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.CategoryTheory.GuitartExact.Basic | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 62
} | [
{
"pp": "case h\nC₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nX₂ X₂' : C₂\nX₃ : C₃\ng : R.obj X₂ ⟶ B.ob... | rw [← CostructuredArrow.w φ, structuredArrowDownwards_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GuitartExact.Basic | {
"line": 299,
"column": 14
} | {
"line": 299,
"column": 33
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nF : C₁ ⥤ C₂\nX₂ : C₁\nX₃ : C₂\n⊢ ∀ (g : F.obj X₂ ... | (g : F.obj X₂ ⟶ X₃) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.CategoryTheory.GuitartExact.Basic | {
"line": 298,
"column": 2
} | {
"line": 308,
"column": 37
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nF : C₁ ⥤ C₂\n⊢ (mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)).Gu... | rw [guitartExact_iff_isConnected_rightwards]
intro X₂ X₃ (g : F.obj X₂ ⟶ X₃)
let Z := StructuredArrowRightwards (TwoSquare.mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)) g
let X₀ : Z := StructuredArrow.mk (Y := CostructuredArrow.mk g) (CostructuredArrow.homMk (𝟙 _))
have φ : ∀ (X : Z), X₀ ⟶ X := fun X =>
StructuredArrow.h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GuitartExact.Basic | {
"line": 298,
"column": 2
} | {
"line": 308,
"column": 37
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nF : C₁ ⥤ C₂\n⊢ (mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)).Gu... | rw [guitartExact_iff_isConnected_rightwards]
intro X₂ X₃ (g : F.obj X₂ ⟶ X₃)
let Z := StructuredArrowRightwards (TwoSquare.mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)) g
let X₀ : Z := StructuredArrow.mk (Y := CostructuredArrow.mk g) (CostructuredArrow.homMk (𝟙 _))
have φ : ∀ (X : Z), X₀ ⟶ X := fun X =>
StructuredArrow.h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.Opposite | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : ModelCategory C\ninst✝ : (cofibrations C).HasFactorization (trivialFibrations C)\n⊢ (trivialCofibrations Cᵒᵖ).HasFactorization (fibrations Cᵒᵖ)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"HomotopicalAlgebra.fi... | rw [trivialCofibrations_op, fibrations_op, ]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.Opposite | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : ModelCategory C\ninst✝ : (cofibrations C).HasFactorization (trivialFibrations C)\n⊢ (trivialCofibrations Cᵒᵖ).HasFactorization (fibrations Cᵒᵖ)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"HomotopicalAlgebra.fi... | rw [trivialCofibrations_op, fibrations_op, ]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 40
} | [
{
"pp": "case «1»\nn✝ : ℕ\nX✝ : Truncated (n✝ + 1)\nm n : ℕ\nX : Truncated (n + 1)\np q : X.Path 1\nh : p.vertex 1 = q.vertex 0\n⊢ (ConcreteCategory.hom (((trunc (n + 1) 1 ⋯).obj X).map (tr (SimplexCategory.δ 1) Path₁._proof_1 Path₁._proof_5).op))\n (![p.arrow 0, q.arrow 0] ((fun i ↦ i) ⟨1, ⋯⟩)) =\n ![p... | exact (q.arrow_src 0).trans h.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 40
} | [
{
"pp": "case «1»\nn✝ : ℕ\nX✝ : Truncated (n✝ + 1)\nm n : ℕ\nX : Truncated (n + 1)\np q : X.Path 1\nh : p.vertex 1 = q.vertex 0\n⊢ (ConcreteCategory.hom (((trunc (n + 1) 1 ⋯).obj X).map (tr (SimplexCategory.δ 1) Path₁._proof_1 Path₁._proof_5).op))\n (![p.arrow 0, q.arrow 0] ((fun i ↦ i) ⟨1, ⋯⟩)) =\n ![p... | exact (q.arrow_src 0).trans h.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 40
} | [
{
"pp": "case «1»\nn✝ : ℕ\nX✝ : Truncated (n✝ + 1)\nm n : ℕ\nX : Truncated (n + 1)\np q : X.Path 1\nh : p.vertex 1 = q.vertex 0\n⊢ (ConcreteCategory.hom (((trunc (n + 1) 1 ⋯).obj X).map (tr (SimplexCategory.δ 1) Path₁._proof_1 Path₁._proof_5).op))\n (![p.arrow 0, q.arrow 0] ((fun i ↦ i) ⟨1, ⋯⟩)) =\n ![p... | exact (q.arrow_src 0).trans h.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 191,
"column": 4
} | {
"line": 193,
"column": 27
} | [
{
"pp": "case hᵥ.h\nn : ℕ\nX : Truncated (n + 1)\nm : ℕ\nhₘ : m ≤ n + 1\nj l : ℕ\nh : j + l ≤ m\nΔ : X.obj (op { obj := ⦋m⦌, property := hₘ })\ni : Fin (l + 1)\n⊢ (X.spine l ⋯ ((ConcreteCategory.hom (X.map (tr (subinterval j l h) ⋯ hₘ).op)) Δ)).vertex i =\n ((X.spine m hₘ Δ).interval j l h).vertex i",
"u... | dsimp only [spine_vertex, Path.interval]
rw [← Functor.map_comp_apply, ← op_comp, ← tr_comp,
const_subinterval_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 191,
"column": 4
} | {
"line": 193,
"column": 27
} | [
{
"pp": "case hᵥ.h\nn : ℕ\nX : Truncated (n + 1)\nm : ℕ\nhₘ : m ≤ n + 1\nj l : ℕ\nh : j + l ≤ m\nΔ : X.obj (op { obj := ⦋m⦌, property := hₘ })\ni : Fin (l + 1)\n⊢ (X.spine l ⋯ ((ConcreteCategory.hom (X.map (tr (subinterval j l h) ⋯ hₘ).op)) Δ)).vertex i =\n ((X.spine m hₘ Δ).interval j l h).vertex i",
