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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Algebra.Subalgebra.Matrix.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Grp.Subobject.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.sym.json

@@ -0,0 +1 @@
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(CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}))) R₀ (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R)) => CategoryTheory.Presheaf.IsLocallyInjective.{u, u, v', succ u, u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} (fun (R : RingCat.{u}) (S : RingCat.{u}) => RingHom.{u, u} (RingCat.carrier.{u} R) (RingCat.carrier.{u} S) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} R) (Ring.toSemiring.{u} (RingCat.carrier.{u} R) (RingCat.ring.{u} R))) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} S) (Ring.toSemiring.{u} (RingCat.carrier.{u} S) (RingCat.ring.{u} S)))) RingCat.carrier.{u} (fun (X : RingCat.{u}) (Y : RingCat.{u}) => RingHom.instFunLike.{u, u} (RingCat.carrier.{u} X) (RingCat.carrier.{u} Y) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} X) ((fun (X : RingCat.{u}) => Ring.toSemiring.{u} (RingCat.carrier.{u} X) (RingCat.ring.{u} X)) X)) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} Y) (Ring.toSemiring.{u} (RingCat.carrier.{u} Y) (RingCat.ring.{u} Y)))) 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(CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}))) R₀ (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R)) α (fun (α : Quiver.Hom.{max u u', max 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(CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}))) R₀ (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R)) => CategoryTheory.Presheaf.IsLocallySurjective.{v', u', u, succ u, u, u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.3 J RingCat.{u} RingCat.instCategory.{u} (fun (R : RingCat.{u}) (S : RingCat.{u}) => RingHom.{u, u} (RingCat.carrier.{u} R) (RingCat.carrier.{u} S) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} R) (Ring.toSemiring.{u} (RingCat.carrier.{u} R) (RingCat.ring.{u} R))) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} S) (Ring.toSemiring.{u} (RingCat.carrier.{u} S) (RingCat.ring.{u} S)))) RingCat.carrier.{u} (fun (X : RingCat.{u}) (Y : RingCat.{u}) => RingHom.instFunLike.{u, u} (RingCat.carrier.{u} X) (RingCat.carrier.{u} Y) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} X) ((fun (X : RingCat.{u}) => Ring.toSemiring.{u} (RingCat.carrier.{u} X) (RingCat.ring.{u} X)) X)) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} Y) (Ring.toSemiring.{u} (RingCat.carrier.{u} Y) (RingCat.ring.{u} Y)))) 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inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification.4138856428._hygCtx._hyg.35)","typeFull":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α α_1 : R₀ ⟶ R.obj) (e_α : α = α_1)\n  [inst_1 : CategoryTheory.Presheaf.IsLocallyInjective J α] [inst_2 : CategoryTheory.Presheaf.IsLocallySurjective J α]\n  [inst_3 : J.WEqualsLocallyBijective AddCommGrpCat] [inst_4 : CategoryTheory.HasWeakSheafify J AddCommGrpCat],\n  PresheafOfModules.sheafification α = PresheafOfModules.sheafification α_1","typeReadable":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α α_1 : R₀ ⟶ R.obj) (e_α : α = α_1)\n  [inst_1 : CategoryTheory.Presheaf.IsLocallyInjective J α] [inst_2 : CategoryTheory.Presheaf.IsLocallySurjective J α]\n  [inst_3 : J.WEqualsLocallyBijective AddCommGrpCat] [inst_4 : CategoryTheory.HasWeakSheafify J AddCommGrpCat],\n  PresheafOfModules.sheafification α = PresheafOfModules.sheafification