module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 779,
"column": 20
} | {
"line": 782,
"column": 23
} | {
"line": 783,
"column": 6
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\nm : Type u_2\nn : Type u_3\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM : CStarMatrix n n A\nv : C⋆ᵐᵒᵈ(A, n → A)\n⊢ ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖ ^ 2 = ‖M * star M‖ * ‖v‖ ^ 2",
"ppTerm": "?m.322",
"... | [] | congr
apply MulOpposite.op_injective
simp only [← toCLMNonUnitalAlgHom_eq_toCLM, map_mul]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 41
} | {
"line": 130,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\n⊢ spectrum ℂ ↑(ℑ a) = (fun x ↦ ↑x.im) '' spectrum ℂ a",
"ppTerm": "?m.40",
"assigned... | [] | rw [← cfc_im_id a, cfc_map_spectrum ..] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 41
} | {
"line": 130,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\n⊢ spectrum ℂ ↑(ℑ a) = (fun x ↦ ↑x.im) '' spectrum ℂ a",
"ppTerm": "?m.40",
"assigned... | [] | rw [← cfc_im_id a, cfc_map_spectrum ..] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 41
} | {
"line": 130,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\n⊢ spectrum ℂ ↑(ℑ a) = (fun x ↦ ↑x.im) '' spectrum ℂ a",
"ppTerm": "?m.40",
"assigned... | [] | rw [← cfc_im_id a, cfc_map_spectrum ..] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 282,
"column": 62
} | {
"line": 282,
"column": 77
} | {
"line": 282,
"column": 77
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Opens X\nhs : Nonempty ↥s\nx : ↥s\nhxe : ↑x ∈ e.source\n⊢ ↑e.symm (↑e ↑x) ∈ ↑s",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Opens X\nhs : Nonempty ↥s\nx : ↥s\nhxe : ↑x ∈ e.source\n⊢ ↑x ∈ ↑s"
] | e.leftInvOn hxe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 206,
"column": 51
} | {
"line": 206,
"column": 72
} | {
"line": 206,
"column": 73
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne e' : Pretrivialization F proj\n⊢ (e'.symm.trans e.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Set... | [
"B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne e' : Pretrivialization F proj\n⊢ (e'.baseSet ∩ e.baseSet) ×ˢ univ = (e.baseSet ∩ e'.baseSet) ×ˢ univ"
] | symm_trans_source_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 950,
"column": 16
} | {
"line": 957,
"column": 38
} | {
"line": 959,
"column": 0
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne'✝ : Trivialization F TotalSpace.proj\nb : B\ny : E b\ne e' : T... | [] | by
rintro p (hp | hp')
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_mem, e.coe_fst] <;> exact hp
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_notMem, e'.coe_fst hp']
simp only [source_eq] at hp' ⊢
exact fun h => H.le_bot ⟨h, hp'⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 614,
"column": 29
} | {
"line": 614,
"column": 71
} | {
"line": 615,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : StarRing 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : CStarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : StarModule 𝕜 A\na : 𝓜(𝕜, A)\nhball : (Metric.... | [] | by simp only [mul_one, nnnorm_fst, le_rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.SeparatedMap | {
"line": 48,
"column": 4
} | {
"line": 49,
"column": 23
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Sort u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nx : X\n⊢ Filter.comap (toPullbackDiag f) (𝓝 (toPullbackDiag f x)) = 𝓝 x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instReflLe",
"SProd.sprod",
"congrArg",
"inf_of_le_left",
"Fu... | [] | simp [nhds_induced, Filter.comap_comap, nhds_prod_eq, Filter.comap_prod, Function.comp_def,
Filter.comap_id'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.SeparatedMap | {
"line": 48,
"column": 4
} | {
"line": 49,
"column": 23
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Sort u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nx : X\n⊢ Filter.comap (toPullbackDiag f) (𝓝 (toPullbackDiag f x)) = 𝓝 x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instReflLe",
"SProd.sprod",
"congrArg",
"inf_of_le_left",
"Fu... | [] | simp [nhds_induced, Filter.comap_comap, nhds_prod_eq, Filter.comap_prod, Function.comp_def,
Filter.comap_id'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.SeparatedMap | {
"line": 48,
"column": 4
} | {
"line": 49,
"column": 23
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Sort u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nx : X\n⊢ Filter.comap (toPullbackDiag f) (𝓝 (toPullbackDiag f x)) = 𝓝 x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instReflLe",
"SProd.sprod",
"congrArg",
"inf_of_le_left",
"Fu... | [] | simp [nhds_induced, Filter.comap_comap, nhds_prod_eq, Filter.comap_prod, Function.comp_def,
Filter.comap_id'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 130,
"column": 34
} | {
"line": 130,
"column": 60
} | {
"line": 130,
"column": 60
