module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Moments.Variance | {
"line": 364,
"column": 2
} | {
"line": 379,
"column": 42
} | {
"line": 381,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\n⊢ eVar[X; μ] = ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ - ENNReal.ofReal ((∫ (x : Ω), X x ∂μ) ^ 2)",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Real.instIsOrde... | [] | by_cases hℒ : MemLp X 2 μ
· rw [← ofReal_variance hℒ, variance_eq_sub hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
congr
simp_rw [← enorm_pow, enorm]
rw [lintegral_coe_eq_integral]
· simp
· simpa using hℒ.abs.integrable_sq
· symm
rw [evariance_eq_top hX hℒ, ENNReal.sub_eq_top_iff]
refine ⟨?_,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Variance | {
"line": 445,
"column": 2
} | {
"line": 445,
"column": 47
} | {
"line": 446,
"column": 2
} | [
{
"pp": "case neg.inr\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\nX : ι → Ω → ℝ\ns : Finset ι\nhs : ∀ i ∈ s, MemLp (X i) 2 μ\nh : (↑s).Pairwise fun i j ↦ X i ⟂ᵢ[μ] X j\nh'' : ¬∀ i ∈ s, X i =ᵐ[μ] 0\nj : ι\nhj1 : j ∈ s\nhj2 : ¬X j =ᵐ[μ] 0\nh' : s.Nontrivial\nk : ι\nhk1 : k ∈ s\nhk2 : k ≠ j... | [
"case neg.inr\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\nX : ι → Ω → ℝ\ns : Finset ι\nhs : ∀ i ∈ s, MemLp (X i) 2 μ\nh : (↑s).Pairwise fun i j ↦ X i ⟂ᵢ[μ] X j\nh'' : ¬∀ i ∈ s, X i =ᵐ[μ] 0\nj : ι\nhj1 : j ∈ s\nhj2 : ¬X j =ᵐ[μ] 0\nh' : s.Nontrivial\nk : ι\nhk1 : k ∈ s\nhk2 : k ≠ j\nthis : IsP... | rw [← covariance_self (hs i hi).aemeasurable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.IdentDistrib | {
"line": 189,
"column": 10
} | {
"line": 189,
"column": 57
} | {
"line": 189,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\ninst✝² : NormedAddCommGroup γ\ninst✝¹ : NormedSpace ℝ γ\ninst✝ : BorelSpace γ\nh : IdentDistrib f g μ ν\nhf : AEStronglyMeasu... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\ninst✝² : NormedAddCommGroup γ\ninst✝¹ : NormedSpace ℝ γ\ninst✝ : BorelSpace γ\nh : IdentDistrib f g μ ν\nhf : AEStronglyMeasurable f μ\n⊢... | aestronglyMeasurable_iff_aemeasurable_separable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 65
} | {
"line": 167,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁴ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝³ : PseudoEMetricSpace E\ninst✝² : BorelSpace E\ninst✝¹ : l.IsCountablyGenerated\ninst✝ : l.NeBot\nX : ι → Ω' → E\nh : TendstoInM... | [] | exact tendstoInDistribution_of_ae_tendsto (by fun_prop) hZ hms2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.FactorsThrough | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 58
} | {
"line": 40,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng : X → Z\ninst✝¹ : MeasurableSpace Z\ninst✝ : MeasurableSingletonClass Z\nhg : Measurable g\nx₁ x₂ : X\nh : f x₁ = f x₂\n⊢ g x₁ ∈ {g x₂}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MeasurableS... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng : X → Z\ninst✝¹ : MeasurableSpace Z\ninst✝ : MeasurableSingletonClass Z\nhg : Measurable g\nx₁ x₂ : X\nh : f x₁ = f x₂\ns : Set Y\nhs : f ⁻¹' s = g ⁻¹' {g x₂}\n⊢ g x₁ ∈ {g x₂}"
] | obtain ⟨s, -, hs⟩ := hg (measurableSet_singleton (g x₂)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 112,
"column": 4
} | {
"line": 116,
"column": 41
} | {
"line": 118,
"column": 0
} | [] | [] | eLpNorm (indicator sᶜᶜ (f i + g i)) p μ
= eLpNorm (indicator s (f i) + indicator s (g i)) p μ := by rw [compl_compl, indicator_add']
_ ≤ ε := le_of_lt <|
hη _ _ ((hf_meas i).indicator hsm) ((hg_meas i).indicator hsm)
(η_cast ▸ hfs i) (η_cast ▸ hgs i) | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Group.ModularCharacter | {
"line": 63,
"column": 4
} | {
"line": 65,
"column": 54
} | {
"line": 66,
"column": 2
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : LocallyCompactSpace G\ninst✝² : MeasurableSpace G\ninst✝¹ : BorelSpace G\nμ : Measure G\ninst✝ : μ.IsHaarMeasure\ng : G\nν : Measure G := haar\nf : G → ℝ\nf_cont : Continuous f\nf_comp : ... | [] | · have j : (fun x ↦ f (x * g)) = (f ∘ (Homeomorph.mulRight g)) := rfl
rw [j]
exact HasCompactSupport.comp_homeomorph f_comp _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 70
} | {
"line": 91,
"column": 2
} | [
{
"pp": "R r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun w ↦ (1 / (circleMap z R w.2 - w.1)) ^ 2) (closedBall z r ×ˢ univ)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"instHDiv... | [
"case hg\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun w ↦ circleMap z R w.2 - w.1) (closedBall z r ×ˢ univ)",
"case h₀\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ∀ x ∈ closedBall z r ×ˢ univ, circleMap z R x.2 - x.1 ≠ 0"
] | apply_rules [ContinuousOn.pow, ContinuousOn.div, continuousOn_const] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 79
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)",
"ppTerm": "?hg.hg.hf",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivisionCommM... | [] | simpa only [inv_pow] using continuousOn_prod_circle_transform_function hr | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 79
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)",
"ppTerm": "?hg.hg.hf",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivisionCommM... | [] | simpa only [inv_pow] using continuousOn_prod_circle_transform_function hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 79
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)",
"ppTerm": "?hg.hg.hf",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivisionCommM... | [] | simpa only [inv_pow] using continuousOn_prod_circle_transform_function hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 564,
"column": 12
} | {
"line": 564,
"column": 14
} | {
"line": 564,
"column": 15
