module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Matrix.Normed | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 49
} | {
"line": 96,
"column": 0
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : SeminormedAddCommGroup α\nr : ℝ\nhr : 0 ≤ r\nA : Matrix m n α\n⊢ ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedA... | [] | simp_rw [norm_def, pi_norm_le_iff_of_nonneg hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Normed | {
"line": 336,
"column": 15
} | {
"line": 336,
"column": 34
} | {
"line": 336,
"column": 35
} | [
{
"pp": "l : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype l\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalSeminormedRing α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (Finset.univ.sup fun i ↦ ∑ k, ∑ j, ‖A i j‖₊ * ‖B j k‖₊) = Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊ * ∑ k, ‖B j k‖₊",
... | [
"l : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype l\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalSeminormedRing α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (Finset.univ.sup fun i ↦ ∑ y, ∑ x, ‖A i y‖₊ * ‖B y x‖₊) = Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊ * ∑ k, ‖B j k‖₊"
] | @Finset.sum_comm m, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 41
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case star_id\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : Is... | [] | rwa [map_star, cfcₙHom_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 41
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case star_id\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : Is... | [] | rwa [map_star, cfcₙHom_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 41
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case star_id\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : Is... | [] | rwa [map_star, cfcₙHom_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 815,
"column": 2
} | {
"line": 815,
"column": 64
} | {
"line": 817,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝ :... | [] | exact ⟨⟨hf.1.mono hks, hf.2⟩, ha.1, subset_of_mem_nhdsSet ha'⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 144,
"column": 8
} | {
"line": 144,
"column": 33
} | {
"line": 144,
"column": 33
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : MeasurableSpace X\nμ : Measure X\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : CompleteSpace A\ninst✝ : NormedSpace ℝ A\nf : X → 𝕜 → ... | [
"X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : MeasurableSpace X\nμ : Measure X\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : CompleteSpace A\ninst✝ : NormedSpace ℝ A\nf : X → 𝕜 → 𝕜\na : A\nh... | mkD_apply_of_continuousOn | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 471,
"column": 18
} | {
"line": 471,
"column": 42
} | {
"line": 472,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝⁵ : Unique n\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Mul A\ninst✝¹ : Star A\ninst✝ : Module R A\nx✝¹ x✝ : A\n⊢ (fun x y ↦ x✝¹ * x✝) = (fun x y ↦ x✝¹) * fun x y ↦ x✝",
"ppTerm": "?m.63",
"as... | [] | by ext; simp [mul_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 302,
"column": 8
} | {
"line": 302,
"column": 33
} | {
"line": 302,
"column": 33
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : MeasurableSpace X\nμ : Measure X\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : NonUnitalContinuousFunctionalCal... | [
"X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : MeasurableSpace X\nμ : Measure X\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p... | mkD_apply_of_continuousOn | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 966,
"column": 2
} | {
"line": 968,
"column": 43
} | {
"line": 969,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalNormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedSpace ℝ A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedR... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalNormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedSpace ℝ A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝... | have : {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}.EqOn (cfcₙ f)
(cfcₙ (fun x : ℝ ↦ f x.toNNReal)) :=
fun a ha ↦ cfcₙ_nnreal_eq_real _ _ ha.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 23
} | {
"line": 30,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : StarModule ℂ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsStarNormal a\nx : ℂ\nhx : x ∈... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 1010,
"column": 2
} | {
"line": 1010,
"column": 64
} | {
"line": 1012,
"column": 0
} | [
{
"pp": "A : Type u_2\ninst✝¹² : NonUnitalNormedRing A\ninst✝¹¹ : StarRing A\ninst✝¹⁰ : NormedSpace ℝ A\ninst✝⁹ : IsScalarTower ℝ A A\ninst✝⁸ : SMulCommClass ℝ A A\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrdered... | [] | exact ⟨⟨hf.1.mono hks, hf.2⟩, ha.1, subset_of_mem_nhdsSet ha'⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 23
} | {
"line": 111,
"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\nhp : IsStarNormal a\nx : ℂ\nhx : x ∈ spectrum ℂ a\n⊢ (x + (starRingEnd ℂ) x) / 2 = (2⁻¹ • 1) • (x + star x)",
... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.Hom | {
"line": 27,
"column": 2
} | {
"line": 27,
"column": 84
} | {
"line": 28,
"column": 2
} | [
{
"pp": "F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CStarAlgebra A\ninst✝³ : CStarAlgebra B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F ℂ A B\ninst✝ : StarHomClass F A B\na : A\nha : IsSelfAdjoint a\nφ : F\nhφ : Function.Injective ⇑φ\n⊢ spectrum ℝ (φ a) = spectrum ℝ a",
