module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 633,
"column": 29
} | {
"line": 633,
"column": 54
} | {
"line": 633,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF₁ F₂ : J ⥤ C\nc₁ : Cocone F₁\nc₂ : Cocone F₂\nh₁ : IsColimit c₁\nh₂ : IsColimit c₂\nf : F₁ ⟶ F₂\nx✝ : ∀ (j : J), IsIso (f.app j)\nφ : c₁.pt ⟶ c₂.pt\nhφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c... | [] | simp [reassoc_of% (hφ j)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 633,
"column": 29
} | {
"line": 633,
"column": 54
} | {
"line": 633,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF₁ F₂ : J ⥤ C\nc₁ : Cocone F₁\nc₂ : Cocone F₂\nh₁ : IsColimit c₁\nh₂ : IsColimit c₂\nf : F₁ ⟶ F₂\nx✝ : ∀ (j : J), IsIso (f.app j)\nφ : c₁.pt ⟶ c₂.pt\nhφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c... | [] | simp [reassoc_of% (hφ j)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 633,
"column": 29
} | {
"line": 633,
"column": 54
} | {
"line": 633,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF₁ F₂ : J ⥤ C\nc₁ : Cocone F₁\nc₂ : Cocone F₂\nh₁ : IsColimit c₁\nh₂ : IsColimit c₂\nf : F₁ ⟶ F₂\nx✝ : ∀ (j : J), IsIso (f.app j)\nφ : c₁.pt ⟶ c₂.pt\nhφ : ∀ (j : J), c₁.ι.app j ≫ φ = f.app j ≫ c... | [] | simp [reassoc_of% (hφ j)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Limits | {
"line": 857,
"column": 50
} | {
"line": 865,
"column": 36
} | {
"line": 867,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasPullbacks C\nP : MorphismProperty C\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nQ : MorphismProperty C\ninst✝ : Q.IsStableUnderBaseChange\n⊢ P.HasOfPostcompProperty Q ↔ Q ≤ P.diagonal",
"ppTerm": "?m.17",
"assigned": true... | [] | by
refine ⟨fun hP X Y f hf ↦ ?_, fun hP ↦ ⟨fun {Y X S} g f hf hcomp ↦ ?_⟩⟩
· exact hP.of_postcomp _ _ (Q.pullback_fst _ _ hf) (by simpa using P.id_mem X)
· set gr : Y ⟶ pullback (g ≫ f) f := pullback.lift (𝟙 Y) g (by simp)
have : g = gr ≫ pullback.snd _ _ := by simp [gr]
rw [this]
apply P.comp_mem
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coalgebra.Convolution | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 37
} | {
"line": 98,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : NonUnitalNonAssocSemiring A\ninst✝¹² : Module R A\ninst✝¹¹ : SMulCommClass R A A\ninst✝¹⁰ : IsScalarTower R A A\ninst✝⁹ : AddCommMonoid C\ninst✝⁸ : Module R C\ninst✝⁷ : CoalgebraStruct R C\nD : Type u_7\ninst✝⁶ ... | [
"R : Type u_1\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : NonUnitalNonAssocSemiring A\ninst✝¹² : Module R A\ninst✝¹¹ : SMulCommClass R A A\ninst✝¹⁰ : IsScalarTower R A A\ninst✝⁹ : AddCommMonoid C\ninst✝⁸ : Module R C\ninst✝⁷ : CoalgebraStruct R C\nD : Type u_7\ninst✝⁶ : AddCommMon... | nth_rw 2 [← comp_assoc, comp_assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 328,
"column": 30
} | {
"line": 328,
"column": 73
} | {
"line": 329,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nA B : C\ninst✝² : GrpObj A\ninst✝¹ : GrpObj B\nf : A ⟶ B\ninst✝ : IsMonHom f\nthis : lift (lift f (ι ≫ f)) f = lift (lift f (f ≫ ι)) f\n⊢ ι ≫ f = f ≫ ι",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
... | [] | by simpa using (this =≫ fst _ _) =≫ snd _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 69
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : MonoidalCategory C\nA₁ A₂ A₃ : C\ninst✝² : Closed A₁\ninst✝¹ : Closed A₂\ninst✝ : Closed A₃\nf : A₁ ⟶ A₂\ng : A₂ ⟶ A₃\n⊢ pre (f ≫ g) = pre g ≫ pre f",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [] | rw [pre, pre, pre, conjugateEquiv_comp, (tensoringLeft C).map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 69
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : MonoidalCategory C\nA₁ A₂ A₃ : C\ninst✝² : Closed A₁\ninst✝¹ : Closed A₂\ninst✝ : Closed A₃\nf : A₁ ⟶ A₂\ng : A₂ ⟶ A₃\n⊢ pre (f ≫ g) = pre g ≫ pre f",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [] | rw [pre, pre, pre, conjugateEquiv_comp, (tensoringLeft C).map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Closed.Basic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 69
} | {
"line": 285,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : MonoidalCategory C\nA₁ A₂ A₃ : C\ninst✝² : Closed A₁\ninst✝¹ : Closed A₂\ninst✝ : Closed A₃\nf : A₁ ⟶ A₂\ng : A₂ ⟶ A₃\n⊢ pre (f ≫ g) = pre g ≫ pre f",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheo... | [] | rw [pre, pre, pre, conjugateEquiv_comp, (tensoringLeft C).map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Subfield | {
"line": 76,
"column": 6
} | {
"line": 81,
"column": 75
} | {
"line": 82,
"column": 4
} | [
{
"pp": "case e'_4\nα : Type u\ns : Set α\ninst✝ : DivisionRing α\n⊢ max #↑s ℵ₀ = max (lift.{0, u} #(Fin 6 ⊕ ↑s)) ℵ₀",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",... | [] | rw [lift_uzero, mk_sum, lift_uzero]
have : lift.{u, 0} #(Fin 6) < ℵ₀ := lift_lt_aleph0.mpr (lt_aleph0_of_finite _)
obtain h | h := lt_or_ge #s ℵ₀
· rw [max_eq_right h.le, max_eq_right]
exact (add_lt_aleph0 this h).le
· rw [max_eq_left h, add_eq_right h (this.le.trans h), max_eq_left h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Subfield | {
"line": 76,
"column": 6
} | {
"line": 81,
"column": 75
} | {
"line": 82,
"column": 4
} | [
{
"pp": "case e'_4\nα : Type u\ns : Set α\ninst✝ : DivisionRing α\n⊢ max #↑s ℵ₀ = max (lift.{0, u} #(Fin 6 ⊕ ↑s)) ℵ₀",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",... | [] | rw [lift_uzero, mk_sum, lift_uzero]
have : lift.{u, 0} #(Fin 6) < ℵ₀ := lift_lt_aleph0.mpr (lt_aleph0_of_finite _)
obtain h | h := lt_or_ge #s ℵ₀
· rw [max_eq_right h.le, max_eq_right]
exact (add_lt_aleph0 this h).le
· rw [max_eq_left h, add_eq_right h (this.le.trans h), max_eq_left h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 264,
"column": 2
} | {
"line": 265,
"column": 32
} | {
"line": 266,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : HasRankNullity.{u, u_2} R\ninst✝⁵ : StrongRankCondition R\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nL : Submodule R M\ninst✝ : Module.Fin... | [
"R : Type u_2\nM : Type u\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : HasRankNullity.{u, u_2} R\ninst✝⁵ : StrongRankCondition R\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nL : Submodule R M\ninst✝ : Module.Finite R ↥L\nf ... | refine LinearMap.injective_domRestrict_iff.mp <| LinearMap.ker_eq_bot.mp <|