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : TopologicalSpace α\nδ : Type u_5\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : Zero δ\nf : γ →C_c δ\ng : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw repr... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : TopologicalSpace α\nδ : Type u_5\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : Zero δ\nf : γ →C_c δ\ng : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\... | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 45
} | {
"line": 217,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n⊢ ∫ᶜ (x : E) in γ.symm, ω x = -∫ᶜ (x : E) in γ, ω x",
"ppTerm": "?m.64",
"... | [] | simp [curveIntegral, curveIntegralFun_symm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 45
} | {
"line": 217,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n⊢ ∫ᶜ (x : E) in γ.symm, ω x = -∫ᶜ (x : E) in γ, ω x",
"ppTerm": "?m.64",
"... | [] | simp [curveIntegral, curveIntegralFun_symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 45
} | {
"line": 217,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n⊢ ∫ᶜ (x : E) in γ.symm, ω x = -∫ᶜ (x : E) in γ, ω x",
"ppTerm": "?m.64",
"... | [] | simp [curveIntegral, curveIntegralFun_symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 776,
"column": 18
} | {
"line": 776,
"column": 30
} | {
"line": 777,
"column": 2
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝ : TopologicalSpace α\nΛ : (α →C_c ℝ) →ₚ[ℝ] ℝ\nf g : α →C_c ℝ≥0\n⊢ NNReal.mk (Λ (toRealLinearMap (f + g))) ⋯ = NNReal.mk (Λ (toRealLinearMap f)) ⋯ + NNReal.mk (Λ (toRealLinearMap g)) ⋯",
"ppTerm": "?m.61",
"assigned": true,
"usedC... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 48
} | {
"line": 91,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b C : ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C\ns : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}\n⊢ ‖f b - f a‖ ≤ C * vol... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b C : ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C\ns : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}\nhsm : MeasurableSet s\n⊢ ‖f b - f a... | have hsm : MeasurableSet s := by measurability | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 55,
"column": 4
} | {
"line": 58,
"column": 63
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case neg.inr\na b : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nthis :\n ∀ {a b : ℝ},\n ContinuousOn f [[a, b]] →\n IntervalIntegrable g μ a b →\n (∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b)... | [] | simp only [not_lt] at hab
obtain ⟨c, c_in_uIcc, that⟩ :=
this (by rwa [uIcc_comm]) hg.symm (by rwa [uIoc_comm]) (by lia) (lt_of_le_of_ne' hab h)
exact ⟨c, by rwa [uIcc_comm], by simpa [integral_symm b a]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 55,
"column": 4
} | {
"line": 58,
"column": 63
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case neg.inr\na b : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nthis :\n ∀ {a b : ℝ},\n ContinuousOn f [[a, b]] →\n IntervalIntegrable g μ a b →\n (∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b)... | [] | simp only [not_lt] at hab
obtain ⟨c, c_in_uIcc, that⟩ :=
this (by rwa [uIcc_comm]) hg.symm (by rwa [uIoc_comm]) (by lia) (lt_of_le_of_ne' hab h)
exact ⟨c, by rwa [uIcc_comm], by simpa [integral_symm b a]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 32
} | {
"line": 185,
"column": 2
} | [
{
"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\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nC : ℝ\nhC₀ : C > 0\nhC : ... | [
"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\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nC : ℝ\nhC₀ : C > 0\nhC : ∀ᶠ (x : E) i... | refine ⟨ε, hε₀, fun y hy ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 49,
"column": 11
} | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nf₁ : X →C_c ℝ≥0\n⊢ ∀ ⦃b : X →C_c ℝ≥0⦄, f₁ ≤ b → Λ f₁ ≤ Λ b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"CompactlySupportedContinuousMap",
"NNReal",
"N... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nf₁ f₂ : X →C_c ℝ≥0\n⊢ f₁ ≤ f₂ → Λ f₁ ≤ Λ f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 68
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u\ns : Set (Set α)\nt✝ : Set α\nht✝ : t✝ ∈ generateMeasurableRec s (ω_ 1)\nt : Set α\nht : t ∈ s\n⊢ ∃ i, ∃ (_ : i < ω_ 1), t ∈ generateMeasurableRec s i",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Ordinal.partialOrder",... | [] | exact ⟨0, omega_pos 1, self_subset_generateMeasurableRec s 0 ht⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 46
} | {
"line": 67,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\n⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] (charFun μ)",
"ppTerm": ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\n⊢ MemLp id (↑0) μ"
] | refine contDiff_zero.1 (contDiff_charFun ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 149,
"column": 4
} | {
"line": 153,
"column": 12
} | {
"line": 155,
"column": 0
} | [
{
"pp": "case h.right.right\nι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⋯\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I... | [] | simp only [EqOn, Pi.one_apply, show (1 : ↥I) = ⊤ from rfl] at hfsU ⊢
conv => equals ∀ x i, x ∈ (V i)ᶜ → ∃ b, fs b x = ⊤ => simp [Finset.sup_eq_top_iff]
intro x i hxi
specialize hfsU i (by tauto_set)
exists i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 149,
"column": 4
} | {
"line": 153,
"column": 12
} | {
"line": 155,
"column": 0
} | [
{
"pp": "case h.right.right\nι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⋯\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I... | [] | simp only [EqOn, Pi.one_apply, show (1 : ↥I) = ⊤ from rfl] at hfsU ⊢
conv => equals ∀ x i, x ∈ (V i)ᶜ → ∃ b, fs b x = ⊤ => simp [Finset.sup_eq_top_iff]
