module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.TrivSqZeroExt.Basic | {
"line": 822,
"column": 34
} | {
"line": 822,
"column": 65
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : DivisionSemiring R\ninst✝³ : AddCommGroup M\ninst✝² : Module Rᵐᵒᵖ M\ninst✝¹ : Module R M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nhx : x.fst ≠ 0\n⊢ 1 * x⁻¹⁻¹ = x * x⁻¹ * x⁻¹⁻¹",
"usedConstants": [
"Eq.mpr",
"TrivSqZeroExt.one",
"NegZeroClass.... | TrivSqZeroExt.mul_inv_cancel hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GradedMonoid | {
"line": 466,
"column": 70
} | {
"line": 473,
"column": 7
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nα : Type u_3\ninst✝¹ : AddMonoid ι\ninst✝ : Monoid R\nl : List α\nfι : α → ι\nfA : α → R\n⊢ l.dProd fι fA = (map fA l).prod",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"List.map_cons",
"Monoi... | by
match l with
| [] =>
rw [List.dProd_nil, List.map_nil, List.prod_nil]
rfl
| head::tail =>
rw [List.dProd_cons, List.map_cons, List.prod_cons, List.dProd_monoid tail _ _]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Basic | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 79
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_1\nR : Type u_2\nA : Type u_3\nσ : Type u_4\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ni : ... | · simp only [mul_zero, decompose_zero, zero_apply, ZeroMemClass.coe_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.GradedAlgebra.Basic | {
"line": 329,
"column": 63
} | {
"line": 331,
"column": 60
} | [
{
"pp": "ι : Type u_1\nA : Type u_3\nσ : Type u_4\ninst✝¹⁰ : Semiring A\ninst✝⁹ : DecidableEq ι\ninst✝⁸ : AddCommMonoid ι\ninst✝⁷ : PartialOrder ι\ninst✝⁶ : CanonicallyOrderedAdd ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\na b : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹... | by
lift b to 𝒜 i using b_mem
rwa [decompose_mul, decompose_coe, coe_mul_of_apply_of_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 502,
"column": 2
} | {
"line": 503,
"column": 61
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_6\nM₂ : Type u_7\nn : Type u_11\nm : Type u_12\nn' : Type u_13\nm' : Type u_14\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : Fintype n\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ ... | simp_rw [← LinearMap.toMatrix_id_eq_basis_toMatrix]
rw [← LinearMap.toMatrix₂_compl₁₂, LinearMap.compl₁₂_id_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.SesquilinearForm | {
"line": 502,
"column": 2
} | {
"line": 503,
"column": 61
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_6\nM₂ : Type u_7\nn : Type u_11\nm : Type u_12\nn' : Type u_13\nm' : Type u_14\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R M₂\ninst✝⁷ : Fintype n\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ ... | simp_rw [← LinearMap.toMatrix_id_eq_basis_toMatrix]
rw [← LinearMap.toMatrix₂_compl₁₂, LinearMap.compl₁₂_id_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 841,
"column": 2
} | {
"line": 843,
"column": 37
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R M\ninst✝² : Module R N\nN' : Type u_8\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nQ : QuadraticMap R M N\n⊢ (f.compQuadraticMap' Q).polarBilin = compr₂ ... | ext
rw [polarBilin_apply_apply, compr₂_apply, polarBilin_apply_apply,
LinearMap.compQuadraticMap_polar] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 841,
"column": 2
} | {
"line": 843,
"column": 37
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R M\ninst✝² : Module R N\nN' : Type u_8\ninst✝¹ : AddCommGroup N'\ninst✝ : Module R N'\nf : N →ₗ[R] N'\nQ : QuadraticMap R M N\n⊢ (f.compQuadraticMap' Q).polarBilin = compr₂ ... | ext
rw [polarBilin_apply_apply, compr₂_apply, polarBilin_apply_apply,
LinearMap.compQuadraticMap_polar] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 991,
"column": 2
} | {
"line": 991,
"column": 11
} | [
{
"pp": "R : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Invertible 2\nQ : QuadraticMap R M N\nhB₁ : associated' Q ≠ 0\nh : ∀ (x : M), Q x = 0\n⊢ False",
"usedConstants": []
}
] | apply hB₁ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.QuadraticForm.Basic | {
"line": 1368,
"column": 2
} | {
"line": 1368,
"column": 46
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nhB₁ : B ≠ 0\nhB₂ : IsSymm B\n⊢ ∃ x, ¬IsOrtho B x x",
"usedConstants": [
"QuadraticMap.canLift",
"IsScalarTower.to_smulCommClass'",
"Semiring.toModu... | lift B to QuadraticForm R M using hB₂ with Q | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.LinearAlgebra.Multilinear.Curry | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 9
} | [
{
"pp": "R : Type uR\nn : ℕ\nM : Fin n.succ → Type v\nM₂ : Type v₂\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : Fin n.succ) → AddCommMonoid (M i)\ninst✝² : AddCommMonoid M₂\ninst✝¹ : (i : Fin n.succ) → Module R (M i)\ninst✝ : Module R M₂\nf : MultilinearMap R (fun i ↦ M i.castSucc) (M (last n) →ₗ[R] M₂)\n⊢ f.uncurry... | ext m x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Algebra.Category.ModuleCat.Free | {
"line": 71,
"column": 2
} | {
"line": 76,
"column": 10
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nhS : S.Exact\nv : ι → ↑S.X₁\nhv : LinearIndependent R v\nu : ι ⊕ ι' → ↑S.X₂\nhw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)\nhm : Mono S.f\nhuv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ ... | rw [linearIndependent_sum]
refine ⟨?_, LinearIndependent.of_comp S.g.hom hw, disjoint_span_sum hS hw huv⟩
rw [huv, LinearMap.linearIndependent_iff S.f.hom]; swap
· rw [LinearMap.ker_eq_bot, ← mono_iff_injective]
infer_instance
exact hv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Free | {
