] [IsCancelMulZero α]
 variable (pf : ∀ a : α, a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a)
 include pf
 
-set_option backward.isDefEq.respectTransparency false in
 theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
   ⟨by
     classical
diff --git a/Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.lean b/Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.lean
index 2442cda2f38211..9abefd4e9dcc53 100644
--- a/Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.lean
+++ b/Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicity.lean
@@ -79,7 +79,6 @@ theorem le_emultiplicity_iff_replicate_le_normalizedFactors {a b : R} {n : ℕ}
     rw [← (prod_normalizedFactors hb).dvd_iff_dvd_right, hu, prod_add, prod_replicate]
     exact (Associated.pow_pow <| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The multiplicity of an irreducible factor of a nonzero element is exactly the number of times
 the normalized factor occurs in the `normalizedFactors`.
 
diff --git a/Mathlib/RingTheory/UniqueFactorizationDomain/Nat.lean b/Mathlib/RingTheory/UniqueFactorizationDomain/Nat.lean
index a48e19eb43a057..06086e026574e4 100644
--- a/Mathlib/RingTheory/UniqueFactorizationDomain/Nat.lean
+++ b/Mathlib/RingTheory/UniqueFactorizationDomain/Nat.lean
@@ -23,7 +23,6 @@ assert_not_exists Field
 
 namespace Nat
 
-set_option backward.isDefEq.respectTransparency false in
 instance instWfDvdMonoid : WfDvdMonoid ℕ where
   wf := by
     refine RelHomClass.wellFounded
diff --git a/Mathlib/RingTheory/Unramified/Basic.lean b/Mathlib/RingTheory/Unramified/Basic.lean
index e355a275dff095..db0cc35ffa5634 100644
--- a/Mathlib/RingTheory/Unramified/Basic.lean
+++ b/Mathlib/RingTheory/Unramified/Basic.lean
@@ -67,7 +67,6 @@ variable {R : Type v} [CommRing R]
 variable {A : Type u} [CommRing A] [Algebra R A]
 variable {B : Type w} [CommRing B] [Algebra R B] (I : Ideal B)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem comp_injective [FormallyUnramified R A] (hI : I ^ 2 = ⊥) :
     Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I) := by
   intro f₁ f₂ e
diff --git a/Mathlib/RingTheory/Unramified/LocalRing.lean b/Mathlib/RingTheory/Unramified/LocalRing.lean
index 974981888213ae..b9d1bcbff96345 100644
--- a/Mathlib/RingTheory/Unramified/LocalRing.lean
+++ b/Mathlib/RingTheory/Unramified/LocalRing.lean
@@ -149,7 +149,6 @@ lemma isUnramifiedAt_iff_map_eq :
 instance [Algebra.IsUnramifiedAt R q] : Algebra.IsSeparable p.ResidueField q.ResidueField :=
   ((Algebra.isUnramifiedAt_iff_map_eq _ _ _).mp inferInstance).1
 
-set_option backward.isDefEq.respectTransparency false in
 instance [Algebra.IsUnramifiedAt R q] : Module.Finite p.ResidueField q.ResidueField :=
   Algebra.FormallyUnramified.finite_of_free _ _
 
diff --git a/Mathlib/SetTheory/Cardinal/ENat.lean b/Mathlib/SetTheory/Cardinal/ENat.lean
index 928f2a91a5ff9f..cc6a8c9a77ef95 100644
--- a/Mathlib/SetTheory/Cardinal/ENat.lean
+++ b/Mathlib/SetTheory/Cardinal/ENat.lean
@@ -176,7 +176,6 @@ lemma toENatAux_gc : GaloisConnection (↑) toENatAux := fun n x ↦ by
   | inl hx => lift x to ℕ using hx; simp
   | inr hx => simp [toENatAux_eq_top hx, (ofENat_le_aleph0 n).trans hx]
 
-set_option backward.isDefEq.respectTransparency false in
 theorem toENatAux_le_nat {x : Cardinal} {n : ℕ} : toENatAux x ≤ n ↔ x ≤ n := by
   cases lt_or_ge x ℵ₀ with
   | inl hx => lift x to ℕ using hx; simp
@@ -187,7 +186,6 @@ lemma toENatAux_eq_nat {x : Cardinal} {n : ℕ} : toENatAux x = n ↔ x = n := b
 
 lemma toENatAux_eq_zero {x : Cardinal} : toENatAux x = 0 ↔ x = 0 := toENatAux_eq_nat
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Projection from cardinals to `ℕ∞`. Sends all infinite cardinals to `⊤`.
 
