module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 160,
"column": 71
} | {
"line": 160,
"column": 86
} | {
"line": 160,
"column": 86
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns t : Set X\nht : MeasurableSet t\nhdisj : Disjoint s t\ni : ι\nhi : MeasurableSet (F i)\n⊢ (μ (s ∩ F i) + μ (t ∩... | [
"G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns t : Set X\nht : MeasurableSet t\nhdisj : Disjoint s t\ni : ι\nhi : MeasurableSet (F i)\n⊢ μ (s ∩ F i) / μ (F i) + μ (t ∩ F ... | ENNReal.add_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 87
} | {
"line": 162,
"column": 0
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns t : Set X\nht : MeasurableSet t\nhdisj : Disjoint s t\ni : ι\nhi : MeasurableSet (F i)\n⊢ μ ((s ∪ t) ∩ F i) / μ... | [] | rw [union_inter_distrib_right,
measure_union (hdisj.inter_left _ |>.inter_right _) (ht.inter hi), ENNReal.add_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 21
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\nf : ℕ → α → β\nhp' : p ≠ ∞\nhf : ∀ (n : ℕ), MemLp (f n) p μ\nhf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε\nε : ℝ≥0\nhε : 0 < ε\nN : ℕ\nhNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε\n... | [] | exact hFε ⟨n, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 21
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\nf : ℕ → α → β\nhp' : p ≠ ∞\nhf : ∀ (n : ℕ), MemLp (f n) p μ\nhf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε\nε : ℝ≥0\nhε : 0 < ε\nN : ℕ\nhNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε\n... | [] | exact hFε ⟨n, hn⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 21
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\nf : ℕ → α → β\nhp' : p ≠ ∞\nhf : ∀ (n : ℕ), MemLp (f n) p μ\nhf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε\nε : ℝ≥0\nhε : 0 < ε\nN : ℕ\nhNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε\n... | [] | exact hFε ⟨n, hn⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 56,
"column": 6
} | {
"line": 56,
"column": 13
} | {
"line": 56,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nz w : ℂ\nf : ℂ → E\nx✝ : ℝ\n⊢ ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (-(circleMap z R x✝ * w * 2) + circleMap z R x✝ ^ 2 + w ^ 2)⁻¹ * (1 / 2)) •\n f (circleMap z R x✝) =\n ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ *... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nz w : ℂ\nf : ℂ → E\nx✝ : ℝ\n⊢ ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (-(circleMap z R x✝ * w * 2) + circleMap z R x✝ ^ 2 + w ^ 2)⁻¹ * (1 / 2)) •\n f (circleMap z R x✝) =\n ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * ((circleMap... | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 214,
"column": 45
} | {
"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",
"... | [] | by
simp [curveIntegral, curveIntegralFun_symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 20
} | {
"line": 141,
"column": 6
} | [
{
"pp": "case e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t :... | [
"case e'_4.e_a.e_s\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ),... | congr 2 with t | Batteries.Tactic._aux_Batteries_Tactic_Congr___macroRules_Batteries_Tactic_congrConfigWith_1 | Batteries.Tactic.congrConfigWith |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 66
} | {
"line": 82,
"column": 2
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\n... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a_1 ... | have hclosure : closure U = Icc 0 1 := by
simp [hU, closure_prod_eq, Prod.mk_zero_zero, Prod.mk_one_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 35,
"column": 14
} | {
"line": 35,
"column": 16
} | {
"line": 35,
"column": 17
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 32,
"column": 4
} | {
"line": 37,
"column": 18
} | {
"line": 38,
"column": 4
} | [
{
"pp": "case mp.ht\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : g =ᵐ[μ... | [
"case mp.htc\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : g =ᵐ[μ.withDensit... | · rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg'
rw [ae_restrict_iff' A]
filter_upwards [hg']
intro a ha h'a
have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne, ENNReal.coe_eq_zero] using h'a
rw [ha this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Regular | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 77
} | {
"line": 36,
"column": 4
} | [
{
"pp": "case a\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nk : Set X\nhk : IsCompact k\nμ : Measure X\ninst✝³ : IsFiniteMeasureOnCompacts μ\ninst✝² : μ.InnerRegularCompactLTTop\ninst✝¹ : LocallyCompactSpace X\ninst✝ : RegularSpace X\nf : X → ℝ\nf_cont : Contin... | [
"case a.f_nonneg\nX : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nk : Set X\nhk : IsCompact k\nμ : Measure X\ninst✝³ : IsFiniteMeasureOnCompacts μ\ninst✝² : μ.InnerRegularCompactLTTop\ninst✝¹ : LocallyCompactSpace X\ninst✝ : RegularSpace X\nf : X → ℝ\nf_cont : Continuou... | apply (f_cont.integrable_of_hasCompactSupport f_comp).measure_le_integral | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 112,
"column": 12
} | {
"line": 112,
"column": 14
} | {
"line": 113,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 22
} | {
"line": 184,
"column": 2
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\nN : ℕ\nN_nonzero : 0 < N\n⊢ |trapezoidal_error f N a b| ≤ (b - a) ^ 3 * ζ... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\nN : ℕ\nN_nonzero : 0 < N\nh : ℝ := (b - a) / ↑N\n⊢ |trapezoidal_error f N a b| ≤ (b -... | let h := (b - a) / N | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 14
} | {
"line": 121,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 132,