"ppTerm": "?m.25",
"a... | [
"F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CStarAlgebra A\ninst✝³ : CStarAlgebra B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F ℂ A B\ninst✝ : StarHomClass F A B\na : A\nha : IsSelfAdjoint a\nφ : F\nhφ : Function.Injective ⇑φ\nh_spec : spectrum ℝ ((StarAlgHom.restrictScalars ℝ ↑φ) a) ⊆ spectrum ℝ a\n⊢ s... | have h_spec := AlgHom.spectrum_apply_subset ((φ : A →⋆ₐ[ℂ] B).restrictScalars ℝ) a | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 25
} | {
"line": 132,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\nf : A →ₚ[ℂ] ℂ\ninst✝ : StarOrderedRing A\na b : A\n⊢ f.leftMulMapPreGNS (a * b) = f.leftMulMapPreGNS a ∘SL f.leftMulMapPreGNS b",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"ContinuousLinearMap.comp"... | [] | ext c; simp [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 25
} | {
"line": 132,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\nf : A →ₚ[ℂ] ℂ\ninst✝ : StarOrderedRing A\na b : A\n⊢ f.leftMulMapPreGNS (a * b) = f.leftMulMapPreGNS a ∘SL f.leftMulMapPreGNS b",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"ContinuousLinearMap.comp"... | [] | ext c; simp [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Projection | {
"line": 126,
"column": 57
} | {
"line": 126,
"column": 89
} | {
"line": 126,
"column": 89
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np q : A\nhp : IsStarProjection p\nhq : IsStarProjection q\nh : p ≤ q\n⊢ p * q = p",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCStarAlgebra.toNonU... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np q : A\nhp : IsStarProjection p\nhq : IsStarProjection q\nh : p ≤ q\n⊢ p = p"
] | hp.le_iff_mul_eq_left hq |>.mp h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.OpenPartialHomeomorph.Composition | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 70
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\ne' : OpenPartialHomeomorph X Y\nhs : IsOpen[inst✝¹] s\nhs' : IsOpen[inst✝] (↑e '' s)\nhs'' : s ⊆ e.source\n⊢ e.target ∩ ↑e.symm ⁻¹' interior s ∩ ↑e.symm ⁻¹' e'.s... | [
"case refine_1\nX : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\ne' : OpenPartialHomeomorph X Y\nhs : IsOpen[inst✝¹] s\nhs' : IsOpen[inst✝] (↑e '' s)\nhs'' : s ⊆ e.source\n⊢ e.target ∩ ↑e.symm ⁻¹' s ∩ ↑e.symm ⁻¹' e'.source = e.target ∩ ↑e... | rw [interior_eq_iff_isOpen.mpr hs', interior_eq_iff_isOpen.mpr hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.OpenPartialHomeomorph.Composition | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 32
} | {
"line": 177,
"column": 32
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\ne' : OpenPartialHomeomorph X Y\nhs : IsOpen[inst✝¹] s\n⊢ IsOpen[inst✝] (e'.target ∩ ↑e'.symm ⁻¹' s)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
... | [
"X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\ne' : OpenPartialHomeomorph X Y\nhs : IsOpen[inst✝¹] s\n⊢ IsOpen[inst✝] (↑e' '' (e'.source ∩ s))"
] | ← image_source_inter_eq' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 364,
"column": 4
} | {
"line": 367,
"column": 25
} | {
"line": 368,
"column": 2
} | [
{
"pp": "X✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorph X✝ Y\n... | [] | intro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, hf.injective.extend_apply e, comp_apply]
congr
exact e.left_inv' hx₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 364,
"column": 4
} | {
"line": 367,
"column": 25
} | {
"line": 368,
"column": 2
} | [
{
"pp": "X✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorph X✝ Y\n... | [] | intro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, hf.injective.extend_apply e, comp_apply]
congr
exact e.left_inv' hx₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 536,
"column": 85
} | {
"line": 536,
"column": 92
} | {
"line": 536,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nf : A →L[𝕜] A\nC : ℝ≥0\nh : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nf : A →L[𝕜] A\nC : ℝ≥0\nh : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊\... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 895,
"column": 63
} | {
"line": 895,
"column": 91
} | {
"line": 895,
"column": 91
} | [
{
"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 rwa [e.frontier_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.FiberBundle.Basic | {
"line": 798,
"column": 26
} | {
"line": 809,
"column": 84
} | {
"line": 810,
"column": 4
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝³ : TopologicalSpace X\nE : B → Type u_5\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\ninst✝ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ a.pretrivializationAtlas\n... | [] | by
refine continuousOn_iff'.mpr fun s hs => ⟨e ⁻¹' s ∩ e.source,
isOpen_iSup_iff.mpr fun e' => ?_, by rw [inter_assoc, inter_self]; rfl⟩
refine isOpen_iSup_iff.mpr fun he' => ?_
rw [isOpen_coinduced, isOpen_induced_iff]
obtain ⟨u, hu1, hu2⟩ := continuousOn_iff'.mp (a.continuous_trivChang... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.SeparatedMap | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 42
} | {
"line": 110,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Sort u_2\nA : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\nf : X → Y\nsep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)\ng : A → Y\n⊢ IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal snd)",
"ppTerm": "?m.26",
"assig... | [
"X : Type u_1\nY : Sort u_2\nA : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\nf : X → Y\nsep : IsClosed[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)\ng : A → Y\n⊢ IsClosed[instTopologicalSpaceSubtype] (Function.PullbackSelf.map_fst ⁻¹' Function.pullbackDiagonal f)"