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ns : Set X\ncont : ∀ x ∈ s, ContinuousAt f x\nx : X\nhx : x ∈ s\ng : OpenPartialHomeomorph Y Z\nhxg : f x ∈ g.source\nhgf✝ : IsLocalHomeomorphOn (↑g ∘ f) s\nhg : IsL... | [] | by apply g.map_source this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Covering.Quotient | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 51
} | {
"line": 327,
"column": 6
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.o... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ne... | rintro ⟨_, ⟨_, ⟨x, hx, rfl⟩, rfl⟩, y, hy, eq⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 33
} | {
"line": 41,
"column": 0
} | [
{
"pp": "z : Circle\n⊢ (↑z).arg = 0 ↔ z = 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Complex.arg_one",
... | [] | simpa using arg_eq_arg (w := 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 33
} | {
"line": 41,
"column": 0
} | [
{
"pp": "z : Circle\n⊢ (↑z).arg = 0 ↔ z = 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Complex.arg_one",
... | [] | simpa using arg_eq_arg (w := 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 33
} | {
"line": 41,
"column": 0
} | [
{
"pp": "z : Circle\n⊢ (↑z).arg = 0 ↔ z = 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Complex.arg_one",
... | [] | simpa using arg_eq_arg (w := 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 152,
"column": 88
} | {
"line": 152,
"column": 93
} | {
"line": 152,
"column": 93
} | [
{
"pp": "case funProp.discharger\nA : Type u_1\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra ℝ A\ninst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\nn : ℕ\na : A\nha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0\nha₁ : IsSelfAdjoint a\nha₂' : ContinuousOn Real.log (spectrum ℝ a)\n⊢ ∀ x ∈ (fun x ↦ x ^ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 152,
"column": 88
} | {
"line": 152,
"column": 93
} | {
"line": 152,
"column": 93
} | [
{
"pp": "case funProp.discharger\nA : Type u_1\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra ℝ A\ninst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\nn : ℕ\na : A\nha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0\nha₁ : IsSelfAdjoint a\nha₂' : ContinuousOn Real.log (spectrum ℝ a)\n⊢ ∀ x ∈ (fun x ↦ x ^ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 152,
"column": 88
} | {
"line": 152,
"column": 93
} | {
"line": 152,
"column": 93
} | [
{
"pp": "case funProp.discharger\nA : Type u_1\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra ℝ A\ninst✝ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\nn : ℕ\na : A\nha₂ : ∀ x ∈ spectrum ℝ a, x ≠ 0\nha₁ : IsSelfAdjoint a\nha₂' : ContinuousOn Real.log (spectrum ℝ a)\n⊢ ∀ x ∈ (fun x ↦ x ^ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Basic | {
"line": 564,
"column": 33
} | {
"line": 564,
"column": 38
} | {
"line": 564,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝ : T2Space E\nhf : IsClosedMap f\nhs : ∀ x ∈ s, (f ⁻¹' {x}).Finite\nh : IsLocalHomeomorphOn f (f ⁻¹' s)\nx : X\nhx : x ∈ s\ne : E\nhe : e ∈ f ⁻¹' {x}\n⊢ e ∈ f ⁻¹' s",
"ppTerm": "?m.42",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Covering.Basic | {
"line": 564,
"column": 33
} | {
"line": 564,
"column": 38
} | {
"line": 564,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝ : T2Space E\nhf : IsClosedMap f\nhs : ∀ x ∈ s, (f ⁻¹' {x}).Finite\nh : IsLocalHomeomorphOn f (f ⁻¹' s)\nx : X\nhx : x ∈ s\ne : E\nhe : e ∈ f ⁻¹' {x}\n⊢ e ∈ f ⁻¹' s",
"ppTerm": "?m.42",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Basic | {
"line": 564,
"column": 33
} | {
"line": 564,
"column": 38
} | {
"line": 564,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝ : T2Space E\nhf : IsClosedMap f\nhs : ∀ x ∈ s, (f ⁻¹' {x}).Finite\nh : IsLocalHomeomorphOn f (f ⁻¹' s)\nx : X\nhx : x ∈ s\ne : E\nhe : e ∈ f ⁻¹' {x}\n⊢ e ∈ f ⁻¹' s",
"ppTerm": "?m.42",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Basic | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 7
} | {
"line": 567,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : T2Space E\nx : X\nhx : x ∈ s\ne : E\nφ : OpenPartialHomeomorph E X\nhφ : e ∈ φ.source\nhf : IsClosedMap ↑φ\nhs : ∀ x ∈ s, (↑φ ⁻¹' {x}).Finite\nh : IsLocalHomeomorphOn (↑φ) (↑φ ⁻¹' s)\nhe : e ∈ ↑φ ⁻¹... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Covering.Basic | {
"line": 594,
"column": 33
} | {
"line": 594,
"column": 38
} | {
"line": 594,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝² : T2Space E\ninst✝¹ : T2Space X\ninst✝ : CompactSpace E\nhf : Continuous[inst✝⁴, inst✝³] f\nh : IsLocalHomeomorphOn f (f ⁻¹' s)\nx : X\nhx : x ∈ s\ne : E\nhe : e ∈ f ⁻¹' {x}\n⊢ e ∈ f ⁻¹' s... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Covering.Basic | {
"line": 594,
"column": 33
} | {
"line": 594,
"column": 38
} | {
"line": 594,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝² : T2Space E\ninst✝¹ : T2Space X\ninst✝ : CompactSpace E\nhf : Continuous[inst✝⁴, inst✝³] f\nh : IsLocalHomeomorphOn f (f ⁻¹' s)\nx : X\nhx : x ∈ s\ne : E\nhe : e ∈ f ⁻¹' {x}\n⊢ e ∈ f ⁻¹' s... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Basic | {