"column": 12
} | {
"line": 132,
"column": 14
} | {
"line": 133,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 100,
"column": 13
} | {
"line": 100,
"column": 23
} | {
"line": 100,
"column": 24
} | [
{
"pp": "case inl.inr\nα : Type u\ns : Set (Set α)\ni : Ordinal.{u_1}\nt✝ : Set α\np : Set α → Prop\nhs : ∀ t ∈ s, p t\nh0 : p ∅\nhc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ\nhn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)\nk✝ ... | [
"case inl.inr\nα : Type u\ns : Set (Set α)\ni : Ordinal.{u_1}\nt✝ : Set α\np : Set α → Prop\nhs : ∀ t ∈ s, p t\nh0 : p ∅\nhc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ\nhn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)\nk✝ k : Ordinal.... | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 17
} | {
"line": 140,
"column": 18
} | [
{
"pp": "case basic\nα : Type u\ns : Set (Set α)\nt u : Set α\nhu : u ∈ s\n⊢ u ∈ generateMeasurableRec s (ω_ 1)",
"ppTerm": "?basic",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"MeasurableSpace.self_subset_generateMeasurableRec",
"PartialOrder.toPreorder",
"... | [] | | basic u hu => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 59
} | {
"line": 180,
"column": 4
} | [
{
"pp": "case hf\na' b' ε : ℝ\nhε : 0 < ε\n⊢ Tendsto (HDiv.hDiv a') atTop (𝓝 0)",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.div_atTop",
"Real",
"Filter.tendsto_id",
"tendsto_const_nhds",
"PseudoMetricSpace.toUniformSpace",
"instOrderTo... | [
"case hg\na' b' ε : ℝ\nhε : 0 < ε\n⊢ Tendsto (fun x ↦ b' + a' / x) atTop (𝓝 (b' + 0))"
] | · exact Tendsto.div_atTop tendsto_const_nhds tendsto_id | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.FiniteMeasureExt | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 60,
"column": 2
} | [
{
"pp": "E : Type u_1\n𝕜 : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nA : StarSubalgebra 𝕜 (E →ᵇ 𝕜)\nhA ... | [
"E : Type u_1\n𝕜 : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nA : StarSubalgebra 𝕜 (E →ᵇ 𝕜)\nhA : (StarSubal... | have h0 : Tendsto (fun ε : ℝ => 6 * √ε) (𝓝[>] 0) (𝓝 0) := by
nth_rewrite 3 [← mul_zero 6]
apply tendsto_nhdsWithin_of_tendsto_nhds (Tendsto.const_mul 6 _)
nth_rewrite 2 [← sqrt_zero]
exact Continuous.tendsto continuous_sqrt 0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 146,
"column": 4
} | {
"line": 152,
"column": 97
} | {
"line": 153,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\n... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a_1 ... | have hfi (s : I) :
∫ t in 0..1, f (s, t) = ∫ᶜ x in ⟨φ.curry s, rfl, rfl⟩, ω x := by
simp only [curveIntegral_def, curveIntegralFun_def]
apply intervalIntegral.integral_congr
rw [uIcc_of_le zero_le_one]
intro t ht
simp [Path.extend, hf (s, t), Prod.le_def, s.2.1, s.2.2, ht.1, ht.2, ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 89,
"column": 15
} | {
"line": 89,
"column": 17
} | {
"line": 90,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\ns : Set X\na : X\nhs : IsClosed[inst✝¹] s\n⊢ a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Not",
"Set"
... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\ns : Set X\na : X\nhs : IsClosed[inst✝¹] s\nha : a ∉ s\n⊢ Disjoint (𝓝ˢ s) (𝓝 a)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 342,
"column": 2
} | {
"line": 348,
"column": 6
} | {
"line": 350,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual ℝ E\n⊢ charFunDual μ L = charFun (Measure.map (⇑L) μ) 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instInnerProd... | [] | rw [charFunDual_apply]
have : ∫ x, cexp (L x * I) ∂μ = ∫ x, cexp (x * I) ∂(μ.map L) := by
rw [integral_map]
· fun_prop
· exact Measurable.aestronglyMeasurable <| by fun_prop
rw [this, charFun_apply]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 342,
"column": 2
} | {
"line": 348,
"column": 6
} | {
"line": 350,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual ℝ E\n⊢ charFunDual μ L = charFun (Measure.map (⇑L) μ) 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instInnerProd... | [] | rw [charFunDual_apply]
have : ∫ x, cexp (L x * I) ∂μ = ∫ x, cexp (x * I) ∂(μ.map L) := by
rw [integral_map]
· fun_prop
· exact Measurable.aestronglyMeasurable <| by fun_prop
rw [this, charFun_apply]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 46
} | {
"line": 72,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nc : ℝ≥0∞\nh : levyProkhorovEDist μ ν < c\nB : Set Ω\nB_mble : MeasurableSet B\n⊢ μ B ≤ ν (thickening c.toReal B) + c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
... | [
"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nc : ℝ≥0∞\nh : levyProkhorovEDist μ ν < c\nB : Set Ω\nB_mble : MeasurableSet B\nc' : ℝ≥0∞\nhc' :\n c' ∈ {ε | ∀ (B : Set Ω), MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε}\nlt_c : c... | obtain ⟨c', ⟨hc', lt_c⟩⟩ := sInf_lt_iff.mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 46
} | {
"line": 78,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nc : ℝ≥0∞\nh : levyProkhorovEDist μ ν < c\nB : Set Ω\nB_mble : MeasurableSet B\n⊢ ν B ≤ μ (thickening c.toReal B) + c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
... | [
"Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nc : ℝ≥0∞\nh : levyProkhorovEDist μ ν < c\nB : Set Ω\nB_mble : MeasurableSet B\nc' : ℝ≥0∞\nhc' :\n c' ∈ {ε | ∀ (B : Set Ω), MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε}\nlt_c : c... | obtain ⟨c', ⟨hc', lt_c⟩⟩ := sInf_lt_iff.mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 539,
"column": 2
} | {
"line": 539,
"column": 75
} | {
"line": 540,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nμ ν : Measure E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nL : StrongDual ℝ E\n⊢ ∫ (x : E), ∫ (y : E), cexp (↑(L (x + y)) * I) ∂ν ∂μ =\n... | [