] | rw [← preimage_map_fst_pullbackDiagonal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 54
} | {
"line": 54,
"column": 6
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nf : X → Y\nh : ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.restrict f)\nx : X\nhx : x ∈ s\nU : Set X\nhU : U ∈ 𝓝 x\nemb : IsOpenEmbedding (U.restrict f)\n⊢ IsOpenEmbedding ((interior U).restrict f)",
"ppTer... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nf : X → Y\nh : ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.restrict f)\nx : X\nhx : x ∈ s\nU : Set X\nhU : U ∈ 𝓝 x\nemb : IsOpenEmbedding (U.restrict f)\n⊢ IsOpen[instTopologicalSpaceSubtype] (Set.range (Set.inclusion ⋯))"... | refine emb.comp ⟨.inclusion interior_subset, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 270,
"column": 2
} | {
"line": 273,
"column": 27
} | {
"line": 274,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsLocalHomeomorph f\n⊢ ∀ u ∈ {U | ∃ V, IsOpen[inst✝] V ∧ ∃ s, f ∘ ⇑s = Subtype.val ∧ Set.range ⇑s = U}, IsOpen[inst✝¹] u",
"ppTerm": "?refine_1",
"assigned": true,
"usedConsta... | [
"case refine_2\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsLocalHomeomorph f\nx : X\nU : Set X\nhx : x ∈ U\nhU : IsOpen[inst✝¹] U\n⊢ ∃ v ∈ {U | ∃ V, IsOpen[inst✝] V ∧ ∃ s, f ∘ ⇑s = Subtype.val ∧ Set.range ⇑s = U}, x ∈ v ∧ v ⊆ U"
] | · rintro _ ⟨U, hU, s, hs, rfl⟩
refine (isOpenEmbedding_of_comp hf (hs ▸ ⟨IsEmbedding.subtypeVal, ?_⟩)
s.continuous).isOpen_range
rwa [Subtype.range_val] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 13
} | {
"line": 104,
"column": 13
} | [
{
"pp": "s : Set ℝ\nhs : ∀ x ∈ s, ∀ y ∈ s, x - y ∈ Ioo (-2 * π) (2 * π)\nt₁ : ℝ\nht₁ : t₁ ∈ s\nt₂ : ℝ\nht₂ : t₂ ∈ s\nheq : exp t₁ = exp t₂\nh1 : -2 * π < t₁ - t₂\nh2 : t₁ - t₂ < 2 * π\n⊢ t₁ = t₂",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Non... | [
"s : Set ℝ\nhs : ∀ x ∈ s, ∀ y ∈ s, x - y ∈ Ioo (-2 * π) (2 * π)\nt₁ : ℝ\nht₁ : t₁ ∈ s\nt₂ : ℝ\nht₂ : t₂ ∈ s\nheq : exp t₁ = exp t₂\nh1 : -(2 * π) < t₁ - t₂\nh2 : t₁ - t₂ < 2 * π\n⊢ t₁ = t₂"
] | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 138,
"column": 7
} | {
"line": 138,
"column": 59
} | {
"line": 138,
"column": 59
} | [
{
"pp": "r : ℝ\nhr : r ≤ π\nt : ℝ\nht : t ∈ {x | |x| < r}\nhtπ : |t| < π\n⊢ |(↑(exp t)).arg| < r",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
... | [
"r : ℝ\nhr : r ≤ π\nt : ℝ\nht : t ∈ {x | |x| < r}\nhtπ : |t| < π\n⊢ |t| < r"
] | arg_exp (neg_lt_of_abs_lt htπ) (lt_of_abs_lt htπ).le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 264,
"column": 2
} | {
"line": 272,
"column": 9
} | {
"line": 274,
"column": 0
} | [
{
"pp": "x y : Circle\nh : x ≠ y\n⊢ Disjoint (⇑(x.path y) '' Ioc 0 1) (⇑(y.path x) '' Ioc 0 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real.partialO... | [] | have hdisj : Disjoint (Ioc x.val.arg (angleDiff x y + x.val.arg))
(Ioc y.val.arg (angleDiff y x + y.val.arg)) := by grind [angleDiff]
rw [path_image_Ioc_of_ne h, path_image_Ioc_of_ne h.symm]
refine Set.disjoint_image_image fun a ha b hb ↦ ?_
refine exp_injOn_Ioc (a := min x.val.arg y.val.arg) (b := min x.va... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 264,
"column": 2
} | {
"line": 272,
"column": 9
} | {
"line": 274,
"column": 0
} | [
{
"pp": "x y : Circle\nh : x ≠ y\n⊢ Disjoint (⇑(x.path y) '' Ioc 0 1) (⇑(y.path x) '' Ioc 0 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real.partialO... | [] | have hdisj : Disjoint (Ioc x.val.arg (angleDiff x y + x.val.arg))
(Ioc y.val.arg (angleDiff y x + y.val.arg)) := by grind [angleDiff]
rw [path_image_Ioc_of_ne h, path_image_Ioc_of_ne h.symm]
refine Set.disjoint_image_image fun a ha b hb ↦ ?_
refine exp_injOn_Ioc (a := min x.val.arg y.val.arg) (b := min x.va... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 441,
"column": 2
} | {
"line": 441,
"column": 9
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case H\nT : ℝ\nhT : T ≠ 0\nz✝ : ℝ\n⊢ Circle.exp (z✝ * (T⁻¹ * (2 * π))) = Circle.exp (2 * π / T * z✝)",
"ppTerm": "?H",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Calculus.AddTorsor.AffineMap | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 33
} | {
"line": 36,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace 𝕜 W\nn : WithTop ℕ∞\nf : V →ᴬ[𝕜] W\n⊢ ContDiff 𝕜 n (⇑f.contLinear + Function.const V (f 0))",
"ppTerm": "?m... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace 𝕜 W\nn : WithTop ℕ∞\nf : V →ᴬ[𝕜] W\n⊢ ContDiff 𝕜 n (Function.const V (f 0))"
] | apply f.contLinear.contDiff.add | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Group.Integral | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 52
} | {
"line": 106,
"column": 2
} | [
{
"pp": "G : Type u_4\nE : Type u_5\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulRightInvariant\nf : G → E\ng : G\n⊢ ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ",
"ppTerm": "?m.26",
"assign... | [
"G : Type u_4\nE : Type u_5\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulRightInvariant\nf : G → E\ng : G\nh_mul : MeasurableEmbedding fun x ↦ x * g\n⊢ ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ"
] | have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 335,
"column": 2
} | {
"line": 338,
"column": 77
} | {
"line": 340,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ninst✝¹ : Fintype ι\np : ι → P\nhi : AffineIndependent k p\nsp : AffineSubspace k P\ninst✝ : FiniteDimensional k ↥sp.direction\nhle : affineSpan k (Set... | [] | classical
rw [← Finset.card_univ] at hc
rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image] at hle ⊢
exact hi.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one hle hc | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 335,
"column": 2
} | {
"line": 338,
"column": 77
} | {
"line": 340,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ninst✝¹ : Fintype ι\np : ι → P\nhi : AffineIndependent k p\nsp : AffineSubspace k P\ninst✝ : FiniteDimensional k ↥sp.direction\nhle : affineSpan k (Set... | [] | classical
rw [← Finset.card_univ] at hc
rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image] at hle ⊢
exact hi.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one hle hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 335,
"column": 2
} | {
"line": 338,
"column": 77
} | {
"line": 340,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ninst✝¹ : Fintype ι\np : ι → P\nhi : AffineIndependent k p\nsp : AffineSubspace k P\ninst✝ : FiniteDimensional k ↥sp.direction\nhle : affineSpan k (Set... | [] | classical
rw [← Finset.card_univ] at hc
rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image] at hle ⊢
exact hi.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one hle hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 21
} | {
"line": 359,
"column": 22
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : FiniteDimensional k V\nn : ℕ\nT : Simplex k V n\nhrank : finrank k V = n\n⊢ Fintype.card (Fin (n + 1)) = finrank k V + 1",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
... | [
"k : Type u_1\nV : Type u_2\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : FiniteDimensional k V\nn : ℕ\nT : Simplex k V n\nhrank : finrank k V = n\n⊢ n + 1 = finrank k V + 1"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 523,
"column": 29
} | {
"line": 523,
"column": 46
} | {
"line": 523,
"column": 47
} | [
{
"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\nhn : 1 < n\np : P\nh : ∀ (i : Fin (n + 1)), p ∈ s.median i\ni₀ : Fin (n + 1) := 0\nhp : p = (p -ᵥ s... | [
"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\nhn : 1 < n\np : P\nh : ∀ (i : Fin (n + 1)), p ∈ s.median i\ni₀ : Fin (n + 1) := 0\nhp : p = (p -ᵥ s.centroid) +... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 102,
"column": 90
} | {
"line": 110,
"column": 31
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Summable fun i ↦ ‖f i‖ ^ p.toReal\n⊢ Memℓp f p",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Norm.norm",
"S... | [] | by
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
exact (Set.Finite.of_summable_const (by simp) H).subset (Set.subset_univ _)
· apply memℓp_infty
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf
simpa using ((Set.... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.SmoothSeries | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 57
} | {
"line": 246,
"column": 4
} | [
{
"pp": "case right\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → ... | [
"case right\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α → ℝ\nN : ℕ... | refine differentiable_tsum (hv _ h'm) A fun n x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 14
} | {
"line": 443,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E p)\nhp : p = 0\n⊢ ℝ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"instAddCommGroupPreLp",
"AddCommGroup.toAddGroup",
... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E 0)\n⊢ ℝ"
] | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 40
} | {
"line": 154,
"column": 4
} | [
{
"pp": "case left\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜]... | [
"case left\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜] G\nx : E\nh... | refine lt_of_le_of_lt ?_ (hxyε y hy) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 133,
"column": 91
} | {
"line": 134,
"column": 32
} | {
"line": 136,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\nhr : 0 < r\n⊢ (univBall c r).target = ball c r",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"OpenPartialHomeom... | [] | by
rw [univBall, dif_pos hr]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 613,
"column": 4
} | {
"line": 613,
"column": 67
} | {
"line": 615,
"column": 0
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\ni : α\nx✝¹ x✝ : ↥(lp E p)\n⊢ ‖↑(x✝¹ - x✝) i‖ ≤ ‖x✝¹ - x✝‖",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"lp.norm_apply_le_norm",
"ENNReal.instIsOrderedRing",
... | [] | exact norm_apply_le_norm (zero_lt_one.trans_le Fact.out).ne' .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 29
} | {
"line": 321,
"column": 29
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜... | [
"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E ... | hasFDerivAt_iff_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 146,
"column": 16
} | {
"line": 146,
"column": 40
} | {
"line": 147,
"column": 6
} | [
{
"pp": "case hbc\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... | [] | exact (F' a).le_opNorm _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convolution | {
"line": 507,
"column": 67
} | {
"line": 507,
"column": 81
} | {
"line": 507,
"column": 81
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : NormedSpace 𝕜 F... | [
"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[�... | notMem_support | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convolution | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 56
} | {
"line": 570,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : 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 : N... | [
"case neg\n𝕜 : 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 �... | let g' : (P × G) → G → E' := fun p x ↦ g p.1 (p.2 - x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convolution | {