Submodule.rank_eq_zero.mp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 63
} | {
"line": 95,
"column": 6
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nW : Submodule K V\nv : V\nhv : v ∉ W\nhW : W ⊔ K ∙ v = ⊤\nthis : W ⊓ K ∙ v = ⊥\nhv₀ : v ≠ 0\n⊢ finrank K (V ⧸ W) = 1",
"ppTerm": "?m.60",
"assigned": true,
"usedConstant... | [
"K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nW : Submodule K V\nv : V\nhv : v ∉ W\nhW : W ⊔ K ∙ v = ⊤\nthis : W ⊓ K ∙ v = ⊥\nhv₀ : v ≠ 0\naux : finrank K ↥(W ⊔ K ∙ v) + finrank K ↥(W ⊓ K ∙ v) = finrank K ↥W + finrank K ↥(K ∙ v)\n⊢ finrank... | have aux := finrank_sup_add_finrank_inf_eq W (span K {v}) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 90
} | {
"line": 183,
"column": 2
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nf : V →ₗ[K] V₂\nh : finrank K V₂ < finrank K V\nh₁ : finrank K ↥f.range + finrank ... | [
"K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nf : V →ₗ[K] V₂\nh : finrank K V₂ < finrank K V\nh₁ : finrank K ↥f.range + finrank K ↥f.ker = f... | have h₂ : finrank K (LinearMap.range f) ≤ finrank K V₂ := (LinearMap.range f).finrank_le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 67,
"column": 86
} | {
"line": 69,
"column": 29
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring R₁\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : Module R₁ M₁\ninst✝⁴ : CommSemiring R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\ninst✝¹ : AddCommMonoid M\ninst✝ : Modu... | [] | by
dsimp only [IsOrtho]
rw [map_zero B, zero_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Contraction | {
"line": 147,
"column": 9
} | {
"line": 147,
"column": 74
} | {
"line": 148,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh : 1 ∈ (dualTensorHom R M M).range\ns : Finset (Dual R M × M)\neq : (dualTensorHom R M M) (∑ i ∈ s, i.1 ⊗ₜ[R] i.2) = 1\nf : (↥s → R) →ₗ[R] M := Fintype.linearCombination R fun x ↦ (↑x).2\nx✝ : M\n⊢ (f ∘ₗ... | [] | simp [f, ← eq, Fintype.linearCombination_apply, ← s.sum_coe_sort] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 13
} | {
"line": 224,
"column": 13
} | [
{
"pp": "K : Type uK\nV : Type uV\ninst✝³ : CommSemiring K\ninst✝² : AddCommMonoid V\ninst✝¹ : Module K V\ninst✝ : Projective K V\n⊢ Nontrivial (Dual K V) ↔ Nontrivial V",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Mathlib.Tactic.Contrapose.cont... | [
"K : Type uK\nV : Type uV\ninst✝³ : CommSemiring K\ninst✝² : AddCommMonoid V\ninst✝¹ : Module K V\ninst✝ : Projective K V\n⊢ Subsingleton (Dual K V) ↔ Subsingleton V"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 94
} | {
"line": 87,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n⊢ PreservesLimit (parallelPair f g) G",
"ppTerm": "?m.62",
"assigned... | [] | apply preservesLimit_of_preserves_limit_cone (equalizerIsEqualizer f g)
apply (isLimitMapConeForkEquiv _ _).symm _
exact @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (parallelPair (G.map f) (G.map g))) i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 94
} | {
"line": 87,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasEqualizer (G.map f) (G.map g)\ni : IsIso (equalizerComparison f g G)\n⊢ PreservesLimit (parallelPair f g) G",
"ppTerm": "?m.62",
"assigned... | [] | apply preservesLimit_of_preserves_limit_cone (equalizerIsEqualizer f g)
apply (isLimitMapConeForkEquiv _ _).symm _
exact @IsLimit.ofPointIso _ _ _ _ _ _ _ (limit.isLimit (parallelPair (G.map f) (G.map g))) i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 130,
"column": 8
} | {
"line": 131,
"column": 54
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\n⊢ Cofork.π\n ((Cocone.precompose (diagramIsoParallelPair (parallelPair f g ⋙ G)).inv).obj (G.mapCocone (Cofork.ofπ h w))) ≫\n (Iso.refl\n ... | [] | dsimp only [Cofork.π, Cofork.ofπ_ι_app]
dsimp; rw [Category.comp_id, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers | {
"line": 130,
"column": 8
} | {
"line": 131,
"column": 54
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\nw : f ≫ h = g ≫ h\n⊢ Cofork.π\n ((Cocone.precompose (diagramIsoParallelPair (parallelPair f g ⋙ G)).inv).obj (G.mapCocone (Cofork.ofπ h w))) ≫\n (Iso.refl\n ... | [] | dsimp only [Cofork.π, Cofork.ofπ_ι_app]
dsimp; rw [Category.comp_id, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 840,
"column": 8
} | {
"line": 840,
"column": 39
} | {
"line": 842,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx :... | [] | simpa [hx, hy] using hs (x + y) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 840,
"column": 8
} | {
"line": 840,
"column": 39
} | {
"line": 842,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx :... | [] | simpa [hx, hy] using hs (x + y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 840,
"column": 8
} | {
"line": 840,
"column": 39
} | {
"line": 842,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nx y : M\naux : ∀ (x y : M), 0 ≤ (B x) x * ((B x) x * (B y) y - (B x) y * (B y) x)\nhx :... | [] | simpa [hx, hy] using hs (x + y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 884,
"column": 2
} | {
"line": 884,
"column": 13
} | {
"line": 885,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\n⊢ (∀ (x : M), (B x) x = 0 ↔ x = 0) ↔ ∀ (x : M), x ≠ 0 → 0 < (B x) x",
"ppTer... | [
"R : Type u_1\nM : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nB : LinearMap.BilinForm R M\nhs : ∀ (x : M), 0 ≤ (B x) x\nhB : IsSymm B\n⊢ (∃ x, (B x) x = 0 ∧ x ≠ 0 ∨ (B x) x ≠ 0 ∧ x = 0) ↔ ∃ x, x ≠ 0 ∧ (B x) x ≤ 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers | {
"line": 87,
"column": 4
} | {
"line": 89,
"column": 19
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : SmallCategory J\nF : J ⥤ C\nc₁ : Fan F.obj\nc₂ : Fan fun f ↦ F.obj f.fst.2\ns t : c₁.pt ⟶ c₂.pt\nhs : ∀ (f : (p : J × J) × (p.1 ⟶ p.2)), s ≫ c₂.π.app { as := f } = c₁.π.app { as := f.fst.1 } ≫ F.map f.snd\nht : ∀ (f : (p : J × J... | [] | · apply t₂.hom_ext
intro ⟨j⟩
simp [hs, ht] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 908,
"column": 4
} | {
"line": 908,
"column": 15
} | {
"line": 909,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : LinearMap.BilinForm R M\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ≤ (B z) z\n⊢ (B x) y *... | [
