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": 300,
"column": 2
} | {
"line": 300,
"column": 77
} | [
{
"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\ng : F.obj X₂ ⟶ X₃\... | let Z := StructuredArrowRightwards (TwoSquare.mk (𝟭 C₁) F F (𝟭 C₂) (𝟙 F)) g | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 106,
"column": 2
} | {
"line": 108,
"column": 48
} | [
{
"pp": "case obj.h.h.mp\nn d : ℕ\nj : Δ[n] _⦋d⦌\n⊢ (∃ j_1, ∀ (x : Fin (d + 1)), ¬(j x.rev).rev = j_1) → ∃ j_1, ∀ (x : Fin (d + 1)), ¬j x = j_1",
"usedConstants": [
"Opposite",
"congrArg",
"CategoryTheory.Functor.category",
"Exists",
"instOfNatNat",
"SSet",
"SSet.st... | all_goals
· rintro ⟨k, hk⟩
exact ⟨k.rev, fun l _ ↦ hk l.rev (by aesop)⟩ | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Subfunctor.Equalizer | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF₁ F₂ : C ⥤ Type w\nA : Subfunctor F₁\nf g : A.toFunctor ⟶ F₂\nG : C ⥤ Type w\nφ : G ⟶ A.toFunctor\n⊢ range (φ ≫ A.ι) ≤ Subfunctor.equalizer f g ↔ φ ≫ f = φ ≫ g",
"usedConstants": [
"Iff.mpr",
"_private.Mathlib.CategoryTheory.Subfunctor.Equalizer.0... | rw [NatTrans.ext_iff]
simp [le_def, Set.subset_def, ConcreteCategory.hom_ext_iff, funext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subfunctor.Equalizer | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 73
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF₁ F₂ : C ⥤ Type w\nA : Subfunctor F₁\nf g : A.toFunctor ⟶ F₂\nG : C ⥤ Type w\nφ : G ⟶ A.toFunctor\n⊢ range (φ ≫ A.ι) ≤ Subfunctor.equalizer f g ↔ φ ≫ f = φ ≫ g",
"usedConstants": [
"Iff.mpr",
"_private.Mathlib.CategoryTheory.Subfunctor.Equalizer.0... | rw [NatTrans.ext_iff]
simp [le_def, Set.subset_def, ConcreteCategory.hom_ext_iff, funext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 26
} | [
{
"pp": "case «0»\nn : ℕ\ni a b : Fin (n + 1)\nhab : a ≤ b\nH : #{i, a, b} ≤ n\nk : Fin (n + 1)\nhk : ¬k = i ∧ ¬k = a ∧ ¬k = b\n⊢ ¬(stdSimplex.edge n a b hab) ((fun i ↦ i) ⟨0, ⋯⟩) = k",
"usedConstants": [
"Opposite",
"Nat.le_refl",
"CategoryTheory.Functor.category",
"Fin.mk",
"... | · exact Ne.symm hk.2.1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.HornColimits | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 46
} | [
{
"pp": "case succ\nX : SSet\nn : ℕ\ni : Fin (n + 1 + 2)\nf : (j : Fin (n + 1 + 2)) → j ≠ i → (Δ[n + 1] ⟶ X)\nhf : horn.IsCompatible f\n⊢ ∀ (a : (MultispanShape.ofLinearOrder ↑{i}ᶜ).L),\n ((CompleteLattice.MulticoequalizerDiagram.multispanIndex ⋯).toLinearOrder.map Subcomplex.toSSetFunctor).fst a ≫\n ... | rintro ⟨⟨⟨a, ha⟩, ⟨b, hb⟩⟩, hab : a < b⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 42
} | [
{
"pp": "case mk\nV : Type u_1\ninst✝ : ReflQuiver V\nmotive : {x y : FreeRefl V} → (x ⟶ y) → Prop\nid : ∀ (x : V), motive (homMk (𝟙rq x))\ncomp_homMk : ∀ {x y z : V} (f : mk x ⟶ mk y) (g : y ⟶ z), motive f → motive (f ≫ homMk g)\ny : FreeRefl V\nx : V\nf : mk x ⟶ y\n⊢ motive f",
"usedConstants": [
"... | induction y using induction with | _ y
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 12
} | [
{
"pp": "V : Type u_1\ninst✝ : ReflQuiver V\nX✝ Y✝ : FreeRefl V\nf : X✝ ⟶ Y✝\nhf : ⊤ f\n⊢ (morphismPropertyHomMk V).multiplicativeClosure f",
"usedConstants": []
}
] | clear hf | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 273,
"column": 11
} | {
"line": 273,
"column": 45
} | [
{
"pp": "case nil\nm₁ m : ℕ\nhL : IsAdmissible (m + 1) []\nj : ℕ\nhj : j < m + 1\nhm₁ : m + [].length + 1 = m₁\n⊢ standardσ (simplicialInsert j []) ⋯ = standardσ [] ⋯ ≫ σ (Fin.ofNat (m + 1) j)",
"usedConstants": [
"instNeZeroNatHAdd_1",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom... | simp [standardσ, simplicialInsert] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 273,
"column": 11
} | {
"line": 273,
"column": 45
} | [
{
"pp": "case nil\nm₁ m : ℕ\nhL : IsAdmissible (m + 1) []\nj : ℕ\nhj : j < m + 1\nhm₁ : m + [].length + 1 = m₁\n⊢ standardσ (simplicialInsert j []) ⋯ = standardσ [] ⋯ ≫ σ (Fin.ofNat (m + 1) j)",
"usedConstants": [
"instNeZeroNatHAdd_1",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom... | simp [standardσ, simplicialInsert] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 273,
"column": 11
} | {
"line": 273,
"column": 45
} | [
{
"pp": "case nil\nm₁ m : ℕ\nhL : IsAdmissible (m + 1) []\nj : ℕ\nhj : j < m + 1\nhm₁ : m + [].length + 1 = m₁\n⊢ standardσ (simplicialInsert j []) ⋯ = standardσ [] ⋯ ≫ σ (Fin.ofNat (m + 1) j)",
"usedConstants": [
"instNeZeroNatHAdd_1",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom... | simp [standardσ, simplicialInsert] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 58,
"column": 26
} | {
"line": 60,
"column": 9
} | [
{
"pp": "ι : Type u_1\nK x✝¹ : PreAbstractSimplicialComplex ι\nx✝ : (fun K ↦ K.faces) K = (fun K ↦ K.faces) x✝¹\n⊢ K = x✝¹",
"usedConstants": [
"PreAbstractSimplicialComplex.faces",
"Finset",
"IsRelLowerSet",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Eq.rec",
"Fi... | by
cases K
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 50
} | [
{
"pp": "case left\nι : Type u_1\nK : PreAbstractSimplicialComplex ι\nx : Finset ι\nx✝ : x ∈ {s | ∃ v, s = {v}}\nv : ι\nhv : x = {v}\n⊢ x.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.singleton_nonempty",