 We define this function as a bundled monotone ring homomorphism. -/
@@ -319,7 +317,6 @@ lemma ofENat_add (m n : ℕ∞) : ofENat (m + n) = m + n := by apply toENat_injO
 
 @[simp] lemma ofENat_add_aleph0 (m : ℕ∞) : m + ℵ₀ = ℵ₀ := by rw [add_comm, aleph0_add_ofENat]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma ofENat_mul_aleph0 {m : ℕ∞} (hm : m ≠ 0) : ↑m * ℵ₀ = ℵ₀ := by
   induction m with
   | top => exact aleph0_mul_aleph0
diff --git a/Mathlib/SetTheory/Cardinal/Finite.lean b/Mathlib/SetTheory/Cardinal/Finite.lean
index 22af94576028a8..f1c4a09831396a 100644
--- a/Mathlib/SetTheory/Cardinal/Finite.lean
+++ b/Mathlib/SetTheory/Cardinal/Finite.lean
@@ -279,7 +279,6 @@ namespace ENat
 def card (α : Type*) : ℕ∞ :=
   toENat (mk α)
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 theorem card_eq_coe_fintype_card [Fintype α] : card α = Fintype.card α := by
   simp [card]
@@ -371,7 +370,6 @@ lemma card_eq_one_iff_unique {α : Type*} : card α = 1 ↔ Nonempty (Unique α)
   rw [unique_iff_subsingleton_and_nonempty α, le_antisymm_iff]
   exact and_congr (card_le_one_iff_subsingleton α) (one_le_card_iff_nonempty α)
 
-set_option backward.isDefEq.respectTransparency false in
 theorem one_lt_card_iff_nontrivial (α : Type*) : 1 < card α ↔ Nontrivial α := by
   rw [← Cardinal.one_lt_iff_nontrivial]
   conv_rhs => rw [← Nat.cast_one]
@@ -387,7 +385,6 @@ lemma exists_ne_ne_of_three_le (h : 3 ≤ ENat.card α) (x y : α) : ∃ z, z 
 theorem card_prod (α β : Type*) : card (α × β) = card α * card β := by
   simp [ENat.card]
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma card_fun {α β : Type*} : card (α → β) = (card β) ^ card α := by
   classical
diff --git a/Mathlib/SetTheory/Cardinal/Ordinal.lean b/Mathlib/SetTheory/Cardinal/Ordinal.lean
index a9fb92415b3ea9..bb6322897107a3 100644
--- a/Mathlib/SetTheory/Cardinal/Ordinal.lean
+++ b/Mathlib/SetTheory/Cardinal/Ordinal.lean
@@ -53,7 +53,6 @@ end Cardinal
 
 namespace Ordinal
 
-set_option backward.isDefEq.respectTransparency false in
 theorem lift_card_iSup_le_sum_card {ι : Type u} [Small.{v} ι] (f : ι → Ordinal.{v}) :
     Cardinal.lift.{u} (⨆ i, f i).card ≤ Cardinal.sum fun i ↦ (f i).card := by
   simp_rw [← mk_toType]
diff --git a/Mathlib/SetTheory/ZFC/Class.lean b/Mathlib/SetTheory/ZFC/Class.lean
index ec7b69092fb62b..84db00306f8d38 100644
--- a/Mathlib/SetTheory/ZFC/Class.lean
+++ b/Mathlib/SetTheory/ZFC/Class.lean
@@ -345,7 +345,6 @@ theorem choice_mem (h : ∅ ∉ x) (y : ZFSet.{u}) (yx : y ∈ x) :
   rw [@map_fval _ (Classical.allZFSetDefinable _) x y yx, Class.coe_mem, Class.coe_apply]
   exact choice_mem_aux x h y yx
 
-set_option backward.isDefEq.respectTransparency false in
 set_option backward.privateInPublic true in
 private lemma coe_equiv_aux {s : Set ZFSet.{u}} (hs : Small.{u} s) :
     (mk <| PSet.mk (Shrink s) fun x ↦ ((equivShrink s).symm x).1.out) = s := by
diff --git a/Mathlib/Tactic/NormNum/Irrational.lean b/Mathlib/Tactic/NormNum/Irrational.lean
index b4d8ab1a39aef6..f7e8d397854c7c 100644
--- a/Mathlib/Tactic/NormNum/Irrational.lean
+++ b/Mathlib/Tactic/NormNum/Irrational.lean
@@ -222,7 +222,6 @@ private theorem irrational_rpow_nat_rat {x y : ℝ} {x_num y_num y_den k : ℕ}
     Irrational (x ^ y) :=
   irrational_rpow_rat_rat_of_num hx_isNat.to_isNNRat hy_isNNRat (by simp) hy_coprime hn1 hn2
 