"line": 71,
"column": 2
} | {
"line": 76,
"column": 10
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nhS : S.Exact\nv : ι → ↑S.X₁\nhv : LinearIndependent R v\nu : ι ⊕ ι' → ↑S.X₂\nhw : LinearIndependent R (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr)\nhm : Mono S.f\nhuv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ ... | rw [linearIndependent_sum]
refine ⟨?_, LinearIndependent.of_comp S.g.hom hw, disjoint_span_sum hS hw huv⟩
rw [huv, LinearMap.linearIndependent_iff S.f.hom]; swap
· rw [LinearMap.ker_eq_bot, ← mono_iff_injective]
infer_instance
exact hv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Free | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 54
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\ninst✝ : Ring R\nS : ShortComplex (ModuleCat R)\nhS : S.Exact\nv : ι → ↑S.X₁\nβ : Type u_4\nu : ι ⊕ β → ↑S.X₂\nhuv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v\nhv : ⊤ ≤ span R (range v)\nhw : ⊤ ≤ span R (range (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr))\nm : ↑S.X₂\na✝ : ... | rw [Finsupp.mem_span_range_iff_exists_finsupp] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 99
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\n⊢ span R (⇑(ιMulti R n) '' {a | range a ⊆ s}) = ⊤",
"usedConstants": [
"AlternatingMap",
"Set.image_image",
"Eq.mpr",
"Submodule",
"RingHomS... | apply LinearMap.map_injective (ker_subtype (⋀[R]^n M))
simpa [LinearMap.map_span, Set.image_image] using ιMulti_span_fixedDegree_of_span_eq_top R n M hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.ExteriorPower.Basic | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 99
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nn : ℕ\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : Set M\nhs : span R s = ⊤\n⊢ span R (⇑(ιMulti R n) '' {a | range a ⊆ s}) = ⊤",
"usedConstants": [
"AlternatingMap",
"Set.image_image",
"Eq.mpr",
"Submodule",
"RingHomS... | apply LinearMap.map_injective (ker_subtype (⋀[R]^n M))
simpa [LinearMap.map_span, Set.image_image] using ιMulti_span_fixedDegree_of_span_eq_top R n M hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.IsConnected | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 30
} | [
{
"pp": "case trans\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nc d x✝ y✝ z✝ : (j : C) × F.obj j\na✝¹ : Relation.EqvGen F.ColimitTypeRel x✝ y✝\na✝ : Relation.EqvGen F.ColimitTypeRel y✝ z✝\nih₁ : Zigzag x✝.fst y✝.fst\nih₂ : Zigzag y✝.fst z✝.fst\n⊢ Zigzag x✝.fst z✝.fst",
"usedConstants": [
"... | | trans _ _ _ _ _ ih₁ ih₂ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.Limits.IsConnected | {
"line": 154,
"column": 11
} | {
"line": 154,
"column": 13
} | [
{
"pp": "case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nx : C\nh : Limits.IsInitial x\nthis : Nonempty C := Nonempty.intro x\nj₁ : C\n⊢ ∀ (j₂ : C), ∃ l, List.IsChain Zag (j₁ :: l) ∧ (j₁ :: l).getLast ⋯ = j₂",
"usedConstants": []
}
] | j₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.IsConnected | {
"line": 164,
"column": 11
} | {
"line": 164,
"column": 13
} | [
{
"pp": "case h\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nx : C\nh : Limits.IsTerminal x\nthis : Nonempty C := Nonempty.intro x\nj₁ : C\n⊢ ∀ (j₂ : C), ∃ l, List.IsChain Zag (j₁ :: l) ∧ (j₁ :: l).getLast ⋯ = j₂",
"usedConstants": []
}
] | j₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Adjunction.PartialAdjoint | {
"line": 97,
"column": 52
} | {
"line": 98,
"column": 30
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : D ⥤ C\nX Y : F.PartialLeftAdjointSource\nf : X ⟶ Y\n⊢ F.partialLeftAdjointHomEquiv (F.partialLeftAdjointMap f) =\n f.hom ≫ F.partialLeftAdjointHomEquiv (𝟙 (F.partialLeftAdjointObj Y))",
"usedConstants": [
... | by
simp [partialLeftAdjointMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\ninst✝¹ : IsCofiltered C\ninst✝ : InitiallySmall C\nR : Cᵒᵖ ⥤ RingCat\ncR : Cocone R\nhcR : IsColimit cR\nM : PresheafOfModules R\ncM : Cocone M.presheaf\nhcM : IsColimit cM\nM'✝ : PresheafOfModules R\ncM'✝ : Cocone M'✝.presheaf\... | let c := (Cocone.precompose ((toPresheaf _).map f)).obj cM' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Pullback | {
"line": 78,
"column": 16
} | {
"line": 78,
"column": 42
} | [
{
"pp": "C D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nF : C ⥤ D\nR : Dᵒᵖ ⥤ RingCat\nS : Cᵒᵖ ⥤ RingCat\nφ : S ⟶ F.op ⋙ R\nX : C\nM N : PresheafOfModules R\ng : M ⟶ N\nf : (free R).obj (yoneda.obj (F.obj X)) ⟶ M\n| freeYonedaEquiv ({ toFun := fun g_1 ↦ g_1 ≫ (pushforward φ).map g } ((freeYoned... | erw [freeYonedaEquiv_comp] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convErw___1 | Lean.Parser.Tactic.Conv.convErw__ |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Pullback | {
"line": 78,
"column": 16
} | {
"line": 78,
"column": 42
} | [
{
"pp": "C D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nF : C ⥤ D\nR : Dᵒᵖ ⥤ RingCat\nS : Cᵒᵖ ⥤ RingCat\nφ : S ⟶ F.op ⋙ R\nX : C\nM N : PresheafOfModules R\ng : M ⟶ N\nf : (free R).obj (yoneda.obj (F.obj X)) ⟶ M\n| freeYonedaEquiv ({ toFun := fun g_1 ↦ g_1 ≫ (pushforward φ).map g } ((freeYoned... | erw [freeYonedaEquiv_comp] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Pullback | {
"line": 78,
"column": 16
} | {
"line": 78,
"column": 42
} | [
{
"pp": "C D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nF : C ⥤ D\nR : Dᵒᵖ ⥤ RingCat\nS : Cᵒᵖ ⥤ RingCat\nφ : S ⟶ F.op ⋙ R\nX : C\nM N : PresheafOfModules R\ng : M ⟶ N\nf : (free R).obj (yoneda.obj (F.obj X)) ⟶ M\n| freeYonedaEquiv ({ toFun := fun g_1 ↦ g_1 ≫ (pushforward φ).map g } ((freeYoned... | erw [freeYonedaEquiv_comp] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Sites.Grothendieck | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 64