"id",
"Finset.Nonempty",
"Finset.inst... | rw [hv]; exact Finset.singleton_nonempty _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 50
} | [
{
"pp": "case left\nι : Type u_1\nK : PreAbstractSimplicialComplex ι\nx : Finset ι\nx✝ : x ∈ {s | ∃ v, s = {v}}\nv : ι\nhv : x = {v}\n⊢ x.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.singleton_nonempty",
"id",
"Finset.Nonempty",
"Finset.inst... | rw [hv]; exact Finset.singleton_nonempty _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 231,
"column": 12
} | {
"line": 231,
"column": 54
} | [
{
"pp": "case left\nι : Type u_1\ns : Set (AbstractSimplicialComplex ι)\nx : Finset ι\nx✝ : x ∈ {t | ∃ v, t = {v}}\nv : ι\nhv : x = {v}\n⊢ x.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.singleton_nonempty",
"id",
"Finset.Nonempty",
"Finset.i... | rw [hv]; exact Finset.singleton_nonempty _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 231,
"column": 12
} | {
"line": 231,
"column": 54
} | [
{
"pp": "case left\nι : Type u_1\ns : Set (AbstractSimplicialComplex ι)\nx : Finset ι\nx✝ : x ∈ {t | ∃ v, t = {v}}\nv : ι\nhv : x = {v}\n⊢ x.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.singleton_nonempty",
"id",
"Finset.Nonempty",
"Finset.i... | rw [hv]; exact Finset.singleton_nonempty _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 261,
"column": 10
} | {
"line": 261,
"column": 52
} | [
{
"pp": "case left\nι : Type u_1\nx : Finset ι\nx✝ : x ∈ {s | ∃ v, s = {v}}\nv : ι\nhv : x = {v}\n⊢ x.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.singleton_nonempty",
"id",
"Finset.Nonempty",
"Finset.instSingleton",
"Singleton.singlet... | rw [hv]; exact Finset.singleton_nonempty _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialComplex.Basic | {
"line": 261,
"column": 10
} | {
"line": 261,
"column": 52
} | [
{
"pp": "case left\nι : Type u_1\nx : Finset ι\nx✝ : x ∈ {s | ∃ v, s = {v}}\nv : ι\nhv : x = {v}\n⊢ x.Nonempty",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.singleton_nonempty",
"id",
"Finset.Nonempty",
"Finset.instSingleton",
"Singleton.singlet... | rw [hv]; exact Finset.singleton_nonempty _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Pairing | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 43
} | [
{
"pp": "case inl\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nx : A.N\nh : x ∈ P.I\n⊢ ∃ y, x = ↑y ∨ x = ↑(P.p y)",
"usedConstants": [
"SSet.Subcomplex.Pairing.p",
"Membership.mem",
"SSet.Subcomplex.N",
"Set.Elem",
"Subtype.mk",
"SSet.Subcomplex.Pairing.I",
"Equiv.su... | obtain ⟨y, hy⟩ := P.p.surjective ⟨x, h⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Pairing | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 24
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nx : A.N\nhx : x ∈ P.I\nhy : ↑⟨x, hx⟩ ∈ P.II\nthis : x ∈ P.I ∩ P.II\n⊢ False",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"False.elim",
"Membership.mem",
"SSet.Subcomplex.N",
"Eq.mp",... | simp [P.inter] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RankNat | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 81
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\ny : ↑P.II\nT : Type (max 0 u) := { x // P.AncestralRel x y }\nU : Type := (d : Fin (↑(P.p y)).dim) × (⦋↑d⦌ ⟶ ⦋(↑(P.p y)).dim⦌)\nψ : U → X.S :=\n fun x ↦\n match x with\n | ⟨d, f⟩ => { dim := ↑d, simplex := (CategoryTheory.ConcreteCategory.hom (X.map f.o... | rw [Subtype.ext_iff, Subtype.ext_iff, N.ext_iff, SSet.N.ext_iff, ← hφ, ← hφ, h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteColimits | {
"line": 56,
"column": 6
} | {
"line": 56,
"column": 34
} | [
{
"pp": "J : Type u_1\ninst✝² : Category.{u_2, u_1} J\ninst✝¹ : HasColimitsOfShape J (Type u)\nF : J ⥤ SSet\nc : Cocone F\nhc : IsColimit c\ninst✝ : Finite J\nh : ∀ (j : J), (F.obj j).Finite\n⊢ c.pt.Finite",
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"Lattice.toSemilatticeSup",... | ← finite_subcomplex_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteColimits | {
"line": 67,
"column": 2
} | {
"line": 68,
"column": 45
} | [
{
"pp": "J : Type u_1\ninst✝³ : Category.{?u.5212, u_1} J\ninst✝² : HasColimitsOfShape J (Type u)\nF : J ⥤ SSet\nc : Cocone F\nhc : IsColimit c\nX Y : SSet\ninst✝¹ : X.Finite\ninst✝ : Y.Finite\n⊢ (X ⨿ Y).Finite",
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfSize",
"CategoryTheory... | apply finite_of_isColimit (coprodIsCoprod X Y)
rintro ⟨_ | _⟩ <;> dsimp <;> infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteColimits | {
"line": 67,
"column": 2
} | {
"line": 68,
"column": 45
} | [
{
"pp": "J : Type u_1\ninst✝³ : Category.{?u.5212, u_1} J\ninst✝² : HasColimitsOfShape J (Type u)\nF : J ⥤ SSet\nc : Cocone F\nhc : IsColimit c\nX Y : SSet\ninst✝¹ : X.Finite\ninst✝ : Y.Finite\n⊢ (X ⨿ Y).Finite",
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfSize",
"CategoryTheory... | apply finite_of_isColimit (coprodIsCoprod X Y)
rintro ⟨_ | _⟩ <;> dsimp <;> infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 186,
"column": 4
} | {
"line": 188,
"column": 37
} | [
{