"line": 646,
"column": 11
} | {
"line": 646,
"column": 24
} | {
"line": 646,
"column": 25
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ... | [
"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace 𝕜 F\nL : E ... | sub_sub_self, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convolution | {
"line": 713,
"column": 10
} | {
"line": 713,
"column": 24
} | {
"line": 713,
"column": 24
} | [
{
"pp": "case neg\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : Normed... | [
"case neg\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : NormedSpace 𝕜 F\n... | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convolution | {
"line": 747,
"column": 10
} | {
"line": 747,
"column": 24
} | {
"line": 747,
"column": 24
} | [
{
"pp": "case neg\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : No... | [
"case neg\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace 𝕜... | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 271,
"column": 2
} | {
"line": 276,
"column": 49
} | {
"line": 277,
"column": 2
} | [
{
"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 ... | [
"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 G\nh : μ.IsE... | have : k ∩ ((z * x⁻¹) • k)ᶜ ∈ 𝓝[k] z := by
apply inter_mem_nhdsWithin k
apply IsOpen.mem_nhds (by simpa using h'k.smul _)
push _ ∈ _
contrapose H
simpa [mem_smul_set_iff_inv_smul_mem] using H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 78
} | {
"line": 104,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nn : ℕ∞\ns : Set E\nhs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nh's : s.Nonempty\nι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ ... | [] | simp only [g, hf.2.2.2.2, mem_support, Ne, one_ne_zero, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 62
} | {
"line": 167,
"column": 8
} | [
{
"pp": "case h'f\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 h'f\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 M'_comp : IsCompact (closure M) := M_comp.closure | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 62
} | {
"line": 198,
"column": 8
} | [
{
"pp": "G : 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... | [
"G : 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✝ : ν.IsOpen... | have M'_comp : IsCompact (closure M) := M_comp.closure | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 389,
"column": 2
} | {
"line": 391,
"column": 84
} | {
"line": 392,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\nμ' μ ν : Measure G\ninst✝³ : μ.IsHaarMeasure\ninst✝² : ν.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nhG : Loc... | [
"case pos\nG : Type u_1\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\nμ' μ ν : Measure G\ninst✝³ : μ.IsHaarMeasure\ninst✝² : ν.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nhG : LocallyCompactS... | simp only [integral_smul_nnreal_measure, smul_smul,
integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' ν g_cont g_comp,
integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ ν g_cont g_comp] at Z | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 30
} | {
"line": 416,
"column": 4
} | [
{
"pp": "case pos.int_nonzero\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nφ : G ≃ₜ* G\nhG : LocallyCompactSpace G\nf : G → ℝ\nf_cont : Continu... | [
"case pos.int_nonzero\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nφ : G ≃ₜ* G\nhG : LocallyCompactSpace G\nf : G → ℝ\nf_cont : Continuous[inst✝⁶, ... | change ∫ x, f (φ x) ∂μ ≠ 0 | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 279,
"column": 4
} | {
"line": 280,
"column": 47
} | {
"line": 281,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nn : ℕ\nf : Fin n → E\nh : ∀ (i : Fin n), ‖f i‖ ≤ 2\nh' : Pairwise fun i j ↦ 1 - goodδ E ≤ ‖f i - f j‖\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : s.card = n\n⊢ ∀ ... | [] | simp only [s, h, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ,
Finset.mem_image, imp_true_iff, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 959,
"column": 36
} | {
"line": 959,
"column": 68
} | {
"line": 959,
"column": 68
} | [
{
"pp": "G : Type u_1\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : Group G\ninst✝¹⁰ : IsTopologicalGroup G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : BorelSpace G\nH : Type u_2\ninst✝⁷ : Group H\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : IsTopologicalGroup H\ninst✝⁴ : CompactSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : Bor... | [] | rw [huniv]; apply measure_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 959,
"column": 36
} | {
"line": 959,
"column": 68
} | {
"line": 959,
"column": 68
} | [
{
"pp": "G : Type u_1\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : Group G\ninst✝¹⁰ : IsTopologicalGroup G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : BorelSpace G\nH : Type u_2\ninst✝⁷ : Group H\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : IsTopologicalGroup H\ninst✝⁴ : CompactSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : Bor... | [] | rw [huniv]; apply measure_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Holder | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 22
} | {
"line": 194,
"column": 22
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < r\n⊢ HolderOnWith (C * D ^ (↑r - ↑s)) s f A",
"ppTerm"... | [
"case inr\nX : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < ↑r\n⊢ HolderOnWith (C * D ^ (↑r - ↑s)) s f A"
] | ← NNReal.coe_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Pi | {
"line": 26,
"column": 4
} | {