"u... | dsimp only [spine_vertex, Path.interval]
rw [← Functor.map_comp_apply, ← op_comp, ← tr_comp,
const_subinterval_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 195,
"column": 4
} | {
"line": 196,
"column": 30
} | [
{
"pp": "case hₐ.h\nn : ℕ\nX : Truncated (n + 1)\nm : ℕ\nhₘ : m ≤ n + 1\nj l : ℕ\nh : j + l ≤ m\nΔ : X.obj (op { obj := ⦋m⦌, property := hₘ })\ni : Fin l\n⊢ (ConcreteCategory.hom (X.map (tr (mkOfSucc i) ⋯ ⋯).op))\n ((ConcreteCategory.hom (X.map (tr (subinterval j l h) ⋯ hₘ).op)) Δ) =\n Path.arrow\n ... | rw [← Functor.map_comp_apply, ← op_comp, ← tr_comp,
mkOfSucc_subinterval_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 150,
"column": 18
} | {
"line": 153,
"column": 26
} | [
{
"pp": "C : Type u_1\ninst✝ : EnrichedCategory Cat C\na✝ b✝ c✝ : CatEnriched C\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\n⊢ (eqToIso ⋯).hom ≫ hComp (𝟙 f) (eqToIso ⋯).hom = hComp (eqToIso ⋯).hom (𝟙 g)",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.CatEnriched.id_hComp_id",
"Catego... | by
generalize_proofs h1 h2 h3; revert h1 h2 h3
generalize 𝟙 _ ≫ g = g, f ≫ 𝟙 _ = f
rintro _ rfl rfl; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 254,
"column": 56
} | {
"line": 254,
"column": 68
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\nX Y : CatEnrichedOrdinary C\nf g : X ⟶ Y\nα : f = g\n⊢ base (eqToHom α) = eqToHom ⋯",
"usedConstants": [
"CategoryTheory.Cat.category",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
... | cases α; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 254,
"column": 56
} | {
"line": 254,
"column": 68
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\nX Y : CatEnrichedOrdinary C\nf g : X ⟶ Y\nα : f = g\n⊢ base (eqToHom α) = eqToHom ⋯",
"usedConstants": [
"CategoryTheory.Cat.category",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
... | cases α; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 287,
"column": 36
} | {
"line": 287,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\na b : CatEnrichedOrdinary C\nf f' : a ⟶ b\nη : f ⟶ f'\n⊢ hComp (𝟙 (𝟙 a)) η ≍ η",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.CatEnrichedOrdinary",
... | simp [id_hComp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 287,
"column": 36
} | {
"line": 287,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\na b : CatEnrichedOrdinary C\nf f' : a ⟶ b\nη : f ⟶ f'\n⊢ hComp (𝟙 (𝟙 a)) η ≍ η",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.CatEnrichedOrdinary",
... | simp [id_hComp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 287,
"column": 36
} | {
"line": 287,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\na b : CatEnrichedOrdinary C\nf f' : a ⟶ b\nη : f ⟶ f'\n⊢ hComp (𝟙 (𝟙 a)) η ≍ η",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.CatEnrichedOrdinary",
... | simp [id_hComp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 325,
"column": 23
} | {
"line": 325,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\n⊢ ∀ {a b : CatEnrichedOrdinary C} {f g : a ⟶ b} (η : f ⟶ g), hComp (𝟙 (𝟙 a)) η = (eqToIso ⋯).hom ≫ η ≫ (eqToIso ⋯).inv",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Ca... | simp [id_hComp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 325,
"column": 23
} | {
"line": 325,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\n⊢ ∀ {a b : CatEnrichedOrdinary C} {f g : a ⟶ b} (η : f ⟶ g), hComp (𝟙 (𝟙 a)) η = (eqToIso ⋯).hom ≫ η ≫ (eqToIso ⋯).inv",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Ca... | simp [id_hComp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 325,
"column": 23
} | {
"line": 325,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : EnrichedOrdinaryCategory Cat C\n⊢ ∀ {a b : CatEnrichedOrdinary C} {f g : a ⟶ b} (η : f ⟶ g), hComp (𝟙 (𝟙 a)) η = (eqToIso ⋯).hom ≫ η ≫ (eqToIso ⋯).inv",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Ca... | simp [id_hComp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 74,
"column": 17
} | {
"line": 74,
"column": 71
} | [
{
"pp": "S T : Truncated 2\nf : S ⟶ T\nx : OneTruncation₂ S\n⊢ Truncated.Edge.map (𝟙rq x) f = 𝟙rq ((ConcreteCategory.hom (f.app (op { obj := ⦋0⦌, property := _proof_1 }))) x)",
"usedConstants": [
"SSet.Truncated.Edge",
"SSet.OneTruncation₂",
"CategoryTheory.ObjectProperty.FullSubcategory... | ext; simp [← NatTrans.naturality_apply, reflQuiver_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 74,
"column": 17
} | {
"line": 74,
"column": 71
} | [
{
"pp": "S T : Truncated 2\nf : S ⟶ T\nx : OneTruncation₂ S\n⊢ Truncated.Edge.map (𝟙rq x) f = 𝟙rq ((ConcreteCategory.hom (f.app (op { obj := ⦋0⦌, property := _proof_1 }))) x)",
"usedConstants": [
"SSet.Truncated.Edge",
"SSet.OneTruncation₂",