"case mp\nR : Type u_1\nM : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : LinearMap.BilinForm R M\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ≤ (B z) z\n⊢ ¬LinearIndependent R ... | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 549,
"column": 2
} | {
"line": 549,
"column": 61
} | {
"line": 551,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nΦ : Subspace K (Dual K V)\n⊢ finrank K ↥(comap (↑(evalEquiv K V)) (dualAnnihilator Φ)) = finrank K ↥(dualAnnihilator Φ)",
"ppTerm": "?m.46",
"assigned": true,
"usedCons... | [] | exact LinearEquiv.finrank_eq (LinearEquiv.ofSubmodule' _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.SesquilinearForm.Basic | {
"line": 924,
"column": 4
} | {
"line": 924,
"column": 15
} | {
"line": 925,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u_1\nM : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : LinearMap.BilinForm R M\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ≤ (B z) z\n⊢ LinearIn... | [
"case mpr\nR : Type u_1\nM : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : LinearMap.BilinForm R M\ninst✝ : IsTorsionFree R M\nhp : ∀ (x : M), x ≠ 0 → 0 < (B x) x\nx y : M\nhle : ∀ (z : M), 0 ≤ (B z) z\n⊢ (B x) x * (B y) y ≤ ... | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 866,
"column": 2
} | {
"line": 866,
"column": 82
} | {
"line": 867,
"column": 2
} | [
{
"pp": "K : Type u_1\nV₁ : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := (mkQ W).prod (mkQ W')\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'",
"ppTerm": "?m.76",
"assigned": true,
"usedConst... | [
"K : Type u_1\nV₁ : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nW W' : Subspace K V₁\nF : V₁ →ₗ[K] (V₁ ⧸ W) × V₁ ⧸ W' := (mkQ W).prod (mkQ W')\nthis : F.ker = W ⊓ W'\n⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'"
] | have : LinearMap.ker F = W ⊓ W' := by simp only [F, LinearMap.ker_prod, ker_mkQ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 894,
"column": 8
} | {
"line": 894,
"column": 20
} | {
"line": 894,
"column": 21
} | [
{
"pp": "case h_option\nK : Type u_1\nV₁ : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\nα : Type u_4\ninst✝ : Fintype α\nh : ∀ (W : α → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)\nW : Option α → Subspace K V₁\n⊢ dualAnnihilator (⨅ i, W i) = ⨆ i, dualAn... | [
"case h_option\nK : Type u_1\nV₁ : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V₁\ninst✝¹ : Module K V₁\nα : Type u_4\ninst✝ : Fintype α\nh : ∀ (W : α → Subspace K V₁), dualAnnihilator (⨅ i, W i) = ⨆ i, dualAnnihilator (W i)\nW : Option α → Subspace K V₁\n⊢ dualAnnihilator (W none ⊓ ⨅ b, W (some b)) = ⨆ i, du... | iInf_option, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic | {
"line": 330,
"column": 25
} | {
"line": 332,
"column": 18
} | {
"line": 334,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y : C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : IsNormalEpiCategory C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Epi f\n⊢ IsRegularEpi f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.NormalEpi",
"CategoryTheory.IsRegularEpi"... | [] | by
have := normalEpiOfEpi f
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers | {
"line": 62,
"column": 59
} | {
"line": 62,
"column": 87
} | {
"line": 62,
"column": 87
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasFiniteProducts C\ninst✝³ : HasKernels C\ninst✝² : IsNormalMonoCategory C\nX Y Z : C\na : X ⟶ Z\nb : Y ⟶ Z\ninst✝¹ : Mono a\ninst✝ : Mono b\nP : C\nf : Z ⟶ P\nhaf : a ≫ f = 0\ni : IsLimit (KernelFork.ofι a haf)\nQ : C\ng : Z ⟶ Q\... | PullbackCone.condition_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 668,
"column": 16
} | {
"line": 668,
"column": 31
} | {
"line": 670,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi f\nR : C\ne : Y ⟶ R\nh : pullback.snd f g ≫ e = 0\nu : X ⊞ Y ⟶ R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0\nthis : IsColimit (Co... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi f\nR : C\ne : Y ⟶ R\nh : pullback.snd f g ≫ e = 0\nu : X ⊞ Y ⟶ R := biprod.desc 0 e\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0\nthis : IsColimit (CokernelCofork... | dsimp [u] at hd | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 701,
"column": 16
} | {
"line": 701,
"column": 31
} | {
"line": 703,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\nR : C\ne : X ⟶ R\nh : pullback.fst f g ≫ e = 0\nu : X ⊞ Y ⟶ R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0\nthis : IsColimit (Co... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\nR : C\ne : X ⟶ R\nh : pullback.fst f g ≫ e = 0\nu : X ⊞ Y ⟶ R := biprod.desc e 0\nhu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0\nthis : IsColimit (CokernelCofork... | dsimp [u] at hd | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 187,
"column": 8
} | {
"line": 188,
"column": 49
} | {
"line": 188,
"column": 50
} | [
{
"pp": "A B : GrpCat\nf : A ⟶ B\nβ : ↑B\nx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ (fun x ↦ β • x) ((fun x ↦ β⁻¹ • x) x) = x",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"GrpCat.Surj... | [] | dsimp only
rw [← mul_smul, mul_inv_cancel, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 187,
"column": 8
} | {
"line": 188,
"column": 49
} | {
"line": 188,
"column": 50
} | [
{
"pp": "A B : GrpCat\nf : A ⟶ B\nβ : ↑B\nx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ (fun x ↦ β • x) ((fun x ↦ β⁻¹ • x) x) = x",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"GrpCat.Surj... | [] | dsimp only
rw [← mul_smul, mul_inv_cancel, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 753,
"column": 4
} | {
"line": 753,
"column": 19
} | {
"line": 754,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : Mono f\nR : C\ne : R ⟶ Z\nh : e ≫ pushout.inr f g = 0\nu : R ⟶ Y ⊞ Z := biprod.lift 0 e\nhu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis : IsLimit (Kerne... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : Mono f\nR : C\ne : R ⟶ Z\nh : e ≫ pushout.inr f g = 0\nu : R ⟶ Y ⊞ Z := biprod.lift 0 e\nhu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis : IsLimit (KernelFork.ofι (b... | dsimp [u] at hd | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 776,
"column": 4
} | {
"line": 776,
"column": 19
} | {