"line": 26,
"column": 33
} | {
"line": 27,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nι : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nx : (i : ι) → E i\nj : ι\ny : E j\n⊢ HasFDerivAt (fun x_1 ↦ Function.update x j x_1 j) (Pi.single j ... | [] | simpa using! hasFDerivAt_id _ | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Calculus.FDeriv.Pi | {
"line": 26,
"column": 4
} | {
"line": 26,
"column": 33
} | {
"line": 27,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nι : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nx : (i : ι) → E i\nj : ι\ny : E j\n⊢ HasFDerivAt (fun x_1 ↦ Function.update x j x_1 j) (Pi.single j ... | [] | simpa using! hasFDerivAt_id _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Pi | {
"line": 26,
"column": 4
} | {
"line": 26,
"column": 33
} | {
"line": 27,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nι : Type u_2\ninst✝³ : DecidableEq ι\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nx : (i : ι) → E i\nj : ι\ny : E j\n⊢ HasFDerivAt (fun x_1 ↦ Function.update x j x_1 j) (Pi.single j ... | [] | simpa using! hasFDerivAt_id _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Pi | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 31
} | {
"line": 28,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : NontriviallyNormedField 𝕜\nx : ι → 𝕜\ni : ι\ny z : 𝕜\nj : ι\nh : ¬j = i\n⊢ 0 j = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 𝕜))) z j",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | · simp [Pi.single_eq_of_ne h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Star | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 41
} | {
"line": 56,
"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\nf : 𝕜 → F\nx : 𝕜\ninst✝ : TrivialStar 𝕜\ns : Set 𝕜\n⊢ derivWithin (fun y ↦ sta... | [
"case pos\n𝕜 : 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\nf : 𝕜 → F\nx : 𝕜\ninst✝ : TrivialStar 𝕜\ns : Set 𝕜\nhxs : UniqueDiffWithinAt 𝕜 ... | by_cases hxs : UniqueDiffWithinAt 𝕜 s x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 379,
"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)\n⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x₀, deriv f b > 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"sub_pos",
"AddGroup.toSubtractionMono... | [] | filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)
rwa [← sub_pos, ← sign_eq_one_iff, ← hx', sign_eq_one_iff] at hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 379,
"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)\n⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x₀, deriv f b > 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"sub_pos",
"AddGroup.toSubtractionMono... | [] | filter_upwards [nhdsLT_le_nhdsNE _ h₀, self_mem_nhdsWithin] with x hx' (hx : x < x₀)
rwa [← sub_pos, ← sign_eq_one_iff, ← hx', sign_eq_one_iff] at hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 795,
"column": 4
} | {
"line": 795,
"column": 29
} | {
"line": 797,
"column": 0
} | [
{
"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... | [] | exact (hF (u n) (Pu n)).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 845,
"column": 69
} | {
"line": 849,
"column": 12
} | {
"line": 850,
"column": 2
} | [
{
"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✝ : SFinite μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nR : α → ℝ\nhR : ∀ x ∈ s,... | [] | by
have I : ∀ p ∈ v, 0 ≤ p.2 := fun p hp => (vg p hp).2.1.le
rw [exists_eq_graphOn]
refine fun x hx y hy heq ↦ v_disj.eq hx hy <| not_disjoint_iff.2 ⟨x.1, ?_⟩
simp [*] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 432,
"column": 6
} | {
"line": 435,
"column": 64
} | {
"line": 436,
"column": 4
} | [
{
"pp": "case a.e'_6\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 ... | [] | · rw [← pow_add]
congr 1
rw [Nat.add_succ, Nat.succ_inj]
exact Nat.add_sub_of_le (Finset.mem_range_succ_iff.1 hi) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 295,
"column": 8
} | {
"line": 295,
"column": 71
} | {
"line": 296,
"column": 8
} | [
{
"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 : ∀ ⦃... | [
"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 : ∀ ⦃c : ℝ⦄, 0 < ... | simp only [sub_apply, add_apply, smul_apply, map_add, map_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 93
} | {
"line": 100,
"column": 2
} | [
{
"pp": "case e'_11\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| •... | [
"case e'_12\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) (t⁻¹ • t... | · rw [norm_smul, ← mul_assoc, norm_eq_abs, ← abs_mul, mul_inv_cancel₀ ht, abs_one, one_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 297,
"column": 25
} | {
"line": 297,
"column": 50
} | {
"line": 297,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx y : F\ns : Set F\n⊢ (starRingEnd 𝕜) ⟪gradientWithin f s y, x⟫ = (starRingEnd 𝕜) ((fderivWithin 𝕜 f s y) x)",
"ppTerm": "?m.35",
"assigned": ... | [
"𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx y : F\ns : Set F\n⊢ (starRingEnd 𝕜) ((fderivWithin 𝕜 f s y) x) = (starRingEnd 𝕜) ((fderivWithin 𝕜 f s y) x)"
] | inner_gradientWithin_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 33
} | {
"line": 68,
"column": 2
} | [
{
"pp": "a b : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhab : a < b\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Ioo a b, g' x ≠ 0\nhfa : Tendsto f (𝓝[>] a) (𝓝 0)\nhga : Tendsto g (𝓝[>] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l\nsub : ∀ x... | [] | exact hg' y (sub x hx hyx) hy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.LHopital | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 58