"CategoryTheory.ObjectProperty.FullSubcategory... | ext; simp [← NatTrans.naturality_apply, reflQuiver_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 286,
"column": 45
} | {
"line": 286,
"column": 57
} | [
{
"pp": "V : Truncated 2\nx₀ x₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge x₀ x₁\ny₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne' : Edge x₀ y₁\nh : e.edge = e'.edge\n⊢ (ConcreteCategory.hom (V.map (δ₂ 0 Edge._proof_1 Edge._proof_3).op)) e.edge = y₁",
"... | ← e'.tgt_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 333,
"column": 8
} | {
"line": 333,
"column": 42
} | [
{
"pp": "V : Truncated 2\nx y : V.HomotopyCategory\nf : x.as ⟶ y.as\nx✝ : ⊤ ((quotientFunctor V).map f)\n⊢ (morphismPropertyHomMk V).multiplicativeClosure ((quotientFunctor V).map f)",
"usedConstants": [
"SSet.OneTruncation₂",
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"SSet.Trunca... | morphismPropertyHomMk_eq_strictMap | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated | {
"line": 171,
"column": 71
} | {
"line": 176,
"column": 45
} | [
{
"pp": "A : Truncated 2\ninst✝ : A.Quasicategory₂\nx y z : A.obj (Opposite.op { obj := { len := 0 }, property := Quasicategory₂._proof_1 })\nf f' : Edge x y\ng g' : Edge y z\nh h' : Edge x z\ns : f.CompStruct g h\ns' : f'.CompStruct g' h'\nhf : HomotopicL f f'\nhg : HomotopicL g g'\n⊢ HomotopicL h h'",
"us... | by
rcases hg.homotopicR with ⟨hg⟩
rcases hf with ⟨hf⟩
let ⟨s₁⟩ := Quasicategory₂.fill32 hf (idComp g') s'
let ⟨s₂⟩ := Quasicategory₂.fill31 (compId f) hg s₁
exact Quasicategory₂.fill31 s (compId g) s₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 153,
"column": 16
} | {
"line": 153,
"column": 19
} | [
{
"pp": "X : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\nk : ℕ\nhk :\n ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),\n i + k = j →\n (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =\n (ConcreteCategory.hom (s.... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 32
} | [
{
"pp": "case inl.inr\nn : ℕ\ni : Fin (n + 2)\nh : i.succ ≤ i.succ\n⊢ δ i.succ ≫ σ i = 𝟙 (mk (n + 1)) ∨ ∃ j j', δ i.succ ≫ σ i = σ j ≫ δ j'",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Fin.succ",
"SimplexCategoryGenRel.mk",
"SimplexCategoryGenR... | exact Or.inl δ_comp_σ_succ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 32
} | [
{
"pp": "case inl.inr\nn : ℕ\ni : Fin (n + 2)\nh : i.succ ≤ i.succ\n⊢ δ i.succ ≫ σ i = 𝟙 (mk (n + 1)) ∨ ∃ j j', δ i.succ ≫ σ i = σ j ≫ δ j'",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Fin.succ",
"SimplexCategoryGenRel.mk",
"SimplexCategoryGenR... | exact Or.inl δ_comp_σ_succ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 32
} | [
{
"pp": "case inl.inr\nn : ℕ\ni : Fin (n + 2)\nh : i.succ ≤ i.succ\n⊢ δ i.succ ≫ σ i = 𝟙 (mk (n + 1)) ∨ ∃ j j', δ i.succ ≫ σ i = σ j ≫ δ j'",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Fin.succ",
"SimplexCategoryGenRel.mk",
"SimplexCategoryGenR... | exact Or.inl δ_comp_σ_succ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.MorphismProperty | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 32
} | [
{
"pp": "case h.mk.mk.zero\nd : ℕ\nW : MorphismProperty (Truncated d)\ninst✝ : W.IsMultiplicative\nδ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 2)), W (δ d i ⋯ ⋯)\nσ_mem : ∀ (n : ℕ) (hn : n < d) (i : Fin (n + 1)), W (σ d i ⋯ ⋯)\na : ℕ\nha : a ≤ d\nb : ℕ\nhb : b ≤ d\nf : { obj := { len := a }, property := ha } ⟶... | obtain rfl : a = 0 := by lia | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 14
} | [
{
"pp": "case neg.«_@».Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne.3003013004._hygCtx._hyg.53.«0»\nn : ℕ\nf : ⦋n⦌ ⟶ ⦋1⦌\nS : Finset (Fin (n + 1)) := {i | (ConcreteCategory.hom f) i = 1}\nhS : ¬S.Nonempty\ni : Fin (⦋n⦌.len + 1)\nhj : (ConcreteCategory.hom f) i = (fun i ↦ i) ⟨0, ⋯⟩\n⊢ 0 = (ConcreteCategory.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 22
} | [
{
"pp": "case w.h\nX X' Y Y' Z : Truncated 2\nx₀ x₁ : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge x₀ x₁\ny : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\n⊢ homMk (Edge.id y) ≫\n 𝟙\n (mk\n ((ConcreteCategory.hom\n ... | exact homMk_id y | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 96,
"column": 2
} | {
"line": 101,
"column": 23
} | [
{
"pp": "case mpr\nn m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\nx : Fin (m + 2)\ny : Fin (n + 1)\nh : x ≤ (f y).castSucc\n⊢ (∀ i < y, (f i).castSucc < x) → ∀ b ∈ finset f x, y.castSucc ≤ b",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"SimplexCategory.II.finset",
"SimplexCategory.II.ca... | · intro h' i hi
obtain ⟨i, rfl⟩ | rfl := i.eq_castSucc_or_eq_last
· simp only [Fin.castSucc_le_castSucc_iff]
by_contra!