intro x i hxi
specialize hfsU i (by tauto_set)
exists i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 48
} | {
"line": 215,
"column": 2
} | [
{
"pp": "case a\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν : Measure Ω\nhLP : levyProkhorovEDist μ ν = 0\ns : Set Ω\ns_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s\nhμs : ∃ δ > 0, μ (thickening δ s) ≠ ∞\nhνs : ∃ δ > ... | [
"case a\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν : Measure Ω\nhLP : levyProkhorovEDist μ ν = 0\ns : Set Ω\ns_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s\nhμs : ∃ δ > 0, μ (thickening δ s) ≠ ∞\nhνs : ∃ δ > 0, ν (thicke... | · exact measure_le_measure_closure_of_levyProkhorovEDist_eq_zero
hLP s_closed.measurableSet hνs |>.trans <|
le_of_eq (congr_arg _ s_closed.closure_eq) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 40
} | {
"line": 235,
"column": 41
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ ε : ℝ≥0∞\nB : Set Ω\nh : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε\nε_... | [
"Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ ε : ℝ≥0∞\nB : Set Ω\nh : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε\nε_gt : δ < ε\n... | tsub_add_cancel_of_le prob_le_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 147,
"column": 10
} | {
"line": 147,
"column": 97
} | {
"line": 148,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : LocallyCompactSpace G\ninst✝ : CompactSpace G\nφ : G ≃ₜ* G\nμ : Measure G := haarMeasure { carrier := univ, isCompact' := ⋯, interior_nonempty' := ⋯ }\... | [] | conv_rhs => rw [isMulInvariant_eq_smul_of_compactSpace μ (map φ μ), Measure.smul_apply] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 147,
"column": 10
} | {
"line": 147,
"column": 97
} | {
"line": 148,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : LocallyCompactSpace G\ninst✝ : CompactSpace G\nφ : G ≃ₜ* G\nμ : Measure G := haarMeasure { carrier := univ, isCompact' := ⋯, interior_nonempty' := ⋯ }\... | [] | conv_rhs => rw [isMulInvariant_eq_smul_of_compactSpace μ (map φ μ), Measure.smul_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 147,
"column": 10
} | {
"line": 147,
"column": 97
} | {
"line": 148,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : LocallyCompactSpace G\ninst✝ : CompactSpace G\nφ : G ≃ₜ* G\nμ : Measure G := haarMeasure { carrier := univ, isCompact' := ⋯, interior_nonempty' := ⋯ }\... | [] | conv_rhs => rw [isMulInvariant_eq_smul_of_compactSpace μ (map φ μ), Measure.smul_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 24
} | {
"line": 577,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\n⊢ ContinuousAt ofMeasure P",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZer... | [
"Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\n⊢ ∀ ε > 0, ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε"
] | rw [continuousAt_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 627,
"column": 77
} | {
"line": 636,
"column": 80
} | {
"line": 637,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n... | [] | by
refine this.trans <| union_subset_union le_rfl ?_
intro ω hω
simp only [mem_Ici, mem_iUnion, exists_prop] at hω
obtain ⟨i, i_large, ω_in_Esi⟩ := hω
by_contra con
simp only [mem_Iio, compl_iUnion, mem_iInter, mem_compl_iff, not_forall, not_not,
exists_prop] at con... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 18
} | {
"line": 638,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n... | [
"Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n)\nEs_bdd : ... | intro ω ω_in_B | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 57
} | {
"line": 274,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | [
"E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1)... | apply tendsto_of_forall_integral_tendsto (fun g ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 57
} | {
"line": 313,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | [
"E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1)... | apply tendsto_of_forall_integral_tendsto (fun g ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 19
} | {
"line": 259,
"column": 20
} | [
{
"pp": "case e'_3.e'_6\nι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p\neq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\neq_zero : ∀ (x : ι → ℂ), ... | [
"case e'_3.e'_6\nι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p\neq_norm : ∀ (x : ι → ℂ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, ‖x i‖ ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\neq_zero : ∀ (x : ι → ℂ), (∑ i, ‖x i‖ ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 382,
"column": 2
} | {
"line": 385,
"column": 73
} | {
"line": 386,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ns : Set E\nω : E → E →L[𝕜] F\ninst✝ : CompleteSpace F\nhs : Convex ℝ s\... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ns : Set E\nω : E → E →L[𝕜] F\ninst✝ : CompleteSpace F\nhs : Convex ℝ s\nhso : IsOpe... | obtain ⟨f, hf⟩ : ∃ f, ∀ a ∈ s, HasFDerivWithinAt f (ω a) s a := by
refine hs.exists_forall_hasFDerivWithinAt_of_fderivWithin_symmetric hω fun a ha x _ y _ ↦ ?_
rw [fderivWithin_eq_fderiv, hdω a ha]
exacts [hso.uniqueDiffOn a ha, hω.differentiableAt (hso.mem_nhds ha)] | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 37
} | {
"line": 202,
"column": 4
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nh : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)\ny : E\nthis : ProperSpace E\nhy : ¬y = 0\nh' : Tends... | [
"E : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nh : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)\ny : E\nthis : ProperSpace E\nhy : ¬y = 0\nh' : Tendsto (fun r ↦ ... | refine measure_mono fun x hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 33