"pp": "case a\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝² : LinearOrder ι\nf : P.RankFunction ι\ninst✝¹ : OrderBot ι\ninst✝ : SuccOrder ι\nm : ι\nhm : Order.IsSuccLimit m\n⊢ ⨆ i, f.filtration ↑i ≤ f.filtration m",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Opposite... | simp only [iSup_le_iff, Subtype.forall, Set.mem_Iio]
intro j hj
exact f.filtration_monotone hj.le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 186,
"column": 4
} | {
"line": 188,
"column": 37
} | [
{
"pp": "case a\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝² : LinearOrder ι\nf : P.RankFunction ι\ninst✝¹ : OrderBot ι\ninst✝ : SuccOrder ι\nm : ι\nhm : Order.IsSuccLimit m\n⊢ ⨆ i, f.filtration ↑i ≤ f.filtration m",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Opposite... | simp only [iSup_le_iff, Subtype.forall, Set.mem_Iio]
intro j hj
exact f.filtration_monotone hj.le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Retracts | {
"line": 35,
"column": 6
} | {
"line": 36,
"column": 38
} | [
{
"pp": "case refine_1\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.{max ?u.47... | obtain ⟨i, g, hg⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ hc (h.r ≫ f)
exact ⟨i, h.i ≫ g, by simp [hg]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Retracts | {
"line": 35,
"column": 6
} | {
"line": 36,
"column": 38
} | [
{
"pp": "case refine_1\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.{max ?u.47... | obtain ⟨i, g, hg⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ hc (h.r ≫ f)
exact ⟨i, h.i ≫ g, by simp [hg]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 439,
"column": 18
} | {
"line": 439,
"column": 63
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ninst✝ : P.IsProper\nj : ι\nd : ℕ\nx : f.Cell j\ng : unop (op ⦋d⦌) ⟶ ⦋x.dim + 1⦌\nhs✝ :\n (ConcreteCategory.hom (x.ιSigmaStdSimplex.app (op ⦋d⦌))) (stdSimplex.objEquiv.symm g) ∈\n (f.sigmaStdSimplex ... | by simp [horn_obj_eq_univ x.index d (by lia)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | {
"line": 137,
"column": 18
} | {
"line": 140,
"column": 28
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\ninst✝ : P.IsRegular\n⊢ ∃ P, P.IsRegular",
"usedConstants": [
"SSet.Subcomplex.toSSet",
"SSet.Subcomplex.range",
"Opposite",
"congrArg",
"CategoryTheory.Subfunctor.range_ι",
"Exists",
"inferInstance",
"Eq.mp",... | by
generalize h : Subcomplex.range A.ι = B
obtain rfl : B = A := by simpa using h.symm
exact ⟨P, inferInstance⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate | {
"line": 183,
"column": 2
} | {
"line": 185,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasCoproducts C\ninst✝¹ : Preadditive C\nX : SSet\nR : C\nn d : ℕ\ninst✝ : X.HasDimensionLT d\nh : d ≤ n\n⊢ IsZero ((X.normalizedChainComplex R).X n)",
"usedConstants": [
"Eq.mpr",
"SSet.normalizedChainComplex_hom_ext",
"Opposite",
... | rw [IsZero.iff_id_eq_zero]
ext x hx
exact (h.not_gt (X.dim_lt_of_nonDegenerate ⟨x, hx⟩ d)).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate | {
"line": 183,
"column": 2
} | {
"line": 185,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasCoproducts C\ninst✝¹ : Preadditive C\nX : SSet\nR : C\nn d : ℕ\ninst✝ : X.HasDimensionLT d\nh : d ≤ n\n⊢ IsZero ((X.normalizedChainComplex R).X n)",
"usedConstants": [
"Eq.mpr",
"SSet.normalizedChainComplex_hom_ext",
"Opposite",
... | rw [IsZero.iff_id_eq_zero]
ext x hx
exact (h.not_gt (X.dim_lt_of_nonDegenerate ⟨x, hx⟩ d)).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 72
} | [
{
"pp": "case succ\nX : SSet\nx : X _⦋0⦌\nn : ℕ\nf : X.PtSimplex (n + 1) x\ni : Fin (n + 1)\nj : Fin (n + 1 + 1)\nhj : i.castSucc.succ < j.succ\n⊢ stdSimplex.δ j.succ ≫ stdSimplex.σ i.castSucc ≫ f.map = const x",
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"SSet.Subcomplex.ofSim... | · rw [stdSimplex.δ_comp_σ_of_gt_assoc (by grind), δ_map, comp_const] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 315,
"column": 2
} | {
"line": 316,
"column": 18
} | [
{
"pp": "X Y : SSet\ni : X ⟶ Y\nd : ℕ\nc : Cell i d\nn : SimplexCategory\nf : n ⟶ ⦋d⦌\n⊢ ↑((ConcreteCategory.hom ((b i d).app (op n)))\n ((ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op n))) (stdSimplex.objEquiv.symm f))) =\n (ConcreteCategory.hom (Y.map f.op)) c.simplex",
"usedConstants": [
... | simp only [← yonedaEquiv_symm_app_objEquiv_symm, ← ι_b_ι]
dsimp +instances | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 315,
"column": 2
} | {
"line": 316,
"column": 18
} | [
{
"pp": "X Y : SSet\ni : X ⟶ Y\nd : ℕ\nc : Cell i d\nn : SimplexCategory\nf : n ⟶ ⦋d⦌\n⊢ ↑((ConcreteCategory.hom ((b i d).app (op n)))\n ((ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op n))) (stdSimplex.objEquiv.symm f))) =\n (ConcreteCategory.hom (Y.map f.op)) c.simplex",
"usedConstants": [
... | simp only [← yonedaEquiv_symm_app_objEquiv_symm, ← ι_b_ι]
dsimp +instances | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SingularHomology.Basic | {
"line": 46,
"column": 19
} | {
"line": 51,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasCoproducts C\ninst✝¹ : Preadditive C\nn : ℕ\ninst✝ : HasPullbacks C\nX : C\nX✝ Y✝ : TopCat\nf : X✝ ⟶ Y✝\nx✝ : Mono f\n⊢ Mono (((singularChainComplexFunctor C).obj X).map f)",