exact (h' i this).not_ge (by simpa using hi)
· apply Fin.le_last | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal | {
"line": 211,
"column": 33
} | {
"line": 214,
"column": 61
} | [
{
"pp": "x✝ y z✝ : AugmentedSimplexCategory\nf✝ g✝ : x✝ ⊗ y ⟶ z✝\nx z : SimplexCategory\nf g : WithInitial.of x ⊗ WithInitial.star ⟶ WithInitial.of z\nh₁ : inl (WithInitial.of x) WithInitial.star ≫ f = inl (WithInitial.of x) WithInitial.star ≫ g\nh₂ : inr (WithInitial.of x) WithInitial.star ≫ f = inr (WithIniti... | by
simp only [inl, Category.assoc, Iso.cancel_iso_inv_left, Limits.IsInitial.to_self,
whiskerLeft_id_star] at h₁
simpa [Category.id_comp f, Category.id_comp g] using h₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 78,
"column": 24
} | {
"line": 81,
"column": 46
} | [
{
"pp": "S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ns : PushoutCocone f g\nm : Pushout f g ⟶ s.pt\nh₁ : inl f g ≫ m = s.inl\nh₂ : inr f g ≫ m = s.inr\n⊢ m =\n (fun s ↦\n ↾Quot.lift\n (fun x ↦\n match x with\n | Sum.inl x₁ => (ConcreteCategory.hom s.inl) x₁\n ... | by
ext ⟨x₁ | x₂⟩
· exact ConcreteCategory.congr_hom h₁ x₁
· exact ConcreteCategory.congr_hom h₂ x₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 23
} | [
{
"pp": "case mp\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nx₁ : X₁\nx₂ : X₂\n⊢ Rel' f g (Sum.inl x₁) (Sum.inr x₂) → ∃ s, x₁ = (ConcreteCategory.hom f) s ∧ x₂ = (ConcreteCategory.hom g) s",
"usedConstants": [
"CategoryTheory.Limits.Types.Pushout.Rel'.inl_inr",
"Sum.ctorIdx",
"CategoryTheor... | · rintro ⟨_⟩
exact ⟨_, rfl, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 45
} | [
{
"pp": "S X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ninst✝ : Mono f\na b : X₁ ⊕ X₂\n⊢ Quot.mk (Rel f g) a = Quot.mk (Rel f g) b ↔ Rel' f g a b",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.Types.Pushout.equivalence_rel'",
"congrArg",
"Sum",
"id",
"CategoryTheory.Lim... | ← (equivalence_rel' f g).quot_mk_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 7
} | [
{
"pp": "case inl.inl\nX₁ X₂ X₃ X₄ X₅ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nk : X₄ ⟶ X₅\nh₁ : IsPushout t l r b\nhr' : Function.Injective ⇑(ConcreteCategory.hom (r ≫ k))\nH :\n ∀ (x₃ y₃ : X₃),\n x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →\n y₃ ∉ Set.range ⇑(ConcreteCategory.hom l... | obtain rfl : x₂ = y₂ := hr' eq
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 7
} | [
{
"pp": "case inl.inl\nX₁ X₂ X₃ X₄ X₅ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nk : X₄ ⟶ X₅\nh₁ : IsPushout t l r b\nhr' : Function.Injective ⇑(ConcreteCategory.hom (r ≫ k))\nH :\n ∀ (x₃ y₃ : X₃),\n x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →\n y₃ ∉ Set.range ⇑(ConcreteCategory.hom l... | obtain rfl : x₂ = y₂ := hr' eq
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 18
} | [
{
"pp": "case notMem.inl\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝⁴ : LinearOrder ι\nf : P.RankFunction ι\ninst✝³ : P.IsProper\ninst✝² : OrderBot ι\ninst✝¹ : SuccOrder ι\ninst✝ : NoMaxOrder ι\nt : ↑P.II\na✝ : (↑t).subcomplex ≤ ⊤\n⊢ (↑t).subcomplex ≤ (↑(P.p { s := t, rank_s := ⋯ }.s)).subcomp... | · exact P.le t | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 345,
"column": 8
} | {
"line": 345,
"column": 22
} | [
{
"pp": "case a.notMem.refine_1.inl\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ninst✝ : P.IsProper\nj : ι\nc : f.Cell j\nt : ↑P.II\nhs : (↑t).subcomplex ≤ c.horn.image c.map\n⊢ (↑t).subcomplex ≤ (↑(P.p { s := t, rank_s := ⋯ }.s)).subcomplex",
"usedCo... | · exact P.le t | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.Retracts | {
"line": 40,
"column": 6
} | {
"line": 40,
"column": 56
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nh : Retract Y X\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalPresentable X κ\nJ : Type w\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nthis✝ : EssentiallySmall.{w, w, w} ... | exact ⟨k, u, by simpa [← cancel_epi h.r] using hj⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 27
} | [
{
"pp": "case inl.inr\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ninst✝ : P.IsProper\nj : ι\nx : f.Cell j\nhd : x.dim ≤ x.dim\nhs✝ :\n (ConcreteCategory.hom (x.ιSigmaStdSimplex.app (op ⦋x.dim⦌))) (stdSimplex.objEquiv.symm (SimplexCategory.δ x.index)) ∈\... | exact ⟨x, Or.inr rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Limits | {
"line": 106,
"column": 41
} | {
"line": 106,
"column": 44
} | [
{
"pp": "case a.a.h\nC : Type u\ninst✝⁴ : Category.{v, u} C\nK : Type u'\ninst✝³ : Category.{v', u'} K\nF : K ⥤ C ⥤ Type w'\nc : Cone F\nhc : (Y : C) → IsLimit (((evaluation C (Type w')).obj Y).mapCone c)\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\nhK : HasCardinalLT (Arrow K) κ\nJ : Type w\ninst✝¹ : SmallCat... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy | {
"line": 136,
"column": 6
} | {
"line": 140,
"column": 9
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nn : ℕ\nα : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H.h x.1\nβ : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun ... | rw [Finset.card_disjUnion, Finset.card_disjUnion, Finset.card_disjUnion,
Finset.card_image_of_injective _ hγ₁, Finset.card_image_of_injective _ hγ₂,
Finset.card_image_of_injective _ hγ₃, Finset.card_image_of_injective _ hγ₄]
simp
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy | {
"line": 136,
"column": 6
} | {
"line": 140,
"column": 9
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nn : ℕ\nα : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H.h x.1\nβ : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun ... | rw [Finset.card_disjUnion, Finset.card_disjUnion, Finset.card_disjUnion,
Finset.card_image_of_injective _ hγ₁, Finset.card_image_of_injective _ hγ₂,
Finset.card_image_of_injective _ hγ₃, Finset.card_image_of_injective _ hγ₄]
simp
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy | {
"line": 165,
"column": 45
} | {
"line": 165,
"column": 73
} | [
{
"pp": "case zero\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH✝ H : Homotopy f g\n⊢ ((alternatingFaceMapComplex C).map f).f 0 =\n (dNext 0) (ToChainHomotopy.hom H) + ToChainHomotopy.hom H 0 1 ≫ ((alternatingFaceMapComplex C).obj Y).d 1 0 +\n (... | dNext_eq_zero _ _ (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 70
} | [
{
"pp": "⊢ anodyneExtensions =\n (transfiniteCompositions.{u, u, u + 1} (coproducts.{u, u, u + 1} modelCategoryQuillen.J).pushouts).retracts",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.pushouts",
"Opposite",
"CategoryThe... | rw [anodyneExtensions_eq_llp_rlp, llp_rlp_of_hasSmallObjectArgument] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 70
} | [
{
"pp": "⊢ anodyneExtensions =\n (transfiniteCompositions.{u, u, u + 1} (coproducts.{u, u, u + 1} modelCategoryQuillen.J).pushouts).retracts",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.pushouts",
"Opposite",
"CategoryThe... | rw [anodyneExtensions_eq_llp_rlp, llp_rlp_of_hasSmallObjectArgument] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 70
} | [
{
"pp": "⊢ anodyneExtensions =\n (transfiniteCompositions.{u, u, u + 1} (coproducts.{u, u, u + 1} modelCategoryQuillen.J).pushouts).retracts",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.pushouts",
"Opposite",
"CategoryThe... | rw [anodyneExtensions_eq_llp_rlp, llp_rlp_of_hasSmallObjectArgument] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nhP : IsPushout f g inl inr\nW : C\nh : X ⟶ W\nk : Y ⟶ W\nw : f ≫ h = g ≫ k\nQ : C\n⊢ Q ◁ inr ≫ Q ◁ hP.desc h k w = Q ◁ k",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← MonoidalCategory.whiskerLeft_comp, IsPushout.inr_desc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nhP : IsPushout f g inl inr\nW : C\nh : X ⟶ W\nk : Y ⟶ W\nw : f ≫ h = g ≫ k\nQ : C\n⊢ Q ◁ inr ≫ Q ◁ hP.desc h k w = Q ◁ k",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← MonoidalCategory.whiskerLeft_comp, IsPushout.inr_desc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nhP : IsPushout f g inl inr\nW : C\nh : X ⟶ W\nk : Y ⟶ W\nw : f ≫ h = g ≫ k\nQ : C\n⊢ Q ◁ inr ≫ Q ◁ hP.desc h k w = Q ◁ k",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← MonoidalCategory.whiskerLeft_comp, IsPushout.inr_desc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nhP : IsPushout f g inl inr\nW : C\nh : X ⟶ W\nk : Y ⟶ W\nw : f ≫ h = g ≫ k\nQ : C\n⊢ inr ▷ Q ≫ hP.desc h k w ▷ Q = k ▷ Q",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← comp_whiskerRight, IsPushout.inr_desc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nhP : IsPushout f g inl inr\nW : C\nh : X ⟶ W\nk : Y ⟶ W\nw : f ≫ h = g ≫ k\nQ : C\n⊢ inr ▷ Q ≫ hP.desc h k w ▷ Q = k ▷ Q",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← comp_whiskerRight, IsPushout.inr_desc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Limits.Shapes.Pullback | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nhP : IsPushout f g inl inr\nW : C\nh : X ⟶ W\nk : Y ⟶ W\nw : f ≫ h = g ≫ k\nQ : C\n⊢ inr ▷ Q ≫ hP.desc h k w ▷ Q = k ▷ Q",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← comp_whiskerRight, IsPushout.inr_desc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 49,
"column": 4
} | {
"line": 52,
"column": 29
} | [
{
"pp": "X : SSet\ni j : ℕ\nh : i ≤ j\n⊢ (fun n ↦ ⨆ i, ⨆ x, Subcomplex.ofSimplex ↑x) i ≤ (fun n ↦ ⨆ i, ⨆ x, Subcomplex.ofSimplex ↑x) j",
"usedConstants": [
"Eq.mpr",