"line": 594,
"column": 33
} | {
"line": 594,
"column": 38
} | {
"line": 594,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝² : T2Space E\ninst✝¹ : T2Space X\ninst✝ : CompactSpace E\nhf : Continuous[inst✝⁴, inst✝³] f\nh : IsLocalHomeomorphOn f (f ⁻¹' s)\nx : X\nhx : x ∈ s\ne : E\nhe : e ∈ f ⁻¹' {x}\n⊢ e ∈ f ⁻¹' s... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Basic | {
"line": 595,
"column": 2
} | {
"line": 595,
"column": 7
} | {
"line": 597,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ns : Set X\ninst✝² : T2Space E\ninst✝¹ : T2Space X\ninst✝ : CompactSpace E\nx : X\nhx : x ∈ s\ne : E\nφ : OpenPartialHomeomorph E X\nhφ : e ∈ φ.source\nhf : Continuous[inst✝⁴, inst✝³] ↑φ\nh : IsLocalHomeomorphOn (↑φ) (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 182,
"column": 49
} | {
"line": 182,
"column": 54
} | {
"line": 182,
"column": 54
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis✝ : ContinuousOn arg (spectrum ℂ ↑u)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\nhy₁ : ‖y‖ = 1\nthis : I * ↑y.arg = log y\n⊢ y ≠ 0",
"ppTerm": "?m.378",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 182,
"column": 49
} | {
"line": 182,
"column": 54
} | {
"line": 182,
"column": 54
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis✝ : ContinuousOn arg (spectrum ℂ ↑u)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\nhy₁ : ‖y‖ = 1\nthis : I * ↑y.arg = log y\n⊢ y ≠ 0",
"ppTerm": "?m.378",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 182,
"column": 49
} | {
"line": 182,
"column": 54
} | {
"line": 182,
"column": 54
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis✝ : ContinuousOn arg (spectrum ℂ ↑u)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\nhy₁ : ‖y‖ = 1\nthis : I * ↑y.arg = log y\n⊢ y ≠ 0",
"ppTerm": "?m.378",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 76
} | {
"line": 103,
"column": 4
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\n⊢ ‖adjointAux A‖ ≤ ‖A‖",
"... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nx : F\n⊢ ‖(adjointAux A) x‖ ≤ ‖A‖ * ‖x‖"
] | refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 76
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\n⊢ ‖adjointAux (adjointAux A)‖ ... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nx : E\n⊢ ‖(adjointAux (adjointAux A)) x‖ ≤... | refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 349,
"column": 2
} | {
"line": 352,
"column": 67
} | {
"line": 354,
"column": 0
} | [
{
"pp": "case mpr\nA : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\n⊢ (∃ l, (List.map expUnitary l).prod = u) → u ∈ pathComponent 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CStarAlgebra.toNonUnitalCStarAlgebra",
"NonUnitalCStarAlgebra.toNonUnitalNormedRing",
... | [] | · rintro ⟨l, rfl⟩
induction l with
| nil => simp
| cons x xs ih => simpa using! (joined_one_expUnitary x).mul ih | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 27
} | {
"line": 183,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\n⊢ adjoint U.orthogonalProjectionOnto = U.subtypeL",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\n⊢ adjoint (adjoint U.subtypeL) = U.subtypeL"
] | ← U.adjoint_subtypeL, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 268,
"column": 26
} | {
"line": 268,
"column": 59
} | {
"line": 269,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\n⊢ ‖adjoint A‖ * ‖A‖ = ‖A‖ * ‖A‖",
"ppTerm... | [] | rw [LinearIsometryEquiv.norm_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 268,
"column": 26
} | {
"line": 268,
"column": 59
} | {
"line": 269,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\n⊢ ‖adjoint A‖ * ‖A‖ = ‖A‖ * ‖A‖",
"ppTerm... | [] | rw [LinearIsometryEquiv.norm_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 268,
"column": 26
} | {
"line": 268,
"column": 59
} | {
"line": 269,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\n⊢ ‖adjoint A‖ * ‖A‖ = ‖A‖ * ‖A‖",
"ppTerm... | [] | rw [LinearIsometryEquiv.norm_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 53
} | {
"line": 573,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\nx : E\ny : F\nthis✝ : CompleteS... | [] | exact ContinuousLinearMap.adjoint_inner_right _ x y | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 36
} | {
"line": 145,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : DecidableEq P\np : ι → P\nhi : AffineIndependent k p\ns : Finset ι\nn : ℕ\nhc : #s = n + 1\nhc' : #(Finset.image p s) = n + 1\np₁ : P\nhi' : L... | [] | exact Nat.pred_eq_of_eq_succ hc' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 19
} | {
"line": 339,
"column": 20
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ -(↑n + 1) • (s.faceOppositeCentroid i -ᵥ s.centroid) = (↑n + 1) • (s.centroid -ᵥ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ - -(↑n + 1) • -(s.faceOppositeCentroid i -ᵥ s.centroid) = (↑n + 1) • (s.centroid -ᵥ s.faceOp... | ← neg_smul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 362,
"column": 64
} | {
"line": 362,
"column": 79
