module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.Gauge | {
"line": 600,
"column": 2
} | {
"line": 601,
"column": 27
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx : E\nhs : Absorbent ℝ s\nhr : 0 ≤ r\nhsr : s ⊆ closedBall 0 r\n⊢ ‖x‖ / r ≤ gauge s x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"gauge",
... | rw [← gauge_closedBall hr]
exact gauge_mono hs hsr _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 319,
"column": 4
} | {
"line": 341,
"column": 24
} | [
{
"pp": "case const\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n (∀ (x :... | by_cases hc : c = 0
· exact ⟨fun _ => 0, by simp [hc, upperSemicontinuous_const]⟩
have μs_lt_top : μ s < ∞ := by simpa [hs, hc, ENNReal.mul_eq_top, lt_top_iff_ne_top] using int_f
have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 319,
"column": 4
} | {
"line": 341,
"column": 24
} | [
{
"pp": "case const\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n (∀ (x :... | by_cases hc : c = 0
· exact ⟨fun _ => 0, by simp [hc, upperSemicontinuous_const]⟩
have μs_lt_top : μ s < ∞ := by simpa [hs, hc, ENNReal.mul_eq_top, lt_top_iff_ne_top] using int_f
have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 42
} | [
{
"pp": "case h_ind\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : Normed... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 27
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\nf : ℝ → E\ng : ℝ → F\nk : Set ℝ\nl : Filter ℝ\ninst✝¹ : l.NeBot\ninst✝ : TendstoIxxClass Icc l l\nhl : k ∈ l\nhd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x\nhf : Tendsto (fun x ↦ ‖f x‖) l ... | rw [← isBigO_norm_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 191,
"column": 81
} | {
"line": 198,
"column": 65
} | [
{
"pp": "a b c : ℝ\n⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b ↔ a = b ∨ c ∉ [[a, b]]",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real",
"Continuous.continuousOn",
"Real.lattice",
"Real.instZero",
"Real.instInv",
"IntervalIntegrab... | by
constructor
· refine fun h => or_iff_not_imp_left.2 fun hne hc => ?_
exact not_intervalIntegrable_of_sub_inv_isBigO_punctured (isBigO_refl _ _) hne hc h
· rintro (rfl | h₀)
· exact IntervalIntegrable.refl
refine ((continuous_sub_right c).continuousOn.inv₀ ?_).intervalIntegrable
exact fun x hx =... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 554,
"column": 32
} | {
"line": 554,
"column": 37
} | [
{
"pp": "case h\nE : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt✝ : ℝ\nht : ‖f (min 0 t✝)‖ₑ ≠ ∞\nh : t✝ < 0\nt : ℝ\n| f (-t)",
"usedConstants": [
"Real",
"congrArg",
"Real.instNeg",
"Eq.... | ← h₁f | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 581,
"column": 32
} | {
"line": 581,
"column": 37
} | [
{
"pp": "case h\nE : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), -f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt✝ : ℝ\nht : ‖f (min 0 t✝)‖ₑ ≠ ∞\nh : t✝ < 0\nt : ℝ\n| f (-t)",
"usedConstants": [
"NegZeroClass.toNeg",
"Real",
"congrArg",
... | ← h₁f | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 209,
"column": 2
} | {
"line": 219,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ AnalyticSet s ↔ ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous f ∧ range f = s",
"usedConstants": [
"Pi.uniformSpace",
"Eq.mpr",
"DiscreteUniformity.instCompleteSpace",
"Continuous",
"instUniformSpaceEmpty",
"... | constructor
· intro h
rw [AnalyticSet] at h
rcases h with h | h
· refine ⟨Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, ?_⟩
rw [h]
exact range_eq_empty _
· exact ⟨ℕ → ℕ, inferInstance, inferInstance, h⟩
· rintro ⟨β, h, h', f, f_cont, f_range⟩
rw [← f_range]
exa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 209,
"column": 2
} | {
"line": 219,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ AnalyticSet s ↔ ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous f ∧ range f = s",
"usedConstants": [
"Pi.uniformSpace",
"Eq.mpr",
"DiscreteUniformity.instCompleteSpace",
"Continuous",
"instUniformSpaceEmpty",
"... | constructor
· intro h