α_1","typeReferences":[["RingHom"],["AddCommGroup","toAddGroup"],["Opposite"],["CategoryTheory","Category"],["RingHom","instFunLike"],["PresheafOfModules"],["PresheafOfModules","instCategory"],["RingCat","carrier"],["Semiring","toNonAssocSemiring"],["Quiver","Hom"],["CategoryTheory","Presheaf","IsLocallySurjective"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["CategoryTheory","Presheaf","IsSheaf"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["Eq"],["AddCommGrpCat","carrier"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["SheafOfModules","instCategory"],["SheafOfModules"],["AddCommGrpCat","instCategory"],["CategoryTheory","Category","opposite"],["CategoryTheory","Functor"],["CategoryTheory","Presheaf","IsLocallyInjective"],["CategoryTheory","HasWeakSheafify"],["CategoryTheory","GrothendieckTopology"],["CategoryTheory","GrothendieckTopology","WEqualsLocallyBijective"],["AddZeroClass","toAddZero"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["PresheafOfModules","sheafification"],["CategoryTheory","Category","toCategoryStruct"],["Ring","toSemiring"],["AddCommGrpCat"],["RingCat"],["AddCommGrpCat","str"],["CategoryTheory","CategoryStruct","toQuiver"],["SubNegMonoid","toAddMonoid"],["RingCat","instCategory"],["CategoryTheory","Functor","category"],["RingCat","ring"],["RingCat","instConcreteCategoryRingHomCarrier"],["CategoryTheory","Sheaf"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["RingHom"],["AddCommGroup","toAddGroup"],["Opposite"],["RingHom","instFunLike"],["PresheafOfModules"],["PresheafOfModules","instCategory"],["RingCat","carrier"],["Semiring","toNonAssocSemiring"],["Quiver","Hom"],["CategoryTheory","Presheaf","IsLocallySurjective"],["AddMonoidHom"],["AddMonoidHom","instFunLike"],["CategoryTheory","Presheaf","IsSheaf"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["Eq"],["Eq","rec"],["AddCommGrpCat","carrier"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["SheafOfModules","instCategory"],["SheafOfModules"],["CategoryTheory","Category","opposite"],["CategoryTheory","Functor"],["AddCommGrpCat","instCategory"],["CategoryTheory","Presheaf","IsLocallyInjective"],["CategoryTheory","HasWeakSheafify"],["CategoryTheory","GrothendieckTopology","WEqualsLocallyBijective"],["AddZeroClass","toAddZero"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["PresheafOfModules","sheafification"],["CategoryTheory","Category","toCategoryStruct"],["Ring","toSemiring"],["AddCommGrpCat"],["RingCat"],["AddCommGrpCat","str"],["CategoryTheory","CategoryStruct","toQuiver"],["SubNegMonoid","toAddMonoid"],["Eq","refl"],["RingCat","instCategory"],["CategoryTheory","Functor","category"],["RingCat","ring"],["RingCat","instConcreteCategoryRingHomCarrier"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["PresheafOfModules","instIsLocalizationSheafOfModulesSheafificationInverseImageFunctorOppositeAbWToPresheaf"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.595633009._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.595633009._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.595633009._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R) α] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.595633009._hygCtx._hyg.28 : CategoryTheory.Presheaf.IsLocallySurjective.{v', u', u, succ u, u, u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.595633009._hygCtx._hyg.3 J RingCat.{u} RingCat.instCategory.{u} (fun (R : RingCat.{u}) (S : RingCat.{u}) => RingHom.{u, u} (RingCat.carrier.{u} R) (RingCat.carrier.{u} S) (Semiring.toNonAssocSemiring.{u} 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CategoryTheory.GrothendieckTopology C}\n  {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.obj)\n  [inst_1 : CategoryTheory.Presheaf.IsLocallyInjective J α] [inst_2 : CategoryTheory.Presheaf.IsLocallySurjective J α]\n  [inst_3 : J.WEqualsLocallyBijective AddCommGrpCat] [inst_4 : CategoryTheory.HasWeakSheafify J AddCommGrpCat],\n  (PresheafOfModules.sheafification α).IsLocalization (J.W.inverseImage (PresheafOfModules.toPresheaf 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{C : Type.{u}} [inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3 : CategoryTheory.Category.{v, u} C] {D : Type.{u_1}} [inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7 : CategoryTheory.Category.{v_1, u_1} D] (P : CategoryTheory.MorphismProperty.{v_1, u_1} D (CategoryTheory.Category.toCategoryStruct.{v_1, u_1} D inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7)) (F : CategoryTheory.Functor.{v, v_1, u, u_1} C inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7) {X : C} {Y : C} (f : Quiver.Hom.{v, u} C (CategoryTheory.CategoryStruct.toQuiver.{v, u} C (CategoryTheory.Category.toCategoryStruct.{v, u} C inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3)) X Y), Eq.{1} Prop (CategoryTheory.MorphismProperty.inverseImage.