} | {
"line": 307,
"column": 2
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\ns : Set ℝ\nhs : Convex ℝ s\nhff' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, HasDerivWithinAt f (f' x) (s \\ {a}) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, HasDerivWithinAt g (g' x) (s \\ {a}) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[s \\ {a}] a) (�... | [
"a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\ns : Set ℝ\nhs : Convex ℝ s\nhgg' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, HasDerivWithinAt g (g' x) (s \\ {a}) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[s \\ {a}] a, g' x ≠ 0\nhfa : Tendsto f (𝓝[s \\ {a}] a) (𝓝 0)\nhga : Tendsto g (𝓝[s \\ {a}] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[s... | replace hff' := h.mp <| hff'.mono fun _ h ↦ h.hasDerivAt | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 149,
"column": 73
} | {
"line": 154,
"column": 40
} | {
"line": 155,
"column": 4
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nr : (... | [] | by
have : f' x ∈ ⋃ z ∈ T, ball (f' (z : E)) (r (f' z)) := by
rw [hT]
refine mem_iUnion.2 ⟨⟨x, xs⟩, ?_⟩
simpa only [mem_ball, Subtype.coe_mk, dist_self] using! (rpos (f' x)).bot_lt
rwa [mem_iUnion₂, bex_def] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 70
} | {
"line": 278,
"column": 2
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\ng : ℝ → F\nH : IntegrableOn g (f '' s) volume\nH' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume\na b c : S... | [
"case pos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\ng : ℝ → F\nH : IntegrableOn g (f '' s) volume\nH' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume\na b c : Set ℝ\nh_unio... | have bc_s : b ∪ c ⊆ s := by rw [← h_union]; exact subset_union_right | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 369,
"column": 49
} | {
"line": 394,
"column": 18
} | {
"line": 396,
"column": 0
} | [
{
"pp": "a b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\n⊢ IntervalIntegrable (fun x ↦ f' x • (g ∘ f) x) volume a b ↔ ... | [] | by
have M : MonotoneOn f (uIcc a b) := by
apply monotoneOn_of_deriv_nonneg (convex_uIcc a b) hf
· rw [uIcc, interior_Icc]
exact fun z hz ↦ (hff' z hz).differentiableAt.differentiableWithinAt
· rw [uIcc, interior_Icc]
intro z hz
simpa [(hff' z hz).deriv] using hf' z hz
simp only [Functi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 314,
"column": 2
} | {
"line": 331,
"column": 36
} | {
"line": 333,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Icc a b, f' x • g (f x) = ∫ (u : ℝ) in Icc (f a) (f b), g u",
... | [] | have M : MonotoneOn f (Icc a b) := by
apply monotoneOn_of_deriv_nonneg (convex_Icc a b) hf
· rw [interior_Icc]
exact fun z hz ↦ (hff' z hz).differentiableAt.differentiableWithinAt
· rw [interior_Icc]
intro z hz
simpa [(hff' z hz).deriv] using hf' z hz
have A : ∫ u in Icc (f a) (f b), g u... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 314,
"column": 2
} | {
"line": 331,
"column": 36
} | {
"line": 333,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Icc a b, f' x • g (f x) = ∫ (u : ℝ) in Icc (f a) (f b), g u",
... | [] | have M : MonotoneOn f (Icc a b) := by
apply monotoneOn_of_deriv_nonneg (convex_Icc a b) hf
· rw [interior_Icc]
exact fun z hz ↦ (hff' z hz).differentiableAt.differentiableWithinAt
· rw [interior_Icc]
intro z hz
simpa [(hff' z hz).deriv] using hf' z hz
have A : ∫ u in Icc (f a) (f b), g u... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 450,
"column": 2
} | {
"line": 457,
"column": 18
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case inr\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ IntervalI... | [] | · rw [IntervalIntegrable.symm_iff, intervalIntegrable_iff_integrableOn_Icc_of_le hab.le,
integrableOn_Icc_deriv_smul_iff_of_deriv_nonpos,
intervalIntegrable_iff_integrableOn_Icc_of_le]
· apply M right_mem_uIcc left_mem_uIcc hab.le
· rwa [uIcc_of_ge hab.le] at hf
· grind
· grind
· exact h... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 944,
"column": 4
} | {
"line": 945,
"column": 77
} | {
"line": 946,
"column": 2
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := ⋯\nhdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x\n⊢ IntegrableOn (fun x ↦ -f' (-x)) (I... | [] | exact ((MeasurePreserving.integrableOn_comp_preimage (Measure.measurePreserving_neg _)
(Homeomorph.neg ℝ).measurableEmbedding).2 f'int.neg).mono_set (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.LineDeriv.Measurable | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 90
} | {
"line": 105,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : LocallyCompactSpace 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : Comp... | [] | exact (stronglyMeasurable_deriv_with_param this).comp_measurable measurable_prodMk_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 61
} | {
"line": 477,
"column": 2
} | [
{
"pp": "E : 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 : MeasurableSet s\nf' ... | [
"E : 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 : MeasurableSet s\nf' : E → E →L[ℝ... | apply ContinuousLinearMap.opNorm_le_bound _ δ.2 fun z => ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 532,
"column": 60
} | {
"line": 535,
"column": 10
} | {
"line": 536,
"column": 4
} | [
{
"pp": "E : 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 : MeasurableSet s\nf' ... | [] | by
congr 1
simp only [FunLike.coe_sub, map_sub, Pi.sub_apply]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ParametricIntervalIntegral | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 27