"E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nμ ν : Measure E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nL : StrongDual ℝ E\n⊢ Integrable (fun v ↦ cexp (↑(L v) * I)) (μ ∗ ν)"
] | · simp [add_mul, Complex.exp_add, integral_const_mul, integral_mul_const] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 195,
"column": 12
} | {
"line": 195,
"column": 14
} | {
"line": 196,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 199,
"column": 12
} | {
"line": 199,
"column": 14
} | {
"line": 200,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b c d : E\nγ₁ : Path a✝ b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 77
} | {
"line": 111,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : LocallyCompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ.Regular\nX : Set G\nφ : G ≃ₜ* G\n⊢ ↑(mulEquivHaarChar φ) * μ (⇑φ ⁻¹' X) = ↑(m... | [] | exact congr_arg _ <| (MeasurableEquiv.map_apply φ.toMeasurableEquiv X).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 264,
"column": 12
} | {
"line": 264,
"column": 14
} | {
"line": 264,
"column": 15
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : Me... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : MeasurableSpac... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 21
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case inr\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ni... | [
"case inr\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : Meas... | · exact (hf4 _).2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 88,
"column": 4
} | {
"line": 99,
"column": 64
} | {
"line": 100,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝⁴ : MeasurableSpace E\ninst✝³ : TopologicalSpace E\ninst✝² : T2Space E\ninst✝¹ : BorelSpace E\ninst✝ : CompactSpace E\nC : ℝ≥0\nf : Ultrafilter (FiniteMeasure E)\nhf : {μ | μ.mass ≤ C} ∈ f\ng : E →C_c ℝ\nμ : FiniteMeasure E\nhμ : μ.mass ≤ C\n⊢ -(↑C * ‖g.toBoundedContin... | [
"case refine_2\nE : Type u_1\ninst✝⁴ : MeasurableSpace E\ninst✝³ : TopologicalSpace E\ninst✝² : T2Space E\ninst✝¹ : BorelSpace E\ninst✝ : CompactSpace E\nC : ℝ≥0\nf : Ultrafilter (FiniteMeasure E)\nhf : {μ | μ.mass ≤ C} ∈ f\ng : E →C_c ℝ\nμ : FiniteMeasure E\nhμ : μ.mass ≤ C\n⊢ ∫ (x : E), g x ∂↑μ ≤ ↑C * ‖g.toBounde... | · calc - (C * ‖g.toBoundedContinuousFunction‖)
_ ≤ ∫ (x : E), - ‖g.toBoundedContinuousFunction‖ ∂μ := by
simp only [integral_const, smul_eq_mul, mul_neg, neg_le_neg_iff]
gcongr
exact hμ
_ ≤ ∫ x, g x ∂μ := by
gcongr
· simp
· exact g.continuous.integrable_of_has... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 71
} | {
"line": 206,
"column": 6
} | [
{
"pp": "case h₁.hf\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nC : ℝ≥0\nK : Set E\nhK : IsCompact K\nf : ↑K → E := Subtype.val\nhf : IsClosedEmbedding f\nrf : range f = K\nF : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f\nT : Set (F... | [
"case h₁.hs\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nC : ℝ≥0\nK : Set E\nhK : IsCompact K\nf : ↑K → E := Subtype.val\nhf : IsClosedEmbedding f\nrf : range f = K\nF : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f\nT : Set (FiniteMeasure... | · exact fun t ht ↦ hf.measurableEmbedding.measurableSet_image' ht | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 24
} | {
"line": 328,
"column": 4
} | [
{
"pp": "case refine_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c : E\ns : Set E\nω : E → E →L[𝕜] F\ndω : E → E →L[ℝ]... | [
"case refine_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c : E\ns : Set E\nω : E → E →L[𝕜] F\ndω : E → E →L[ℝ] E →L[𝕜] F\... | intro x hx u hu v hv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 401,
"column": 12
} | {
"line": 401,
"column": 14
} | {
"line": 402,
"column": 4
} | [
{
"pp": "case hω\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : 𝕜 → E\ns : Set 𝕜\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\na : 𝕜\n⊢ a ∈ s → HasFDerivWi... | [
"case hω\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : 𝕜 → E\ns : Set 𝕜\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\na : 𝕜\nha : a ∈ s\n⊢ HasFDerivWithinAt (... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 208,
"column": 34
} | {
"line": 208,
"column": 53
} | {
"line": 210,
"column": 0
} | [
{
"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... | [] | by simp [hμS, h_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 136,
"column": 4
} | {
"line": 162,
"column": 33
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ∀ (f : ℕ → Set α),\n Pairwise (Function.onFun Disjoint ... | [] | intro f f_disj f_meas hf ε εpos
rcases ENNReal.exists_pos_sum_of_countable' (ENNReal.half_pos εpos.ne').ne' ℕ with ⟨δ, δpos, hδ⟩
have A i : ∃ t ∈ C, μ (t ∆ (f i)) < δ i := hf i _ (δpos i)
choose! t tC ht using A
have : Tendsto (fun n ↦ μ (⋃ i ∈ Ici n, f i)) atTop (𝓝 0) :=
tendsto_measure_biUnion_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 136,
"column": 4
} | {
"line": 162,
"column": 33
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ ∀ (f : ℕ → Set α),\n Pairwise (Function.onFun Disjoint ... | [] | intro f f_disj f_meas hf ε εpos
rcases ENNReal.exists_pos_sum_of_countable' (ENNReal.half_pos εpos.ne').ne' ℕ with ⟨δ, δpos, hδ⟩
have A i : ∃ t ∈ C, μ (t ∆ (f i)) < δ i := hf i _ (δpos i)
choose! t tC ht using A
have : Tendsto (fun n ↦ μ (⋃ i ∈ Ici n, f i)) atTop (𝓝 0) :=
tendsto_measure_biUnion_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 47
} | {
"line": 96,
"column": 0
} | [
{