{v, u, v_1, u_1} C inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7 P F X Y f) (P (CategoryTheory.Functor.obj.{v, v_1, u, u_1} C inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7 F X) (CategoryTheory.Functor.obj.{v, v_1, u, u_1} C inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7 F Y) (CategoryTheory.Functor.map.{v, v_1, u, u_1} C inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.3 D inst._@.Mathlib.CategoryTheory.MorphismProperty.Basic.3448162710._hygCtx._hyg.7 F X Y f))","typeFull":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} D]\n  (P : CategoryTheory.MorphismProperty D) (F : CategoryTheory.Functor C D) {X Y : C} (f : X ⟶ Y),\n  P.inverseImage F f = P (F.map f)","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u_1} [inst_1 : CategoryTheory.Category.{v_1, u_1} D]\n  (P : CategoryTheory.MorphismProperty D) (F : CategoryTheory.Functor C D) {X Y : C} (f : X ⟶ Y),\n  P.inverseImage F f = P (F.map 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(Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R)) [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.24 : CategoryTheory.Presheaf.IsLocallyInjective.{u, u, v', succ u, u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} (fun (R : RingCat.{u}) (S : RingCat.{u}) => RingHom.{u, u} (RingCat.carrier.{u} R) (RingCat.carrier.{u} S) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} R) (Ring.toSemiring.{u} (RingCat.carrier.{u} R) (RingCat.ring.{u} R))) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} S) (Ring.toSemiring.{u} (RingCat.carrier.{u} S) (RingCat.ring.{u} S)))) RingCat.carrier.{u} (fun (X : RingCat.{u}) (Y : RingCat.{u}) => RingHom.instFunLike.{u, u} (RingCat.carrier.{u} X) (RingCat.carrier.{u} Y) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} X) ((fun (X : RingCat.{u}) => Ring.toSemiring.{u} (RingCat.carrier.{u} X) (RingCat.ring.{u} X)) X)) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} Y) (Ring.toSemiring.{u} (RingCat.carrier.{u} Y) (RingCat.ring.{u} Y)))) RingCat.instConcreteCategoryRingHomCarrier.{u} J R₀ (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R) α] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.28 : CategoryTheory.Presheaf.IsLocallySurjective.{v', u', u, succ u, u, u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3 J RingCat.{u} RingCat.instCategory.{u} (fun (R : RingCat.{u}) (S : RingCat.{u}) => RingHom.{u, u} (RingCat.carrier.{u} R) (RingCat.carrier.{u} S) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} R) (Ring.toSemiring.{u} (RingCat.carrier.{u} R) (RingCat.ring.{u} R))) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} S) (Ring.toSemiring.{u} (RingCat.carrier.{u} S) (RingCat.ring.{u} S)))) RingCat.carrier.{u} (fun (X : RingCat.{u}) (Y : RingCat.{u}) => RingHom.instFunLike.{u, u} (RingCat.carrier.{u} X) (RingCat.carrier.{u} Y) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} X) ((fun (X : RingCat.{u}) => Ring.toSemiring.{u} (RingCat.carrier.{u} X) (RingCat.ring.{u} X)) X)) (Semiring.toNonAssocSemiring.{u} (RingCat.carrier.{u} Y) (Ring.toSemiring.{u} (RingCat.carrier.{u} Y) (RingCat.ring.{u} Y)))) RingCat.instConcreteCategoryRingHomCarrier.{u} R₀ (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R) α] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.32 : CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective.{v, v, v', succ v, u', v} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.3 J AddCommGrpCat.{v} AddCommGrpCat.instCategory.{v} (fun (x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6 : AddCommGrpCat.{v}) (x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6 : AddCommGrpCat.{v}) => AddMonoidHom.