rw [AnalyticSet] at h
rcases h with h | h
· refine ⟨Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, ?_⟩
rw [h]
exact range_eq_empty _
· exact ⟨ℕ → ℕ, inferInstance, inferInstance, h⟩
· rintro ⟨β, h, h', f, f_cont, f_range⟩
rw [← f_range]
exa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 739,
"column": 50
} | {
"line": 739,
"column": 88
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nh : a ≤ b\n⊢ ∫ (x : ℝ) in Ι a b, ‖f x‖ ∂μ = ∫ (x : ℝ) in a..b, ‖f x‖ ∂μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Set.Ioc",
"InnerProductSpace.toNormedSpace",
"R... | by rw [uIoc_of_le h, integral_of_le h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 332,
"column": 71
} | {
"line": 334,
"column": 53
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nt : TopologicalSpace α\ninst✝⁵ : PolishSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\ntβ : TopologicalSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : OpensMeasurableSpace β\nf : α → β\ninst✝ : SecondCountableTopology ↑(range f)\nhf : Measurable f\nb : Set (Set ↑(r... | by
apply MeasurableSet.isClopenable
exact hf.subtype_mk (hb.isOpen s.2).measurableSet | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\n⊢ MeasurablySeparable (range f) (range g)",
"usedConstants": [
"Classical.byCon... | by_contra hfg | Batteries.Tactic._aux_Batteries_Tactic_Init___macroRules_Batteries_Tactic_byContra_1 | Batteries.Tactic.byContra |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 544,
"column": 6
} | {
"line": 544,
"column": 99
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NoAtoms μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous (Function.uncurry f)\nq : X\nb₀ ε : ℝ\nεpos : ε > 0\na : ℝ\na_lt : a < a₀ ∧ a < b... | filter_upwards [(tendsto_order.1 I).1 _ a_lt.2, (tendsto_order.1 J).2 _ lt_b.2] with δ hδ h'δ | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 493,
"column": 4
} | {
"line": 499,
"column": 56
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ ... | · refine Disjoint.mono_left ?_ huv.symm
change g '' cylinder y n ⊆ v
rw [image_subset_iff]
apply Subset.trans _ hεy
intro z hz
rw [mem_cylinder_iff_dist_le] at hz
exact hz.trans_lt (hn.trans_le (min_le_right _ _)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 134,
"column": 28
} | {
"line": 140,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝¹⁰ : Group G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : PolishSpace G\nΓ : Subgroup G\ninst✝⁴ : Γ.Normal\ninst✝³ : T2Space (G ⧸ Γ)\ninst✝² : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure... | by
ext A hA
obtain ⟨x₁, h⟩ := @Quotient.exists_rep _ (QuotientGroup.leftRel Γ) x
convert measure_preimage_smul μ x₁ A using 1
· rw [← h, Measure.map_apply (measurable_const_mul _) hA]
simp [← MulAction.Quotient.coe_smul_out, ← Quotient.mk''_eq_mk]
exact smulInvariantMeasure_quotient ν | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 9
} | [
{
"pp": "G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Mea... | ext U _ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 51
} | [
{
"pp": "G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\ninst✝⁵ ... | obtain ⟨K⟩ := PositiveCompacts.nonempty' (α := G) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 114,
"column": 20
} | {
"line": 114,
"column": 85
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\n| if ε < T / 2 then Metric.closedBall 0 ε else I",
"usedConstants": [
"Real",
"instHDiv",
"Real.instZero",
... | ← if_ctx_congr (Iff.rfl : ε < T / 2 ↔ ε < T / 2) h₁ fun _ => rfl, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 55
} | [
{
"pp": "case neg\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\nh₂ : QuotientAddGroup.mk ⁻¹' Metric.closedBall 0 ε ∩ I = if ε < T / 2 then Metric.closedBall 0 ε else I\nhε : ¬ε < T / 2\n⊢... | simp [I, hε, min_eq_left (by linarith : T ≤ 2 * ε)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 55
} | [
{