"le_refl",
"SSet.Subcomplex.ofSimplex",
"Opposite",
"iSup",
"PartialOrder.toPreorder",
"SSet.nonDege... | simp only [iSup_le_iff]
intro k x
exact le_trans (by exact le_trans (by rfl) (le_iSup _ x))
(le_iSup _ ⟨k, by lia⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 49,
"column": 4
} | {
"line": 52,
"column": 29
} | [
{
"pp": "X : SSet\ni j : ℕ\nh : i ≤ j\n⊢ (fun n ↦ ⨆ i, ⨆ x, Subcomplex.ofSimplex ↑x) i ≤ (fun n ↦ ⨆ i, ⨆ x, Subcomplex.ofSimplex ↑x) j",
"usedConstants": [
"Eq.mpr",
"le_refl",
"SSet.Subcomplex.ofSimplex",
"Opposite",
"iSup",
"PartialOrder.toPreorder",
"SSet.nonDege... | simp only [iSup_le_iff]
intro k x
exact le_trans (by exact le_trans (by rfl) (le_iSup _ x))
(le_iSup _ ⟨k, by lia⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 336,
"column": 29
} | {
"line": 336,
"column": 53
} | [
{
"pp": "case refine_1\nX Y : SSet\ni : X ⟶ Y\nd : ℕ\nx✝¹ : SimplexCategoryᵒᵖ\nn : ℕ\nx✝ : (fun X ↦ X) ↑(((skeletonOfMono i) d).obj (op ⦋n⦌))\nc : Cell i d\ny : Δ[d] _⦋n⦌\nhx : (ConcreteCategory.hom (c.map.app (op ⦋n⦌))) y ∈ ((skeletonOfMono i) d).obj (op ⦋n⦌)\n⊢ y ∈ (((skeletonOfMono i) d).preimage c.map).obj ... | Subcomplex.preimage_obj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 12
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nh✝ : v x = 0\n⊢ v x ≤ v y ↔ w x ≤ w y",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 12
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nh✝ : v x = 0\n⊢ v x ≤ v y ↔ w x ≤ w y",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 12
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nh✝ : v x = 0\n⊢ v x ≤ v y ↔ w x ≤ w y",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 109,
"column": 70
} | {
"line": 109,
"column": 78
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ 0 < v x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 109,
"column": 70
} | {
"line": 109,
"column": 78
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ 0 < v x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 109,
"column": 70
} | {
"line": 109,
"column": 78
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ 0 < v x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 110,
"column": 26
} | {
"line": 110,
"column": 34
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ 0 < w x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 110,
"column": 26
} | {
"line": 110,
"column": 34
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ 0 < w x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 110,
"column": 26
} | {
"line": 110,
"column": 34
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : Field R\ninst✝² : Semifield S\ninst✝¹ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝ : IsStrictOrderedRing S\nh : ∀ (x : R), v x < 1 ↔ w x < 1\nx y : R\nhy₀ : v x ≠ 0\n⊢ 0 < w x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemirin... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 122,
"column": 58
} | {
"line": 122,
"column": 66
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : Field R\ninst✝⁴ : Semifield S\ninst✝³ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝² : IsStrictOrderedRing S\ninst✝¹ : Archimedean S\ninst✝ : ExistsAddOfLE S\nhv✝ : v.IsNontrivial\nh : ∀ (x : R), v x < 1 → w x < 1\na : R\nha₀ : a ≠ 0\nhw : w a < 1\nx₀ : R\nhx₀ : x... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 122,
"column": 58
} | {
"line": 122,
"column": 66
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : Field R\ninst✝⁴ : Semifield S\ninst✝³ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝² : IsStrictOrderedRing S\ninst✝¹ : Archimedean S\ninst✝ : ExistsAddOfLE S\nhv✝ : v.IsNontrivial\nh : ∀ (x : R), v x < 1 → w x < 1\na : R\nha₀ : a ≠ 0\nhw : w a < 1\nx₀ : R\nhx₀ : x... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 122,
"column": 58
} | {
"line": 122,
"column": 66
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : Field R\ninst✝⁴ : Semifield S\ninst✝³ : LinearOrder S\nv w : AbsoluteValue R S\ninst✝² : IsStrictOrderedRing S\ninst✝¹ : Archimedean S\ninst✝ : ExistsAddOfLE S\nhv✝ : v.IsNontrivial\nh : ∀ (x : R), v x < 1 → w x < 1\na : R\nha₀ : a ≠ 0\nhw : w a < 1\nx₀ : R\nhx₀ : x... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 183,
"column": 60
} | {
"line": 183,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E → F\ninst✝ : CompleteSpace E\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 183,
"column": 60
} | {
"line": 183,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E → F\ninst✝ : CompleteSpace E\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 269,
"column": 48
} | {