} | {
"line": 363,
"column": 4
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ -(↑n • (s.centroid -ᵥ s.faceOppositeCentroid i)) = ↑n • (s.faceOppositeCentroid ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ -(-↑n • -(s.centroid -ᵥ s.faceOppositeCentroid i)) = ↑n • (s.faceOppositeCentroid i -ᵥ s.cen... | ← neg_smul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 108,
"column": 2
} | {
"line": 109,
"column": 21
} | {
"line": 111,
"column": 0
} | [
{
"pp": "E : Type u_1\nc : E\nf : ContDiffBump c\n⊢ 1 < f.rOut / f.rIn",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"GroupWithZero.toD... | [] | rw [one_lt_div f.rIn_pos]
exact f.rIn_lt_rOut | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 108,
"column": 2
} | {
"line": 109,
"column": 21
} | {
"line": 111,
"column": 0
} | [
{
"pp": "E : Type u_1\nc : E\nf : ContDiffBump c\n⊢ 1 < f.rOut / f.rIn",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"GroupWithZero.toD... | [] | rw [one_lt_div f.rIn_pos]
exact f.rIn_lt_rOut | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 477,
"column": 6
} | {
"line": 477,
"column": 33
} | {
"line": 478,
"column": 6
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ (↑n)⁻¹ • (s.centroid -ᵥ s.points i) = (-1 / ↑n) • (s.points i -ᵥ s.centroid)",
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ (↑n)⁻¹ • -(s.points i -ᵥ s.centroid) = (-1 / ↑n) • (s.points i -ᵥ s.centroid)"
] | rw [← neg_vsub_eq_vsub_rev] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 18
} | {
"line": 420,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\nh : Collinear k s ↔ ↑(finrank k ↥(vectorSpan k s)) ≤ 1\n⊢ Collinear k s ↔ finrank k ↥(vectorSpan k s) ≤ 1"... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 7
} | {
"line": 672,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p : P\nhcol : Collinear k {p₂, p, p₃}\nhne1 : p₂ ≠ p\nhne2 : p₂ ≠ p₃\nh : AffineIndependent k ![p₁, p₂, p₃]\n⊢ {p₂, p₃, p} = {p₂, p, p₃}",
"ppTerm": "?m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 694,
"column": 2
} | {
"line": 694,
"column": 18
} | {
"line": 696,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : Set P\ninst✝ : FiniteDimensional k ↥(vectorSpan k s)\nh : Coplanar k s ↔ ↑(finrank k ↥(vectorSpan k s)) ≤ 2\n⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2",
... | [] | exact mod_cast h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 302,
"column": 4
} | {
"line": 304,
"column": 59
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nf : (i : α) → E i\nc : 𝕜\nhf : Memℓp f 0\n⊢ Memℓp (c • f) 0",
"ppTerm": "?inl",
"assigned... | [] | apply memℓp_zero
refine hf.finite_dsupport.subset fun i => (?_ : ¬c • f i = 0 → ¬f i = 0)
exact not_imp_not.mpr fun hf' => hf'.symm ▸ smul_zero c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 302,
"column": 4
} | {
"line": 304,
"column": 59
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nf : (i : α) → E i\nc : 𝕜\nhf : Memℓp f 0\n⊢ Memℓp (c • f) 0",
"ppTerm": "?inl",
"assigned... | [] | apply memℓp_zero
refine hf.finite_dsupport.subset fun i => (?_ : ¬c • f i = 0 → ¬f i = 0)
exact not_imp_not.mpr fun hf' => hf'.symm ▸ smul_zero c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 520,
"column": 6
} | {
"line": 520,
"column": 33
} | {
"line": 521,
"column": 6
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E p)\nh : ‖f‖ = 0\nhp : 0 < p.toReal\n⊢ HasSum (fun i ↦ ‖↑f i‖ ^ p.toReal) 0",
"ppTerm": "?m.265",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real",
... | [
"α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E p)\nh : ‖f‖ = 0\nhp : 0 < p.toReal\nthis : HasSum (fun i ↦ ‖↑f i‖ ^ p.toReal) (‖f‖ ^ p.toReal)\n⊢ HasSum (fun i ↦ ‖↑f i‖ ^ p.toReal) 0"
] | have := lp.hasSum_norm hp f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 725,
"column": 4
} | {
"line": 725,
"column": 44
} | {
"line": 726,
"column": 4
} | [
{
"pp": "case inr.inr\n𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : ↥(lp E p)\nhp : 0 < p.toReal\ninst : NNNorm ↥(lp E p)\ncoe_... | [
"case inr.inr\n𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : ↥(lp E p)\nhp : 0 < p.toReal\ninst : NNNorm ↥(lp E p)\ncoe_nnnorm : ∀ (... | refine hasSum_mono hLHS hRHS fun i => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1105,
"column": 12
} | {
"line": 1105,
"column": 29
} | {
"line": 1105,
"column": 29
} | [
{
"pp": "case top.refine_1.inl\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis : Nonempty α\nhp : 0 < ∞\n⊢ ‖Pi.single i x i‖ ≤ ‖x‖",
"ppTerm": "?top.refine_1.inl",
"assigned": true,
"usedConstants": [
"Norm.nor... | [
"case top.refine_1.inl\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis : Nonempty α\nhp : 0 < ∞\n⊢ ‖x‖ ≤ ‖x‖"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1108,
"column": 10
} | {
"line": 1108,
"column": 27
} | {
"line": 1108,
"column": 27
} | [
{
"pp": "case top.refine_2\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis : Nonempty α\nhp : 0 < ∞\nn : ℝ\nhn : n < ‖x‖\n⊢ ‖x‖ = ‖Pi.single i x i‖",