"line": 777,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : Mono g\nR : C\ne : R ⟶ Y\nh : e ≫ pushout.inl f g = 0\nu : R ⟶ Y ⊞ Z := biprod.lift e 0\nhu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis : IsLimit (Kerne... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPushouts C\nW X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\ninst✝ : Mono g\nR : C\ne : R ⟶ Y\nh : e ≫ pushout.inl f g = 0\nu : R ⟶ Y ⊞ Z := biprod.lift e 0\nhu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0\nthis : IsLimit (KernelFork.ofι (b... | dsimp [u] at hd | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 745,
"column": 56
} | {
"line": 745,
"column": 90
} | {
"line": 746,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ : S₁ ⟶ S₂\nφ₂ : S₂ ⟶ S₃\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nh₃ : S₃.RightHomologyData\nγ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂\nγ₂ : RightHomologyMapDat... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ : S₁ ⟶ S₂\nφ₂ : S₂ ⟶ S₃\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nh₃ : S₃.RightHomologyData\nγ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂\nγ₂ : RightHomologyMapData φ₂ h₂ h₃ :... | (γ₁.comp γ₂).rightHomologyMap'_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 836,
"column": 2
} | {
"line": 837,
"column": 16
} | {
"line": 839,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ S₄ : ShortComplex C\nφ✝ : S₁ ⟶ S₂\nh₁✝ : S₁.HomologyData\nh₂✝ : S₂.HomologyData\nφ : S₁ ⟶ S₂\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\n⊢ IsIso (homologyMap' φ h₁... | [] | dsimp only [homologyMap']
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 836,
"column": 2
} | {
"line": 837,
"column": 16
} | {
"line": 839,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ S₄ : ShortComplex C\nφ✝ : S₁ ⟶ S₂\nh₁✝ : S₁.HomologyData\nh₂✝ : S₂.HomologyData\nφ : S₁ ⟶ S₂\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\n⊢ IsIso (homologyMap' φ h₁... | [] | dsimp only [homologyMap']
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 945,
"column": 2
} | {
"line": 946,
"column": 39
} | {
"line": 947,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\n⊢ S₁.homologyπ ≫ homologyMap φ = cyclesMap φ ≫ S₂.homologyπ",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\n⊢ S₁.homologyπ ≫ S₁.leftHomologyIso.inv ≫ leftHomologyMap φ = cyclesMap φ ≫ S₂.leftHomologyπ"
] | simp only [← cancel_mono S₂.leftHomologyIso.inv, assoc, ← leftHomologyIso_inv_naturality φ,
homologyπ_comp_leftHomologyIso_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.Limits | {
"line": 56,
"column": 56
} | {
"line": 56,
"column": 61
} | {
"line": 56,
"column": 61
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\ninst✝ : HasZeroMorphisms C\nF : J ⥤ ShortComplex C\nc : Cone F\nh₁ : IsLimit (π₁.mapCone c)\nh₂ : IsLimit (π₂.mapCone c)\nh₃ : IsLimit (π₃.mapCone c)\ns : Cone F\nj : J\neq₂ : ∀ (j : J), h₂.lift (π₂.mapCone s) ≫... | [
"J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\ninst✝ : HasZeroMorphisms C\nF : J ⥤ ShortComplex C\nc : Cone F\nh₁ : IsLimit (π₁.mapCone c)\nh₂ : IsLimit (π₂.mapCone c)\nh₃ : IsLimit (π₃.mapCone c)\ns : Cone F\nj : J\neq₂ : ∀ (j : J), h₂.lift (π₂.mapCone s) ≫ (c.π.app j)... | eq₂₃' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 453,
"column": 14
} | {
"line": 453,
"column": 78
} | {
"line": 454,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ + φ₃).τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂",
"ppTerm": "?m.168",
"assigned": true,
"usedConstant... | [] | rw [add_τ₂, add_τ₂, h.comm₂, h'.comm₂, comp_add, add_comp]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 453,
"column": 14
} | {
"line": 453,
"column": 78
} | {
"line": 454,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ + φ₃).τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂",
"ppTerm": "?m.168",
"assigned": true,
"usedConstant... | [] | rw [add_τ₂, add_τ₂, h.comm₂, h'.comm₂, comp_add, add_comp]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 476,
"column": 11
} | {
"line": 476,
"column": 87
} | {
"line": 477,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nψ : S₃ ⟶ S₁\n⊢ (ψ ≫ φ₁).τ₂ = S₃.g ≫ ψ.τ₃ ≫ h.h₂ + (ψ.τ₂ ≫ h.h₁) ≫ S₂.f + (ψ ≫ φ₂).τ₂",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"E... | [] | by rw [comp_τ₂, comp_τ₂, h.comm₂, comp_add, comp_add, assoc, ψ.comm₂₃_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.LeftExact | {
"line": 111,
"column": 4
} | {
"line": 112,
"column": 44
} | {
"line": 113,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : Preadditive D\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F\nK : WalkingParallelPa... | [
"C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : Preadditive D\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F\nK : WalkingParallelPair ⥤ C\nthis... | let := preservesEqualizer_of_preservesKernels F (K.map WalkingParallelPairHom.left)
(K.map WalkingParallelPairHom.right) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.ShortComplex.Ab | {
"line": 117,
"column": 2
} | {
"line": 119,
"column": 19
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case mpr\nS : ShortComplex Ab\n⊢ (∀ (x₂ : ↑S.X₂), (ConcreteCategory.hom S.g) x₂ = 0 → ∃ x₁, (ConcreteCategory.hom S.f) x₁ = x₂) →\n Function.Surjective ⇑S.abToCycles",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.abToCycles",
"AddCommGrp... | [] | · rintro h ⟨x₂, hx₂⟩
obtain ⟨x₁, rfl⟩ := h x₂ hx₂
exact ⟨x₁, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 277,
"column": 26
} | {
"line": 290,
"column": 23
} | {
"line": 292,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\ninst✝ : HasZeroObject C\nhf : S.f = 0\n⊢ S.Exact ↔ Mono S.g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.exact_iff",
"CategoryTheory.S... | [] | by
constructor
· intro h
have := h.hasHomology
simp only [exact_iff_isZero_homology] at h
have := S.isIso_pOpcycles hf
have := mono_of_isZero_kernel' _ S.homologyIsKernel h
rw [← S.p_fromOpcycles]
apply mono_comp
· intro
rw [(HomologyData.ofIsLimitKernelFork S hf _
(KernelFork.Is... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 68,
"column": 6
} | {
"line": 69,
"column": 79
} | {
"line": 69,
"column": 79
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y : C\nP✝ : Subobject Y\nf : X ⟶ Y\nP Q : MonoOver Y\nh : P ≅ Q\n⊢ Q.Factors f → P.Factors f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [] | · rintro ⟨i, w⟩