-set_option backward.isDefEq.respectTransparency false in
 private theorem irrational_sqrt_rat_of_num {x : ℝ} {num den num_k : ℕ}
     (hx_isNNRat : IsNNRat x num den)
     (hx_coprime : Nat.Coprime num den)
@@ -234,7 +233,6 @@ private theorem irrational_sqrt_rat_of_num {x : ℝ} {num den num_k : ℕ}
     hn1 hn2
   exact ⟨Invertible.mk (1/2) (by simp) (by simp), by simp⟩
 
-set_option backward.isDefEq.respectTransparency false in
 private theorem irrational_sqrt_rat_of_den {x : ℝ} {num den den_k : ℕ}
     (hx_isNNRat : IsNNRat x num den)
     (hx_coprime : Nat.Coprime num den)
diff --git a/Mathlib/Topology/Algebra/AffineSubspace.lean b/Mathlib/Topology/Algebra/AffineSubspace.lean
index 498b17d043be3b..eb390c38681362 100644
--- a/Mathlib/Topology/Algebra/AffineSubspace.lean
+++ b/Mathlib/Topology/Algebra/AffineSubspace.lean
@@ -29,7 +29,6 @@ namespace AffineSubspace
 variable {R V P : Type*} [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace P]
   [AddTorsor V P]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Embedding of an affine subspace to the ambient space, as a continuous affine map. -/
 def subtypeA (s : AffineSubspace R P) [Nonempty s] : s →ᴬ[R] P where
   toAffineMap := s.subtype
@@ -38,7 +37,6 @@ def subtypeA (s : AffineSubspace R P) [Nonempty s] : s →ᴬ[R] P where
 @[simp] lemma coe_subtypeA (s : AffineSubspace R P) [Nonempty s] : ⇑s.subtypeA = Subtype.val :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 @[simp] lemma subtypeA_toAffineMap (s : AffineSubspace R P) [Nonempty s] :
     s.subtypeA.toAffineMap = s.subtype :=
   rfl
diff --git a/Mathlib/Topology/Algebra/Module/Equiv.lean b/Mathlib/Topology/Algebra/Module/Equiv.lean
index 5421134b855c43..eb05d7a3087fc1 100644
--- a/Mathlib/Topology/Algebra/Module/Equiv.lean
+++ b/Mathlib/Topology/Algebra/Module/Equiv.lean
@@ -992,7 +992,6 @@ theorem map_neg (e : M ≃SL[σ₁₂] M₂) (x : M) : e (-x) = -e x :=
 
 variable [Module R M₂] [IsTopologicalAddGroup M]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A pair of continuous linear maps such that `f₁ ∘ f₂ = id` generates a continuous
 linear equivalence `e` between `M` and `M₂ × f₁.ker` such that `(e x).2 = x` for `x ∈ f₁.ker`,
 `(e x).1 = f₁ x`, and `(e (f₂ y)).2 = 0`. The map is given by `e x = (f₁ x, x - f₂ (f₁ x))`. -/
@@ -1356,7 +1355,6 @@ variable {R : Type*} [Ring R] {M : Type*} [TopologicalSpace M] [AddCommGroup M]
 
 open ContinuousLinearMap
 
-set_option backward.isDefEq.respectTransparency false in
 /-- If `p` is a closed complemented submodule,
 then there exists a submodule `q` and a continuous linear equivalence `M ≃L[R] (p × q)` such that
 `e (x : p) = (x, 0)`, `e (y : q) = (0, y)`, and `e.symm x = x.1 + x.2`.
diff --git a/Mathlib/Topology/Algebra/Module/FiniteDimension.lean b/Mathlib/Topology/Algebra/Module/FiniteDimension.lean
index 0c0ff4b6ac3125..33b23aff3b5587 100644
--- a/Mathlib/Topology/Algebra/Module/FiniteDimension.lean
+++ b/Mathlib/Topology/Algebra/Module/FiniteDimension.lean
@@ -203,7 +203,6 @@ theorem LinearMap.continuous_of_nonzero_on_open (l : E →ₗ[𝕜] 𝕜) (s : S
 