"ppTerm": "?top.refine_2",
"assigned": true,
"usedConstants": [
... | [
"case top.refine_2\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis : Nonempty α\nhp : 0 < ∞\nn : ℝ\nhn : n < ‖x‖\n⊢ ‖x‖ = ‖x‖"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1112,
"column": 37
} | {
"line": 1112,
"column": 54
} | {
"line": 1112,
"column": 54
} | [
{
"pp": "case coe\nα : Type u_3\nE : α → Type u_4\np✝ : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis✝ : Nonempty α\np : ℝ≥0\nhp : 0 < ↑p\nthis : 0 < (↑p).toReal\n⊢ ‖Pi.single i x i‖ = ‖x‖",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
... | [
"case coe\nα : Type u_3\nE : α → Type u_4\np✝ : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis✝ : Nonempty α\np : ℝ≥0\nhp : 0 < ↑p\nthis : 0 < (↑p).toReal\n⊢ ‖x‖ = ‖x‖",
"α : Type u_3\nE : α → Type u_4\np✝ : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ :... | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 43
} | {
"line": 155,
"column": 4
} | [
{
"pp": "case pos.h_lim\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbou... | [
"case pos.h_lim\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\n... | rw [tendsto_zero_iff_norm_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.ProperSpace.Lemmas | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 62
} | {
"line": 72,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nH : ∀ (x : α), IsProperMap (dist x)\nx : α\nr : ℝ\n⊢ IsCompact (closedBall x r)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"HEq.refl",
"PseudoMetricSpace.toUniformSpace... | [
"α : Type u_1\ninst✝ : PseudoMetricSpace α\nH : ∀ (x : α), IsProperMap (dist x)\nx : α\nr : ℝ\n⊢ closedBall x r = dist x ⁻¹' closedBall 0 r"
] | convert! (H x).isCompact_preimage (isCompact_closedBall 0 r) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Convolution | {
"line": 544,
"column": 73
} | {
"line": 584,
"column": 96
} | {
"line": 586,
"column": 0
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace... | [] | by
/- First get rid of the case where the space is not locally compact. Then `g` vanishes everywhere
and the conclusion is trivial. -/
by_cases! H : ∀ p ∈ s, ∀ x, g p x = 0
· apply (continuousOn_const (c := 0)).congr
rintro ⟨p, x⟩ ⟨hp, -⟩
apply integral_eq_zero_of_ae (Eventually.of_forall (fun y ↦ ?_))
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 40
} | {
"line": 280,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nk : Set ... | [] | exact lt_irrefl _ (C.le.trans_lt this) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 72
} | {
"line": 140,
"column": 2
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvar... | [
"case inr\nG : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst✝ ... | have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 132,
"column": 59
} | {
"line": 132,
"column": 78
} | {
"line": 132,
"column": 79
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : HasContDiffBump E\ninst✝⁴ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsAddHaarMeasure\nK : ℝ\nh : f.rOu... | [
"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : HasContDiffBump E\ninst✝⁴ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsAddHaarMeasure\nK : ℝ\nh : f.rOut ≤ K * f.rI... | mul_comm _ (K ^ _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 473,
"column": 69
} | {
"line": 473,
"column": 82
} | {
"line": 473,
"column": 83
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nR : ℝ\nhR : 1 < R\n⊢ 0 < (R - 1) ... | [
"case ha\nE✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := ⋯\nthis✝ : BorelSpace E\nR : ℝ\nhR : 1 < R\n⊢ 0 < R - 1",
"case h... | apply div_pos | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 508,
"column": 47
} | {
"line": 508,
"column": 60
} | {
"line": 508,
"column": 61
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nIR : ∀ (R : ℝ), 1 < R → 0 < (R - ... | [
"case ha\nE✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := ⋯\nthis✝ : BorelSpace E\nIR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R +... | apply div_pos | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 420,
"column": 2
} | {
"line": 444,
"column": 83
} | {
"line": 446,
"column": 2
} | [
{
"pp": "case ind\nα : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (... | [
"case ind\nα : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.index... | let sc : SatelliteConfig α N p.τ :=
{ c := fun k => p.c (p.index (G k))
r := fun k => p.r (p.index (G k))
rpos := fun k => p.rpos (p.index (G k))
h := by
intro a b a_ne_b
wlog G_le : G a ≤ G b generalizing a b
· exact (this a_ne_b.symm (le_of_not_ge G_le)).symm
have... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 355,
"column": 4
} | {
"line": 356,
"column": 57
} | {
"line": 357,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\ni j : Fin N.succ\ninej : i ≠ j\nhi : ‖a.c i‖ ≤ 2\nhj : 2 < ‖a.c j‖\nah : Pair... | [
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\ni j : Fin N.succ\ninej : i ≠ j\nhi : ‖a.c i‖ ≤ 2\nhj : 2 < ‖a.c j‖\nah : Pairwise fun i j... | · apply (a.hlast i H).1.trans