"line": 269,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\ninst✝¹ : CompleteSpace E\ninst✝ : CharZero 𝕜\nh : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤\nthis : analyticOrderAt (deriv f) x = ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.Order | {
"line": 269,
"column": 48
} | {
"line": 269,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\ninst✝¹ : CompleteSpace E\ninst✝ : CharZero 𝕜\nh : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤\nthis : analyticOrderAt (deriv f) x = ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Order | {
"line": 269,
"column": 48
} | {
"line": 269,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\ninst✝¹ : CompleteSpace E\ninst✝ : CharZero 𝕜\nh : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ⊤\nthis : analyticOrderAt (deriv f) x = ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 277,
"column": 29
} | {
"line": 277,
"column": 77
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\ninst✝¹ : CompleteSpace E\ninst✝ : CharZero 𝕜\nr : ℕ\nh : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = 0\nhr : r = 0\n⊢ False",
"use... | AnalyticAt.analyticOrderAt_eq_zero (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Order | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 14
} | [
{
"pp": "case coe.a\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\nhf' : deriv f x ≠ 0\nr : ℕ\nh : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ↑r\nF : 𝕜 → E\nhFa : AnalyticAt 𝕜 F x\nhFne : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 64
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nn : ℕ\nh : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) x\nthis : ∀ᶠ (y : 𝕜) in 𝓝[≠] x, ContinuousAt (fun z ↦ (z - x) ^ n • f z) y\ny : 𝕜\nhy : Continuo... | have : ContinuousAt (fun z ↦ ((z - x) ^ n)⁻¹) y :=
ContinuousAt.inv₀ (by fun_prop) (by simp [sub_eq_zero, h'y]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 349,
"column": 8
} | {
"line": 349,
"column": 33
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nx : 𝕜\ninst✝ : CompleteSpace E\nf : 𝕜 → E\nn : ℕ\nh : ∀ᶠ (y : 𝕜) in 𝓝 x, AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) y\n⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[≠] x, AnalyticAt 𝕜 ... | eventually_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 362,
"column": 2
} | {
"line": 364,
"column": 29
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nx✝ : MeromorphicAt f x\nn : ℕ\nhn : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) x\n⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝 x, x_1 ∈ {x}ᶜ → f x_1 = (x_1 - x) ^ (-↑n) ... | · filter_upwards with z hz
match_scalars
simp [sub_ne_zero.mpr hz] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 450,
"column": 4
} | {
"line": 450,
"column": 40
} | [
{
"pp": "case neg\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜'\ninst✝⁴ : NormedAlgebra 𝕜 𝕜'\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nx : 𝕜\nf : 𝕜' → F\ng :... | set j := fun z ↦ (z - g x) ^ r • f z | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 62
} | [
{
"pp": "case inl\n⊢ (deriv fun x ↦ ↑x ^ 0) =O[atTop] fun x ↦ x ^ (re 0 - 1)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"Complex.instNormedAddCommGroup",
"deriv_const'",
"Real.denselyNormedField",
"congrArg",
... | simp_rw [cpow_zero, deriv_const', Asymptotics.isBigO_zero] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 62
} | [
{
"pp": "case inl\n⊢ (deriv fun x ↦ ↑x ^ 0) =O[atTop] fun x ↦ x ^ (re 0 - 1)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"Complex.instNormedAddCommGroup",
"deriv_const'",
"Real.denselyNormedField",
"congrArg",
... | simp_rw [cpow_zero, deriv_const', Asymptotics.isBigO_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 62
} | [
{
"pp": "case inl\n⊢ (deriv fun x ↦ ↑x ^ 0) =O[atTop] fun x ↦ x ^ (re 0 - 1)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"Complex.instNormedAddCommGroup",
"deriv_const'",
"Real.denselyNormedField",
"congrArg",
... | simp_rw [cpow_zero, deriv_const', Asymptotics.isBigO_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 619,
"column": 4
} | {
"line": 619,
"column": 15
} | [
{
"pp": "case a\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nx : 𝕜\nh₁f : AnalyticOnNhd 𝕜 f U\nh₂f : analyticOrderAt f x = 0\nhx : x ∈ U\nhU : IsConnected U\na : 𝕜\nha : analyticOrderAt f a = 0 ∨ analyticOr... | use ⟨x, hx⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Analytic.Order | {
"line": 620,
"column": 4
} | {
"line": 620,
"column": 12
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nx : 𝕜\nh₁f : AnalyticOnNhd 𝕜 f U\nh₂f : analyticOrderAt f x = 0\nhx : x ∈ U\nhU : IsConnected U\na : 𝕜\nha : analyticOrderAt f a = 0 ∨ analyticOr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Analytic.IteratedFDeriv | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : ContinuousMultilinearMap 𝕜 (fun i ↦ E) F\nx : E\nv : Fin n → E\ng : E →L[𝕜] Fin n → E := Contin... | ← sum_comp (Equiv.embeddingEquivOfFinite (Fin n)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 174,