{v, v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} 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inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Localization.4243574590._hygCtx._hyg.35))","typeFull":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.obj)\n  [inst_1 : CategoryTheory.Presheaf.IsLocallyInjective J α] [inst_2 : CategoryTheory.Presheaf.IsLocallySurjective J α]\n  [inst_3 : J.WEqualsLocallyBijective AddCommGrpCat] [inst_4 : CategoryTheory.HasWeakSheafify J AddCommGrpCat],\n  J.W.inverseImage (PresheafOfModules.toPresheaf R₀) =\n    (CategoryTheory.MorphismProperty.isomorphisms (SheafOfModules R)).inverseImage (PresheafOfModules.sheafification α)","typeReadable":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  {R₀ : CategoryTheory.Functor Cᵒᵖ RingCat} {R : CategoryTheory.Sheaf J 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α)","typeReferences":[["RingHom"],["Opposite"],["AddCommGroup","toAddGroup"],["PresheafOfModules","instCategory"],["RingCat","carrier"],["Semiring","toNonAssocSemiring"],["CategoryTheory","Presheaf","IsLocallySurjective"],["AddMonoidHom"],["CategoryTheory","Presheaf","IsSheaf"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["CategoryTheory","Functor"],["SheafOfModules"],["CategoryTheory","GrothendieckTopology","WEqualsLocallyBijective"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["AddZeroClass","toAddZero"],["CategoryTheory","Category","toCategoryStruct"],["PresheafOfModules","sheafification"],["Ring","toSemiring"],["RingCat","ring"],["CategoryTheory","Sheaf"],["AddMonoid","toAddZeroClass"],["CategoryTheory","Category"],["RingHom","instFunLike"],["PresheafOfModules"],["Quiver","Hom"],["PresheafOfModules","toPresheaf"],["Eq"],["AddCommGrpCat","carrier"],["SheafOfModules","instCategory"],["CategoryTheory","Category","opposite"],["AddCommGrpCat","instCategory"],["CategoryTheory","Presheaf","IsLocallyInjective"],["CategoryTheory","GrothendieckTopology"],["CategoryTheory","HasWeakSheafify"],["AddCommGrpCat"],["RingCat"],["CategoryTheory","CategoryStruct","toQuiver"],["AddCommGrpCat","str"],["SubNegMonoid","toAddMonoid"],["Ab"],["CategoryTheory","MorphismProperty"],["CategoryTheory","MorphismProperty","isomorphisms"],["RingCat","instCategory"],["RingCat","instConcreteCategoryRingHomCarrier"],["CategoryTheory","Functor","category"],["CategoryTheory","GrothendieckTopology","W"],["CategoryTheory","MorphismProperty","inverseImage"]],"valueReferences":[["Eq","trans"],["Opposite"],["CategoryTheory","ObjectProperty","FullSubcategory","category"],["CategoryTheory","IsIso"],["PresheafOfModules","instCategory"],["CategoryTheory","isIso_iff_of_reflects_iso"],["CategoryTheory","Functor","map"],["CategoryTheory","Functor","mapIso"],["Eq","symm"],["CategoryTheory","Presheaf","IsSheaf"],["SheafOfModules"],["CategoryTheory","Functor"],["_private","Mathlib","Algebra","Category","ModuleCat","Sheaf","Localization",0,"PresheafOfModules","inverseImage_W_toPresheaf_eq_inverseImage_isomorphisms","_simp_1_1"],["CategoryTheory","Functor","mapArrowFunctor"],["PresheafOfModules","sheafification"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","MorphismProperty","ext"],["CategoryTheory","GrothendieckTopology","W_eq_inverseImage_isomorphisms"],["SheafOfModules","toSheaf"],["Iff"],["id"],["Eq","mpr"],["CategoryTheory","instCategoryArrow"],["CategoryTheory","Sheaf"],["CategoryTheory","Functor","comp"],["PresheafOfModules"],["CategoryTheory","Functor","obj"],["congrArg"],["PresheafOfModules","toPresheaf"],["CategoryTheory","presheafToSheaf"],["congr"],["PresheafOfModules","instReflectsIsomorphismsSheafOfModulesSheafAddCommGrpCatToSheaf_1"],["CategoryTheory","Arrow"],["Eq"],["propext"],["SheafOfModules","instCategory"],["CategoryTheory","Category","opposite"],["AddCommGrpCat","instCategory"],["_private","Mathlib","Algebra","Category","ModuleCat","Sheaf","Localization",0,"PresheafOfModules","inverseImage_W_toPresheaf_eq_inverseImage_isomorphisms","_simp_1_2"],["AddCommGrpCat"],["CategoryTheory","Iso","app"],["Ab"],["PresheafOfModules","sheafificationCompToSheaf"],["CategoryTheory","MorphismProperty","RespectsIso","isomorphisms"],["CategoryTheory","MorphismProperty","isomorphisms"],["CategoryTheory","MorphismProperty"],["CategoryTheory","GrothendieckTopology","W"],["CategoryTheory","Functor","category"],["CategoryTheory","Arrow","mk"],["CategoryTheory","MorphismProperty","inverseImage"],["CategoryTheory","MorphismProperty","arrow_mk_iso_iff"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","presheafToSheaf","congr_simp"],"typeFallback":"forall {C : Type.{u₁}} [inst._@.Mathlib.CategoryTheory.Sites.Sheafification.2150463039._hygCtx._hyg.3 : CategoryTheory.Category.{v₁, u₁} C] (J : CategoryTheory.GrothendieckTopology.{v₁, u₁} C inst._@.Mathlib.CategoryTheory.Sites.Sheafification.2150463039._hygCtx._hyg.3) (A : Type.