exact ⟨i ≫ h.inv.hom.left, by rw [Category.assoc, Over.w h.inv.hom, w]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 657,
"column": 20
} | {
"line": 657,
"column": 38
} | {
"line": 657,
"column": 38
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS S₁ S₂ : ShortComplex C\ns : S₁.Splitting\ne : S₁ ≅ S₂\neq : e.inv.τ₂ ≫ (s.r ≫ S₁.f + S₁.g ≫ s.s) ≫ e.hom.τ₂ = 𝟙 S₂.X₂\n⊢ e.inv.τ₂ ≫ s.r ≫ S₁.f ≫ e.hom.τ₂ + S₂.g ... | [
"C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS S₁ S₂ : ShortComplex C\ns : S₁.Splitting\ne : S₁ ≅ S₂\neq : e.inv.τ₂ ≫ (s.r ≫ S₁.f + S₁.g ≫ s.s) ≫ e.hom.τ₂ = 𝟙 S₂.X₂\n⊢ e.inv.τ₂ ≫ s.r ≫ S₁.f ≫ e.hom.τ₂ + S₂.g ≫ e.inv.τ₃ ≫... | e.inv.comm₂₃_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.MonoOver | {
"line": 132,
"column": 29
} | {
"line": 132,
"column": 43
} | {
"line": 132,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nf g : MonoOver X\nh : f.obj.left ≅ g.obj.left\nw : h.hom ≫ g.arrow = f.arrow\n⊢ h.inv ≫ f.arrow = g.arrow",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nf g : MonoOver X\nh : f.obj.left ≅ g.obj.left\nw : h.hom ≫ g.arrow = f.arrow\n⊢ f.arrow = h.hom ≫ g.arrow"
] | h.inv_comp_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 684,
"column": 8
} | {
"line": 684,
"column": 61
} | {
"line": 685,
"column": 6
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : X ≅ Y\nA B : Subobject X\nh : (map e.inv).obj ((map e.hom).obj A) ≤ (map e.inv).obj ((map e.hom).obj B)\n⊢ A ≤ B",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Catego... | [] | simpa only [← map_comp, e.hom_inv_id, map_id] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 684,
"column": 8
} | {
"line": 684,
"column": 61
} | {
"line": 685,
"column": 6
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : X ≅ Y\nA B : Subobject X\nh : (map e.inv).obj ((map e.hom).obj A) ≤ (map e.inv).obj ((map e.hom).obj B)\n⊢ A ≤ B",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Catego... | [] | simpa only [← map_comp, e.hom_inv_id, map_id] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 684,
"column": 8
} | {
"line": 684,
"column": 61
} | {
"line": 685,
"column": 6
} | [
{
"pp": "case mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\ne : X ≅ Y\nA B : Subobject X\nh : (map e.inv).obj ((map e.hom).obj A) ≤ (map e.inv).obj ((map e.hom).obj B)\n⊢ A ≤ B",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Catego... | [] | simpa only [← map_comp, e.hom_inv_id, map_id] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.FunctorCategory | {
"line": 88,
"column": 2
} | {
"line": 92,
"column": 33
} | {
"line": 94,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type w\ninst✝¹ : Category.{z, w} D\ninst✝ : Abelian D\nF G : C ⥤ D\nα : F ⟶ G\nX : C\n⊢ IsIso (coimageImageComparison α)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.FunctorCategory.coimageImageComparison... | [] | have : ∀ X : C, IsIso ((Abelian.coimageImageComparison α).app X) := by
intros
rw [coimageImageComparison_app']
infer_instance
apply NatIso.isIso_of_isIso_app | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.FunctorCategory | {
"line": 88,
"column": 2
} | {
"line": 92,
"column": 33
} | {
"line": 94,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type w\ninst✝¹ : Category.{z, w} D\ninst✝ : Abelian D\nF G : C ⥤ D\nα : F ⟶ G\nX : C\n⊢ IsIso (coimageImageComparison α)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.FunctorCategory.coimageImageComparison... | [] | have : ∀ X : C, IsIso ((Abelian.coimageImageComparison α).app X) := by
intros
rw [coimageImageComparison_app']
infer_instance
apply NatIso.isIso_of_isIso_app | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 674,
"column": 2
} | {
"line": 681,
"column": 10
} | {
"line": 682,
"column": 2
} | [
{
"pp": "case f\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : LocallySmall.{w, v₁, u₁} C\ninst✝² : WellPowered.{w, v₁, u₁} C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ g ∈ s, g ≤ f\n⊢ underlying.obj (sSup s) ⟶ underlying.obj f",
"ppTerm": "?f"... | [
"case w\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : LocallySmall.{w, v₁, u₁} C\ninst✝² : WellPowered.{w, v₁, u₁} C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ g ∈ s, g ≤ f\n⊢ ((underlyingIso (image.ι (smallCoproductDesc s))).hom ≫\n image.lift... | · refine (underlyingIso _).hom ≫ image.lift ⟨_, f.arrow, ?_, ?_⟩
· refine Sigma.desc ?_
rintro ⟨g, m⟩
refine underlying.map (homOfLE (k _ ?_))
simpa using m
· ext
dsimp [smallCoproductDesc]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 20
} | {
"line": 231,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\nX : Type v₂\nα : Type u_1\nZ : α → C\ninst✝ : HasProduct Z\nf : Fan Z\nhf : IsLimit f\ne : (Discrete.functor fun x ↦ Opposite.op (Z x)) ≅ (Discrete.opposite α).inverse ⋙ (Discrete.functor Z).op :=\n Discrete.natIso f... | [] | simp [e, Fan.proj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products | {
"line": 342,
"column": 23
} | {
"line": 342,
"column": 46
} | {
"line": 342,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ coprod.inl = prod.fst.op ≫ (opProdIsoCoprod A B).hom",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.opProdIsoCoprod",
"Opposite",
"Category... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA B : C\ninst✝ : HasBinaryProduct A B\n⊢ coprod.inl = coprod.inl"
] | fst_opProdIsoCoprod_hom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Constructions.Filtered | {
"line": 287,
"column": 13
} | {
"line": 287,
"column": 41
} | {
"line": 289,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nα : Type w\ninst✝ : HasFiniteProducts C\nI : Finset (Discrete α)\nX✝ Y✝ : Discrete α ⥤ C\nx✝ : X✝ ⟶ Y✝\n⊢ (liftToFinset C α ⋙ (evaluation (Finset (Discrete α))ᵒᵖ C).obj (op I)).map x✝ ≫\n (HasLimit.isoOfNatIso (Discrete.natIso fun x ↦ Iso.refl ((Discrete.fun... | [] | by dsimp; ext; simp [Pi.map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Functor.KanExtension.Basic | {
"line": 244,
"column": 43
} | {
"line": 251,
"column": 55
} | {
"line": 253,
"column": 0
} | [
{
"pp": "C : Type u_1\nH : Type u_3\nD : Type u_4\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_3, u_3} H\ninst✝¹ : Category.{v_4, u_4} D\nF' F'' : D ⥤ H\nφ : F' ⟶ F''\nL : C ⥤ D\nF : C ⥤ H\nα : F ⟶ L ⋙ F'\nα' : F ⟶ L ⋙ F''\ncomm : α ≫ L.whiskerLeft φ = α'\ninst✝ : F'.IsLeftKanExtension α\n⊢ F''.IsLeftK... | [] | by