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nι : Type u_1\nX : C\nZ : ι → C\nf : (i : ι) → Z i ⟶ X\nR : (i : ι) → Presieve (Z i)\nh : Sieve.ofArrows Z f ∈ J X\nhR : ∀ (i : ι), Sieve.generate (R i) ∈ J (Z i)\n⊢ (Sieve.bind (Sieve.ofArrows Z f).arrows fun x x_1 hg ↦\n Sieve.pul... | exact J.bind_covering h fun _ _ _ ↦ J.pullback_stable _ (hR _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Pretopology | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 73
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nf : Y ⟶ X\nZ : C\ng : Z ⟶ X\ni : IsIso g\n⊢ IsIso (pullback.snd g f)",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Limits.pullback",
"CategoryTheory.Limits.pullback.lift... | refine ⟨⟨pullback.lift (f ≫ inv g) (𝟙 _) (by simp), ⟨?_, by simp⟩⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Pretopology | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 30
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nf : Y ⟶ X\nZ : C\ng : Z ⟶ X\ni : IsIso g\n⊢ pullbackArrows f (Presieve.singleton g) = Presieve.singleton (pullback.snd g f)",
"usedConstants": [
"CategoryTheory.Limits.WidePullbackShape.category",
"C... | · apply pullback_singleton | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.h.a.mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nf : Y ⟶ X\ng : Z ⟶ X\ninst✝ : HasPullback g f\nW : C\nh : W ⟶ Y\n⊢ singleton (pullback.snd g f) h → pullbackArrows f (singleton g) h",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Presieve.sing... | rintro ⟨_⟩
exact pullbackArrows.mk Z g singleton.mk | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 44
} | [
{
"pp": "case h.h.a.mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nf : Y ⟶ X\ng : Z ⟶ X\ninst✝ : HasPullback g f\nW : C\nh : W ⟶ Y\n⊢ singleton (pullback.snd g f) h → pullbackArrows f (singleton g) h",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Presieve.sing... | rintro ⟨_⟩
exact pullbackArrows.mk Z g singleton.mk | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.SheafOfTypes | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\ns : Cocone S.arrows.diagram\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nhf₁ : S.arrows f₁\nhf₂ : S.arrows f₂\nhgf : g₁ ≫ f₁ = g₂ ≫ f₂\nHs :\n ∀ ⦃X_1 Y : S.arrows.category⦄ (f : X_1 ⟶ Y),\n S.arrows.diagram.map f ≫ s.ι.... | let F₂ : (Over.mk (g₂ ≫ f₂) : Over X) ⟶ (Over.mk f₂ : Over X) := Over.homMk g₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Sites.SheafOfTypes | {
"line": 226,
"column": 46
} | {
"line": 251,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nS : Sieve X\n⊢ (∀ (W : C), IsSheafFor (yoneda.obj W) S.arrows) ↔ Nonempty (IsColimit S.arrows.cocone)",
"usedConstants": [
"CategoryTheory.Presieve.yonedaFamilyOfElements_fromCocone",
"Eq.mpr",
"CategoryTheory.Limits.IsColimit.fac",
... | by
constructor
· intro H
refine Nonempty.intro ?_
exact
{ desc := fun s => H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
(yonedaFamily_fromCocone_compatible S s) |>.choose
fac := by
intro s f
replace H := H s.pt (yonedaFamilyOfElements_fromCocone S.arrows s)
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1119,
"column": 35
} | {
"line": 1124,
"column": 30
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\nX✝ Y Z : C\nf : Y ⟶ X✝\nS R : Sieve X✝\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\nG : D ⥤ E\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nX : C\n⊢ GaloisCoinsertion (functorPushforward F) (functorPullback F)",
... | by
apply (functor_galoisConnection F X).toGaloisCoinsertion
rintro S Y f ⟨Z, g, h, h₁, h₂⟩
rw [← F.map_preimage h, ← F.map_comp] at h₂
rw [F.map_injective h₂]
exact S.downward_closed h₁ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1290,
"column": 4
} | {
"line": 1292,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R✝ : Sieve X\nR : Cᵒᵖ ⥤ Type (max w v₁)\nf : R ⟶ uliftYoneda.{w, v₁, u₁}.obj X\n⊢ ∀ {Y Z : C} {f_1 : Y ⟶ X},\n (∃ t, (hom (f.app (Opposite.op Y))) t = { down := f_1 }) →\n ∀ ... | intro Y Z _ ⟨t, ht⟩ g
refine ⟨R.map g.op t, ?_⟩
simp [ht] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1290,
"column": 4
} | {
"line": 1292,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf✝ : Y ⟶ X\nS R✝ : Sieve X\nR : Cᵒᵖ ⥤ Type (max w v₁)\nf : R ⟶ uliftYoneda.{w, v₁, u₁}.obj X\n⊢ ∀ {Y Z : C} {f_1 : Y ⟶ X},\n (∃ t, (hom (f.app (Opposite.op Y))) t = { down := f_1 }) →\n ∀ ... | intro Y Z _ ⟨t, ht⟩ g
refine ⟨R.map g.op t, ?_⟩
simp [ht] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Sieves | {
"line": 1307,
"column": 2
} | {
"line": 1310,
"column": 22
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nι : Type u_1\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nY : C\ng : Y ⟶ S\nP : ι → C\np₁ : (i : ι) → P i ⟶ Y\np₂ : (i : ι) → P i ⟶ X i\nh : ∀ (i : ι), IsPullback (p₁ i) (p₂ i) g (f i)\n⊢ ofArrows P p₁ ≤ pullback g (ofArrows X f)",
"usedConstant... | · rw [Sieve.ofArrows, Sieve.generate_le_iff]
rintro - - ⟨i⟩
use X i, p₂ i, f i, ⟨i⟩
exact (h i).w.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 400,
"column": 6
} | {
"line": 400,
"column": 41
} | [
{
"pp": "case mpr.refine_1\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S ∈ J Y\nz : Presieve.FamilyOfElements (P ⋙ coyoneda.obj (op X)) S.arrows\nhz : z.Compatible\nx : unop (op X)\n⊢ Presieve.FamilyOfEl... | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 397,
"column": 4
} | {
"line": 414,
"column": 63
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\n⊢ Presieve.IsSheaf J P → Presheaf.IsSheaf J P",
"usedConstants": [
"CategoryTheory.Presieve.IsSheaf",
"Eq.mpr",