simpa only [dist_eq_norm, lastc, sub_zero] using hi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.BumpFunction.Convolution | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 39
} | {
"line": 71,
"column": 0
} | [
{
"pp": "G : Type uG\nE' : Type uE'\ninst✝⁹ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁸ : MeasurableSpace G\nμ : Measure G\ninst✝⁷ : NormedSpace ℝ E'\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace ℝ G\ninst✝⁴ : CompleteSpace E'\nφ : ContDiffBump 0\ninst✝³ : BorelSpace G\ninst✝² : FiniteDimensional ℝ G\nin... | [] | rw [convolution_eq_right' _ φ.support_normed_eq.subset hg]
exact integral_normed_smul φ μ (g x₀) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.Convolution | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 39
} | {
"line": 71,
"column": 0
} | [
{
"pp": "G : Type uG\nE' : Type uE'\ninst✝⁹ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁸ : MeasurableSpace G\nμ : Measure G\ninst✝⁷ : NormedSpace ℝ E'\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace ℝ G\ninst✝⁴ : CompleteSpace E'\nφ : ContDiffBump 0\ninst✝³ : BorelSpace G\ninst✝² : FiniteDimensional ℝ G\nin... | [] | rw [convolution_eq_right' _ φ.support_normed_eq.subset hg]
exact integral_normed_smul φ μ (g x₀) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.Convolution | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 74
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case hg\nG : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : NormedSpace ℝ E'\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace ℝ G\ninst✝³ : CompleteSpace E'\ninst✝² : BorelSpace G\ninst✝¹ : FiniteDimensional ℝ G\ninst✝ : μ.IsA... | [] | apply tendsto_const_nhds.congr (fun i ↦ ?_)
rw [← integral_neg_eq_self]
simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.Convolution | {
"line": 123,
"column": 4
} | {
"line": 125,
"column": 74
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case hg\nG : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : NormedSpace ℝ E'\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace ℝ G\ninst✝³ : CompleteSpace E'\ninst✝² : BorelSpace G\ninst✝¹ : FiniteDimensional ℝ G\ninst✝ : μ.IsA... | [] | apply tendsto_const_nhds.congr (fun i ↦ ?_)
rw [← integral_neg_eq_self]
simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 390,
"column": 10
} | {
"line": 390,
"column": 26
} | {
"line": 390,
"column": 27
} | [
{
"pp": "case h₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\ni j : Fin N.succ\ninej : i ≠ j\nhi : ‖a.c i‖ ≤ 2\nhj : 2 < ‖a.c j‖\nah : Pairw... | [
"case h₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\ni j : Fin N.succ\ninej : i ≠ j\nhi : ‖a.c i‖ ≤ 2\nhj : 2 < ‖a.c j‖\nah : Pairwise fun i j ... | abs_of_nonneg A, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 590,
"column": 4
} | {
"line": 590,
"column": 41
} | {
"line": 591,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : MetricSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : MeasurableSpace α\ninst✝¹ : OpensMeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\ns : Set α\nr : α → ℝ\nrpos : ∀ x ∈ s, 0 < r x\nrle : ∀ x ∈ s, ... | [
"α : Type u_1\ninst✝⁴ : MetricSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : MeasurableSpace α\ninst✝¹ : OpensMeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\ns : Set α\nr : α → ℝ\nrpos : ∀ x ∈ s, 0 < r x\nrle : ∀ x ∈ s, r x ≤ 1\nhμs... | apply ENNReal.exists_le_of_sum_le _ S | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.ConformalLinearMap | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 55
} | {
"line": 36,
"column": 4
} | [
{
"pp": "case mp\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nc₁ : ℝ\nhc₁ : c₁ ≠ 0\nli : E →ₗᵢ[ℝ] F\n⊢ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪(c₁ • li.toContinuousLinearMap) u, (c₁ • li.toContinuousLinearMap) v⟫ = c... | [
"case mp\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nc₁ : ℝ\nhc₁ : c₁ ≠ 0\nli : E →ₗᵢ[ℝ] F\nu v : E\n⊢ ⟪(c₁ • li.toContinuousLinearMap) u, (c₁ • li.toContinuousLinearMap) v⟫ = c₁ * c₁ * ⟪u, v⟫"
] | refine ⟨c₁ * c₁, mul_self_pos.2 hc₁, fun u v => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 459,
"column": 4
} | {
"line": 459,
"column": 29
} | {
"line": 459,
"column": 30
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) •... | [
"case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i\ni : ... | by_cases hi : ‖a.c i‖ = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 82,
"column": 2
} | {
"line": 84,
"column": 62
} | {
"line": 86,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | [] | have h' : ContDiffWithinAt 𝕜' n f Set.univ x := h
convert! (h'.restrictScalars_iteratedFDerivWithin_eventuallyEq _ trivial)
<;> simp [iteratedFDerivWithin_univ.symm, uniqueDiffOn_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 82,
"column": 2
} | {
"line": 84,
"column": 62
} | {
"line": 86,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | [] | have h' : ContDiffWithinAt 𝕜' n f Set.univ x := h