} | {
"line": 379,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | [
"case refine_1\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finse... | refine ⟨μ, ⟨?_, fun n ↦ ?_⟩, L⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 169,
"column": 4
} | {
"line": 179,
"column": 76
} | {
"line": 180,
"column": 4
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁹ : (i : ι) → TopologicalSpace (X i)\ninst✝⁸ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁶ : (j : κ) → TopologicalSpac... | [
"ι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁹ : (i : ι) → TopologicalSpace (X i)\ninst✝⁸ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁶ : (j : κ) → TopologicalSpace (Y j)\nins... | have {μ : Measure ((Π i, X i) × Π j, Y j)} :
(∫⁻ p, (∏ i, (f i (p.1 i) : ℝ≥0∞)) * ∏ j, (g j (p.2 j) : ℝ≥0∞) ∂μ).toReal =
∫ p, (∏ i, (f i (p.1 i)).toReal) * ∏ j, (g j (p.2 j)).toReal ∂μ := by
rw [integral_eq_lintegral_of_nonneg_ae]
· simp [Finset.prod_nonneg, ofReal_prod_of_nonneg]
· ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 17
} | {
"line": 177,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\nn : ℕ\nhn : ¬n = 0\nε' : ℝ≥0\nhε' : 0 < ε'\nhsnetop : preVariationFun f (⋃ i, s i) < ∞\n⊢ ∑ i ∈ Finset.range n, preVariationFun f (s i) ≤ preVariation... | [
"case neg\nX : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\nn : ℕ\nhn : ¬n = 0\nε' : ℝ≥0\nhε' : 0 < ε'\nhsnetop : preVariationFun f (⋃ i, s i) < ∞\nε : ℝ≥0 := ε' / ↑n\n⊢ ∑ i ∈ Finset.range n, preVariationFun f (s i) ≤ preV... | let ε := ε' / n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 474,
"column": 2
} | {
"line": 478,
"column": 47
} | {
"line": 480,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\n⊢ IsCompact {μ | μ.mass = C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}",
... | [] | have : {μ : FiniteMeasure E | μ.mass = C ∧ ∀ n, μ (K n)ᶜ ≤ u n} =
{μ | μ.mass ≤ C ∧ ∀ n, μ (K n)ᶜ ≤ u n} ∩ {μ | μ.mass = C} := by ext; grind
rw [this]
apply IsCompact.inter_right (isCompact_setOf_finiteMeasure_mass_le_compl_isCompact_le C hu hK h)
exact isClosed_eq (by fun_prop) (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 474,
"column": 2
} | {
"line": 478,
"column": 47
} | {
"line": 480,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\n⊢ IsCompact {μ | μ.mass = C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n}",
... | [] | have : {μ : FiniteMeasure E | μ.mass = C ∧ ∀ n, μ (K n)ᶜ ≤ u n} =
{μ | μ.mass ≤ C ∧ ∀ n, μ (K n)ᶜ ≤ u n} ∩ {μ | μ.mass = C} := by ext; grind
rw [this]
apply IsCompact.inter_right (isCompact_setOf_finiteMeasure_mass_le_compl_isCompact_le C hu hK h)
exact isClosed_eq (by fun_prop) (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 136,
"column": 8
} | {
"line": 136,
"column": 33
} | {
"line": 137,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ ... | [] | simpa [hμAn] using! hsome | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 136,
"column": 8
} | {
"line": 136,
"column": 33
} | {
"line": 137,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ ... | [] | simpa [hμAn] using! hsome | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 136,
"column": 8
} | {
"line": 136,
"column": 33
} | {
"line": 137,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ ... | [] | simpa [hμAn] using! hsome | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 27
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis✝ : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ... | [
"case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis✝ : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ → Set α := ... | rw [← hx₀, hBn] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 467,
"column": 6
} | {
"line": 467,
"column": 73
} | {
"line": 469,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : IsSeparable μ\ninst✝ : SeparableSpace E\n𝒜 : Set (Set X)\ncount_𝒜 : 𝒜.Countable\nh𝒜 : μ.Me... | [
"case refine_2.refine_1\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : IsSeparable μ\ninst✝ : SeparableSpace E\n𝒜 : Set (Set X)\ncount_𝒜 : 𝒜.Countable\nh𝒜 : μ.MeasureDense �... | have μs_pow_nonneg : 0 ≤ μ.real s ^ (1 / p.toReal) := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 61
} | {
"line": 124,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\na b... | have A : hf.exists_vectorMeasure_le_measureAux.choose (Ioc a b) =
f.rightLim b - f.rightLim a :=
hf.exists_vectorMeasure_le_measureAux.choose_spec.1 a b h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 24
} | {
"line": 170,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [] | exact self_mem_Iic | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 230,
"column": 8
} | {
"line": 230,
"column": 38
} | {
"line": 231,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nm : AddContent E C\nhC : IsSetRing C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε :... | [] | simpa [hs, m'] using! hBound _ | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 230,
"column": 8
} | {
"line": 230,
"column": 38
} | {
"line": 231,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nm : AddContent E C\nhC : IsSetRing C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε :... | [] | simpa [hs, m'] using! hBound _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 230,
"column": 8
} | {
"line": 230,
"column": 38
} | {
"line": 231,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nm : AddContent E C\nhC : IsSetRing C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε :... | [] | simpa [hs, m'] using! hBound _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 21
} | {
"line": 213,
"column": 0
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nh : ∀ t ⊆ s, MeasurableSet t → μ t = 0\nt : Set X\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ ‖μ t‖ₑ ≤ 0 t",