"pp": "case inl.inl.inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsZeroOneMeasure μ\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nthis✝ : μ (s ∩ t) ≤ μ s\nthis : μ (s ∩ t) ≤ μ t\nh✝² : μ s = 0\nh✝¹ : μ t = 0\nh✝ : μ (s ∩ t) = 0\n⊢ μ (s ∩ t) = μ s * μ t",
"ppTerm": "?inl.in... | [] | all_goals try simp_all [measure_inter_eq_one] | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 368,
"column": 4
} | {
"line": 370,
"column": 55
} | {
"line": 371,
"column": 4
} | [
{
"pp": "case refine_1\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\nh : (countableGeneratingSet X).Countable\nhgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m\n𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpann... | [
"case refine_2\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\nh : (countableGeneratingSet X).Countable\nhgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m\n𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpanningSetsIn.se... | · rw [← hgen]
exact generateFrom_mono <| le_trans self_subset_generateSetAlgebra <|
generateSetAlgebra_mono <| subset_union_left .. | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.OuterMeasure.OfAddContent | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 17
} | {
"line": 127,
"column": 18
} | [
{
"pp": "case basic\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\nm : AddContent ℝ≥0∞ C\ns u : Set α\nhu : u ∈ C\n⊢ (inducedOuterMeasure (fun x x_1 ↦ m x) ⋯ ⋯).IsCaratheodory u",
"ppTerm": "?basic",
"assigned": true,
"usedConstants": [
"MeasureTheory.AddContent.isCaratheodory_inducedOu... | [] | | basic u hu => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 178,
"column": 14
} | {
"line": 178,
"column": 16
} | {
"line": 178,
"column": 17
} | [
{
"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\nm : Measure X\nhs : MeasurableSet s\nh : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E\ni : Finpartition ⟨s, ⋯⟩\na : Subtype Mea... | [
"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\nm : Measure X\nhs : MeasurableSet s\nh : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E\ni : Finpartition ⟨s, ⋯⟩\na : Subtype MeasurableSet\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 67
} | {
"line": 166,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L... | [] | rw [← setIntegral_add_compl (μ := μ) hs hfi, add_sub_cancel_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 67
} | {
"line": 166,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L... | [] | rw [← setIntegral_add_compl (μ := μ) hs hfi, add_sub_cancel_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 67
} | {
"line": 166,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L... | [] | rw [← setIntegral_add_compl (μ := μ) hs hfi, add_sub_cancel_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 44
} | {
"line": 151,
"column": 2
} | [
{
"pp": "case neg\nα : 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✝ : CompleteS... | [
"case neg\nα : 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 : ... | replace u_lt_a n : u n < a := (u_lt_a n).2 | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 63
} | {
"line": 207,
"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 ... | [] | simpa [hx] using (hu.eventually (Ici_mem_atTop x)).exists | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 32
} | {
"line": 67,
"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 t : Set X\nhst : s ⊆ t\nℓ : StrongDual ℝ E\nhℓ : ℓ ∈ {ℓ | ‖ℓ‖ₑ ≤ 1}\n⊢ (μ.mapRange ↑ℓ ⋯).variation s ≤ ⨆ ℓ ∈ {ℓ | ‖ℓ‖ₑ ≤ 1}, (μ.mapRange ↑ℓ ⋯).variation t",
"ppTerm":... | [
"X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns t : Set X\nhst : s ⊆ t\nℓ : StrongDual ℝ E\nhℓ : ℓ ∈ {ℓ | ‖ℓ‖ₑ ≤ 1}\n⊢ (μ.mapRange ↑ℓ ⋯).variation t ≤ ⨆ ℓ ∈ {ℓ | ‖ℓ‖ₑ ≤ 1}, (μ.mapRange ↑ℓ ⋯).variation t"
] | apply (measure_mono hst).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 19
} | {
"line": 250,
"column": 0
} | [
{
"pp": "case neg\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 :... | [] | · simp [m', hs] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 263,
"column": 4
} | {
"line": 264,
"column": 70
} | {
"line": 265,
"column": 4
} | [
{
"pp": "α : 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 : IsSetSemiring C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε : ℝ≥0∞)... | [
"α : 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 : IsSetSemiring C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε : ℝ≥0∞), Measurable... | rw [hm₀, AddContent.supClosure_apply_finpartition hC _ PC, Finset.sup_set_eq_biUnion,
measure_biUnion_finset P.disjoint (fun b hb ↦ hCmeas _ (PC hb))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 268,
"column": 2
} | {
"line": 272,
"column": 70
} | {
"line": 273,
"column": 2
} | [
{
"pp": "α : 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 : IsSetSemiring C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε : ℝ≥0∞)... | [
"α : 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 : IsSetSemiring C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε : ℝ≥0∞), Measurable... | have B (s) (hs : s ∈ supClosure C) : MeasurableSet s := by
rw [hC.mem_supClosure_iff] at hs
rcases hs with ⟨P, PC⟩
rw [← P.sup_parts, Finset.sup_set_eq_biUnion]
exact Finset.measurableSet_biUnion _ (fun b hb ↦ hCmeas _ (PC hb)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.Prod | {
"line": 73,
"column": 17
} | {
"line": 76,
"column": 8
} | {
"line": 78,
"column": 0
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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 : VectorMeas... | [] | by
refine ⟨(μ.prod ν B).map Prod.swap, fun s t hs ht ↦ ?_⟩
rw [map_apply _ (by fun_prop) (hs.prod ht)]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Variation.SignedMeasure | {
"line": 64,