constructor
· intro
rw [F'.hom_ext_of_isLeftKanExtension α φ (leftKanExtensionUnique _ α _ α').hom
(by simp [comm])]
infer_instance
· intro
exact isLeftKanExtension_of_iso (asIso φ) α α' comm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Comma.Presheaf.Basic | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 68
} | {
"line": 473,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nA F : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nX : C\n⊢ unitBackward η X ∘ unitForward η X = id",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.OverPresheafAux.unitForward",
"CategoryTheory.Functor",
"CategoryTheory.OverPres... | [
"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nA F : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nX : C\np : YonedaCollection (restrictedYonedaObj η) X\n⊢ ((unitBackward η X ∘ unitForward η X) p).fst = (id p).fst",
"case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nA F : Cᵒᵖ ⥤ Type v\nη : F ⟶ A\nX : C\np : YonedaCollecti... | refine funext fun p => YonedaCollection.ext ?_ (OverArrows.ext ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Abelian.Subobject | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 92
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case refine_1.refine_5\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nA : Cᵒᵖ\nf : A ⟶ op X\nhf : Mono f\n⊢ { hom := (epiDesc f.unop (cokernel.π (kernel.ι f.unop)) ⋯).op, inv := (cokernel.desc (kernel.ι f.unop) f.unop ⋯).op,\n hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n ... | [] | exact Quiver.Hom.unop_inj (by simp only [unop_comp, Quiver.Hom.unop_op, comp_epiDesc]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.Subobject | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 92
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case refine_1.refine_5\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nA : Cᵒᵖ\nf : A ⟶ op X\nhf : Mono f\n⊢ { hom := (epiDesc f.unop (cokernel.π (kernel.ι f.unop)) ⋯).op, inv := (cokernel.desc (kernel.ι f.unop) f.unop ⋯).op,\n hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n ... | [] | exact Quiver.Hom.unop_inj (by simp only [unop_comp, Quiver.Hom.unop_op, comp_epiDesc]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Subobject | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 92
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case refine_1.refine_5\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nA : Cᵒᵖ\nf : A ⟶ op X\nhf : Mono f\n⊢ { hom := (epiDesc f.unop (cokernel.π (kernel.ι f.unop)) ⋯).op, inv := (cokernel.desc (kernel.ι f.unop) f.unop ⋯).op,\n hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n ... | [] | exact Quiver.Hom.unop_inj (by simp only [unop_comp, Quiver.Hom.unop_op, comp_epiDesc]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 204,
"column": 71
} | {
"line": 206,
"column": 55
} | {
"line": 208,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝² : Category.{v₂, u₂} ℰ\nA : C ⥤ ℰ\ninst✝¹ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nL : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ\nα : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L\ninst✝ : L.IsLeftKanExtension α\nP : Cᵒᵖ ⥤ Type (max w ... | [] | by
simp [uliftYonedaAdjunction, restrictedULiftYonedaHomEquiv,
restrictedULiftYonedaHomEquiv', IsColimit.homEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 461,
"column": 6
} | {
"line": 464,
"column": 55
} | {
"line": 465,
"column": 6
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝³ : Category.{v₂, u₂} ℰ\nA✝ A : C ⥤ ℰ\ninst✝² : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P\nX Y : C... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nℰ : Type u₂\ninst✝³ : Category.{v₂, u₂} ℰ\nA✝ A : C ⥤ ℰ\ninst✝² : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝ : ∀ (P : Cᵒᵖ ⥤ Type (max w v₁ v₂)), F.op.HasLeftKanExtension P\nX Y : C\nf : X ⟶ Y\... | have eq₁ := ConcreteCategory.congr_hom
((uliftYoneda.{max w v₁}.obj (F.obj Y)).descOfIsLeftKanExtension_fac_app
(uliftYonedaMap F Y) (F.op.lan.obj (uliftYoneda.obj Y))
(F.op.lanUnit.app (uliftYoneda.obj Y)) _) ⟨f⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 42
} | {
"line": 171,
"column": 42
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\n⊢ ∀ {X Y : WalkingParallelFamily (ULift.{w, 0} Bool)} (f : X ⟶ Y),\n (𝟭 (WalkingParallelFamily (ULift.{w, 0} Bool))).map f ≫ (eqToIso ⋯).hom =\n (eqToIso ⋯).hom ≫\n ((parallelFamily fun p ↦ bif p.down then Walk... | [] | rintro _ _ (_ | ⟨_ | _⟩) <;> cat_disch | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 42
} | {
"line": 171,
"column": 42
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\n⊢ ∀ {X Y : WalkingParallelFamily (ULift.{w, 0} Bool)} (f : X ⟶ Y),\n (𝟭 (WalkingParallelFamily (ULift.{w, 0} Bool))).map f ≫ (eqToIso ⋯).hom =\n (eqToIso ⋯).hom ≫\n ((parallelFamily fun p ↦ bif p.down then Walk... | [] | rintro _ _ (_ | ⟨_ | _⟩) <;> cat_disch | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 42
} | {
"line": 171,
"column": 42
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\n⊢ ∀ {X Y : WalkingParallelFamily (ULift.{w, 0} Bool)} (f : X ⟶ Y),\n (𝟭 (WalkingParallelFamily (ULift.{w, 0} Bool))).map f ≫ (eqToIso ⋯).hom =\n (eqToIso ⋯).hom ≫\n ((parallelFamily fun p ↦ bif p.down then Walk... | [] | rintro _ _ (_ | ⟨_ | _⟩) <;> cat_disch | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 43
} | {
"line": 196,
"column": 43
} | [
{
"pp": "case w\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\nhf : Mono f\nq : S.obj (unop (Subobject.underlying.obj... | [] | exact Quiver.Hom.unop_inj (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 43
} | {
"line": 196,
"column": 43
} | [
{
"pp": "case w\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\nhf : Mono f\nq : S.obj (unop (Subobject.underlying.obj... | [] | exact Quiver.Hom.unop_inj (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 43
} | {
"line": 196,
"column": 43
} | [
{
"pp": "case w\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\nhf : Mono f\nq : S.obj (unop (Subobject.underlying.obj... | [] | exact Quiver.Hom.unop_inj (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Periodic | {
"line": 321,
"column": 57
} | {
"line": 325,
"column": 42