"ppTerm... | [] | simp [h t hts ht] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 229,
"column": 2
} | {
"line": 240,
"column": 33
} | {
"line": 242,
"column": 0
} | [
{
"pp": "case a\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\n⊢ μ.variation.restrict s ≤ (μ.restrict s).variation",
"ppTerm": "?a✝",
"assigned": true,
"used... | [] | · apply Measure.le_iff.2 (fun t ht ↦ ?_)
simp only [ht, Measure.restrict_apply]
calc μ.variation (t ∩ s)
_ ≤ (μ.restrict s).variation (t ∩ s) := by
apply variation_apply_le_of_forall_enorm_le (ht.inter hs) (fun u u_meas hu ↦ ?_)
have : μ u = μ.restrict s u :=
(VectorMeasure.restrict_eq_s... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 15
} | {
"line": 117,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\nt : Set X\nts : t ⊆ s\nt_meas : MeasurableSet t\nht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ\n⊢ 2 * ‖μ s‖ₑ + 2 ≤ 2 * ‖μ t... | [] | exact ht.le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 10
} | {
"line": 343,
"column": 2
} | [
{
"pp": "case h\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\nn : ℕ\nih : ∀ m < n, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\na : M\nt : Finset M\nhn : t.card = n\n⊢ IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))",
"ppTerm": "?h",
"assigned": true,
... | [
"case h\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\n⊢ IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 14
} | {
"line": 358,
"column": 6
} | [
{
"pp": "case mp.h\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nt' : Finset M\nht' : t' ⊆ t\nf : M → ℕ\ni : M\nhi : i ∈ t'\nhfi : 0 < f i\nheq : ∑ x ∈ t', f x • x = ∑ x ∈... | [
"case mp.h\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nt' : Finset M\nht' : t' ⊆ t\nf : M → ℕ\ni : M\nhi : i ∈ t'\nhfi : 0 < f i\nheq : ∑ x ∈ t', f x • x = ∑ x ∈ t \\ t', f ... | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 362,
"column": 8
} | {
"line": 362,
"column": 59
} | {
"line": 363,
"column": 8
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nt' : Finset M\nht' : t' ⊆ t\nf : M → ℕ\ni : M\nhi : i ∈ t'\nhfi : 0 < f i\nheq : ∑ x ∈ t', f x • x = ∑ x ∈ t \\ t', f... | [
"M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nt' : Finset M\nht' : t' ⊆ t\nf : M → ℕ\ni : M\nhi : i ∈ t'\nhfi : 0 < f i\nheq : ∑ x ∈ t', f x • x = ∑ x ∈ t \\ t', f x • x\ng : ... | conv_lhs => rw [← Finset.union_sdiff_of_subset ht'] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 46
} | {
"line": 201,
"column": 2
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\n⊢ IsLinearSet s ↔ ∃ v n A, s = {x | ∃ x_1, v + A *ᵥ x_1 = x}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.addCommMonoid",
"instVAddOfAdd",
"congrArg",
"Matrix",
"AddMonoid.toAddZeroClass",
"s... | [
"ι : Type u_3\ns : Set (ι → ℕ)\n⊢ (∃ a n f, s = a +ᵥ range ⇑f) ↔ ∃ v n A, s = {x | ∃ x_1, v + A *ᵥ x_1 = x}"
] | rw [isLinearSet_iff_exists_fin_addMonoidHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 424,
"column": 6
} | {
"line": 424,
"column": 16
} | {
"line": 424,
"column": 16
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\n⊢ ⊤ ≤ span ℚ (⇑toRatVec '' range ⇑(Pi.basisFun ℕ ι))",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"Semiring... | [
"ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\n⊢ span ℚ (⇑toRatVec '' range ⇑(Pi.basisFun ℕ ι)) = ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 221,
"column": 6
} | {
"line": 226,
"column": 66
} | {
"line": 227,
"column": 6
} | [
{
"pp": "case a.inr\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E ... | [
"case a.inr\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | have : IsFiniteMeasure ((μ.restrict s).variation.restrict p) := by
constructor
rw [variation_restrict hs, Measure.restrict_restrict pmeas,
MeasureTheory.Measure.restrict_apply_univ]
apply lt_of_le_of_lt ?_ (g.integrable_iff.1 (memLp_one_iff_integrable.1 gmem) i h'i)
exact measu... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 561,
"column": 22
} | {
"line": 567,
"column": 6
} | {
"line": 569,
"column": 0
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : x ∈ hs.fundamentalDomain\n⊢ hs.fract x = x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Iff.mpr",
"Finsupp.instFunLike",
"Rat.instSub",
... | [] | by
rw [← toRatVec_inj, hs.toRatVec_fract_eq]
conv_lhs =>
enter [2, 2, i]
rw [Int.fract_eq_self.2 (hx i), ← hs.basis_apply]
rw [hs.basis.sum_repr]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 747,
"column": 4
} | {
"line": 751,
"column": 43
} | {
"line": 753,
"column": 0
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\ni : ↑hs.basisSet\nhi : i ∈ {i | ↑i ∉ hs.periods}\nthis : Fintype ι\n⊢ IsSemilinearSet\n {a |\n ((a ∘ Sum.inl) ∘ Sum.inl) ∘ Sum.inl + a ∘ Sum.inr =\n hs.base + ↑i + ((a ∘ Sum.inl) ∘ Sum.inl) ∘ Sum.inr + (a ∘ Sum.in... | [] | convert!
Nat.isSemilinearSet_setOf_mulVec_eq (κ := ((ι ⊕ ι) ⊕ ι) ⊕ ι) 0 (hs.base + i.1)
(Matrix.fromCols (Matrix.fromCols (Matrix.fromCols 1 0) 0) 1)
(Matrix.fromCols (Matrix.fromCols (Matrix.fromCols 0 1) 1) 0) using 4
<;> simp [add_assoc, fromCols_mulVec] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.ModelTheory.Complexity | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 63
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case ex\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nn : ℕ\ninst✝ : Nonempty M\nψ : L.BoundedFormula α n\nv : α → M\nn✝ : ℕ\nφ✝ : L.BoundedFormula α (n✝ + 1)\nh✝ : φ✝.IsPrenex\nih :\n ∀ {φ : L.BoundedFormula α (n✝ + 1)},\n φ.IsQF → ∀ {xs : Fin (n✝ + 1) → M}, (φ.toPrenexImpRight φ... | [