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.coyon... | intro hP X Y S hS z hz
refine ⟨↾fun x => (hP S hS).amalgamate (fun Z f hf ↦
(ConcreteCategory.hom (z f hf)) x) ?_, ?_, ?_⟩
· intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ h
exact (ConcreteCategory.congr_hom (hz g₁ g₂ hf₁ hf₂ h)) x
· intro Z f hf
apply ConcreteCategory.hom_ext
intro x
simp ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Sheaf | {
"line": 397,
"column": 4
} | {
"line": 414,
"column": 63
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type w\n⊢ Presieve.IsSheaf J P → Presheaf.IsSheaf J P",
"usedConstants": [
"CategoryTheory.Presieve.IsSheaf",
"Eq.mpr",
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.coyon... | intro hP X Y S hS z hz
refine ⟨↾fun x => (hP S hS).amalgamate (fun Z f hf ↦
(ConcreteCategory.hom (z f hf)) x) ?_, ?_, ?_⟩
· intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ h
exact (ConcreteCategory.congr_hom (hz g₁ g₂ hf₁ hf₂ h)) x
· intro Z f hf
apply ConcreteCategory.hom_ext
intro x
simp ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subfunctor.Image | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 21
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nF F' : C ⥤ Type w\nG : Subfunctor F\np : F' ⟶ F\nhp : Epi p\n⊢ (G.preimage p).image p ≤ G",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Subfunctor.image",
"le_refl",
"congrArg",
"PartialOrder.toPreorder",
"CategoryT... | rw [image_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Subfunctor.Image | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 21
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nF F' : C ⥤ Type w\nG : Subfunctor F\np : F' ⟶ F\nhp : Epi p\n⊢ (G.preimage p).image p ≤ G",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Subfunctor.image",
"le_refl",
"congrArg",
"PartialOrder.toPreorder",
"CategoryT... | rw [image_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subfunctor.Image | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 21
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nF F' : C ⥤ Type w\nG : Subfunctor F\np : F' ⟶ F\nhp : Epi p\n⊢ (G.preimage p).image p ≤ G",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Subfunctor.image",
"le_refl",
"congrArg",
"PartialOrder.toPreorder",
"CategoryT... | rw [image_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subfunctor.Sieves | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nU : Cᵒᵖ\ns : F.obj U\n⊢ (G.familyOfElementsOfSection s).Compatible",
"usedConstants": []
}
] | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ e | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF : Cᵒᵖ ⥤ Type w\nG : Subfunctor F\nhF : Presieve.IsSheaf J F\nU : C\nS : Sieve U\nhS : S ∈ J U\nx : Presieve.FamilyOfElements (sheafify J G).toFunctor S.arrows\nhx : x.Compatible\nS' : Sieve U := Sieve.bind S.arrows fun Y f hf ↦ G.siev... | choose W i₁ i₂ hi₂ h₁ h₂ using this | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.CategoryTheory.Sites.Subsheaf | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 73
} | [
{
"pp": "case h.toFun.h\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subfunctor F\nf : G.toFunctor ⟶ F'\nh : Presieve.IsSheaf J F'\nU V : Cᵒᵖ\ni : U ⟶ V\ns : (sheafify J G).toFunctor.obj U\nW : C\nj : W ⟶ unop V\nhj : (G.sieveOfSection ↑((ConcreteCategory.h... | refine Eq.trans ?_ (Presieve.IsSheafFor.valid_glue (h _ s.2)
((G.family_of_elements_compatible s.1).map f) (j ≫ i.unop) ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 37
} | [
{
"pp": "case w.refine_1\nJ : Type v\ninst✝ : Category.{w, v} J\nX Y : TopCat\nS : Cone (pair X Y)\nm : S.pt ⟶ (X.prodBinaryFan Y).pt\nh : ∀ (j : Discrete WalkingPair), m ≫ (X.prodBinaryFan Y).π.app j = S.π.app j\nx : ↑S.pt\n⊢ ((ConcreteCategory.hom m) x).1 =\n ((ConcreteCategory.hom\n (ofHom\n ... | specialize h ⟨WalkingPair.left⟩ | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 272,
"column": 12
} | {
"line": 272,
"column": 44
} | [
{
"pp": "X Y : TopCat\nc : BinaryCofan X Y\nh₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl)\nh₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))\n⊢ ∀ (x : ↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := W... | eq_compl_iff_isCompl.mpr h₃.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 271,
"column": 6
} | {
"line": 273,
"column": 29
} | [
{
"pp": "case mpr\nX Y : TopCat\nc : BinaryCofan X Y\nh₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl)\nh₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))\n⊢ Nonempty (IsColimit c)",
"usedConstants": [
"Iff.mpr... | have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by
rw [eq_compl_iff_isCompl.mpr h₃.symm]
exact fun _ => or_not | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Category.TopCat.Limits.Products | {
"line": 297,
"column": 19
} | {
"line": 297,
"column": 51
} | [
{
"pp": "X Y : TopCat\nc : BinaryCofan X Y\nh₁ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inl)\nh₂ : IsOpenEmbedding ⇑(ConcreteCategory.hom c.inr)\nh₃ : IsCompl (range ⇑(ConcreteCategory.hom c.inl)) (range ⇑(ConcreteCategory.hom c.inr))\nthis :\n ∀ (x : ↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj {... | eq_compl_iff_isCompl.mpr h₃.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.TopCat.Limits.Pullbacks | {
"line": 333,
"column": 2
} | {
"line": 339,
"column": 69
} | [
{
"pp": "X Y S : TopCat\nf : X ⟶ S\ng : Y ⟶ S\nhg : IsEmbedding ⇑(ConcreteCategory.hom g)\nH : Set.range ⇑(ConcreteCategory.hom f) ⊆ Set.range ⇑(ConcreteCategory.hom g)\n⊢ IsIso (pullback.fst f g)",
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Set.mem_range_self",
... | let esto : (pullback f g : TopCat) ≃ₜ X :=
(fst_isEmbedding_of_right f hg).toHomeomorph.trans