} | {
"line": 327,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : AddGroup α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nn : ℤ\n⊢ f (x + n • c) = ↑n.negOnePow • f x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | by
rcases Int.even_or_odd' n with ⟨k, rfl | rfl⟩
· rw [h.even_zsmul_periodic, Int.negOnePow_two_mul, Units.val_one, one_zsmul]
· rw [h.odd_zsmul_antiperiodic, Int.negOnePow_two_mul_add_one, Units.val_neg,
Units.val_one, neg_zsmul, one_zsmul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Interval.Set.Group | {
"line": 214,
"column": 67
} | {
"line": 216,
"column": 45
} | {
"line": 218,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Ring α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedRing α\na : α\n⊢ Pairwise (Disjoint on fun n ↦ Ioc (a + ↑n) (a + ↑n + 1))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Int.cast",
"Set.Ioc",
"instHSMul",
"AddMonoid.toAddSemigroup"... | [] | by
simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using
pairwise_disjoint_Ioc_add_zsmul a (1 : α) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Injective | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 20
} | {
"line": 294,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nr r' : ... | [
"R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nr r' : R\neq1 : r •... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Injective | {
"line": 320,
"column": 35
} | {
"line": 320,
"column": 49
} | {
"line": 320,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nx : ↥(s... | [
"R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nx : ↥(supExtensionO... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.Prod | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 34
} | {
"line": 107,
"column": 2
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : Preorder α\ninst✝ : Preorder β\np : α × β → Prop\nhp : ∀ᶠ (x : α × β) in atTop, p x\n⊢ ∀ᶠ (k : α) in atTop, ∀ᶠ (l : β) in atTop, p (k, l)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"SProd.sprod",
"congrArg",
"Filter.Event... | [
"α : Type u_3\nβ : Type u_4\ninst✝¹ : Preorder α\ninst✝ : Preorder β\np : α × β → Prop\nhp : ∀ᶠ (x : α × β) in atTop ×ˢ atTop, p x\n⊢ ∀ᶠ (k : α) in atTop, ∀ᶠ (l : β) in atTop, p (k, l)"
] | rw [← prod_atTop_atTop_eq] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.AtTopBot.Finite | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 51
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u_3\nf : Filter α\ns : ℕ → Set α\nhs : f.HasAntitoneBasis s\nr : ℕ → ℕ → Prop\nhr : ∀ (m : ℕ), ∀ᶠ (n : ℕ) in atTop, r m n\nthis : ∀ (t : Set ℕ), t.Finite → ∀ᶠ (n : ℕ) in atTop, ∀ m ∈ t, m < n ∧ r m n\nφ : ℕ → ℕ\nhφ : (∀ (x : ℕ) ⦃x_1 : ℕ⦄, x_1 < x → φ x_1 < φ x) ∧ ∀ (x : ℕ) ⦃x_1 : ℕ⦄, x_1 < x →... | [] | exact ⟨φ, forall_comm.2 hφ.1, forall_comm.2 hφ.2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.AtTopBot.CountablyGenerated | {
"line": 76,
"column": 58
} | {
"line": 87,
"column": 16
} | {
"line": 89,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝³ : Preorder α\ninst✝² : Nonempty α\ninst✝¹ : IsDirectedOrder α\ninst✝ : atTop.IsCountablyGenerated\n⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Monotone",
"Filter.at... | [] | by
obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
choose c hleft hright using exists_ge_ge (α := α)
set xs : ℕ → α := fun n => (List.range n).foldl (fun x n ↦ c x (ys n)) (ys 0)
have hsucc (n : ℕ) : xs (n + 1) = c (xs n) (ys n) := by simp [xs, List.range_succ]
refine ⟨xs, ?_, ?_⟩
· refine monotone_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 22
} | {
"line": 81,
"column": 22
} | [
{
"pp": "X : Type u\ns₁ s₂ : Set X\ninst✝ : TopologicalSpace X\nh₁ : IsOpen[inst✝] s₁\nh₂ : IsOpen[inst✝] s₂\n⊢ IsOpen[inst✝] (s₁ ∪ s₂)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Bool",
"IsO... | [
"X : Type u\ns₁ s₂ : Set X\ninst✝ : TopologicalSpace X\nh₁ : IsOpen[inst✝] s₁\nh₂ : IsOpen[inst✝] s₂\n⊢ IsOpen[inst✝] (⋃ b, bif b then s₁ else s₂)"
] | rw [union_eq_iUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Basic | {
"line": 86,
"column": 40
} | {
"line": 86,
"column": 52
} | {
"line": 86,
"column": 52
} | [
{
"pp": "X : Type u\nα : Type u_1\ns : Set X\ninst✝ : TopologicalSpace X\nf : α → Set X\nho : ∀ (i : α), IsOpen[inst✝] (f i)\nhU : ⋃ i, f i = univ\nh : ∀ (i : α), IsOpen[inst✝] (f i ∩ s)\n⊢ IsOpen[inst✝] ((⋃ i, f i) ∩ s)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u\nα : Type u_1\ns : Set X\ninst✝ : TopologicalSpace X\nf : α → Set X\nho : ∀ (i : α), IsOpen[inst✝] (f i)\nhU : ⋃ i, f i = univ\nh : ∀ (i : α), IsOpen[inst✝] (f i ∩ s)\n⊢ IsOpen[inst✝] (⋃ i, f i ∩ s)"
] | iUnion_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Neighborhoods | {
"line": 61,
"column": 17
} | {
"line": 61,
"column": 49
} | {
"line": 63,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nx : X\ns : Set X\nf : Filter X\nh : x ∈ s\no : IsOpen[inst✝] s\nsf : 𝓟 s ≤ f\n⊢ ⨅ s ∈ {s | x ∈ s ∧ IsOpen[inst✝] s}, 𝓟 s ≤ f",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Filter.instCompleteLatticeFilter",
... | [] | exact iInf₂_le_of_le s ⟨h, o⟩ sf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.NhdsSet | {
"line": 189,
"column": 58
} | {
"line": 189,
"column": 82
} | {
"line": 191,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\ns t : Set X\nh : IsClosed[inst✝] t\n⊢ 𝓝ˢ s ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓟 tᶜ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsClosed.nhdsSet_le_sup"
],
"usedFVars": [
"X",
"inst✝",
"s",
"t",
"h"
],
... | [] | exact h.nhdsSet_le_sup s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order | {
"line": 970,
"column": 49
} | {
"line": 970,
"column": 73
} | {
"line": 972,
"column": 0
} | [
{
"pp": "⊢ ⨅ s ∈ {s | False ∈ s ∧ s ∈ {{True}}}, 𝓟 s = ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"iInf_eq_top._simp_1",
"Pure.pure",
"False",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
"Set.ofPred",
"Filter.i... | [] | simp [@and_comm (_ ∈ _)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order | {
"line": 970,
"column": 49
} | {
"line": 970,
"column": 73
} | {
"line": 972,
"column": 0
} | [
{
"pp": "⊢ ⨅ s ∈ {s | False ∈ s ∧ s ∈ {{True}}}, 𝓟 s = ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"iInf_eq_top._simp_1",
"Pure.pure",