"case ex\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nn : ℕ\ninst✝ : Nonempty M\nψ : L.BoundedFormula α n\nv : α → M\nn✝ : ℕ\nφ✝ : L.BoundedFormula α (n✝ + 1)\nh✝ : φ✝.IsPrenex\nih :\n ∀ {φ : L.BoundedFormula α (n✝ + 1)},\n φ.IsQF → ∀ {xs : Fin (n✝ + 1) → M}, (φ.toPrenexImpRight φ✝).Realize v... | refine _root_.trans (exists_congr fun _ => ih hφ.liftAt) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.FinitelyGenerated | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 35
} | {
"line": 122,
"column": 2
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝ : L.Structure M\nN : L.Substructure M\nh : N.FG\n⊢ N.CG",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"PartialOrder.toPreorder",
"FirstOrder.Language.Substructure.fg_def",
... | [
"L : Language\nM : Type u_1\ninst✝ : L.Structure M\ns : Set M\nhf : s.Finite\nh : ((closure L).toFun s).FG\n⊢ ((closure L).toFun s).CG"
] | obtain ⟨s, hf, rfl⟩ := fg_def.1 h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.ModelTheory.DirectLimit | {
"line": 323,
"column": 49
} | {
"line": 323,
"column": 58
} | {
"line": 323,
"column": 58
} | [
{
"pp": "case h\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nS : L.Substructure (DirectLimit G f)\nS_fg : S.FG\nA : Fin... | [
"case h\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nS : L.Substructure (DirectLimit G f)\nS_fg : S.FG\nA : Finset (DirectL... | A_closure | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 322,
"column": 8
} | {
"line": 323,
"column": 37
} | {
"line": 324,
"column": 8
} | [
{
"pp": "case hP₂.inl\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : ... | [
"case hP₂.inr\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L... | · rw [← hij] at hj
exact Pdisj (g i) hi hj hpq | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 491,
"column": 2
} | {
"line": 496,
"column": 42
} | {
"line": 498,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → E\nι : Type u_8\nμ : ι → VectorMeasure X F\ns : Finset ι\nh : ∀ i ∈ s, (μ i).Integrable f\n⊢ (∑ i ∈ s, μ i).Integrable f",
"ppTerm": "?m.133",
"assigned": true,... | [] | induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
simp only [Finset.mem_insert, forall_eq_or_imp, ha, not_false_eq_true,
Finset.sum_insert] at h ⊢
exact h.1.add_vectorMeasure (ih h.2) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 491,
"column": 2
} | {
"line": 496,
"column": 42
} | {
"line": 498,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → E\nι : Type u_8\nμ : ι → VectorMeasure X F\ns : Finset ι\nh : ∀ i ∈ s, (μ i).Integrable f\n⊢ (∑ i ∈ s, μ i).Integrable f",
"ppTerm": "?m.133",
"assigned": true,... | [] | induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
simp only [Finset.mem_insert, forall_eq_or_imp, ha, not_false_eq_true,
Finset.sum_insert] at h ⊢
exact h.1.add_vectorMeasure (ih h.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 491,
"column": 2
} | {
"line": 496,
"column": 42
} | {
"line": 498,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → E\nι : Type u_8\nμ : ι → VectorMeasure X F\ns : Finset ι\nh : ∀ i ∈ s, (μ i).Integrable f\n⊢ (∑ i ∈ s, μ i).Integrable f",
"ppTerm": "?m.133",
"assigned": true,... | [] | induction s using Finset.induction_on with
| empty => simp
| insert a s ha ih =>
simp only [Finset.mem_insert, forall_eq_or_imp, ha, not_false_eq_true,
Finset.sum_insert] at h ⊢
exact h.1.add_vectorMeasure (ih h.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 48
} | {
"line": 523,
"column": 4
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ] G\nf : X →... | [
"X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ] G\nf : X → E\nc : ℝ\nh... | simp only [transpose_smul, FunLike.coe_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 642,
"column": 4
} | {
"line": 642,
"column": 28
} | {
"line": 644,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\nB : E →L[... | [] | simp [integral_undef hf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 642,
"column": 4
} | {
"line": 642,
"column": 28
} | {
"line": 644,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\nB : E →L[... | [] | simp [integral_undef hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 642,
"column": 4
} | {
"line": 642,
"column": 28
} | {
"line": 644,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\nB : E →L[... | [] | simp [integral_undef hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 68
} | {
"line": 351,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | [
"X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\ninst✝ :... | refine le_antisymm (ContinuousLinearMap.le_opNorm (B.flip y) x) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Fraisse | {
"line": 392,
"column": 4
} | {
"line": 393,
"column": 29
} | {
"line": 394,
"column": 2
} | [
{
"pp": "L : Language\nK : Set (Bundled L.Structure)\nM : Type w\ninst✝⁴ : L.Structure M\nN : Type w\ninst✝³ : L.Structure N\ninst✝² : Countable ((l : ℕ) × L.Functions l)\ninst✝¹ : Countable M\ninst✝ : Countable N\nhM : IsFraisseLimit K M\nhN : IsFraisseLimit K N\nS : L.Substructure M := ⊥\nS_fg : Structure.FG ... | [] | rw [hN.age, ← hM.age]
exact ⟨S_fg, ⟨subtype _⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Fraisse | {
"line": 392,
"column": 4
} | {
"line": 393,
"column": 29
} | {
"line": 394,
"column": 2
} | [
{
"pp": "L : Language\nK : Set (Bundled L.Structure)\nM : Type w\ninst✝⁴ : L.Structure M\nN : Type w\ninst✝³ : L.Structure N\ninst✝² : Countable ((l : ℕ) × L.Functions l)\ninst✝¹ : Countable M\ninst✝ : Countable N\nhM : IsFraisseLimit K M\nhN : IsFraisseLimit K N\nS : L.Substructure M := ⊥\nS_fg : Structure.FG ... | [] | rw [hN.age, ← hM.age]
exact ⟨S_fg, ⟨subtype _⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Baire.LocallyCompactRegular | {
"line": 25,
"column": 10
} | {
"line": 25,
"column": 12
} | {
"line": 25,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : R1Space X\ninst✝ : LocallyCompactSpace X\nf : ℕ → Set X\n⊢ (∀ (n : ℕ), IsOpen (f n)) → (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Nat",