{ toFun := Subtype.val
invFun := fun x =>
⟨x, by
rw [pullback_fst_range]
exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩ } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 110,
"column": 6
} | {
"line": 112,
"column": 19
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B : Type u\n⊢ Mono (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Mono",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"Categor... | rw [mono_iff_injective]
intro a₁ a₂ h
simpa using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.MonoCoprod | {
"line": 110,
"column": 6
} | {
"line": 112,
"column": 19
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nA B : Type u\n⊢ Mono (BinaryCofan.mk (↾Sum.inl) (↾Sum.inr)).inl",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Mono",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"Categor... | rw [mono_iff_injective]
intro a₁ a₂ h
simpa using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nι : Type u_1\nX : ι → C\nc : Cofan X\nhc : IsColimit c\ninst✝ : ∀ (i : ι), Mono (c.inj i)\ns : {i j : ι} → i ≠ j → PullbackCone (c.inj i) (c.inj j)\nhs : {i j : ι} → (hij : i ≠ j) → IsLimit (s hij)\nH : {i j : ι} → (hij : i ≠ j) → IsInitial (s hij).pt\nd : Cofan ... | rw [show d.inj i = c.inj i ≫ (hd.uniqueUpToIso hc).inv.hom by simp]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nι : Type u_1\nX : ι → C\nc : Cofan X\nhc : IsColimit c\ninst✝ : ∀ (i : ι), Mono (c.inj i)\ns : {i j : ι} → i ≠ j → PullbackCone (c.inj i) (c.inj j)\nhs : {i j : ι} → (hij : i ≠ j) → IsLimit (s hij)\nH : {i j : ι} → (hij : i ≠ j) → IsInitial (s hij).pt\nd : Cofan ... | rw [show d.inj i = c.inj i ≫ (hd.uniqueUpToIso hc).inv.hom by simp]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 49
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' : H.IsVanKampen\nX' Y' Z' : C\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\ninst✝ : HasPullback αX f\nhh : IsPullback h' αX αZ h\nhi : IsPullback i' α... | refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 94,
"column": 12
} | {
"line": 94,
"column": 14
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : H✝.IsVanKampen\n⊢ IsVanKampenColimit (PushoutCocone.mk h i ⋯)",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Limits.WalkingSpan",
"C... | F' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Extensive | {
"line": 205,
"column": 7
} | {
"line": 205,
"column": 52
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteCoproducts C\ninst✝ : HasPullbacksOfInclusions C\nT : C\nHT : IsTerminal T\nc₀ : BinaryCofan T T\nhc₀ : IsColimit c₀\nH :\n ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ T) (αY : Y' ⟶ T) (f : c'.pt ⟶ c₀.pt),\n αX ≫ c₀.inl = c'.inl ≫ f →\... | by rw [← reassoc_of% hY, hd', Category.assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 17
} | [
{
"pp": "J : Type u₁\nK : Type u₂\ninst✝⁴ : SmallCategory J\ninst✝³ : Category.{v₂, u₂} K\ninst✝² : Small.{v, u₂} K\ninst✝¹ : FinCategory J\nF : J × K ⥤ Type v\ninst✝ : IsFiltered K\nx : limit (curry.obj F ⋙ colim)\nk : J → K\ny : (j : J) → F.obj (j, k j)\ne :\n ∀ (j : J),\n (ConcreteCategory.hom (colimit.ι... | j₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 236,
"column": 26
} | {
"line": 236,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW E X Z : C\nc : BinaryCofan W E\ninst✝¹ : FinitaryExtensive C\ninst✝ : HasPullbacks C\nhc : IsColimit c\nf : W ⟶ X\nh : X ⟶ Z\ni : c.pt ⟶ Z\nH : IsPushout f c.inl h i\nhc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh... | pullback.lift_snd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 239,
"column": 26
} | {
"line": 239,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nW E X Z : C\nc : BinaryCofan W E\ninst✝¹ : FinitaryExtensive C\ninst✝ : HasPullbacks C\nhc : IsColimit c\nf : W ⟶ X\nh : X ⟶ Z\ni : c.pt ⟶ Z\nH : IsPushout f c.inl h i\nhc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh... | pullback.lift_snd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 399,
"column": 12
} | {
"line": 399,
"column": 68
} | [
{
"pp": "case h'_w.h'_w\nJ : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z✝ : C\nf✝ : W ⟶ X\ng✝ : W ⟶ Y\nh : X ⟶ Z✝\ni : Y ⟶ Z✝\ninst✝² : Adhesive C\nZ A B : C\na : A ⟶ Z\nb : B ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nK : C\nf g : K ⟶ pushout (fst a b) (snd a b)\nw : f ≫ p... | ← cancel_mono (u ≫ pushout.desc a b pullback.condition), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adhesive.Basic | {
"line": 414,
"column": 12
} | {
"line": 414,
"column": 25
} | [
{
"pp": "case i'_w.h'_w\nJ : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nW X Y Z✝ : C\nf✝ : W ⟶ X\ng✝ : W ⟶ Y\nh : X ⟶ Z✝\ni : Y ⟶ Z✝\ninst✝² : Adhesive C\nZ A B : C\na : A ⟶ Z\nb : B ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nK : C\nf g : K ⟶ pushout (fst a b) (snd a b)\nw : f ≫ p... | sq₂₁.w_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Opposites | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ Dᵒᵖ\nG : D ⥤ Cᵒᵖ\na : F ⊣ G.leftOp\n⊢ a.leftOp = (opOpEquivalence D).symm.toAdjunction.comp a.op",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"CategoryTheory.CategoryStruc... | ext X; simp [Equivalence.unit] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Opposites | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ Dᵒᵖ\nG : D ⥤ Cᵒᵖ\na : F ⊣ G.leftOp\n⊢ a.leftOp = (opOpEquivalence D).symm.toAdjunction.comp a.op",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"CategoryTheory.CategoryStruc... | ext X; simp [Equivalence.unit] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Opposites | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : Cᵒᵖ ⥤ D\nG : Dᵒᵖ ⥤ C\na : F.rightOp ⊣ G\n⊢ a.rightOp = (opOpEquivalence D).symm.toAdjunction.comp a.op",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"CategoryTheory.CategoryStr... | ext X; simp [Equivalence.unit] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Opposites | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 32