"column": 2
} | {
"line": 177,
"column": 47
} | [
{
"pp": "a : ℕ\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) 0 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Algebra.to_smulCommClass",
... | convert! one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero (a + 1) using 3 with z n
· norm_cast
· rw [eq_comm, add_right_comm, add_sub_cancel_right, ← Nat.cast_add,
Ring.choose_natCast, Nat.choose_symm_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Binomial | {
"line": 174,
"column": 2
} | {
"line": 177,
"column": 47
} | [
{
"pp": "a : ℕ\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) 0 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Algebra.to_smulCommClass",
... | convert! one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero (a + 1) using 3 with z n
· norm_cast
· rw [eq_comm, add_right_comm, add_sub_cancel_right, ← Nat.cast_add,
Ring.choose_natCast, Nat.choose_symm_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 158,
"column": 60
} | {
"line": 158,
"column": 90
} | [
{
"pp": "case inl.inr.inl\ny : ℝ\ny_neg : y < 0\n⊢ ⊥ = ↑y * ⊤",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"instTopEReal",
"id",
"Bot.bot",
"Top.top",
"Eq",
"instBotEReal",
"EReal.coe_mul_top_of_neg",
"Real.toEReal",... | EReal.coe_mul_top_of_neg y_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 248,
"column": 16
} | {
"line": 248,
"column": 24
} | [
{
"pp": "case convert_2\na : ℝ\nH : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)\nthis : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1\nx : ℝ\nhx : x ∈ Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ)\n⊢ (⇑Complex.reCLM ∘ (fun x ↦ (1 +... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 53
} | [
{
"pp": "u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\n⊢ linearGrowthInf u ≤ linearGrowthInf v",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"PartialOrder.toPreorder",
"EReal",
"AddMonoidWithOne.toNatCast",
... | apply (linearGrowthInf_eventually_monotone h).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : NormedField 𝕜\ninst✝ : Norm E\na : 𝕜\nf : 𝕜 → E\nh : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝[≠] a) (norm ∘ f)\n⊢ Tendsto (fun x ↦ ‖x - a‖⁻¹) (𝓝[≠] a) atTop",
"usedConstants": [
"tendsto_norm_sub_self_nhdsNE",
"Norm.norm",
"Real",
"C... | exact (tendsto_norm_sub_self_nhdsNE a).inv_tendsto_nhdsGT_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 35
} | [
{
"pp": "u : ℕ → ℝ≥0∞\na : EReal\n⊢ a ≤ expGrowthSup u ↔ ∀ b < a, ∃ᶠ (n : ℕ) in atTop, (b * ↑n).exp ≤ u n",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"congrArg",
"Filter.is... | rw [expGrowthSup, le_limsup_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 83
} | [
{
"pp": "case h\nu : ℕ → EReal\nv : ℕ → ℕ\nhu : ∃ᶠ (n : ℕ) in atTop, 0 ≤ u n\nhv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0\nhv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤\nhv₂ : Tendsto v atTop atTop\nv_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)\na : EReal\nv_a : a > linearGrowthSup fun n ↦ ↑(v n)\nb : EReal\nu_b : b > l... | replace vn_a := ((div_lt_iff (Nat.cast_pos'.2 n_0) (natCast_ne_top n)).1 vn_a).le | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 195,
"column": 47
} | {
"line": 195,
"column": 73
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nu : ℕ → E\nl : E\nh : Tendsto u atTop (𝓝 l)\nthis : (fun n ↦ ∑ i ∈ range n, (u i - l)) =o[atTop] fun n ↦ ↑n\nn : ℕ\nnpos : n ∈ Set.Ici 1\nnposℝ : 0 < ↑n\n⊢ ((↑n)⁻¹ * ↑n) • l = l",
"usedConstants": [
"Eq.mpr",
... | inv_mul_cancel₀ nposℝ.ne', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 303,
"column": 65
} | {
"line": 306,
"column": 32
} | [
{
"pp": "α : Type u_1\nu : α → ℕ → ℝ≥0∞\ns : Set α\nhs : s.Finite\n⊢ expGrowthInf (⨅ x ∈ s, u x) = ⨅ x ∈ s, expGrowthInf (u x)",
"usedConstants": [
"ExpGrowth.expGrowthInfTopHom",
"ExpGrowth.expGrowthInf",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
... | by
have := map_finset_inf expGrowthInfTopHom hs.toFinset u
simpa only [expGrowthInfTopHom, InfTopHom.coe_mk, InfHom.coe_mk, Finset.inf_eq_iInf,
hs.mem_toFinset, comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℕ), IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun a ↦ |k a ^ z * f a|\nz : ℕ\nm : β\nhm ... | refine (hk.eventually_ne_atTop 0).mono fun x hk0 hx => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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