 variable [CompleteSpace 𝕜]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- This version imposes `ι` and `E` to live in the same universe, so you should instead use
 `continuous_equivFun_basis` which gives the same result without universe restrictions. -/
 private theorem continuous_equivFun_basis_aux [T2Space E] {ι : Type v} [Finite ι]
@@ -516,7 +515,6 @@ theorem FiniteDimensional.complete [FiniteDimensional 𝕜 E] : CompleteSpace E
 
 variable {𝕜 E}
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A finite-dimensional subspace is complete. -/
 theorem Submodule.complete_of_finiteDimensional (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 s] :
     IsComplete (s : Set E) :=
@@ -531,7 +529,6 @@ variable {𝕜 E F : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
   [AddCommGroup F] [TopologicalSpace F] [T2Space F] [IsTopologicalAddGroup F] [Module 𝕜 F]
   [ContinuousSMul 𝕜 F]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A finite-dimensional subspace is closed. -/
 theorem Submodule.closed_of_finiteDimensional
     [T2Space E] (s : Submodule 𝕜 E) [FiniteDimensional 𝕜 s] :
@@ -540,7 +537,6 @@ theorem Submodule.closed_of_finiteDimensional
   haveI : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
   s.complete_of_finiteDimensional.isClosed
 
-set_option backward.isDefEq.respectTransparency false in
 /-- An injective linear map with finite-dimensional domain is a closed embedding. -/
 theorem LinearMap.isClosedEmbedding_of_injective [T2Space E] [FiniteDimensional 𝕜 E] {f : E →ₗ[𝕜] F}
     (hf : LinearMap.ker f = ⊥) : IsClosedEmbedding f :=
diff --git a/Mathlib/Topology/Algebra/Module/UniformConvergence.lean b/Mathlib/Topology/Algebra/Module/UniformConvergence.lean
index 197b0fc6d95df6..39818c9bd1aa8e 100644
--- a/Mathlib/Topology/Algebra/Module/UniformConvergence.lean
+++ b/Mathlib/Topology/Algebra/Module/UniformConvergence.lean
@@ -109,7 +109,6 @@ lemma UniformOnFun.continuousSMul_induced_of_image_bounded (φ : hom) (hφ : IsI
   refine UniformFun.continuousSMul_induced_of_range_bounded 𝕜 s E H φ' ⟨rfl⟩ fun u ↦ ?_
   simpa only [Set.image_eq_range] using h u s hs
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Let `E` be a TVS, `𝔖 : Set (Set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
 `S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `Bornology.IsVonNBounded`), then `H`,
 equipped with the topology of `𝔖`-convergence, is a TVS.
diff --git a/Mathlib/Topology/Compactification/OnePoint/Sphere.lean b/Mathlib/Topology/Compactification/OnePoint/Sphere.lean
index 695eedc0a63cbb..2c54099bb3920d 100644
--- a/Mathlib/Topology/Compactification/OnePoint/Sphere.lean
+++ b/Mathlib/Topology/Compactification/OnePoint/Sphere.lean
@@ -29,7 +29,6 @@ def onePointHyperplaneHomeoUnitSphere
   OnePoint.equivOfIsEmbeddingOfRangeEq _ _
     (isOpenEmbedding_stereographic_symm hv).toIsEmbedding (range_stereographic_symm hv)
 
-set_option backward.isDefEq.respectTransparency false in
 /-- A homeomorphism from the one-point compactification of a finite-dimensional real vector space to
 the sphere. -/
 def onePointEquivSphereOfFinrankEq {ι V : Type*} [Fintype ι]
diff --git a/Mathlib/Topology/MetricSpace/Kuratowski.lean b/Mathlib/Topology/MetricSpace/Kuratowski.lean
index 88704da6fa8ea8..d7d62e15bc42d6 100644
--- a/Mathlib/Topology/MetricSpace/Kuratowski.lean
+++ b/Mathlib/Topology/MetricSpace/Kuratowski.lean
@@ -47,7 +47,6 @@ def embeddingOfSubset : ℓ^∞(ℕ) :=
 theorem embeddingOfSubset_coe : embeddingOfSubset x a n = dist a (x n) - dist (x 0) (x n) :=
   rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The embedding map is always a semi-contraction. -/
 theorem embeddingOfSubset_dist_le (a b : α) :
     dist (embeddingOfSubset x a) (embeddingOfSubset x b) ≤ dist a b := by