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : Cᵒᵖ ⥤ D\nG : Dᵒᵖ ⥤ C\na : F.rightOp ⊣ G\n⊢ a.rightOp = (opOpEquivalence D).symm.toAdjunction.comp a.op",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"CategoryTheory.CategoryStr... | ext X; simp [Equivalence.unit] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.LeftExact | {
"line": 255,
"column": 2
} | {
"line": 256,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝¹³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹² : Category.{t, w} D\ninst✝¹¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\ninst✝¹⁰ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝⁹ : (X Y ... | haveI : HasLimitsOfShape (AsSmall.{t} (FinCategory.AsType K)) D :=
Limits.hasLimitsOfShape_of_equivalence e | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 17
} | [
{
"pp": "case left\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\nZ : C\nx✝ : P Z\n⊢ Function.Injective fun g ↦ f ≫ g",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Category.toCategoryS... | intro g₁ g₂ _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 169,
"column": 33
} | {
"line": 172,
"column": 13
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nP : ObjectProperty C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : P.isColocal g\nhfg : P.isColocal (f ≫ g)\nX : C\nhX : P X\n⊢ Function.Bijective fun f_1 ↦ f_1 ≫ f",
"usedConstants": [
"Eq.mpr",
... | by
rw [← Function.Bijective.of_comp_iff' (hg X hX)]
convert! hfg X hX
cat_disch | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Bousfield | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 17
} | [
{
"pp": "case left\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : ObjectProperty C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\nZ : C\nx✝ : P Z\n⊢ Function.Injective fun f_1 ↦ f_1 ≫ f",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.Category.toCateg... | intro g₁ g₂ _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 384,
"column": 4
} | {
"line": 386,
"column": 74
} | [
{
"pp": "J : Type v'\ninst✝⁸ : Category.{u', v'} J\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u_2\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasColimitsOfShape J C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsVanKampenColimit c\ninst... | convert!
@IsColimit.coconePointUniqueUpToIso_hom_desc _ _ _ _ ((F' ⋙ Gr) ⋙ Gl)
(Gl.mapCocone ⟨_, (whiskerRight α' Gr ≫ c.2 :)⟩) _ _ hl hr using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Sites.PreservesSheafification | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u_1\nB : Type u_2\ninst✝³ : Category.{v_1, u_1} A\ninst✝² : Category.{v_2, u_2} B\nF : A ⥤ B\ninst✝¹ : J.PreservesSheafification F\ninst✝ : HasWeakSheafify J B\nP₁ P₂ : Cᵒᵖ ⥤ A\nf : P₁ ⟶ P₂\nhf : J.W f\n⊢ IsIso ((presheafToShe... | rw [← W_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.ChangeOfRings | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 34
} | [
{
"pp": "case a\nC : Type u'\ninst✝¹ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR R' : Cᵒᵖ ⥤ RingCat\nα : R ⟶ R'\nM₁ M₂ : PresheafOfModules R'\nhM₂ : Presheaf.IsSheaf J M₂.presheaf\ninst✝ : Presheaf.IsLocallySurjective J α\ng : (restrictScalars α).obj M₁ ⟶ (restrictScalars α).obj M₂\nX : Cᵒᵖ\nr' : ↑(R'.... | erw [← (g.app _).hom.map_smul] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 754,
"column": 4
} | {
"line": 758,
"column": 53
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nι : Type u_3\nS B : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nf : (i : ι) → X i ⟶ S\nu : a.pt ⟶ S\nv : B ⟶ S\ns : (i : ι) → PullbackCone v (f i)\nhs : (i : ι) → IsLimit (s i)\nt : PullbackCone v u\nht : IsLimit t\nd : Cofan fun i ↦ (s i... | simp only [Discrete.functor_obj_eq_as, Cofan.mk_pt, Functor.const_obj_obj, Cofan.mk_ι_app,
Discrete.natTrans_app]
rw [← Cofan.inj]
refine IsPullback.of_right ?_ (by simp) (IsPullback.of_isLimit ht)
simpa [hu] using (IsPullback.of_isLimit (hs j.1)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 754,
"column": 4
} | {
"line": 758,
"column": 53
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nι : Type u_3\nS B : C\nX : ι → C\na : Cofan X\nhau : IsUniversalColimit a\nf : (i : ι) → X i ⟶ S\nu : a.pt ⟶ S\nv : B ⟶ S\ns : (i : ι) → PullbackCone v (f i)\nhs : (i : ι) → IsLimit (s i)\nt : PullbackCone v u\nht : IsLimit t\nd : Cofan fun i ↦ (s i... | simp only [Discrete.functor_obj_eq_as, Cofan.mk_pt, Functor.const_obj_obj, Cofan.mk_ι_app,
Discrete.natTrans_app]
rw [← Cofan.inj]
refine IsPullback.of_right ?_ (by simp) (IsPullback.of_isLimit ht)
simpa [hu] using (IsPullback.of_isLimit (hs j.1)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify | {
"line": 188,
"column": 2
} | {
"line": 191,
"column": 68
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.obj\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrpCat\nφ : M₀.presheaf ⟶ A.obj\ninst✝¹ : Pr... | have hr₀ : (r₀.map (whiskerRight α (forget _))).IsAmalgamation r := by
rw [Presieve.FamilyOfElements.restrict_map]
apply Presieve.isAmalgamation_restrict
apply Presieve.FamilyOfElements.isAmalgamation_map_localPreimage | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 386,
"column": 4
} | {
"line": 393,
"column": 38
} | [
{
"pp": "case refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\nX : E\ninst✝¹ : RepresentablyFlat F\ninst✝ : IsCofiltered (StructuredArrow X G)\nT : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow... | let U := IsCofiltered.min (T.obj A) (T.obj B)