"usedConstants": [
"CategoryTheory.Limits.instPreservesMonomorphis... | by
dsimp [singularChainComplexFunctor, SSet.chainComplexFunctor]
apply +allowSynthFailures Functor.map_mono
apply +allowSynthFailures Functor.map_mono
dsimp [SSet, SimplicialObject.whiskering, SimplicialObject]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 43
} | [
{
"pp": "case refine_2.succ.zero.«1»\nX Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, pro... | · ext; apply hσ'₁ f₀ f₁ hδ₁ hδ₀ hσ hY | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.Analytic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝³ : CompleteSpace 𝕜\ninst✝² : CharZero 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nhf : AnalyticAt 𝕜 f x\nhf' : deriv f x ≠ 0\nr : 𝕜 → 𝕜 := HasStrictDerivAt.localInverse f (deriv... | exact (hg.comp hra).congr <| .fun_comp (HasStrictDerivAt.eventually_right_inverse ..) g | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 276,
"column": 6
} | {
"line": 277,
"column": 19
} | [
{
"pp": "case refine_2.inr.inr.inr\nR : Type u_1\nS : Type u_2\ninst✝⁷ : Field R\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\ninst✝¹ : OrderTopology S\nι✝ : Type u_3\ninst✝ : Finite ι✝\nv✝ : ι✝ → AbsoluteValue R S\nthis : Fintype... | let ⟨c, hc⟩ := exists_one_lt_lt_one_pi_of_one_lt ha.1 ha.2 ha_gt hb.1 (hb.2 ⟨j, .inr rfl⟩
(by grind)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 342,
"column": 4
} | {
"line": 345,
"column": 48
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\nh : v.IsEquiv w\nhw : ¬w.IsNontrivial\n⊢ ∃ c, 0 < c ∧ (fun x ↦ v x ^ c) = ⇑w",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"False",
"Real.partialOrder",
"Real",
"e... | exact ⟨1, zero_lt_one,
funext fun x ↦ by
rcases eq_or_ne x 0 with rfl | h₀ <;>
aesop (add simp [h.isNontrivial_congr])⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 342,
"column": 4
} | {
"line": 345,
"column": 48
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\nh : v.IsEquiv w\nhw : ¬w.IsNontrivial\n⊢ ∃ c, 0 < c ∧ (fun x ↦ v x ^ c) = ⇑w",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"False",
"Real.partialOrder",
"Real",
"e... | exact ⟨1, zero_lt_one,
funext fun x ↦ by
rcases eq_or_ne x 0 with rfl | h₀ <;>
aesop (add simp [h.isNontrivial_congr])⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 342,
"column": 4
} | {
"line": 345,
"column": 48
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\nh : v.IsEquiv w\nhw : ¬w.IsNontrivial\n⊢ ∃ c, 0 < c ∧ (fun x ↦ v x ^ c) = ⇑w",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"False",
"Real.partialOrder",
"Real",
"e... | exact ⟨1, zero_lt_one,
funext fun x ↦ by
rcases eq_or_ne x 0 with rfl | h₀ <;>
aesop (add simp [h.isNontrivial_congr])⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 333,
"column": 2
} | {
"line": 345,
"column": 48
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\n⊢ v.IsEquiv w ↔ ∃ c, 0 < c ∧ (fun x ↦ v x ^ c) = ⇑w",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq... | refine ⟨fun h ↦ ?_, fun ⟨t, ht, h⟩ ↦ isEquiv_iff_lt_one_iff.2
fun x ↦ h ▸ (rpow_lt_one_iff' (v.nonneg x) ht).symm⟩
by_cases hw : w.IsNontrivial
· let ⟨a, ha₀, ha₁⟩ := hw
refine ⟨(w a).log / (v a).log, h.log_div_log_pos ha₀ ha₁, funext fun b ↦ ?_⟩
rcases eq_or_ne b 0 with rfl | hb₀; · simp [zero_rpow (by... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 333,
"column": 2
} | {
"line": 345,
"column": 48
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\n⊢ v.IsEquiv w ↔ ∃ c, 0 < c ∧ (fun x ↦ v x ^ c) = ⇑w",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq... | refine ⟨fun h ↦ ?_, fun ⟨t, ht, h⟩ ↦ isEquiv_iff_lt_one_iff.2
fun x ↦ h ▸ (rpow_lt_one_iff' (v.nonneg x) ht).symm⟩
by_cases hw : w.IsNontrivial
· let ⟨a, ha₀, ha₁⟩ := hw
refine ⟨(w a).log / (v a).log, h.log_div_log_pos ha₀ ha₁, funext fun b ↦ ?_⟩
rcases eq_or_ne b 0 with rfl | hb₀; · simp [zero_rpow (by... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 371,
"column": 46
} | {
"line": 371,
"column": 67
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\nh : v.IsEquiv w\nU : Set (WithAbs v)\nhU : U ∈ 𝓝 0\n⊢ ⇑(WithAbs.congr v w (RingEquiv.refl F)) ⁻¹' ⇑(WithAbs.congr v w (RingEquiv.refl F)).symm ⁻¹' U ⊆ U",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Ring... | Set.preimage_preimage | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Analytic | {
"line": 57,
"column": 2
} | {
"line": 62,
"column": 33
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nx : E\ns : Set E\nfa : AnalyticWithinAt ℂ f s x\nga : AnalyticWithinAt ℂ g s x\nm : f x ∈ slitPlane\n⊢ AnalyticWithinAt ℂ (fun z ↦ f z ^ g z) s x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Inner... | have e : (fun z ↦ f z ^ g z) =ᶠ[𝓝[insert x s] x] fun z ↦ exp (log (f z) * g z) := by
filter_upwards [(fa.continuousWithinAt_insert.eventually_ne (slitPlane_ne_zero m))]
intro z fz
simp only [fz, cpow_def, if_false]
apply AnalyticWithinAt.congr_of_eventuallyEq_insert _ e
exact ((fa.clog m).mul ga).cexp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Analytic | {
"line": 57,
"column": 2
} | {
"line": 62,
"column": 33