"column": 94
} | {
"line": 79,
"column": 47
} | {
"line": 81,
"column": 0
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : SignedMeasure X\n⊢ μ.totalVariation = VectorMeasure.variation μ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.VectorMeasure.Variation.SignedMeasure.0.MeasureTheory.SignedMeasure.totalVariation_eq_var... | [] | by
ext r hr
apply le_antisymm
· obtain ⟨s, hs, hpos, hneg, hposPart, hnegPart⟩ := μ.toJordanDecomposition_spec
calc μ.totalVariation r
_ = ‖μ (s ∩ r)‖ₑ + ‖μ (sᶜ ∩ r)‖ₑ := by
rw [totalVariation, Measure.add_apply, hposPart, hnegPart,
μ.toMeasureOfZeroLE_apply_eq_enorm hs hpos hr,
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 523,
"column": 54
} | {
"line": 523,
"column": 87
} | {
"line": 525,
"column": 0
} | [
{
"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\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\n⊢ ∫ᵛ (x : X), f x ∂... | [] | simp [integral, FunLike.coe_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 523,
"column": 54
} | {
"line": 523,
"column": 87
} | {
"line": 525,
"column": 0
} | [
{
"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\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\n⊢ ∫ᵛ (x : X), f x ∂... | [] | simp [integral, FunLike.coe_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 523,
"column": 54
} | {
"line": 523,
"column": 87
} | {
"line": 525,
"column": 0
} | [
{
"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\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\n⊢ ∫ᵛ (x : X), f x ∂... | [] | simp [integral, FunLike.coe_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 35
} | {
"line": 585,
"column": 0
} | [
{
"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\nf : X → E\nμ : VectorMeasure X F\n⊢ ∫ᵛ (x : X), f x ∂... | [] | simp [integral, FunLike.coe_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 35
} | {
"line": 585,
"column": 0
} | [
{
"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\nf : X → E\nμ : VectorMeasure X F\n⊢ ∫ᵛ (x : X), f x ∂... | [] | simp [integral, FunLike.coe_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 35
} | {
"line": 585,
"column": 0
} | [
{
"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\nf : X → E\nμ : VectorMeasure X F\n⊢ ∫ᵛ (x : X), f x ∂... | [] | simp [integral, FunLike.coe_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 707,
"column": 6
} | {
"line": 709,
"column": 67
} | {
"line": 710,
"column": 4
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\ninst✝¹ : MeasurableSpace X\ninst✝ : Comple... | [] | apply integral_congr_ae
simp only [variation_dirac]
exact Measure.ae_smul_measure (ae_eq_dirac' hfm.measurable) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 707,
"column": 6
} | {
"line": 709,
"column": 67
} | {
"line": 710,
"column": 4
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\ninst✝¹ : MeasurableSpace X\ninst✝ : Comple... | [] | apply integral_congr_ae
simp only [variation_dirac]
exact Measure.ae_smul_measure (ae_eq_dirac' hfm.measurable) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 571,
"column": 4
} | {
"line": 571,
"column": 52
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case refine_2\nX : Type u_2\nE : Type u_3\nF✝ : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F✝\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F✝\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F✝\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ]... | [] | filter_upwards [hFi] with i hi using hi.restrict | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 571,
"column": 4
} | {
"line": 571,
"column": 52
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case refine_2\nX : Type u_2\nE : Type u_3\nF✝ : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F✝\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F✝\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F✝\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ]... | [] | filter_upwards [hFi] with i hi using hi.restrict | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 571,
"column": 4
} | {
"line": 571,
"column": 52
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case refine_2\nX : Type u_2\nE : Type u_3\nF✝ : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F✝\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F✝\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F✝\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ]... | [] | filter_upwards [hFi] with i hi using hi.restrict | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 79,
"column": 20
} | {
"line": 79,
"column": 22
} | {
"line": 79,
"column": 23
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : AddCommMonoid M\ninst✝³ : PartialOrder M\ninst✝² : WellQuasiOrderedLE M\ninst✝¹ : IsOrderedCancelAddMonoid M\ninst✝ : CanonicallyOrderedAdd M\ns : Set M\nhs : IsSlice s\nhs' : s.Nonempty\nx : M\nhx : x ∈ s\na b : M\n⊢ a ∈ {y | x + y ∈ s} → b ∈ {y | x + y ∈ s} → a + b ∈ {y | x + y... | [
"M : Type u_1\ninst✝⁴ : AddCommMonoid M\ninst✝³ : PartialOrder M\ninst✝² : WellQuasiOrderedLE M\ninst✝¹ : IsOrderedCancelAddMonoid M\ninst✝ : CanonicallyOrderedAdd M\ns : Set M\nhs : IsSlice s\nhs' : s.Nonempty\nx : M\nhx : x ∈ s\na b : M\nha : a ∈ {y | x + y ∈ s}\n⊢ b ∈ {y | x + y ∈ s} → a + b ∈ {y | x + y ∈ s}"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 132,
"column": 21
} | {
"line": 132,
"column": 31
} | {
"line": 132,
"column": 32
} | [
{
"pp": "α : Type u_1\nA : Set ℕ\ninst✝ : Finite α\nn✝ : ℕ\nthis : Fintype α\nn : ℕ\nφ : presburger[[↑A]].BoundedFormula α (n + 1)\ne : (α ⊕ Fin n) ⊕ Fin 1 ≃ α ⊕ Fin (n + 1) :=\n (Equiv.sumAssoc α (Fin n) (Fin 1)).trans ((_root_.Equiv.refl α).sumCongr finSumFinEquiv)\nih : IsSemilinearSet (⇑(LinearEquiv.funCon... | [