"pp": "case neg\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\nh₂ : QuotientAddGroup.mk ⁻¹' Metric.closedBall 0 ε ∩ I = if ε < T / 2 then Metric.closedBall 0 ε else I\nhε : ¬ε < T / 2\n⊢... | simp [I, hε, min_eq_left (by linarith : T ≤ 2 * ε)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 55
} | [
{
"pp": "case neg\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\nh₂ : QuotientAddGroup.mk ⁻¹' Metric.closedBall 0 ε ∩ I = if ε < T / 2 then Metric.closedBall 0 ε else I\nhε : ¬ε < T / 2\n⊢... | simp [I, hε, min_eq_left (by linarith : T ≤ 2 * ε)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 26
} | [
{
"pp": "case h.e'_7\nE✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (m... | simp [a, Nat.cast_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 560,
"column": 73
} | {
"line": 561,
"column": 57
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nn : ℕ\n⊢ ‖cauchyPowerSeries f c R n‖ = (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖",
"usedConstants": [
"ContinuousMultilinearMap.norm_mkPiRing",
"AddGroup.toSubtractio... | by
simp [cauchyPowerSeries, norm_smul, Real.pi_pos.le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 458,
"column": 16
} | {
"line": 458,
"column": 51
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nh0✝ : 0 ≤ R\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\nhE : CompleteSpace E\nh0 : 0 < R\n⊢ (2 * ↑π * I) • (c - c) • f c = 0",
"used... | rw [sub_self, zero_smul, smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 458,
"column": 16
} | {
"line": 458,
"column": 51
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nh0✝ : 0 ≤ R\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\nhE : CompleteSpace E\nh0 : 0 < R\n⊢ (2 * ↑π * I) • (c - c) • f c = 0",
"used... | rw [sub_self, zero_smul, smul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 458,
"column": 16
} | {
"line": 458,
"column": 51
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nh0✝ : 0 ≤ R\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\nhE : CompleteSpace E\nh0 : 0 < R\n⊢ (2 * ↑π * I) • (c - c) • f c = 0",
"used... | rw [sub_self, zero_smul, smul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.Analytic | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 10
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : CharZero 𝕜\nhf : AnalyticAt 𝕜 f x\nhf' : deriv f x ≠ 0\ni : 𝕜 ≃L[𝕜] 𝕜 := ⋯\nhfd : HasStrictFDerivAt f (↑i) x\nR : OpenPartialHomeomorph 𝕜 𝕜 := ⋯\nhx : x ∈ R.source\n⊢ ((FormalMulti... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 240,
"column": 6
} | {
"line": 241,
"column": 35
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Box.Icc I)\nHd : ∀ x ∈ Box.Ioo I... | rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,
← le_div_iff₀ (hvol_pos _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 45
} | [
{
"pp": "case inl\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Set.Icc a... | rw [this, setIntegral_empty, sum_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 71
} | [
{
"pp": "f : ℂ → ℂ\nf' x : ℂ\ns : Set ℂ\nh₁ : HasDerivWithinAt f f' s x\nh₂ : f x ∈ slitPlane\n⊢ HasDerivWithinAt (fun t ↦ log (f t)) ((f x)⁻¹ * f') s x",
"usedConstants": [
"Complex.log",
"Complex.instNormedAddCommGroup",
"NormedAlgebra.id",
"Complex.instDenselyNormedField",
"... | exact (hasStrictDerivAt_log h₂).hasDerivAt.comp_hasDerivWithinAt x h₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Order | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ analyticOrderAt f z₀ ≠ ⊤ ↔\n ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • g z",
... | simp only [← ENat.coe_toNat_eq_self, Eq.comm, EventuallyEq, ← hf.analyticOrderAt_eq_natCast,
analyticOrderNatAt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Order | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ analyticOrderAt f z₀ ≠ ⊤ ↔\n ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • g z",
... | simp only [← ENat.coe_toNat_eq_self, Eq.comm, EventuallyEq, ← hf.analyticOrderAt_eq_natCast,
analyticOrderNatAt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Order | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\n⊢ analyticOrderAt f z₀ ≠ ⊤ ↔\n ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • g z",