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nx : E\ns : Set E\nfa : AnalyticWithinAt ℂ f s x\nga : AnalyticWithinAt ℂ g s x\nm : f x ∈ slitPlane\n⊢ AnalyticWithinAt ℂ (fun z ↦ f z ^ g z) s x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Inner... | have e : (fun z ↦ f z ^ g z) =ᶠ[𝓝[insert x s] x] fun z ↦ exp (log (f z) * g z) := by
filter_upwards [(fa.continuousWithinAt_insert.eventually_ne (slitPlane_ne_zero m))]
intro z fz
simp only [fz, cpow_def, if_false]
apply AnalyticWithinAt.congr_of_eventuallyEq_insert _ e
exact ((fa.clog m).mul ga).cexp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 259,
"column": 2
} | {
"line": 260,
"column": 87
} | [
{
"pp": "case inr\n𝕜 : 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 : ApproximatesLinearO... | have xmem : x ∈ closedBall b ε :=
isClosed_closedBall.mem_of_tendsto hx (Eventually.of_forall fun n => C n _ (D n).2) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 383,
"column": 2
} | {
"line": 384,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : 𝕜 → E\nx : 𝕜\nn : ℤ\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z\nthis : (deriv fun z ↦ (z - x) ^ n •... | rw [MeromorphicAt.meromorphicAt_congr (Filter.EventuallyEq.nhdsNE_deriv h₂g),
MeromorphicAt.meromorphicAt_congr this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 415,
"column": 51
} | {
"line": 419,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → 𝕜\nx : 𝕜\nf : 𝕜 → E\nhg : AnalyticAt 𝕜 g x\nhg' : g x ≠ 0\n⊢ MeromorphicAt (g • f) x ↔ MeromorphicAt f x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero... | by
refine ⟨fun hfg ↦ ?_, hg.meromorphicAt.smul⟩
refine (hg.inv hg').meromorphicAt.smul hfg |>.congr ?_
filter_upwards [(hg.continuousAt.mono_left nhdsWithin_le_nhds).eventually_ne hg'] with z hz
simp [inv_smul_smul₀ hz] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Order | {
"line": 490,
"column": 8
} | {
"line": 490,
"column": 57
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f (g z₀)\nhg : AnalyticAt 𝕜 g z₀\nhg_nc : ¬EventuallyConst g (𝓝 z₀)\nhf' : analyticOrderAt f (g z₀) = ⊤\n⊢ analytic... | AnalyticAt.analyticOrderAt_ne_zero (by fun_prop), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 399,
"column": 88
} | {
"line": 402,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nn k : ℕ\n⊢ choose (↑n) k = ↑(n.choose k)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Nat.choose",
"HMul.hMul",
"co... | by
rw [← nsmul_right_inj (Nat.factorial_ne_zero k),
← descPochhammer_eq_factorial_smul_choose, nsmul_eq_mul, ← Nat.cast_mul,
← Nat.descFactorial_eq_factorial_mul_choose, ← descPochhammer_smeval_eq_descFactorial] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.RadiusLiminf | {
"line": 45,
"column": 72
} | {
"line": 45,
"column": 90
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nn : ℕ\nhn : 0 < n\nthis : 0 < ↑n\n| (r ^ ↑n) ^ (↑n)⁻¹ * ‖p n‖₊ ... | ← NNReal.mul_rpow, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 35
} | [
{
"pp": "case h\na : ℂ\nB : Set ℂ := Metric.ball 0 1\nn : ℕ\nih :\n Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))\n B\nz : ℂ\n⊢ iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B z =\n derivWithin (iteratedDerivWithin n (fun x ↦ (1 ... | rw [iteratedDerivWithin_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 129,
"column": 17
} | {
"line": 129,
"column": 18
} | [
{
"pp": "α : Type u_1\nu v : α → ℝ\nl : Filter α\nhv : 0 ≤ v\nh : u ~[l] v\nr : ℝ\nφ : α → ℝ\nhφ : Tendsto φ l (𝓝 1)\nhuφv : u =ᶠ[l] φ * v\nhφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)\n| u ^ r =ᶠ[l] (fun x ↦ x ^ r) ∘ φ * v ^ r",
"usedConstants": [
"Real.instPow",
"Real",
"HMul.hMul",
... | 3 | Lean.Elab.Tactic.Conv.evalEnter | null |
Mathlib.MeasureTheory.Covering.VitaliFamily | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 38
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝ : PseudoMetricSpace X\nm0 : MeasurableSpace X\nμ : Measure X\nv : VitaliFamily μ\nδ : ℝ\nδpos : 0 < δ\ns : Set X\nf : X → Set (Set X)\nfset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}\nffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f ... | refine False.elim (h't.2.2 ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.AEEqOfLIntegral | {
"line": 164,
"column": 33
} | {
"line": 164,
"column": 52
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set (Set α)\nf g : α → ℝ≥0∞\nh_eq✝ : m0 = MeasurableSpace.generateFrom s\nh_inter : IsPiSystem s\nbasic : ∀ t ∈ s, ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ\nh_univ : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), g x ∂μ\nhf_int : ∫⁻ (x : α), f x ∂μ ≠ ... | exact hf_int.lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 131,
"column": 10
} | {
"line": 131,
"column": 21
} | [
{
"pp": "case a.refine_1.refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nh_meas_s : MeasurableSet s\nh_nonempty : {d | μ ≤ d + ν}.Nonempty\nν' : Measure α\nh_ν'_in : μ.restrict s ≤ ν' + ν.restrict s\nt : Set α\nh_meas_t : MeasurableSet t\n⊢ μ (t ∩ sᶜ) ≤ (ν' + ν) (t ∩ sᶜ) + ⊤ (t ∩ sᶜ)"... | ← add_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 128,
"column": 53
} | {
"line": 128,
"column": 93