let A' : StructuredArrow U.right F := .mk (IsCofiltered.minToLeft (T.obj A) (T.obj B)).right
let B' : StructuredArrow U.right F := .mk (IsCofiltered.minToRight (T.obj A) (T.obj B)).right
refine ⟨.mk <| U.hom ≫ G.map (IsCofiltered.min A' B').hom,
Structured... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 386,
"column": 4
} | {
"line": 393,
"column": 38
} | [
{
"pp": "case refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\nX : E\ninst✝¹ : RepresentablyFlat F\ninst✝ : IsCofiltered (StructuredArrow X G)\nT : StructuredArrow X (F ⋙ G) ⥤ StructuredArrow... | let U := IsCofiltered.min (T.obj A) (T.obj B)
let A' : StructuredArrow U.right F := .mk (IsCofiltered.minToLeft (T.obj A) (T.obj B)).right
let B' : StructuredArrow U.right F := .mk (IsCofiltered.minToRight (T.obj A) (T.obj B)).right
refine ⟨.mk <| U.hom ≫ G.map (IsCofiltered.min A' B').hom,
Structured... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 23
} | [
{
"pp": "case a.inl\nC : Type u\ninst✝ : Category.{v, u} C\nS : C\nE : PreZeroHypercover S\nF : PreZeroHypercover S\nY : C\ni : E.I₀\n⊢ Presieve.ofArrows (E.sum F).X (E.sum F).f (E.f i)",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"CategoryTheory.Presieve.ofArrows.mk",
"Sum",... | exact ⟨Sum.inl i⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 23
} | [
{
"pp": "case a.inl\nC : Type u\ninst✝ : Category.{v, u} C\nS : C\nE : PreZeroHypercover S\nF : PreZeroHypercover S\nY : C\ni : E.I₀\n⊢ Presieve.ofArrows (E.sum F).X (E.sum F).f (E.f i)",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"CategoryTheory.Presieve.ofArrows.mk",
"Sum",... | exact ⟨Sum.inl i⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 23
} | [
{
"pp": "case a.inl\nC : Type u\ninst✝ : Category.{v, u} C\nS : C\nE : PreZeroHypercover S\nF : PreZeroHypercover S\nY : C\ni : E.I₀\n⊢ Presieve.ofArrows (E.sum F).X (E.sum F).f (E.f i)",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"CategoryTheory.Presieve.ofArrows.mk",
"Sum",... | exact ⟨Sum.inl i⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 453,
"column": 2
} | {
"line": 455,
"column": 87
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nf : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\n𝒢' : Sheaf K... | apply (restrictHomEquivHom (G := G)).symm.injective
simpa only [Equiv.symm_apply_apply] using
(restrictHomEquivHom_naturality_right_symm (G := G) (restrictHomEquivHom f) g).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 453,
"column": 2
} | {
"line": 455,
"column": 87
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nA : Type u_4\ninst✝² : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nℱ : Dᵒᵖ ⥤ A\nℱ' : Sheaf K A\nf : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\n𝒢' : Sheaf K... | apply (restrictHomEquivHom (G := G)).symm.injective
simpa only [Equiv.symm_apply_apply] using
(restrictHomEquivHom_naturality_right_symm (G := G) (restrictHomEquivHom f) g).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 721,
"column": 2
} | {
"line": 722,
"column": 39
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C ⥤ D\nA : Type u_4\ninst✝³ : Category.{v_4, u_4} A\ninst✝² : IsDenseSubsite J K G\ninst✝¹ : (G.sheafPushforwardContinuous A J K).IsEquivalence\ninst✝ ... | · congr 1
apply adj₁.homEquiv_naturality_left | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Over | {
"line": 305,
"column": 8
} | {
"line": 305,
"column": 22
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nX Y : C\nf : X ⟶ Y\nU : Over X\nS : Sieve ((Over.map f).obj U)\nhS : S ∈ (J.over Y) ((Over.map f).obj U)\n⊢ Sieve.functorPullback (Over.map f) S ∈ (J.over X) U",
"usedConstants": [
"CategoryTheory.Over.map",
"CategoryThe... | J.mem_over_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Over | {
"line": 342,
"column": 2
} | {
"line": 352,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ✝ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nX : C\ninst✝ : F.PreservesOneHypercovers J K\n⊢ (Over.post F).PreservesOneHypercovers (J.over X) (K.over (F.obj X))",
... | intro Y E
let E' := (E.map (Over.forget X) J).map F K
refine ⟨?_, ?_⟩
· dsimp [-Over.post_obj]
rw [PreZeroHypercover.sieve₀_map, GrothendieckTopology.mem_over_iff,
Sieve.functorPushforward_ofArrows, Sieve.overEquiv_ofArrows]
exact E'.mem₀
· intro i₁ i₂ W p₁ p₂ w
simp_rw [GrothendieckTopology.m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Over | {
"line": 342,
"column": 2
} | {
"line": 352,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ✝ : GrothendieckTopology C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nX : C\ninst✝ : F.PreservesOneHypercovers J K\n⊢ (Over.post F).PreservesOneHypercovers (J.over X) (K.over (F.obj X))",
... | intro Y E
let E' := (E.map (Over.forget X) J).map F K
refine ⟨?_, ?_⟩
· dsimp [-Over.post_obj]
rw [PreZeroHypercover.sieve₀_map, GrothendieckTopology.mem_over_iff,
Sieve.functorPushforward_ofArrows, Sieve.overEquiv_ofArrows]
exact E'.mem₀
· intro i₁ i₂ W p₁ p₂ w
simp_rw [GrothendieckTopology.m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX : C\n⊢ (x.familyOfElements X).Compatible",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.H... | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ ⟨i₁, ⟨φ₁⟩⟩ ⟨i₂, ⟨φ₂⟩⟩ _
simpa [hx.familyOfElements_apply f₁ i₁ φ₁,
hx.familyOfElements_apply f₂ i₂ φ₂] using hx Z i₁ i₂ (g₁ ≫ φ₁) (g₂ ≫ φ₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.CoversTop.Basic | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nI : Type u_1\nY : I → C\nx : FamilyOfElementsOnObjects F Y\nhx : x.IsCompatible\nX : C\n⊢ (x.familyOfElements X).Compatible",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.H... | intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ ⟨i₁, ⟨φ₁⟩⟩ ⟨i₂, ⟨φ₂⟩⟩ _