"α : Type u_1\nA : Set ℕ\ninst✝ : Finite α\nn✝ : ℕ\nthis : Fintype α\nn : ℕ\nφ : presburger[[↑A]].BoundedFormula α (n + 1)\ne : (α ⊕ Fin n) ⊕ Fin 1 ≃ α ⊕ Fin (n + 1) :=\n (Equiv.sumAssoc α (Fin n) (Fin 1)).trans ((_root_.Equiv.refl α).sumCongr finSumFinEquiv)\nih : IsSemilinearSet (⇑(LinearEquiv.funCongrLeft ℕ ℕ e... | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 27
} | {
"line": 176,
"column": 2
} | [
{
"pp": "A : Set ℕ\nhmul : A.Definable presburger {v | v 0 = v 1 * v 2}\nk p : ℕ\nhp : p > 0\nx : ℕ\nh₁ : x * x = max k p * max k p + p\n⊢ False",
"ppTerm": "?m.215",
"assigned": true,
"usedConstants": [
"False",
"LE.le",
"instLENat",
"dite",
"Max.max",
"Nat",
... | [
"case pos\nA : Set ℕ\nhmul : A.Definable presburger {v | v 0 = v 1 * v 2}\nk p : ℕ\nhp : p > 0\nx : ℕ\nh₁ : x * x = max k p * max k p + p\nh₂ : x ≤ max k p\n⊢ False",
"case neg\nA : Set ℕ\nhmul : A.Definable presburger {v | v 0 = v 1 * v 2}\nk p : ℕ\nhp : p > 0\nx : ℕ\nh₁ : x * x = max k p * max k p + p\nh₂ : ¬x ... | by_cases h₂ : x ≤ max k p | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.ModelTheory.DirectLimit | {
"line": 197,
"column": 2
} | {
"line": 200,
"column": 69
} | {
"line": 202,
"column": 0
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nα : Type u_1\ninst✝ : Finite α\nx y : α → Σˣ f\nxy : x ≈ y\n⊢ ∃ i... | [] | obtain ⟨i, hi⟩ := Finite.bddAbove_range (Sum.elim (fun a => (x a).1) fun a => (y a).1)
rw [Sum.elim_range, upperBounds_union] at hi
simp_rw [← Function.comp_apply (f := Sigma.fst)] at hi
exact ⟨i, hi.1, hi.2, funext fun a => (equiv_iff G f _ _).1 (xy a)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.DirectLimit | {
"line": 197,
"column": 2
} | {
"line": 200,
"column": 69
} | {
"line": 202,
"column": 0
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nα : Type u_1\ninst✝ : Finite α\nx y : α → Σˣ f\nxy : x ≈ y\n⊢ ∃ i... | [] | obtain ⟨i, hi⟩ := Finite.bddAbove_range (Sum.elim (fun a => (x a).1) fun a => (y a).1)
rw [Sum.elim_range, upperBounds_union] at hi
simp_rw [← Function.comp_apply (f := Sigma.fst)] at hi
exact ⟨i, hi.1, hi.2, funext fun a => (equiv_iff G f _ _).1 (xy a)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.PartialEquiv | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 29
} | {
"line": 119,
"column": 30
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\nh : f ≤ g\n⊢ g.toEquiv.toEmbedding.comp (inclusion ⋯) = (inclusion ⋯).comp f.toEquiv.toEmbedding",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.Partia... | [
"L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\nh : f ≤ g\n⊢ g.cod.subtype.comp (g.toEquiv.toEmbedding.comp (inclusion ⋯)) =\n g.cod.subtype.comp ((inclusion ⋯).comp f.toEquiv.toEmbedding)"
] | ← (subtype _).comp_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.PartialEquiv | {
"line": 192,
"column": 77
} | {
"line": 198,
"column": 32
} | {
"line": 200,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\n⊢ f = g ↔\n ∃ (h_dom : f.dom = g.dom),\n ∀ (x : M) (h : x ∈ f.dom), f.cod.subtype (f.toEquiv ⟨x, h⟩) = g.cod.subtype (g.toEquiv ⟨x, ⋯⟩)",
"ppTerm": "?m.53",
"assigned": true,
"used... | [] | by
constructor
· intro h_eq
rcases f with ⟨dom_f, cod_f, equiv_f⟩
cases h_eq
exact ⟨rfl, fun _ _ ↦ rfl⟩
· rintro ⟨h, H⟩; exact ext h H | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.DirectLimit | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 28
} | {
"line": 348,
"column": 2
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nP : Type u₁\ninst✝ : L.Structure P\ng : (i : ι) → G i ↪[L] P\nHg ... | [] | exact (g i).injective xy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.ModelTheory.DirectLimit | {
"line": 416,
"column": 2
} | {
"line": 426,
"column": 92
} | {
"line": 428,
"column": 0
} | [
{
"pp": "L : Language\nι : Type u_1\ninst✝⁵ : Countable ι\ninst✝⁴ : Preorder ι\ninst✝³ : IsDirectedOrder ι\ninst✝² : Nonempty ι\nG : ι → Type w\ninst✝¹ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\nh : ∀ (i : ι), CG L (G i)\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\n⊢ CG L (DirectL... | [] | refine ⟨⟨⋃ i, DirectLimit.of L ι G f i '' Classical.choose (h i).out, ?_, ?_⟩⟩
· exact Set.countable_iUnion fun i => Set.Countable.image (Classical.choose_spec (h i).out).1 _
· rw [eq_top_iff, Substructure.closure_iUnion]
simp_rw [← Embedding.coe_toHom, Substructure.closure_image]
rw [le_iSup_iff]
intro... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.DirectLimit | {
"line": 416,
"column": 2
} | {
"line": 426,
"column": 92
} | {
"line": 428,
"column": 0
} | [
{
"pp": "L : Language\nι : Type u_1\ninst✝⁵ : Countable ι\ninst✝⁴ : Preorder ι\ninst✝³ : IsDirectedOrder ι\ninst✝² : Nonempty ι\nG : ι → Type w\ninst✝¹ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\nh : ∀ (i : ι), CG L (G i)\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\n⊢ CG L (DirectL... | [] | refine ⟨⟨⋃ i, DirectLimit.of L ι G f i '' Classical.choose (h i).out, ?_, ?_⟩⟩
· exact Set.countable_iUnion fun i => Set.Countable.image (Classical.choose_spec (h i).out).1 _
· rw [eq_top_iff, Substructure.closure_iUnion]
simp_rw [← Embedding.coe_toHom, Substructure.closure_image]
rw [le_iSup_iff]
intro... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 604,
"column": 6
} | {
"line": 604,
"column": 66
} | {
"line": 606,
"column": 0
} | [
{
"pp": "case e'_3.a\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx₁ : hs.fract x = hs.base\nhx₂ : ∀ (i : ↑hs.basisSet), 0 ≤ hs.floor x i ∧ (↑i ∉ hs.periods → hs.floor x i = 0)\ni : ↑hs.basisSet\na✝ : i ∈ Finset.univ\n⊢ 0 = (-hs.floor x i).toNat • ↑i",