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\n⊢ ⨍⁻ (x : α) in s, f x ∂μ = (∫⁻ (x : α) in s, f x ∂μ) / μ s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"instHDiv",
"MeasureTheory.laverage",
"congrArg",
"Set.univ",
"... | by rw [laverage_eq, restrict_apply_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 147,
"column": 29
} | {
"line": 147,
"column": 68
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_2\ninst✝¹ : Countable ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν\n⊢ (sum fun i ↦ (μ i).singularPart ν, ∑' (i : ι), (μ i).rnDeriv ν).1 ⟂ₘ ν",
"usedConstants": [
"MeasureTheory.Measure.MutuallySing... | by simp [mutuallySingular_singularPart] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Average | {
"line": 364,
"column": 46
} | {
"line": 364,
"column": 95
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\ns : Set α\nf g : α → E\nhs : MeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x\n⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in s, g x ∂μ",
"usedConstants": [
"Real",
"... | simp only [average_eq, setIntegral_congr_ae hs h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Average | {
"line": 364,
"column": 46
} | {
"line": 364,
"column": 95
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\ns : Set α\nf g : α → E\nhs : MeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x\n⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in s, g x ∂μ",
"usedConstants": [
"Real",
"... | simp only [average_eq, setIntegral_congr_ae hs h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Average | {
"line": 364,
"column": 46
} | {
"line": 364,
"column": 95
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\ns : Set α\nf g : α → E\nhs : MeasurableSet s\nh : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x\n⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in s, g x ∂μ",
"usedConstants": [
"Real",
"... | simp only [average_eq, setIntegral_congr_ae hs h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Average | {
"line": 390,
"column": 31
} | {
"line": 390,
"column": 58
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\nf : α → E\ns t : Set α\nhd : AEDisjoint μ s t\nht : NullMeasurableSet t μ\nhsμ : μ s ≠ ∞\nhtμ : μ t ≠ ∞\nhfs : IntegrableOn f s μ\nhft : IntegrableOn f t μ\nthis : Fact (μ s < ∞)\n... | haveI := Fact.mk htμ.lt_top | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.MeasureTheory.Integral.Average | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 30
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : AEMeasurable f (μ.restrict s)\nh : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞\nthis : 0 < (μ.restrict s) ({a | (f a).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ} \\ {x | f x = ∞})\nx : α\nhfx : (f x)... | simp_rw [ae_iff, not_ne_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.Average | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 32
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : μ s ≠ 0\nhs : NullMeasurableSet s μ\nhμ₁ : μ s ≠ ∞\ng : α → ℝ≥0∞\nhint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞\nhg : Measurable g\nhgf : g ≤ f\nhfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ\nhfg' : ⨍⁻ (a : α) i... | simp_rw [ae_iff, not_ne_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 36
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : ν.HaveLebesgueDecomposition μ\nr : ℝ≥0∞\nhr : r ≠ 0\nhr_ne_top : r ≠ ∞\n⊢ ν.rnDeriv (r.toNNReal • μ) =ᶠ[ae μ] r.toNNReal⁻¹ • ν.rnDeriv μ",
"usedConstants": [
"MeasureTheory.Measure.rnDeriv_smul_right",
... | refine rnDeriv_smul_right ν μ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Average | {
"line": 785,
"column": 6
} | {
"line": 785,
"column": 47
} | [
{
"pp": "case h.hf\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : CompleteSpace E\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 ... | apply Integrable.sub _ (hig.smul_const _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Average | {
"line": 786,
"column": 6
} | {
"line": 788,
"column": 51
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : CompleteSpace E\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int :... | have A : Function.support (fun y ↦ g i y • f y) ⊆ a i := by
apply Subset.trans _ hisupp
exact Function.support_smul_subset_left _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np : ℝ≥0\ns : Set α\nh : s ⊆ {x | v.limRatioMe... | have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nq : ℝ≥0\ns : Set α\nh : s ⊆ {x | ↑q < v.limRa... | have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 563,
"column": 52
} | {
"line": 574,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht ... | by
gcongr
rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']
apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
intro x hx
apply lt_of_lt_of_le _ hx.2.1
rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,
zpow_add₀ t_ne_zero']... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.StarOrdered | {
"line": 88,
"column": 48
} | {
"line": 88,
"column": 63