"IsOpen"
],
"... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : R1Space X\ninst✝ : LocallyCompactSpace X\nf : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen (f n)\n⊢ (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n)"
] | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.ADEInequality | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 39
} | {
"line": 189,
"column": 2
} | [
{
"pp": "p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p < 3\n⊢ Admissible {p, q, r}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"instLinearOrderPNat",
"Finset",
"PartialOrder.toPreorder",
"Finset.I... | [
"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 1 < sumInv {p, q, r}\nhp3 : p ∈ Finset.Iio 3\n⊢ Admissible {p, q, r}"
] | replace hp3 := Finset.mem_Iio.mpr hp3 | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 78
} | {
"line": 113,
"column": 4
} | [
{
"pp": "case refine_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : a.Coprime b\nha : ∑ a ∈ a.divisors with IsPrimePow a, Real.log ↑a.minFac = Real.log ↑a\nhb : ∑ a ∈ b.divisors with IsPrimePow a, Real.log ↑a.minFac = Real.log ↑b\n⊢ Real.log ↑a + Real.log ↑b = Real.log ↑(a * b)",
"ppTerm": "?refine_3",
"... | [
"case refine_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : a.Coprime b\nha : ∑ a ∈ a.divisors with IsPrimePow a, Real.log ↑a.minFac = Real.log ↑a\nhb : ∑ a ∈ b.divisors with IsPrimePow a, Real.log ↑a.minFac = Real.log ↑b\n⊢ Real.log ↑a + Real.log ↑b = Real.log (↑a * ↑b)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Exp | {
"line": 74,
"column": 17
} | {
"line": 74,
"column": 29
} | {
"line": 74,
"column": 30
} | [
{
"pp": "A : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑n + 1 = (algebraMap ℚ A) ↑n + (algebraMap ℚ A) 1",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClas... | [
"A : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ↑n + 1 = ↑n + (algebraMap ℚ A) 1"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Exp | {
"line": 75,
"column": 38
} | {
"line": 75,
"column": 51
} | {
"line": 75,
"column": 52
} | [
{
"pp": "A : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nkey : ↑n + 1 = (algebraMap ℚ A) (↑n + 1)\n⊢ (algebraMap ℚ A) (1 / ↑((n + 1) * n !) * (↑n + 1)) = (algebraMap ℚ A) (1 / ↑n !)",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
... | [
"A : Type u_3\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nkey : ↑n + 1 = (algebraMap ℚ A) (↑n + 1)\n⊢ (algebraMap ℚ A) (1 / (↑(n + 1) * ↑n !) * (↑n + 1)) = (algebraMap ℚ A) (1 / ↑n !)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 174,
"column": 20
} | {
"line": 174,
"column": 33
} | {
"line": 174,
"column": 34
} | [
{
"pp": "case refine_2.hnot\np : ℕ\nhp : Nat.Prime p\nm : ℕ\nhm0 : m ≠ 0\nhpm : m + 2 ≤ p * m\na : ℤ\nha : ¬↑p ∣ a\nn✝ n : ℕ\nthis✝ : Fact (Nat.Prime p)\nthis : ∀ (m_1 : ℕ), ∃ y, (1 + ↑p ^ m * ↑a) ^ p ^ m_1 = 1 + ↑p ^ m_1 * ↑p ^ m * (↑a + ↑p * y)\ny : ZMod (p ^ (n + 1 + m))\nhy : (1 + ↑p ^ m * ↑a) ^ p ^ n = 1 +... | [
"case refine_2.hnot\np : ℕ\nhp : Nat.Prime p\nm : ℕ\nhm0 : m ≠ 0\nhpm : m + 2 ≤ p * m\na : ℤ\nha : ¬↑p ∣ a\nn✝ n : ℕ\nthis✝ : Fact (Nat.Prime p)\nthis : ∀ (m_1 : ℕ), ∃ y, (1 + ↑p ^ m * ↑a) ^ p ^ m_1 = 1 + ↑p ^ m_1 * ↑p ^ m * (↑a + ↑p * y)\ny : ZMod (p ^ (n + 1 + m))\nhy : (1 + ↑p ^ m * ↑a) ^ p ^ n = 1 + ↑p ^ n * ↑p... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 16
} | {
"line": 184,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\np : A[X]\ng : A⟦X⟧\nhg : HasSubst g\n⊢ (d⁄dX A) (subst g ↑p) = subst g ((d⁄dX A) ↑p) * (d⁄dX A) g",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Derivation",
"Polynomial.derivative",
"CommRing",
"instHSMul",
"Semiri... | [] | simp [subst_coe hg, derivative_coe, Derivation.comp_aeval_eq (a := g) (derivative A) p,
smul_eq_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 16
} | {
"line": 184,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\np : A[X]\ng : A⟦X⟧\nhg : HasSubst g\n⊢ (d⁄dX A) (subst g ↑p) = subst g ((d⁄dX A) ↑p) * (d⁄dX A) g",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Derivation",
"Polynomial.derivative",
"CommRing",
"instHSMul",
"Semiri... | [] | simp [subst_coe hg, derivative_coe, Derivation.comp_aeval_eq (a := g) (derivative A) p,
smul_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 16
} | {
"line": 184,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\np : A[X]\ng : A⟦X⟧\nhg : HasSubst g\n⊢ (d⁄dX A) (subst g ↑p) = subst g ((d⁄dX A) ↑p) * (d⁄dX A) g",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Derivation",
"Polynomial.derivative",
"CommRing",
"instHSMul",
"Semiri... | [] | simp [subst_coe hg, derivative_coe, Derivation.comp_aeval_eq (a := g) (derivative A) p,
smul_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 198,
"column": 29
} | {
"line": 198,
"column": 98
} | {
"line": 200,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\nf g : A⟦X⟧\nhg : HasSubst g\nn m : ℕ\nhm : ∀ (b : ℕ), m ≤ b → ∀ n' ≤ n + 1, (coeff n') (g ^ b) = 0\nthis : (coeff (n + 1)) (subst g f) = (coeff (n + 1)) (subst g ↑((trunc (m + 1)) f))\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : (i, j) ∈ Finset.antidiagonal n\nd : ℕ\n⊢ ((if d + 1 < m +... | [] | by split_ifs <;> simp (disch := grind [Finset.mem_antidiagonal]) [hm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 139,
"column": 59
} | {
"line": 139,
"column": 72
} | {
"line": 139,
"column": 73
} | [
{
"pp": "case a\nn x✝¹ : ℕ\na✝¹ : x✝¹ ∈ range (n + 1 - 0)\nx✝ : ℕ\na✝ : x✝ ∈ range (n + 1 - x✝¹)\n| (monomial x✝¹) (↑((n + 1).choose x✝¹ * (n + 1 - x✝¹).choose (x✝¹ + x✝ - x✝¹)) * _root_.bernoulli x✝)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWith... | [
"case a\nn x✝¹ : ℕ\na✝¹ : x✝¹ ∈ range (n + 1 - 0)\nx✝ : ℕ\na✝ : x✝ ∈ range (n + 1 - x✝¹)\n| (monomial x✝¹) (↑((n + 1).choose x✝¹) * ↑((n + 1 - x✝¹).choose (x✝¹ + x✝ - x✝¹)) * _root_.bernoulli x✝)"