... | simp only [← ENat.coe_toNat_eq_self, Eq.comm, EventuallyEq, ← hf.analyticOrderAt_eq_natCast,
analyticOrderNatAt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : CharZero 𝕜\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nhg : AnalyticAt 𝕜 g x\nhg' : deriv g x ≠ 0\nhf : MeromorphicAt (f ∘ g) x\nr : 𝕜 → 𝕜 := HasStric... | rw [← this] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nz₀ : 𝕜\nf : 𝕜 → 𝕜\nhf : AnalyticAt 𝕜 f z₀\nhg : AnalyticAt 𝕜 g z₀\n⊢ analyticOrderAt (f • g) z₀ = analyticOrderAt f z₀ + analyticOrderAt g z₀",
"usedConstants"... | by_cases hf' : analyticOrderAt f z₀ = ⊤ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Analytic.Order | {
"line": 324,
"column": 8
} | {
"line": 324,
"column": 17
} | [
{
"pp": "case coe.a\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\nr : ℕ\nh : analyticOrderAt (fun x_1 ↦ f x_1 - f x) x = ↑r\nF : 𝕜 → E\nhFa : AnalyticAt 𝕜 F x\nhFne : F x ≠ 0\nhf' : 1 < r\... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 20
} | [
{
"pp": "case succ\nR : Type u_2\ninst✝³ : NonAssocSemiring R\ninst✝² : Pow R ℕ\ninst✝¹ : NatPowAssoc R\ninst✝ : BinomialRing R\nn : ℕ\nih : multichoose 2 n = ↑n + 1\n⊢ 1 + (↑n + 1) = ↑(n + 1) + 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.cast_succ",
... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 211,
"column": 8
} | {
"line": 211,
"column": 22
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nr : R\nn : ℕ\nih : (descPochhammer ℤ n).smeval r = (ascPochhammer ℕ n).smeval (r - ↑n + 1)\n⊢ (descPochhammer ℤ (n + 1)).smeval r = (ascPochhammer ℕ (n + 1)).smeval (r - ↑(n + 1) + 1)",
"usedConstants": [
... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 416,
"column": 14
} | {
"line": 416,
"column": 28
} | [
{
"pp": "R : Type u_2\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : NatPowAssoc R\ninst✝ : BinomialRing R\nn : ℕ\n⊢ multichoose (0 - ↑(n + 1) + 1) (n + 1) = 0",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"AddMonoid.toAddSemigroup",
"AddCommMonoidWithOne.toAddCommMonoid",
... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Order | {
"line": 407,
"column": 2
} | {
"line": 419,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace E\nf : 𝕜 → E\nhf : AnalyticAt 𝕜 f 0\nn : ℕ\n⊢ ∃ F,\n AnalyticAt 𝕜 F 0 ∧ ∀ (z : 𝕜), f z = ∑ i ∈ Finset.range n, (z ^ i / ↑i.facto... | classical
obtain ⟨F, hFa, hF⟩ := hf.exists_eventuallyEq_sum_add_pow_mul n
obtain ⟨U, hU0, hU'⟩ := by rwa [eventually_iff_exists_mem] at hF
refine ⟨fun z ↦ if z ∈ U then F z else (z ^ n)⁻¹ • (f z
- (∑ i ∈ .range n, (z ^ i / i.factorial) • iteratedDeriv i f 0)), ?_, fun z ↦ ?_⟩
· exact hFa.congr (by filter_... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Analytic.Order | {
"line": 407,
"column": 2
} | {
"line": 419,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace E\nf : 𝕜 → E\nhf : AnalyticAt 𝕜 f 0\nn : ℕ\n⊢ ∃ F,\n AnalyticAt 𝕜 F 0 ∧ ∀ (z : 𝕜), f z = ∑ i ∈ Finset.range n, (z ^ i / ↑i.facto... | classical
obtain ⟨F, hFa, hF⟩ := hf.exists_eventuallyEq_sum_add_pow_mul n
obtain ⟨U, hU0, hU'⟩ := by rwa [eventually_iff_exists_mem] at hF
refine ⟨fun z ↦ if z ∈ U then F z else (z ^ n)⁻¹ • (f z
- (∑ i ∈ .range n, (z ^ i / i.factorial) • iteratedDeriv i f 0)), ?_, fun z ↦ ?_⟩
· exact hFa.congr (by filter_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Order | {
"line": 407,
"column": 2
} | {
"line": 419,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace E\nf : 𝕜 → E\nhf : AnalyticAt 𝕜 f 0\nn : ℕ\n⊢ ∃ F,\n AnalyticAt 𝕜 F 0 ∧ ∀ (z : 𝕜), f z = ∑ i ∈ Finset.range n, (z ^ i / ↑i.facto... | classical
obtain ⟨F, hFa, hF⟩ := hf.exists_eventuallyEq_sum_add_pow_mul n
obtain ⟨U, hU0, hU'⟩ := by rwa [eventually_iff_exists_mem] at hF
refine ⟨fun z ↦ if z ∈ U then F z else (z ^ n)⁻¹ • (f z
- (∑ i ∈ .range n, (z ^ i / i.factorial) • iteratedDeriv i f 0)), ?_, fun z ↦ ?_⟩
· exact hFa.congr (by filter_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 76