"ppTerm": "?... | [] | simp [fun i => Int.toNat_eq_zero.2 (neg_nonpos.2 (hx₂ i).1)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 604,
"column": 6
} | {
"line": 604,
"column": 66
} | {
"line": 606,
"column": 0
} | [
{
"pp": "case e'_3.a\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx₁ : hs.fract x = hs.base\nhx₂ : ∀ (i : ↑hs.basisSet), 0 ≤ hs.floor x i ∧ (↑i ∉ hs.periods → hs.floor x i = 0)\ni : ↑hs.basisSet\na✝ : i ∈ Finset.univ\n⊢ 0 = (-hs.floor x i).toNat • ↑i",
"ppTerm": "?... | [] | simp [fun i => Int.toNat_eq_zero.2 (neg_nonpos.2 (hx₂ i).1)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 604,
"column": 6
} | {
"line": 604,
"column": 66
} | {
"line": 606,
"column": 0
} | [
{
"pp": "case e'_3.a\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx₁ : hs.fract x = hs.base\nhx₂ : ∀ (i : ↑hs.basisSet), 0 ≤ hs.floor x i ∧ (↑i ∉ hs.periods → hs.floor x i = 0)\ni : ↑hs.basisSet\na✝ : i ∈ Finset.univ\n⊢ 0 = (-hs.floor x i).toNat • ↑i",
"ppTerm": "?... | [] | simp [fun i => Int.toNat_eq_zero.2 (neg_nonpos.2 (hx₂ i).1)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Fraisse | {
"line": 378,
"column": 8
} | {
"line": 378,
"column": 15
} | {
"line": 378,
"column": 16
} | [
{
"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\nf : M ≃ₚ[L] N\nf_FG : f.dom.FG\nm : M\nS : L.Su... | [
"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\nf : M ≃ₚ[L] N\nf_FG : f.dom.FG\nm : M\nS : L.Substructure M... | hN.age, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Fraisse | {
"line": 393,
"column": 8
} | {
"line": 393,
"column": 15
} | {
"line": 393,
"column": 16
} | [
{
"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 ... | [
"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 L ↥S\n⊢ { α ... | hN.age, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Order | {
"line": 541,
"column": 91
} | {
"line": 545,
"column": 49
} | {
"line": 547,
"column": 0
} | [
{
"pp": "M₁ M₂ : Language.order.dlo.ModelType\nh₁ : #↑M₁ = ℵ₀\nh₂ : #↑M₂ = ℵ₀\n⊢ Nonempty (↑M₁ ≃[Language.order] ↑M₂)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Cardinal",
"FirstOrder.Language.Theory.ModelType.struc",
"Set.ofPred",
"FirstOrder.... | [] | by
obtain ⟨_⟩ := denumerable_iff.2 h₁
obtain ⟨_⟩ := denumerable_iff.2 h₂
exact (isFraisseLimit_of_countable_nonempty_dlo M₁).nonempty_equiv
(isFraisseLimit_of_countable_nonempty_dlo M₂) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Topology.Types | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 48
} | {
"line": 78,
"column": 2
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\n⊢ ∃ x ∈ univ, ↑F ≤ nhds x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Set.univ",
"PartialOrder.toPreorder",
"FirstOrder.Languag... | [
"case refine_1\nL : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\n⊢ (L.lhomWithConstants α).onTheory T ⊆ {φ | T.typesWith φ ∈ F}",
"case refine_2\nL : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal u... | refine ⟨⟨{φ | T.typesWith φ ∈ F}, ?_, ?_⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.ModelTheory.Topology.Types | {
"line": 84,
"column": 55
} | {
"line": 84,
"column": 65
} | {
"line": 84,
"column": 65
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\nx : Finset L[[α]].Sentence\nhx : ↑x ⊆ {φ | T.typesWith φ ∈ F}\n⊢ ∀ φ ∈ x, T.typesWith φ ∈ ↑F",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Finset",
"Member... | [
"L : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\nx : Finset L[[α]].Sentence\nhx : ↑x ⊆ {φ | T.typesWith φ ∈ F}\nφ : L[[α]].Sentence\nhφ : φ ∈ x\n⊢ T.typesWith φ ∈ ↑F"
] | intro φ hφ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.AlmostPrime | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 59
} | {
"line": 59,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ IsAlmostPrime 0 n ↔ n = 1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
"and_self",
"id",
"one_ne_zero._simp_1... | [] | rw [IsAlmostPrime, ArithmeticFunction.cardFactors_eq_zero_iff_eq_zero_or_one]
exact ⟨fun h ↦ h.2.resolve_left h.1, fun h ↦ by simp [h]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AlmostPrime | {
"line": 56,
"column": 2
} | {
"line": 57,
"column": 59
} | {
"line": 59,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ IsAlmostPrime 0 n ↔ n = 1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instOne",
"congrArg",
"and_self",
"id",
"one_ne_zero._simp_1... | [] | rw [IsAlmostPrime, ArithmeticFunction.cardFactors_eq_zero_iff_eq_zero_or_one]
exact ⟨fun h ↦ h.2.resolve_left h.1, fun h ↦ by simp [h]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Carmichael | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 22
} | {
"line": 142,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : n ≠ 2\n⊢ λ (2 ^ n) = 2 ^ (n - 2)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"Nat.instMonoid",
"HSub.hSub",
"instSubNat",
"instOfNatNat",
"LE.le",
"instLENa... | [
"case pos\nn : ℕ\nhn : n ≠ 2\nhn' : n ≤ 2\n⊢ λ (2 ^ n) = 2 ^ (n - 2)",
"case neg\nn : ℕ\nhn : n ≠ 2\nhn' : ¬n ≤ 2\n⊢ λ (2 ^ n) = 2 ^ (n - 2)"
] | by_cases hn' : n ≤ 2 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 318,
"column": 8
} | {
"line": 318,
"column": 40
} | {
"line": 319,
"column": 8
} | [
{
"pp": "case hP₂\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... | [
"case hP₂\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] ... | rcases ht with ⟨p, ⟨i, hi⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 126,
"column": 12
} | {
"line": 126,
"column": 14
} | {
"line": 126,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nx a : R\nha : u = v * a\nb : R\nhb : u ^ p = ↑p * u * v * b\ni : ℕ\nhi : ¬i = p ∧ ¬i = 1 ∧ ¬i = 0 ∧ i < p + 1\nhi' : 2 ≤ i\n⊢ u ^ 2 * x ^ 2 * (u * x) ^ (i - 2) * ↑p * ↑(p.choose i / p) =\n ↑p * u * (v * (a * x ^ 2 * ((u * x) ^ (... | [