] | Nat.cast_mul, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.Primorial | {
"line": 129,
"column": 6
} | {
"line": 134,
"column": 62
} | {
"line": 135,
"column": 2
} | [] | [] | (m + m + 1)# = (m + 1 + m)# := by rw [add_right_comm]
_ ≤ (m + 1)# * choose (m + 1 + m) (m + 1) := primorial_add_le m.le_succ
_ = (m + 1)# * choose (2 * m + 1) m := by rw [choose_symm_add, two_mul, add_right_comm]
_ < 4 ^ (m + 1) * 4 ^ m :=
Nat.mul_lt_mul_of_lt_of_le (ihn _ (by lia) (by lia)) ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.NumberTheory.AbelSummation | {
"line": 293,
"column": 4
} | {
"line": 295,
"column": 21
} | {
"line": 296,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | exact Iff.mp integrableOn_Ici_iff_integrableOn_Ioi
<| (locallyIntegrableOn_mul_sum_Icc c le_rfl hf_int).integrableOn_of_isBigO_atTop
hg_dom hg_int | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Bertrand | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 74
} | {
"line": 153,
"column": 2
} | [
{
"pp": "case hf.inl\nn : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\nh : ¬Nat.Prime x\n⊢ x ^ n.centralBinom.factorization x = 1",
"ppTerm": "?hf.inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Finsupp.i... | [] | rw [factorization_eq_zero_of_not_prime n.centralBinom h, Nat.pow_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Bertrand | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 74
} | {
"line": 153,
"column": 2
} | [
{
"pp": "case hf.inl\nn : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\nh : ¬Nat.Prime x\n⊢ x ^ n.centralBinom.factorization x = 1",
"ppTerm": "?hf.inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Finsupp.i... | [] | rw [factorization_eq_zero_of_not_prime n.centralBinom h, Nat.pow_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Bertrand | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 74
} | {
"line": 153,
"column": 2
} | [
{
"pp": "case hf.inl\nn : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\nh : ¬Nat.Prime x\n⊢ x ^ n.centralBinom.factorization x = 1",
"ppTerm": "?hf.inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Finsupp.i... | [] | rw [factorization_eq_zero_of_not_prime n.centralBinom h, Nat.pow_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 60
} | {
"line": 65,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nι : Type u_2\ninst✝ : Finite ι\nε : ℝ\nhε : 0 < ε\nb : R\nhb : b ≠ 0\nA : ι → R\nh : abv.IsAdmissible\nn : ℕ\ne : ι ≃ Fin n\n⊢ ∃ t, ∀ (i₀ i₁ : ι), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε",
"ppTerm": "?m.57",
"assi... | [
"R : Type u_1\ninst✝¹ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nι : Type u_2\ninst✝ : Finite ι\nε : ℝ\nhε : 0 < ε\nb : R\nhb : b ≠ 0\nA : ι → R\nh : abv.IsAdmissible\nn : ℕ\ne : ι ≃ Fin n\nt : Fin n → Fin (h.card ε)\nht : ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑(abv ((A ∘ ⇑e.symm) i₁ % b - (A ∘ ⇑e.symm) i₀ % b)) < ab... | obtain ⟨t, ht⟩ := h.exists_partition' n hε hb (A ∘ e.symm) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Bernoulli | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 33
} | {
"line": 467,
"column": 2
} | [
{
"pp": "k p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\n⊢ ∑ q ∈ vonStaudtPrimes k, 1 / ↑q = vonStaudtIndicator (2 * k) p / ↑p + ∑ q ∈ (vonStaudtPrimes k).erase p, 1 / ↑q",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"_private.Math... | [
"case pos\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhdvd : p - 1 ∣ 2 * k\n⊢ ∑ q ∈ vonStaudtPrimes k, 1 / ↑q = vonStaudtIndicator (2 * k) p / ↑p + ∑ q ∈ (vonStaudtPrimes k).erase p, 1 / ↑q",
"case neg\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhdvd : ¬p - 1 ∣ 2 * k\n⊢ ∑ q ∈ vonStaudtPrimes k, 1 / ↑q... | by_cases hdvd : (p - 1) ∣ 2 * k | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 53
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nι : Type u_2\ninst✝ : Fintype ι\nε : ℝ\nhε : 0 < ε\nb : R\nhb : b ≠ 0\nh✝ : abv.IsAdmissible\nA : Fin (h✝.card ε ^ Fintype.card ι).succ → ι → R\ne : ι ≃ Fin (Fintype.card ι) := Fintype.equivFin ι\ni₀ i₁ : Fin (h✝.card ε ^ Fintype.card ι... | [] | convert! h (e k) <;> simp only [e.symm_apply_apply] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.Chebyshev | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 80
} | {
"line": 318,
"column": 2
} | [
{
"pp": "n k : ℕ\nhkn : k ≤ n\n⊢ n.choose k ∣ n.lcmUpto",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.lcmUpto_ne_zero",
"Nat.instMulZeroClass",
"Nat.Prime",
"Dvd.dvd",
"Nat.choose",
"congrArg",
"Na... | [
"n k : ℕ\nhkn : k ≤ n\n⊢ ∀ (p : ℕ), Nat.Prime p → (n.choose k).factorization p ≤ n.lcmUpto.factorization p"
] | rw [← factorization_prime_le_iff_dvd (choose_ne_zero hkn) (lcmUpto_ne_zero n)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Adjoin.Polynomial.Bivariate | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 52
} | {
"line": 47,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\nhx : Transcendental R x\np : R[X][Y]\n⊢ (algEquivAdjoin hx) (swap p) = (aeval (C ⟨x, ⋯⟩)) p",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Polynomi... | [] | simp [algEquivAdjoin, Bivariate.aveal_eq_map_swap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Adjoin.Polynomial.Bivariate | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 52
} | {
"line": 47,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\nhx : Transcendental R x\np : R[X][Y]\n⊢ (algEquivAdjoin hx) (swap p) = (aeval (C ⟨x, ⋯⟩)) p",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Polynomi... | [] | simp [algEquivAdjoin, Bivariate.aveal_eq_map_swap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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