convert! (h'.restrictScalars_iteratedFDerivWithin_eventuallyEq _ trivial)
<;> simp [iteratedFDerivWithin_univ.symm, uniqueDiffOn_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Star | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 33
} | {
"line": 79,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : StarRing 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : StarAddMonoid F\ninst✝² : StarModule 𝕜 F\ninst✝¹ : ContinuousStar F\nx : 𝕜\ninst✝ : NormedStarGroup 𝕜\nf : 𝕜 → F\nf' : F\nhf : HasDerivAt f f' x\n⊢ ... | [
"𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : StarRing 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : StarAddMonoid F\ninst✝² : StarModule 𝕜 F\ninst✝¹ : ContinuousStar F\nx : 𝕜\ninst✝ : NormedStarGroup 𝕜\nf : 𝕜 → F\nf' : F\nhf : HasDerivAt f f' x\n⊢ HasFDerivAt ... | rw [hasDerivAt_iff_hasFDerivAt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 737,
"column": 36
} | {
"line": 737,
"column": 50
} | {
"line": 737,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nN : ℕ\nτ : ℝ\nhτ... | [
"α : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN... | ← sdiff_sdiff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 66
} | {
"line": 376,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx₀ : ℝ\nh₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x - x₀)\nx : ℝ\nhx' : sign (deriv f x) = sign (x - x₀)\nhx : x₀ < x\n⊢ deriv f x > 0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"sub_pos... | [] | rwa [← sub_pos, ← sign_eq_one_iff, ← hx', sign_eq_one_iff] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 66
} | {
"line": 382,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx₀ : ℝ\nh₀ : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x₀ - x)\nx : ℝ\nhx' : sign (deriv f x) = sign (x₀ - x)\nhx : x < x₀\n⊢ deriv f x > 0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"sub_pos... | [] | rwa [← sub_pos, ← sign_eq_one_iff, ← hx', sign_eq_one_iff] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 420,
"column": 6
} | {
"line": 420,
"column": 20
} | {
"line": 421,
"column": 6
} | [
{
"pp": "case hbc\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nFu : Type u\ninst✝³ : NormedAddCommGroup Fu\ninst✝² : NormedSpace 𝕜 Fu\nf : E → Fu\ns : Set E\nt : Set Fu\nx : E\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : M... | [
"case h₂\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nFu : Type u\ninst✝³ : NormedAddCommGroup Fu\ninst✝² : NormedSpace 𝕜 Fu\nf : E → Fu\ns : Set E\nt : Set Fu\nx : E\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\n... | · exact I i hi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 49
} | {
"line": 269,
"column": 6
} | [
{
"pp": "case hf.hf.hf\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : ... | [
"case hf.hf.hf.refine_1\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈... | refine (hf _ ?_).comp_hasDerivWithinAt _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.VectorField | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 53
} | {
"line": 172,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W V₁ : E → E\ns : Set E\nx : E\nhV : DifferentiableWithinAt 𝕜 V s x\nhV₁ : DifferentiableWithinAt 𝕜 V₁ s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ lieBracketWithin 𝕜 (V + V₁) W s x... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W V₁ : E → E\ns : Set E\nx : E\nhV : DifferentiableWithinAt 𝕜 V s x\nhV₁ : DifferentiableWithinAt 𝕜 V₁ s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ (fderivWithin 𝕜 W s x) (V x) + (fderivWithin ... | simp only [lieBracketWithin, Pi.add_apply, map_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.VectorField | {
"line": 177,
"column": 50
} | {
"line": 180,
"column": 6
} | {
"line": 182,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W V₁ : E → E\nx : E\nhV : DifferentiableAt 𝕜 V x\nhV₁ : DifferentiableAt 𝕜 V₁ x\n⊢ lieBracket 𝕜 (V + V₁) W x = lieBracket 𝕜 V W x + lieBracket 𝕜 V₁ W x",
"ppTerm": "?m.4... | [] | by
simp only [lieBracket, Pi.add_apply, map_add]
rw [fderiv_add hV hV₁, add_apply]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.VectorField | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 53
} | {
"line": 207,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W W₁ : E → E\ns : Set E\nx : E\nhW : DifferentiableWithinAt 𝕜 W s x\nhW₁ : DifferentiableWithinAt 𝕜 W₁ s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ lieBracketWithin 𝕜 V (W + W₁) s x... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W W₁ : E → E\ns : Set E\nx : E\nhW : DifferentiableWithinAt 𝕜 W s x\nhW₁ : DifferentiableWithinAt 𝕜 W₁ s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ (fderivWithin 𝕜 (W + W₁) s x) (V x) - ((fderi... | simp only [lieBracketWithin, Pi.add_apply, map_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.LineDeriv.Basic | {
"line": 412,
"column": 2
} | {
"line": 413,
"column": 78
} | {