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f (g z₀)\nhg : AnalyticAt 𝕜 g z₀\nhg_nc : ¬analyticOrderAt (fun x ↦ g x - g z₀) z₀ = ⊤\nhf' : ¬analyticOrderAt f (g ... | rw [← hr, ← hs, ← ENat.coe_mul, (hf.comp hg).analyticOrderAt_eq_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 239,
"column": 52
} | {
"line": 239,
"column": 61
} | [
{
"pp": "case neg\nf : ℂ → ℂ\ns : Set ℂ\nx : ℂ\nhf : DifferentiableWithinAt ℂ f s x\nc : ℂ\nh : ¬UniqueDiffWithinAt ℂ s x\n⊢ 0 = log c * 0 * c ^ f x",
"usedConstants": [
"Eq.mpr",
"Complex.log",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Complex.instNormedAd... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 528,
"column": 67
} | {
"line": 528,
"column": 99
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : BinomialRing R\nr s : R\nk : ℕ\nh : Commute r s\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =\n ↑(k.choose x.1) * ↑x.1.factorial * ↑x.2.factorial * (choose r x.1 * choose s x.2)",
... | mul_assoc _ (x.2.factorial : R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 530,
"column": 4
} | {
"line": 530,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : BinomialRing R\nr s : R\nk : ℕ\nh : Commute r s\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =\n ↑(k.choose x.1) * ↑x.1.factorial * (choose r x.1 * ↑x.2.factorial * choose s x.2)",
... | mul_assoc _ (x.2.factorial : R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 740,
"column": 6
} | {
"line": 740,
"column": 15
} | [
{
"pp": "case neg\nf : ℝ → ℝ\nx : ℝ\ns : Set ℝ\na : ℝ\nha : 0 < a\nhf : DifferentiableWithinAt ℝ f s x\nhxs : ¬UniqueDiffWithinAt ℝ s x\n⊢ 0 = log a * 0 * a ^ f x",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul"... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 224,
"column": 2
} | {
"line": 231,
"column": 25
} | [
{
"pp": "w x : ℂ\nhw : w ≠ x\n⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / (z - w) ^ 2 - 1 / w ^ 2)\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2)) - Nat.casesOn i (w ^ (-2)) 0) x\n ‖w - x‖ₑ",
"usedConstants": [
"zpow_natCast",
"AddGroup.toSubtractionMonoid",
... | rw [← Pi.sub_def, ← Pi.sub_def, FormalMultilinearSeries.ofScalars_sub]
refine .sub ?_ ?_
· simpa only [sub_sub_sub_cancel_right, zero_add, sub_sq_comm w, zpow_neg, zpow_natCast, mul_comm]
using (one_div_sub_sq_hasFPowerSeriesOnBall_zero
(z := w - x) (by simp [sub_eq_zero, hw])).comp_sub x
· convert ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Binomial | {
"line": 224,
"column": 2
} | {
"line": 231,
"column": 25
} | [
{
"pp": "w x : ℂ\nhw : w ≠ x\n⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / (z - w) ^ 2 - 1 / w ^ 2)\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2)) - Nat.casesOn i (w ^ (-2)) 0) x\n ‖w - x‖ₑ",
"usedConstants": [
"zpow_natCast",
"AddGroup.toSubtractionMonoid",
... | rw [← Pi.sub_def, ← Pi.sub_def, FormalMultilinearSeries.ofScalars_sub]
refine .sub ?_ ?_
· simpa only [sub_sub_sub_cancel_right, zero_add, sub_sq_comm w, zpow_neg, zpow_natCast, mul_comm]
using (one_div_sub_sq_hasFPowerSeriesOnBall_zero
(z := w - x) (by simp [sub_eq_zero, hw])).comp_sub x
· convert ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.IteratedFDeriv | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nh : HasFPowerSeriesWithinOnBall f... | rw [Finset.sum_eq_single_of_mem n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 63,
"column": 86
} | {
"line": 64,
"column": 35
} | [
{
"pp": "x : ℝ\nhx : 0 < x\n⊢ (ENNReal.ofReal x).log = ↑(Real.log x)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
"EReal",
"Preorder.toLE",
"ENNReal.log",
"id",
"Bot.bot",
"LE.le",... | by
rw [log_ofReal, if_neg hx.not_ge] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 139,
"column": 10
} | {
"line": 139,
"column": 19
} | [
{
"pp": "case inr.inl.inl\n⊢ (∞ * 0).log = ⊤ + log 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"EReal",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"ENNReal.log",
"instTopEReal",
"id",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\np q : ℕ\nhpq : q < p\n⊢ Tendsto (fun x ↦ x ^ p / x ^ q) atTop atTop",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"PartialOrder.toPreorder",