"R : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nx a : R\nha : u = v * a\nb : R\nhb : u ^ p = ↑p * u * v * b\ni : ℕ\nhi : ¬i = p ∧ ¬i = 1 ∧ ¬i = 0 ∧ i < p + 1\nhi' : 2 ≤ i\n⊢ (v * a) ^ 2 * x ^ 2 * (v * a * x) ^ (i - 2) * ↑p * ↑(p.choose i / p) =\n ↑p * (v * a) * (v * (a * x ^ 2 * ((v * a ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 89,
"column": 77
} | {
"line": 90,
"column": 72
} | {
"line": 92,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\n⊢ Λ p = Real.log ↑p",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Eq.mpr",
"Real",
"ArithmeticFunction.instFunLikeNat",
"Nat.Prime.minFac_eq",
"Real.instZero",
"congrArg",
... | [] | by
rw [vonMangoldt_apply, Prime.minFac_eq hp, if_pos hp.prime.isPrimePow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 109,
"column": 24
} | {
"line": 109,
"column": 26
} | {
"line": 109,
"column": 27
} | [
{
"pp": "case refine_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : a.Coprime b\n⊢ ∑ i ∈ a.divisors, Λ i = Real.log ↑a →\n ∑ i ∈ b.divisors, Λ i = Real.log ↑b → ∑ i ∈ (a * b).divisors, Λ i = Real.log ↑(a * b)",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.v... | [
"case refine_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : a.Coprime b\nha : ∑ i ∈ a.divisors, Λ i = Real.log ↑a\n⊢ ∑ i ∈ b.divisors, Λ i = Real.log ↑b → ∑ i ∈ (a * b).divisors, Λ i = Real.log ↑(a * b)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Basic | {
"line": 42,
"column": 56
} | {
"line": 42,
"column": 71
} | {
"line": 42,
"column": 72
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\na b : R\nk : ℕ\nih : ↑p ^ (k + 1) ∣ a ^ p ^ k - b ^ p ^ k\nf : R →+* R ⧸ span {↑p} := mk (span {↑p})\nh : f a = f b\nhf : ∀ (r : R), ↑p ∣ r ↔ f r = 0\n⊢ ∑ i ∈ Finset.range p, (f b ^ p ^ k) ^ i * (f b ^ p ^ k) ^ (p - 1 - i) = 0",
"ppTerm": "?succ",... | [
"case succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\na b : R\nk : ℕ\nih : ↑p ^ (k + 1) ∣ a ^ p ^ k - b ^ p ^ k\nf : R →+* R ⧸ span {↑p} := mk (span {↑p})\nh : f a = f b\nhf : ∀ (r : R), ↑p ∣ r ↔ f r = 0\n⊢ ↑p * (f b ^ p ^ k) ^ (p - 1) = 0"
] | geom_sum₂_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PrimeCounting | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 56
} | {
"line": 101,
"column": 0
} | [
{
"pp": "⊢ ¬BddAbove (Set.range π')",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.range_eq_univ",
"False",
"congrArg",
"Set.univ",
"Nat.primeCounting'",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPar... | [] | simp [Set.range_eq_univ.mpr surjective_primeCounting'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.PrimeCounting | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 66
} | {
"line": 216,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ n.primesBelow = filter Prime (Ioo 1 n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finset.mem_range._simp_1",
"Finset.mem_filter._simp_1",
"Nat.Prime",
"Preorder.toLT",
"congrArg",
"Finset",
"Finset.ext",
"Nat.instLocall... | [] | ext p
simp +contextual [primesBelow_eq_filter_range, Nat.Prime.one_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PrimeCounting | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 66
} | {
"line": 216,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ n.primesBelow = filter Prime (Ioo 1 n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finset.mem_range._simp_1",
"Finset.mem_filter._simp_1",
"Nat.Prime",
"Preorder.toLT",
"congrArg",
"Finset",
"Finset.ext",
"Nat.instLocall... | [] | ext p
simp +contextual [primesBelow_eq_filter_range, Nat.Prime.one_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 34
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case e'_5\nR : 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... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 34
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case e'_6\nR : 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... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Bernoulli | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 30
} | {
"line": 498,
"column": 4
} | [
{
"pp": "case pos\np d : ℕ\ninst✝ : Fact (Nat.Prime p)\nhd : d ≥ 2\nhcase : p = 2 ∧ d = 2\n⊢ (d + 1).factorization p ≤ d - 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Nat.Prime",
"HSub.hSub",
"Fact",
... | [
"case pos\ninst✝ : Fact (Nat.Prime 2)\nhd : 2 ≥ 2\n⊢ (2 + 1).factorization 2 ≤ 2 - 1"
] | obtain ⟨rfl, rfl⟩ := hcase | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 124,
"column": 23
} | {
"line": 124,
"column": 53
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case e'_5\nR : 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 ^ Fin... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 124,
"column": 23
} | {
"line": 124,
"column": 53
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case e'_6\nR : 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 ^ Fin... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 43,
"column": 2
} | {
"line": 56,
"column": 48
} | {
"line": 58,
"column": 0
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : b.natDegree ≤ d\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants":... | [] | set f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff j
have : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ) := by
simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m)
-- Therefore, the differences have all coefficients higher than `deg b - d` equal.
obtain ⟨i₀, i₁, i_ne, i_eq⟩ := Fintype.exists_ne_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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