{u₂}) [inst._@.Mathlib.CategoryTheory.Sites.Sheafification.2150463039._hygCtx._hyg.9 : CategoryTheory.Category.{v₂, u₂} A] [inst._@.Mathlib.CategoryTheory.Sites.Sheafification.2150463039._hygCtx._hyg.12 : CategoryTheory.HasWeakSheafify.{v₁, v₂, u₁, u₂} C inst._@.Mathlib.CategoryTheory.Sites.Sheafification.2150463039._hygCtx._hyg.3 J A inst._@.Mathlib.CategoryTheory.Sites.Sheafification.2150463039._hygCtx._hyg.9], Eq.{succ (max (max u₁ v₂) (max (max u₂ u₁) v₂) v₁)} (CategoryTheory.Functor.{max u₁ v₂, max u₁ v₂, max (max (max u₂ u₁) v₂) v₁, max (max (max u₂ u₁) v₂) v₁} (CategoryTheory.Functor.{v₁, v₂, u₁, u₂} (Opposite.{succ u₁} C) (CategoryTheory.Category.opposite.{v₁, u₁} C 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.ContinuedFractions.Determinant.sym.json

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1 (One.toOfNat1.{u_1} K (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Determinant.3004083484._hygCtx._hyg.3)))))))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))])","typeFull":"∀ {K : Type u_1} [inst : Field K] {s : SimpContFract K} (n : ℕ) (gp : GenContFract.Pair K),\n  ((↑s).contsAux n).a * ((↑s).contsAux (n + 1)).b - ((↑s).contsAux n).b * ((↑s).contsAux (n + 1)).a = (-1) ^ n →\n    ((↑s).contsAux (n + 1)).a * (((↑s).contsAux n).b + gp.b * ((↑s).contsAux (n + 1)).b) -\n        ((↑s).contsAux (n + 1)).b * (((↑s).contsAux n).a + gp.b * ((↑s).contsAux (n + 1)).a) =\n      (-1) ^ (n + 1)","typeReadable":"∀ {K : Type u_1} [inst : Field K] {s : SimpContFract K} (n : ℕ) (gp : GenContFract.Pair K),\n  ((↑s).contsAux n).a * ((↑s).contsAux (n + 1)).b - ((↑s).contsAux n).b * ((↑s).contsAux (n + 1)).a = (-1) ^ n →\n    ((↑s).contsAux (n + 1)).a * (((↑s).contsAux n).b + gp.b * ((↑s).contsAux (n + 1)).b) -\n        ((↑s).contsAux (n + 1)).b * (((↑s).contsAux n).a + gp.b * ((↑s).contsAux (n + 1)).a) =\n      (-1) ^ (n + 1)","typeReferences":[["instAddNat"],["SubtractionMonoid","toSubNegZeroMonoid"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["Subtype","val"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Monoid","toPow"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["SimpContFract"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["instHPow"],["GenContFract","IsSimpContFract"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["GenContFract","Pair","a"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toSemiring"],["GenContFract","Pair","b"],["Nat"],["AddMonoidWithOne","toOne"],["instHMul"],["GenContFract","Pair"],["Field"],["SubtractionCommMonoid","toSubtractionMonoid"],["CommRing","toNonUnitalCommRing"],["SubNegZeroMonoid","toNegZeroClass"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Eq"],["Distrib","toAdd"],["Field","toCommRing"],["instHAdd"],["Distrib","toMul"],["DivisionRing","toRing"],["Field","toDivisionRing"],["Semiring","toMonoidWithZero"],["GenContFract"],["HPow","hPow"],["OfNat","ofNat"],["Ring","toAddCommGroup"],["HAdd","hAdd"],["AddGroupWithOne","toAddGroup"],["GenContFract","contsAux"],["NegZeroClass","toNeg"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["Field","toSemifield"],["instHSub"]],"valueReferences":[["instAddNat"],["SubtractionMonoid","toSubNegZeroMonoid"],["Lean","Grind","CommRing","Stepwise","simp"],["eagerReduce"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["CommGroupWithZero","toDivisionCommMonoid"],["Lean","Grind","Field","toCommRing"],["Lean","Grind","CommRing","Mon","mult"],["Subtype","val"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Monoid","toPow"],["Lean","Grind","Semiring","npow"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq","symm"],["HSub","hSub"],["Lean","Grind","ne_of_ne_of_eq_right"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Lean","Grind","CommRing","Power","mk"],["Bool","true"],["instHPow"],["GenContFract","IsSimpContFract"],["Lean","Grind","CommRing","Stepwise","mul"],["Lean","Grind","CommSemiring","toSemiring"],["InvOneClass","toInv"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["GenContFract","Pair","a"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Lean","Grind","CommRing","diseq_to_eq"],["DivisionSemiring","toSemiring"],["GenContFract","Pair","b"],["Int","instNegInt"],["Nat"],["instOfNat"],["Lean","Grind","CommRing","Stepwise","core"],["Eq","refl"],["Lean","Grind","CommRing","Poly","num"],["AddMonoidWithOne","toOne"],["Classical","byContradiction"],["id"],["Lean","Grind","CommRing","Expr","var"],["instHMul"],["Lean","Grind","CommRing","Expr","neg"],["DivisionMonoid","toDivInvOneMonoid"],["Lean","Grind","CommRing","Expr","add"],["Bool"],["Lean","Grind","Semiring","pow_one"],["Lean","Grind","CommRing","Expr","mul"],["SubtractionCommMonoid","toSubtractionMonoid"],["CommRing","toNonUnitalCommRing"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["Lean","Grind","CommRing","one_eq_zero_unsat"],["Lean","RArray","leaf"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Lean","RArray","branch"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["Inv","inv"],["Distrib","toAdd"],["Field","toCommRing"],["Lean","Grind","Semiring","pow_add_congr"],["instHAdd"],["DivisionRing","toRing"],["Distrib","toMul"],["Field","toDivisionRing"],["Semiring","toMonoidWithZero"],["Lean","Grind","CommRing","Mon","unit"],["GenContFract"],["HPow","hPow"],["Lean","Grind","CommRing","Expr","sub"],["OfNat","ofNat"],["Int"],["Lean","Grind","CommRing","Expr","num"],["HAdd","hAdd"],["Ring","toAddCommGroup"],["DivInvOneMonoid","toInvOneClass"],["GenContFract","contsAux"],["AddGroupWithOne","toAddGroup"],["NegZeroClass","toNeg"],["Lean","Grind","CommRing","toCommSemiring"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["Field","toGrindField"],["Field","toSemifield"],["False"],["Lean","Grind","CommRing","Poly","add"],["instHSub"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["SimpContFract","determinant_aux"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Determinant.3004083484._hygCtx._hyg.3 : Field.{u_1} K] {s : SimpContFract.{u_1} K (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Determinant.3004083484._hygCtx._hyg.3)))))} {n : Nat}, (Or (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Not (GenContFract.TerminatedAt.{u_1} K (Subtype.val.{succ u_1} (GenContFract.{u_1} K) (fun (g : GenContFract.{u_1} K) => GenContFract.IsSimpContFract.{u_1} K g (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (DivisionRing.toRing.{u_1} K (Field.toDivisionRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Determinant.3004083484._hygCtx._hyg.3)))))) s) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) -> 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.FreeMonoid.Count.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Commute.Hom.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Pointwise.Finset.BigOperators.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Pointwise.Set.Small.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Divisibility.sym.json

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[inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.3841508168._hygCtx._hyg.3 : CommMonoidWithZero.{u_1} α] [inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.3841508168._hygCtx._hyg.8 : Nontrivial.{u_1} α], Not (IsRelPrime.{u_1} α (MonoidWithZero.toMonoid.{u_1} α (CommMonoidWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.3841508168._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (CommMonoidWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.3841508168._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (CommMonoidWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Divisibility.3841508168._hygCtx._hyg.3)))))))","typeFull":"∀ {α : Type u_1} [inst : 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Idempotent.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Units.Lemmas.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.sym.json

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CochainComplex.HomComplex.Cochain.EqUpTo.{v, u} C inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.6 K L d (Subtype.val.{succ v} (CochainComplex.HomComplex.Cochain.{v, u} C inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.6 K L d) (fun (x : CochainComplex.HomComplex.Cochain.{v, u} C inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.6 K L d) => Membership.mem.{v, v} (CochainComplex.HomComplex.Cochain.{v, u} C inst._@.Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction.1634025895._hygCtx._hyg.3 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