simpa [hx.familyOfElements_apply f₁ i₁ φ₁,
hx.familyOfElements_apply f₂ i₂ φ₂] using hx Z i₁ i₂ (g₁ ≫ φ₁) (g₂ ≫ φ₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 785,
"column": 18
} | {
"line": 785,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS : C\nE✝¹ E✝ : PreOneHypercover S\nF✝ : PreOneHypercover S\nG : PreOneHypercover S\nE F : PreOneHypercover S\nf : E ≅ F\nx✝ : F.multicospanShape.R\n⊢ (Hom.mapMulticospan f.inv ⋙ Hom.mapMulticospan f.hom).obj (WalkingM... | dsimp; congr 1; apply PreOneHypercover.I₁'.ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 785,
"column": 18
} | {
"line": 785,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nS : C\nE✝¹ E✝ : PreOneHypercover S\nF✝ : PreOneHypercover S\nG : PreOneHypercover S\nE F : PreOneHypercover S\nf : E ≅ F\nx✝ : F.multicospanShape.R\n⊢ (Hom.mapMulticospan f.inv ⋙ Hom.mapMulticospan f.hom).obj (WalkingM... | dsimp; congr 1; apply PreOneHypercover.I₁'.ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Sheaf.PushforwardContinuous | {
"line": 195,
"column": 4
} | {
"line": 196,
"column": 72
} | [
{
"pp": "case w.h.h.h.hf.h\nC✝ : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C✝\nD✝ : Type u₂\ninst✝⁸ : Category.{v₂, u₂} D✝\nD' : Type u₃\ninst✝⁷ : Category.{v₃, u₃} D'\nD'' : Type u₄\ninst✝⁶ : Category.{v₄, u₄} D''\nJ✝ : GrothendieckTopology C✝\nK✝ : GrothendieckTopology D✝\nF✝ : C✝ ⥤ D✝\nS✝ : Sheaf J✝ RingCat\nR : S... | suffices X.val.presheaf.map (α.hom.app U.unop).op ≫
X.val.presheaf.map (α.inv.app U.unop).op = 𝟙 _ from congr($this x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Sheaves.Presheaf | {
"line": 306,
"column": 8
} | {
"line": 311,
"column": 44
} | [
{
"pp": "C✝ : Type u\ninst✝⁴ : Category.{v, u} C✝\nX✝ : TopCat\nC : Type u\ninst✝³ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nℱ : Presheaf C Y\nU : Opens ↑X\nH... | fapply CostructuredArrow.homMk
· change op (unop _) ⟶ op (⟨_, H⟩ : Opens _)
refine (homOfLE ?_).op
apply (Set.image_mono s.pt.hom.unop.le).trans
exact Set.image_preimage.l_u_le (SetLike.coe s.pt.left.unop)
· simp [eq_iff_true_of_subsingleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.Presheaf | {
"line": 306,
"column": 8
} | {
"line": 311,
"column": 44
} | [
{
"pp": "C✝ : Type u\ninst✝⁴ : Category.{v, u} C✝\nX✝ : TopCat\nC : Type u\ninst✝³ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nℱ : Presheaf C Y\nU : Opens ↑X\nH... | fapply CostructuredArrow.homMk
· change op (unop _) ⟶ op (⟨_, H⟩ : Opens _)
refine (homOfLE ?_).op
apply (Set.image_mono s.pt.hom.unop.le).trans
exact Set.image_preimage.l_u_le (SetLike.coe s.pt.left.unop)
· simp [eq_iff_true_of_subsingleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.MonCat.Colimits | {
"line": 221,
"column": 10
} | {
"line": 221,
"column": 19
} | [
{
"pp": "case hf.h.h.one\nJ : Type v\ninst✝ : Category.{u, v} J\nF : J ⥤ MonCat\ns : Cocone F\nm : (colimitCocone F).pt ⟶ s.pt\nw : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j\n⊢ (Hom.hom m) (Quot.mk (⇑(colimitSetoid F)) one) = (Hom.hom (descMorphism F s)) (Quot.mk (⇑(colimitSetoid F)) one)",
"used... | quot_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Presentable.Finite | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nX : C\ninst✝ : IsFinitelyPresentable X\n⊢ PreservesFilteredColimitsOfSize.{w, w, v, v, u, v + 1} (coyoneda.obj (op X))",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Opposite",
"Cat... | rw [← isFinitelyPresentable_iff_preservesFilteredColimitsOfSize]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Finite | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nX : C\ninst✝ : IsFinitelyPresentable X\n⊢ PreservesFilteredColimitsOfSize.{w, w, v, v, u, v + 1} (coyoneda.obj (op X))",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Opposite",
"Cat... | rw [← isFinitelyPresentable_iff_preservesFilteredColimitsOfSize]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Ring.LinearAlgebra | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 27
} | [
{
"pp": "A B C D : CommRingCat\nhA : IsField ↑A\nf : A ⟶ B\ng : A ⟶ C\ninl : B ⟶ D\ninr : C ⟶ D\ninst✝¹ : Nontrivial ↑B\ninst✝ : Nontrivial ↑C\nh : IsPushout f g inl inr\nthis : Field ↑A := hA.toField\n⊢ Nontrivial ↑D",
"usedConstants": [
"CommRingCat.Hom.hom",
"CommRingCat.carrier",
"Comm... | algebraize [f.hom, g.hom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.TensorProduct.Pi | {
"line": 40,
"column": 2
} | {
"line": 44,
"column": 24
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nι : Type u_4\nB : ι → Type u_5\ninst✝¹ : (i : ι) → Semiring (B i)\ninst✝ : (i : ι) ... | · induction y
· simp
· ext j
simp
· simp_all [mul_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 332,
"column": 20
} | {
"line": 332,
"column": 38
} | [
{
"pp": "case h\nT : Type u_1\ninst✝¹ : Category.{v_1, u_1} T\nP : MorphismProperty T\nX : T\ninst✝ : P.ContainsIdentities\nY : P.Under ⊤ X\na : Under.mk ⊤ (𝟙 X) ⋯ ⟶ Y\n⊢ a.right = (Under.homMk Y.hom ⋯ ⋯).right",
"usedConstants": [
"CategoryTheory.MorphismProperty.Under",
"CategoryTheory.Morphi... | simp [← Under.w a] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.CharP.Algebra | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : NonAssocSemiring A\nf : R →+* A\nh : Function.Injective ⇑f\nq : ℕ\nhR : ExpChar R q\n⊢ ExpChar A q",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HEq.refl",
"ExpChar",
"AddCommMonoidWithOne.to... | rcases hR with _ | hprime | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
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