"HSub.hSub",
"DivInvMo... | tendsto_congr' pow_div_pow_eventuallyEq_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : ℕ\nhpq : p < q\n⊢ Tendsto (fun x ↦ x ^ p / x ^ q) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.to... | tendsto_congr' pow_div_pow_eventuallyEq_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 387,
"column": 2
} | {
"line": 397,
"column": 33
} | [
{
"pp": "u : ℕ → EReal\nv : ℕ → ℕ\nhu : 0 ≤ᶠ[atTop] u\nhv : Tendsto v atTop atTop\n⊢ (linearGrowthInf fun n ↦ ↑(v n)) * linearGrowthInf u ≤ linearGrowthInf (u ∘ v)",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"_private.Mathlib.An... | have uv_0 : 0 ≤ linearGrowthInf (u ∘ v) := by
rw [← linearGrowthInf_const zero_ne_bot zero_ne_top]
exact linearGrowthInf_eventually_monotone (hv.eventually hu)
apply EReal.mul_le_of_forall_lt_of_nonneg (linearGrowthInf_natCast_nonneg v) uv_0
refine fun a ⟨_, a_v⟩ b ⟨b_0, b_u⟩ ↦ Eventually.le_linearGrowthInf... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 387,
"column": 2
} | {
"line": 397,
"column": 33
} | [
{
"pp": "u : ℕ → EReal\nv : ℕ → ℕ\nhu : 0 ≤ᶠ[atTop] u\nhv : Tendsto v atTop atTop\n⊢ (linearGrowthInf fun n ↦ ↑(v n)) * linearGrowthInf u ≤ linearGrowthInf (u ∘ v)",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"_private.Mathlib.An... | have uv_0 : 0 ≤ linearGrowthInf (u ∘ v) := by
rw [← linearGrowthInf_const zero_ne_bot zero_ne_top]
exact linearGrowthInf_eventually_monotone (hv.eventually hu)
apply EReal.mul_le_of_forall_lt_of_nonneg (linearGrowthInf_natCast_nonneg v) uv_0
refine fun a ⟨_, a_v⟩ b ⟨b_0, b_u⟩ ↦ Eventually.le_linearGrowthInf... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 301,
"column": 4
} | {
"line": 305,
"column": 26
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝³ : NormedField β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nh : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z\nz : ℤ\n⊢ Tendsto (fun a ↦ k a ^ ... | suffices (fun a : α => k a ^ z * f a) =O[l] fun a : α => (k a)⁻¹ from
IsBigO.trans_tendsto this hk.inv_tendsto_atTop
refine ((isBigO_refl (fun a => k a ^ z) l).mul (h (-(z + 1)))).trans ?_
refine .of_bound' <| hk0.mono fun a ha0 => ?_
simp [← zpow_add₀ ha0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 301,
"column": 4
} | {
"line": 305,
"column": 26
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝³ : NormedField β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nh : ∀ (z : ℤ), f =O[l] fun a ↦ k a ^ z\nz : ℤ\n⊢ Tendsto (fun a ↦ k a ^ ... | suffices (fun a : α => k a ^ z * f a) =O[l] fun a : α => (k a)⁻¹ from
IsBigO.trans_tendsto this hk.inv_tendsto_atTop
refine ((isBigO_refl (fun a => k a ^ z) l).mul (h (-(z + 1)))).trans ?_
refine .of_bound' <| hk0.mono fun a ha0 => ?_
simp [← zpow_add₀ ha0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 593,
"column": 4
} | {
"line": 593,
"column": 85
} | [
{
"pp": "u : ℕ → EReal\nm : ℕ\nh : Monotone u\nhm : m ≠ 0\n⊢ Tendsto (fun n ↦ ↑(m * n) / ↑n) atTop (𝓝 ↑m)",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"EReal.instTopologicalSpace",
"EReal"... | refine tendsto_nhds_of_eventually_eq ((eventually_gt_atTop 0).mono fun x hx ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 601,
"column": 4
} | {
"line": 601,
"column": 85
} | [
{
"pp": "u : ℕ → EReal\nm : ℕ\nh : Monotone u\nhm : m ≠ 0\n⊢ Tendsto (fun n ↦ ↑(m * n) / ↑n) atTop (𝓝 ↑m)",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"EReal.instTopologicalSpace",
"EReal"... | refine tendsto_nhds_of_eventually_eq ((eventually_gt_atTop 0).mono fun x hx ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 99,
"column": 8
} | {
"line": 99,
"column": 39
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nB : ι → Set α\nt : Set ι\nδ : ι → ℝ\nτ : ℝ\nhτ : 1 < τ\nδnonneg : ∀ a ∈ t, 0 ≤ δ a\nR : ℝ\nδle : ∀ a ∈ t, δ a ≤ R\nhne : ∀ a ∈ t, (B a).Nonempty\nT : Set (Set ι) :=\n {u |\n u ⊆ t ∧\n u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B ... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 153,