"line": 415,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nv : E\nf : E → F\nf' : F\nx₀ : E\nhf : HasLineDerivAt 𝕜 f f' x₀ v\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip... | [] | refine hf.le_of_lip' C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.LineDeriv.Basic | {
"line": 412,
"column": 2
} | {
"line": 413,
"column": 78
} | {
"line": 415,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nv : E\nf : E → F\nf' : F\nx₀ : E\nhf : HasLineDerivAt 𝕜 f f' x₀ v\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip... | [] | refine hf.le_of_lip' C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 415,
"column": 8
} | {
"line": 415,
"column": 33
} | {
"line": 415,
"column": 33
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : Has... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithin... | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 522,
"column": 54
} | {
"line": 528,
"column": 45
} | {
"line": 530,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nm : ℕ\nhm : minSmoothness 𝕜 ↑m ≤ n\n⊢ ∃ n', minSmoothness 𝕜 ↑m ≤ n' ∧ n' ≤ n ∧ n' ≠ ∞",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"ENat.natCast_ne_coe_top._simp_1",
"Eq.mpr",
"False",
"WithTo... | [] | by
simp only [minSmoothness] at hm ⊢
split_ifs with h
· simp only [h, ↓reduceIte] at hm
exact ⟨m, le_rfl, hm, by simp⟩
· simp only [h, ↓reduceIte] at hm
refine ⟨ω, le_rfl, by simp [hm], by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 32
} | {
"line": 593,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x\nhn : minSmoothness 𝕜 2 ≤ n\nhs : Un... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x\nhn : minSmoothness 𝕜 2 ≤ n\nhs : UniqueDiffOn �... | change (z ∈ s) = (z ∈ s ∩ u) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 11
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGro... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 54
} | {
"line": 105,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 54
} | {
"line": 105,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 54
} | {
"line": 105,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.LHopital | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 13
} | {
"line": 171,
"column": 13
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Iio a, g' x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\nhdnf : ∀ x ∈ -Iio a, HasDerivAt (f ∘ N... | [
"a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Iio a, g' x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\nhdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘ Neg.neg) (f... | neg_Iio | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 137,
"column": 76
} | {
"line": 137,
"column": 78
} | {
"line": 137,
"column": 78
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y... | J2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 155,
"column": 76
} | {
"line": 155,
"column": 78
} | {
"line": 155,
"column": 78
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y... | J2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 77
} | {
"line": 505,
"column": 0
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\nhg2' : I... | [] | simpa [mul_comm] using integral_deriv_smul_comp''' hf hff' hg_cont hg1 hg2' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.LineDeriv.Measurable | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 71
} | {
"line": 96,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : LocallyCompactSpace 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : OpensMeasurableSpace E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : Co... | [] | exact (measurable_deriv_with_param this).comp measurable_prodMk_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 999,
"column": 2
} | {
"line": 999,
"column": 44
} | {
"line": 1001,
"column": 0
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont✝ : ContinuousWithinAt f (Iic a) a\nhderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nhf : Tendsto f atBot (𝓝 m)\nhcont : ContinuousOn f (... | [] | exact f'int.mono (fun y hy => hy.2) le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 540,
"column": 6
} | {
"line": 540,
"column": 43
} | {
"line": 541,
"column": 4
} | [
{
"pp": "case h₂\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nδ : ℝ≥0\nhf : ApproximatesLinearOn f A s δ\nhs : MeasurableS... | [] | · apply ContinuousLinearMap.le_opNorm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 76
} | {
"line": 456,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : LocallyCompactSpace β\nh : ∀ (K : Set β), IsCompact K → HasProdUniformlyOn f g K\n⊢ HasProdLocallyUniformly f g",
"ppTerm": "?m.18",
"assigned... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : LocallyCompactSpace β\nh : ∀ (K : Set β), IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ (K : Set β), IsCompact K → TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2)... | rw [HasProdLocallyUniformly, tendstoLocallyUniformly_iff_forall_isCompact] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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