} | {
"line": 112,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\na b : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\ns : Set 𝕜\nhs : s ∈ 𝓝 x₀\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\na b : ℝ\nbound : ℝ → ℝ\nF F' : 𝕜 → ℝ → E\nx₀ : 𝕜\ns : Set 𝕜\nhs : s ∈ 𝓝 x₀\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_mea... | have := hasDerivAt_integral_of_dominated_loc_of_deriv_le hs hF_meas hF_int hF'_meas h_bound
bound_integrable h_diff | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 62
} | {
"line": 174,
"column": 4
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ StructureGroupoid.maximalAtlas M G\nhe' : e' ∈ atlas H M\ns : Set M\nhs : IsOpen[inst✝²] s... | [
"H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ StructureGroupoid.maximalAtlas M G\nhe' : e' ∈ atlas H M\ns : Set M\nhs : IsOpen[inst✝²] s\n⊢ IsOpen[i... | rw [isOpen_image_iff_of_subset_source _ inter_subset_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 189,
"column": 8
} | {
"line": 189,
"column": 32
} | {
"line": 189,
"column": 32
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ StructureGroupoid.maximalAtlas M G\nhe' : e' ∈ atlas H M\ns : Set M\nhs : IsOpen[inst✝²] s... | [
"H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ StructureGroupoid.maximalAtlas M G\nhe' : e' ∈ atlas H M\ns : Set M\nhs : IsOpen[inst✝²] s\nhs'' : IsO... | ← image_source_inter_eq' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 381,
"column": 6
} | {
"line": 385,
"column": 15
} | {
"line": 385,
"column": 16
} | [
{
"pp": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace M'\ninst✝² : TopologicalSpace M''\nG : StructureGroupoid H\ninst✝¹ : ChartedSpace H M'\ninst✝ : ChartedSpace H M''\ne : ... | [] | intro c c' hc hc'
have : (c'.symm ≫ₕ e.toHomeomorph.toOpenPartialHomeomorph ≫ₕ c).symm ∈ G :=
G.symm (e.mem_groupoid c' c hc' hc)
rwa [trans_symm_eq_symm_trans_symm, trans_symm_eq_symm_trans_symm, symm_symm, trans_assoc]
at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 381,
"column": 6
} | {
"line": 385,
"column": 15
} | {
"line": 385,
"column": 16
} | [
{
"pp": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace M'\ninst✝² : TopologicalSpace M''\nG : StructureGroupoid H\ninst✝¹ : ChartedSpace H M'\ninst✝ : ChartedSpace H M''\ne : ... | [] | intro c c' hc hc'
have : (c'.symm ≫ₕ e.toHomeomorph.toOpenPartialHomeomorph ≫ₕ c).symm ∈ G :=
G.symm (e.mem_groupoid c' c hc' hc)
rwa [trans_symm_eq_symm_trans_symm, trans_symm_eq_symm_trans_symm, symm_symm, trans_assoc]
at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 80
} | {
"line": 87,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n⊢ (f.extend I).source = f.source",... | [] | rw [extend, PartialEquiv.trans_source, I.source_eq, preimage_univ, inter_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 80
} | {
"line": 87,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n⊢ (f.extend I).source = f.source",... | [] | rw [extend, PartialEquiv.trans_source, I.source_eq, preimage_univ, inter_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 80
} | {
"line": 87,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n⊢ (f.extend I).source = f.source",... | [] | rw [extend, PartialEquiv.trans_source, I.source_eq, preimage_univ, inter_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 151,
"column": 49
} | {
"line": 154,
"column": 88
} | {
"line": 156,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\nx : M\nhx : x ∈ f.source\n⊢ ↑I '' ... | [] | by
rw [← f.map_extend_nhds hx, Filter.mem_map,
f.extend_coe, Set.preimage_comp, I.preimage_image f.target]
exact (f.continuousAt hx).preimage_mem_nhds (f.open_target.mem_nhds (f.map_source hx)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 251,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 265,
"column": 0
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne : OpenPartialHomeomorph M H\nP : (H → H') → Set H → H → Prop\ns : Set M\nx : M\nhG : G.Local... | [] | rw [← hG.right_invariance (compatible_of_mem_maximalAtlas_right (x := x) he)]; swap
· simp [xe]
simp only [OpenPartialHomeomorph.trans_apply, mem_chart_source, OpenPartialHomeomorph.left_inv]
apply hG.congr_set_fun
· refine (eventually_of_mem ?_ fun y (hy : y ∈ e.symm ⁻¹' (chartAt H x).source) ↦ ?_).set_eq
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 251,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 265,
"column": 0
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne : OpenPartialHomeomorph M H\nP : (H → H') → Set H → H → Prop\ns : Set M\nx : M\nhG : G.Local... | [] | rw [← hG.right_invariance (compatible_of_mem_maximalAtlas_right (x := x) he)]; swap
· simp [xe]
simp only [OpenPartialHomeomorph.trans_apply, mem_chart_source, OpenPartialHomeomorph.left_inv]
apply hG.congr_set_fun
· refine (eventually_of_mem ?_ fun y (hy : y ∈ e.symm ⁻¹' (chartAt H x).source) ↦ ?_).set_eq
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 52
} | {
"line": 474,
"column": 0
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H' M'\nP : (H → H') → Set H → H → Prop\ng : M → M'\ns : Set M\nx : M\nmono : ∀ ... | [] | rw [← liftPropWithinAt_univ] at h
exact liftPropWithinAt_mono mono h (subset_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 52
} | {
"line": 474,
"column": 0
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H' M'\nP : (H → H') → Set H → H → Prop\ng : M → M'\ns : Set M\nx : M\nmono : ∀ ... | [] | rw [← liftPropWithinAt_univ] at h
exact liftPropWithinAt_mono mono h (subset_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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