"column": 6
} | {
"line": 154,
"column": 33
} | [
{
"pp": "case neg.inr\nα : Type u_1\nι : Type u_2\nB : ι → Set α\nt : Set ι\nδ : ι → ℝ\nτ : ℝ\nhτ : 1 < τ\nδnonneg : ∀ a ∈ t, 0 ≤ δ a\nR : ℝ\nδle : ∀ a ∈ t, δ a ≤ R\nhne : ∀ a ∈ t, (B a).Nonempty\nT : Set (Set ι) :=\n {u |\n u ⊆ t ∧\n u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ... | · rw [← not_disjoint_iff_nonempty_inter] at hcb
exact (hcb (H _ H')).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 63
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), μ s ≠ ∞\nhν : ∀ (s : Set α), ν s ≠ ∞\nto... | simp_rw [f, Nat.Ico_succ_singleton, Finset.inf_singleton] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.Average | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 29
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\nf : α → E\ns t : Set α\nhd : AEDisjoint μ s t\nht : NullMeasurableSet t μ\nhsμ : μ s ≠ ∞\nhtμ : μ t ≠ ∞\nhfs : IntegrableOn f s μ\nhft : IntegrableOn f t μ\n⊢ ⨍ (x : α) in s ∪ t, f... | haveI := Fact.mk hsμ.lt_top | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 51
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nh : ¬μ.HaveLebesgueDecomposition ν\n⊢ Measurable (μ.rnDeriv ν)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Measurable",
"MeasureTheory.Measure.rnDeriv",
"ENNReal.measurableSpace",
"id",
"Pi.ins... | rw [rnDeriv_of_not_haveLebesgueDecomposition h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 51
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nhμν : μ ⟂ₘ ν\nh : ¬μ.HaveLebesgueDecomposition ν\n⊢ μ.rnDeriv ν =ᶠ[ae ν] 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.rnDeriv",
"Fi... | rw [rnDeriv_of_not_haveLebesgueDecomposition h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 51
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nhμν : μ ⟂ₘ ν\nh : ¬μ.HaveLebesgueDecomposition ν\n⊢ μ.rnDeriv ν =ᶠ[ae ν] 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.rnDeriv",
"Fi... | rw [rnDeriv_of_not_haveLebesgueDecomposition h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 51
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nhμν : μ ⟂ₘ ν\nh : ¬μ.HaveLebesgueDecomposition ν\n⊢ μ.rnDeriv ν =ᶠ[ae ν] 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.rnDeriv",
"Fi... | rw [rnDeriv_of_not_haveLebesgueDecomposition h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.OneDim | {
"line": 53,
"column": 2
} | {
"line": 56,
"column": 43
} | [
{
"pp": "case refine_2\nx : ℝ\n⊢ ∀ ε > 0, ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instClosedIicTopology",
"Real.partialOrder",
"Real",
"Mathlib.Meta.N... | · intro ε εpos
filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy
rw [closedBall_eq_Icc]
exact Icc_subset_Icc hy.1 (by linarith) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Monotone.Extension | {
"line": 48,
"column": 10
} | {
"line": 48,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : ConditionallyCompleteLinearOrder β\nf : α → β\ns : Set α\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : β\nha : a ∈ lowerBounds (f '' s)\nhu' : ∀ (x : α), BddAbove (f '' (Iic x ∩ s))\ng : α → β := fun x ↦ if Disjoint (Iic x) s then... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 328,
"column": 6
} | {
"line": 328,
"column": 56
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np q : ℝ≥0\nhpq : p < q\ns : Se... | · exact inter_subset_right.trans subset_union_left | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Monotone | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 50
} | [
{
"pp": "case h1\nf : StieltjesFunction ℝ\nx : ℝ\nhx : Tendsto (fun a ↦ f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))\nh'x : f.measure.rnDeriv volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 : Tendsto (fun y ↦ (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnD... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.Monotone | {
"line": 159,
"column": 8
} | {
"line": 159,
"column": 52
} | [
{