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : Zero α\nR : Type u_2\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : CommSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : IsTopologicalSemiring R\ninst✝¹ : ContinuousStar R\ninst... | congrm($(hp) x) | Mathlib.Tactic._aux_Mathlib_Tactic_CongrM___elabRules_Mathlib_Tactic_congrM_1 | Mathlib.Tactic.congrM |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 36
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\n⊢ eVariationOn ... | refine ENNReal.iSup_add_iSup_le ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 556,
"column": 23
} | {
"line": 556,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\na : α\nh : (𝓝[s ∩ Ioi a] a).NeBot\nh' : ContinuousWithinAt f (s ∩ Ici a) a\n⊢ eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' (s ∩ Ioi a)) = eVariation... | ← comp_ofDual f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.StarSubalgebra | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 96
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : TopologicalSpace A\ninst✝¹⁰ : Semiring A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarRing A\ninst✝⁷ : StarModule R A\ninst✝⁶ : IsSemitopologicalSemiring A\ninst✝⁵ : ContinuousStar A\ninst✝⁴ : TopologicalSpace B\... | have : DenseRange (Set.inclusion (le_topologicalClosure S)) := by simp [-SetLike.coe_sort_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1064,
"column": 2
} | {
"line": 1064,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\nf : α → E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nhf : BoundedVariationOn f univ\na x : α\nhx : ContinuousWithinAt f (Ici x) x\nthis : variationOnFromTo f univ a = fun y ↦ variationOnFromTo f univ a x + va... | rw [eVariationOn.subsingleton _ (by grind [Set.Subsingleton])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 473,
"column": 47
} | {
"line": 475,
"column": 7
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R A... | by
rw [cfc_apply f a, ← map_pow, cfc_apply _ a]
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 526,
"column": 28
} | {
"line": 526,
"column": 49
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalSemiring R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : Algebra R A\ninstCFC : ContinuousFunctio... | ← smul_one_smul R s _ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 851,
"column": 20
} | {
"line": 851,
"column": 32
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : Semifield R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : Algebra R A\ninst✝² : ContinuousFunctionalCalculus R A ... | cfc_inv_id _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 260,
"column": 8
} | {
"line": 260,
"column": 31
} | [
{
"pp": "case refine_4\nn k : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (a : ℚ),\n ∀ x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1),\n eval 1 ((⇑derivative)^[n - k] x) = 0 → eval 1 ((⇑derivative)^[n - k] (a • x)) = 0",
"usedConstant... | intro a x _ h; simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 260,
"column": 8
} | {
"line": 260,
"column": 31
} | [
{
"pp": "case refine_4\nn k : ℕ\nh : k ≤ n\np : ℚ[X]\nm : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)\n⊢ ∀ (a : ℚ),\n ∀ x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1),\n eval 1 ((⇑derivative)^[n - k] x) = 0 → eval 1 ((⇑derivative)^[n - k] (a • x)) = 0",
"usedConstant... | intro a x _ h; simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 417,
"column": 28
} | {
"line": 417,
"column": 49
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : Nontrivial R\ninst✝¹⁴ : StarRing R\ninst✝¹³ : MetricSpace R\ninst✝¹² : IsTopologicalSemiring R\ninst✝¹¹ : ContinuousStar R\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Modul... | ← smul_one_smul R s _ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 181,
"column": 4
} | {
"line": 187,
"column": 82
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\nA : Type u_2\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : Algebra ℝ A\ninst✝² : TopologicalSpace A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\ns : Set ℝ≥0\nhs : CompactSpace ↑s\nφ ψ : C(↑s, ℝ≥0) →⋆ₐ[ℝ≥0] A\nhφ : Continuous[_, inst✝²] ⇑φ\nhψ : Cont... | have (ξ : C(s, ℝ≥0) →⋆ₐ[ℝ≥0] A) (hξ : Continuous ξ) :
(let ξ' := ξ.realContinuousMapOfNNReal.comp <| ContinuousMap.compStarAlgHom' ℝ ℝ e
Continuous ξ' ∧ ξ' (.restrict s' <| .id ℝ) = ξ (.restrict s <| .id ℝ≥0)) := by
intro ξ'
refine ⟨ξ.continuous_realContinuousMapOfNNReal hξ |>.comp <|
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 259,
"column": 35
} | {
"line": 259,
"column": 53
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : Zero X\nr : ℝ≥0\nf : C(X, ℝ)₀\n⊢ (↑r • -f).toNNReal = r • (-f).toNNReal",
"usedConstants": [
"NNReal.instTopologicalSpace",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMon... | ← NNReal.smul_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 423,
"column": 77
} | {
"line": 426,
"column": 35
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\np : A → Prop\nq : B → Prop\ninst✝²⁹ : CommSemiring R\ninst✝²⁸ : Nontrivial R\ninst✝²⁷ : StarRing R\ninst✝²⁶ : MetricSpace R\ninst✝²⁵ : IsTopologicalSemiring R\ninst✝²⁴ : ContinuousStar R\ninst✝²³ : CommRing S\ninst✝²² : Algebra R S\n... | by
have hf' : ContinuousOn f (quasispectrum R (ψ a)) := hf.mono h_spec
rw [cfcₙ_apply .., cfcₙ_apply ..]