"False",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
"Set.ofPred",
"Filter.i... | [] | simp [@and_comm (_ ∈ _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order | {
"line": 970,
"column": 49
} | {
"line": 970,
"column": 73
} | {
"line": 972,
"column": 0
} | [
{
"pp": "⊢ ⨅ s ∈ {s | False ∈ s ∧ s ∈ {{True}}}, 𝓟 s = ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"iInf_eq_top._simp_1",
"Pure.pure",
"False",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
"Set.ofPred",
"Filter.i... | [] | simp [@and_comm (_ ∈ _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions.SumProd | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 49
} | {
"line": 211,
"column": 2
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\ntas : Set (TopologicalSpace X)\ntbs : Set (TopologicalSpace Y)\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntc : TopologicalSpace Z\nhX : tX ∈ tas\nhY : tY ∈ tbs\nhf : Continuous[instTopologicalSpaceProd, tc] fun p ↦ f p.1 p.2\n⊢ Continuous... | [
"X : Type u_5\nY : Type u_6\nZ : Type u_7\nf : X → Y → Z\ntas : Set (TopologicalSpace X)\ntbs : Set (TopologicalSpace Y)\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntc : TopologicalSpace Z\nhX✝ : tX ∈ tas\nhY : tY ∈ tbs\nhf : Continuous[instTopologicalSpaceProd, tc] fun p ↦ f p.1 p.2\nhX : Continuous[sInf ta... | have hX := continuous_sInf_dom hX continuous_id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Maps.Basic | {
"line": 863,
"column": 11
} | {
"line": 863,
"column": 33
} | {
"line": 863,
"column": 34
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nf : X → Y\ng : Y → Z\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nhg : IsClosedEmbedding g\n⊢ IsClosedEmbedding (g ∘ f) ↔ IsClosedEmbedding f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\nf : X → Y\ng : Y → Z\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nhg : IsClosedEmbedding g\n⊢ IsEmbedding (g ∘ f) ∧ IsClosed[inst✝] (range (g ∘ f)) ↔ IsEmbedding f ∧ IsClosed[inst✝¹] (range f)"
] | isClosedEmbedding_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.ContinuousOn | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 37
} | {
"line": 313,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nx : α\nh : ¬AccPt x (𝓟 {x}ᶜ)\n⊢ ContinuousAt f x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ContinuousWithinAt",
"continuousWithinAt_compl_self",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nx : α\nh : ¬AccPt x (𝓟 {x}ᶜ)\n⊢ ContinuousWithinAt f {x}ᶜ x"
] | ← continuousWithinAt_compl_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructions | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 59
} | {
"line": 505,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nx : ↑s\n⊢ map Subtype.val (𝓝 x) = 𝓝[s] ↑x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"nhdsWithin",
"Membership.mem",
"nhds",
"id",
"Subtype... | [] | rw [IsInducing.subtypeVal.map_nhds_eq, Subtype.range_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Constructions | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 59
} | {
"line": 505,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nx : ↑s\n⊢ map Subtype.val (𝓝 x) = 𝓝[s] ↑x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"nhdsWithin",
"Membership.mem",
"nhds",
"id",
"Subtype... | [] | rw [IsInducing.subtypeVal.map_nhds_eq, Subtype.range_val] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructions | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 59
} | {
"line": 505,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nx : ↑s\n⊢ map Subtype.val (𝓝 x) = 𝓝[s] ↑x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"nhdsWithin",
"Membership.mem",
"nhds",
"id",
"Subtype... | [] | rw [IsInducing.subtypeVal.map_nhds_eq, Subtype.range_val] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 89,
"column": 22
} | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 41
} | [
{
"pp": "case refine_2\nα : Type u\ns : Set (Set α)\nthis : TopologicalSpace α := generateFrom s\n⊢ ⋃ a ∈ {f | f.Finite ∧ f ⊆ s}, ⋂₀ a = univ",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.ofPred",
"Set.univ",
"Set.Finite",
... | [
"case refine_2\nα : Type u\ns : Set (Set α)\nthis : TopologicalSpace α := generateFrom s\n⊢ ∀ (a : α), ∃ i, ∃ (_ : i ∈ {f | f.Finite ∧ f ⊆ s}), a ∈ ⋂₀ i"
] | iUnion₂_eq_univ_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Bases | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 19
} | {
"line": 157,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\nhs : s ∈ b\n⊢ IsOpen[t] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"TopologicalSpace.GenerateOpen.basic",
"id",
"TopologicalSpace.ge... | [] | rw [hb.eq_generateFrom]
exact .basic s hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 19
} | {
"line": 157,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\nhs : s ∈ b\n⊢ IsOpen[t] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"TopologicalSpace.GenerateOpen.basic",
"id",
"TopologicalSpace.ge... | [] | rw [hb.eq_generateFrom]
exact .basic s hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 16
} | {
"line": 686,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : ∀ (U : Set X), IsOpen[inst✝] U → U.Nonempty → (U ∩ s).Nonempty\nu : Set ↑s\nhud : ∀ (U : Set ↑s), IsOpen[instTopologicalSpaceSubtype] U → U.Nonempty → (U ∩ u).Nonempty\nv : Set X\nhv1 : IsOpen[inst✝] v\nhv2 : Subtype.val ⁻¹' v = u\n⊢ ∀ (U : Set X)... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : ∀ (U : Set X), IsOpen[inst✝] U → U.Nonempty → (U ∩ s).Nonempty\nu : Set ↑s\nhud : ∀ (U : Set ↑s), IsOpen[instTopologicalSpaceSubtype] U → U.Nonempty → (U ∩ u).Nonempty\nv : Set X\nhv1 : IsOpen[inst✝] v\nhv2 : Subtype.val ⁻¹' v = u\nt : Set X\nht : IsOpen[inst... | intro t ht ht' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.SetAccumulate | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | {
"line": 115,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\n⊢ (∃ n, accumulate s n = univ) → ∃ n, s n = univ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Set.univ",
"Exists",
"id",
... | [
"α : Type u_1\ns : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\n⊢ (∀ (n : ℕ), s n ≠ univ) → ∀ (n : ℕ), accumulate s n ≠ univ"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
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