exact DFunLike.congr_fun this _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 165,
"column": 4
} | {
"line": 168,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : DenselyNormedField 𝕜\ninst✝⁸ : NonUnitalNormedRing E\ninst✝⁷ : StarRing E\ninst✝⁶ : CStarRing E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : IsScalarTower 𝕜 E E\ninst✝³ : SMulCommClass 𝕜 E E\ninst✝² : StarRing 𝕜\ninst✝¹ : StarModule 𝕜 E\ninst✝ : CStarRing 𝕜\nx : Unit... | have h₃ : ‖(Unitization.splitMul 𝕜 E (star x * x)).fst‖
= ‖(Unitization.splitMul 𝕜 E x).fst‖ ^ 2 := by
simp only [Unitization.splitMul_apply, Unitization.fst_mul, Unitization.fst_star,
norm_mul, norm_star, sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 477,
"column": 4
} | {
"line": 479,
"column": 11
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\np : A → Prop\nq : B → Prop\ninst✝²⁴ : CommSemiring R\ninst✝²³ : StarRing R\ninst✝²² : MetricSpace R\ninst✝²¹ : IsTopologicalSemiring R\ninst✝²⁰ : ContinuousStar R\ninst✝¹⁹ : Ring A\ninst✝¹⁸ : StarRing A\ninst✝¹⁷ : TopologicalSpace A\... | trans cfcHom hψa (.restrict (spectrum R (ψ a)) (.id R))
· simp [cfcHom_id]
· congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 477,
"column": 4
} | {
"line": 479,
"column": 11
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\np : A → Prop\nq : B → Prop\ninst✝²⁴ : CommSemiring R\ninst✝²³ : StarRing R\ninst✝²² : MetricSpace R\ninst✝²¹ : IsTopologicalSemiring R\ninst✝²⁰ : ContinuousStar R\ninst✝¹⁹ : Ring A\ninst✝¹⁸ : StarRing A\ninst✝¹⁷ : TopologicalSpace A\... | trans cfcHom hψa (.restrict (spectrum R (ψ a)) (.id R))
· simp [cfcHom_id]
· congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 124,
"column": 2
} | {
"line": 128,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : CommSemiring 𝕜\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : ContinuousAdd 𝕜\ninst✝⁵ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : TopologicalSpace A\ninst✝² : Module 𝕜 A\ninst✝¹ : T2Space 𝕜\ninst✝ : ContinuousMul 𝕜\n⊢ IsClosed (charact... | simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 124,
"column": 2
} | {
"line": 128,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : CommSemiring 𝕜\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : ContinuousAdd 𝕜\ninst✝⁵ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : TopologicalSpace A\ninst✝² : Module 𝕜 A\ninst✝¹ : T2Space 𝕜\ninst✝ : ContinuousMul 𝕜\n⊢ IsClosed (charact... | simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 48
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → Set β\ns : Set α\nx : α\nγ : Type u_5\ninst✝ : TopologicalSpace γ\ni : γ → β\nhf : UpperHemicontinuousWithinAt f s x\nhi : IsInducing i\nh_cl : IsClosed[inst✝¹] (range i)\nv : Set β\nhv : IsOpen[inst✝¹] v\nhu ... | exact hf.inter h_cl |>.forall_isOpen v hv hifu | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 90
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → Set β\nx₀ : α\ninst✝ : (𝓝 x₀).IsCountablyGenerated\nK : Set β\nhK : IsSeqCompact K\nhf : ∀ᶠ (x : α) in 𝓝 x₀, f x ⊆ K\nt : Set β\nht : IsClosed[inst✝¹] t\nhft : ∃ᶠ (x' : α) in 𝓝 x₀, (f x' ∩ t).Nonempty\nx : ... | exact ⟨y₀, h, ht.closure_eq ▸ mem_closure_of_tendsto hyφ <| .of_forall fun n ↦ (hy _).2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 425,
"column": 6
} | {
"line": 427,
"column": 51
} | [] | dist (f x) 0 ≤ dist (g.toBCF x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
_ < dist g.toBCF f + ε / 2 := add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx
_ ≤ ε := by grw [mem_ball.1 hg, add_halves ε] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 141,
"column": 10
} | {
"line": 141,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\ninst✝² : Clo... | isometry_inr (𝕜 := 𝕜) |>.isEmbedding.continuous_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Units | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 23
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : NormedField 𝕜\ninst✝ : CompleteSpace 𝕜\nf : C(X, 𝕜)\n⊢ ⇑(RingHom.id 𝕜) ⁻¹' Set.range ⇑f = Set.range ⇑f",
"usedConstants": [
"ContinuousMap",
"Set.preimage_id",
"NormedDivisionRing.toNormedRing",
"PseudoMe... | exact Set.preimage_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 487,
"column": 2
} | {
"line": 493,
"column": 16
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nf : ℝ≥0 → ℝ≥0\na : A\nc : ℝ≥0\nhc : 0 < c\nh : ∀ x ∈ σ ℝ≥0 a, f x... | obtain (_ | _) := subsingleton_or_nontrivial A
· rw [Subsingleton.elim (cfc f a) 0]
simpa
· refine cfc_cases (‖·‖₊ < c) a f (by simpa) fun hf ha ↦ ?_
simp only [← cfc_apply f a, (IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.lt_iff)]
rintro - ⟨x, hx, rfl⟩
exact h x hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 487,
"column": 2
} | {
"line": 493,
"column": 16
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nf : ℝ≥0 → ℝ≥0\na : A\nc : ℝ≥0\nhc : 0 < c\nh : ∀ x ∈ σ ℝ≥0 a, f x... | obtain (_ | _) := subsingleton_or_nontrivial A
· rw [Subsingleton.elim (cfc f a) 0]
simpa
· refine cfc_cases (‖·‖₊ < c) a f (by simpa) fun hf ha ↦ ?_
simp only [← cfc_apply f a, (IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.lt_iff)]
rintro - ⟨x, hx, rfl⟩
exact h x hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UrysohnsLemma | {
"line": 386,
"column": 2
} | {
"line": 388,
"column": 41
} | [
{
"pp": "case h.refine_1\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\ng : C(X, ℝ)\nhgs : EqOn (⇑g) 0 s\nhgt : EqOn (⇑g) 1 t\nhicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1\n⊢ EqOn (⇑(1 - g)) 0 ... | · intro x hx
simp only [ContinuousMap.sub_apply, ContinuousMap.one_apply, Pi.zero_apply]
exact sub_eq_zero_of_eq (hgt.symm hx) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UrysohnsLemma | {
"line": 444,
"column": 4
} | {
"line": 444,
"column": 65
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nh's : IsGδ s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nU : ℕ → Set X\nU_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhU : s = ⋂ n, U n\nm : Set X\nm_comp : IsCompact m\nsm : s... | exact subset_inter ((hU.subset.trans (iInter_subset U n))) sm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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