"pp": "case h1\nf : ℝ → ℝ\nhf : Monotone f\nx : ℝ\nhx :\n Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)\n (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧\n Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)\n (𝓝 (hf.stieltjesFunction.meas... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.Monotone | {
"line": 185,
"column": 8
} | {
"line": 185,
"column": 52
} | [
{
"pp": "case h1\nf : ℝ → ℝ\nhf : Monotone f\nx : ℝ\nhx :\n Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)\n (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧\n Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)\n (𝓝 (hf.stieltjesFunction.meas... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 59,
"column": 11
} | {
"line": 59,
"column": 28
} | [
{
"pp": "case h.e'_3\nR : Type u_1\ninst✝ : CommSemiring R\ns : Multiset R\nk : ℕ\nh : k ≤ s.card\n⊢ s.esymm (s.card - k) = (∑ j ∈ Finset.range (s.card + 1), C (s.esymm j) * X ^ (s.card - j)).coeff k",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"HMul.hMul",
"congrArg",
"CommS... | finset_sum_coeff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 84,
"column": 22
} | {
"line": 84,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\nx : Multiset R\nhx : x ∈ powersetCard k s\n⊢ (prod ∘ map Neg.neg) x = (map (fun i ↦ Function.const R (-1) i * i) x).prod",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"Multiset.map",
"Monoid.t... | map_congr rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 719,
"column": 2
} | {
"line": 719,
"column": 59
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nhs : MeasurableSet s\nthis : IsLocallyFiniteMeasure (μ.restrict s)\nx : α\nhx : Tendsto ... | simpa only [h'x, restrict_apply' hs, inter_comm] using hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 715,
"column": 2
} | {
"line": 719,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.i... | haveI : IsLocallyFiniteMeasure (μ.restrict s) :=
isLocallyFiniteMeasure_of_le restrict_le_self
filter_upwards [ae_tendsto_rnDeriv v (μ.restrict s), rnDeriv_restrict_self μ hs]
intro x hx h'x
simpa only [h'x, restrict_apply' hs, inter_comm] using hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 715,
"column": 2
} | {
"line": 719,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nhs : MeasurableSet s\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a ↦ μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.i... | haveI : IsLocallyFiniteMeasure (μ.restrict s) :=
isLocallyFiniteMeasure_of_le restrict_le_self
filter_upwards [ae_tendsto_rnDeriv v (μ.restrict s), rnDeriv_restrict_self μ hs]
intro x hx h'x
simpa only [h'x, restrict_apply' hs, inter_comm] using hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 780,
"column": 2
} | {
"line": 780,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\n⊢ ... | let A := MeasureTheory.Measure.finiteSpanningSetsInOpen' μ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 56
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCa... | · exact cfc_apply (0 : R → R) a ▸ map_zero (cfcHom ha) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 712,
"column": 20
} | {
"line": 718,
"column": 46
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R A... | by
rw [Commute, SemiconjBy]
by_cases h : ContinuousOn f (spectrum R a)
· rw [← cfc_star, ← cfc_mul .., ← cfc_mul ..]
congr! 2
exact mul_comm _ _
· simp [cfc_apply_of_not_continuousOn a h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1008,
"column": 4
} | {
"line": 1008,
"column": 31
} | [
{
"pp": "α✝ : Type u_1\ninst✝³ : LinearOrder α✝\nE✝ : Type u_2\ninst✝² : PseudoEMetricSpace E✝\nf✝ : α✝ → E✝\ns✝ : Set α✝\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\na b : α\nha : a ∈ s\nhb : b ∈ s\nh : variationOnF... | apply eVariationOn.edist_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1050,
"column": 2
} | {
"line": 1053,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\nb : α\nbs : b ∈ s\n⊢ AntitoneOn (fun a ↦ variationOnFromTo f s a b) s",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instAddMonoid",
"con... | rintro a as c cs ac
dsimp only
rw [← variationOnFromTo.add hf as cs bs]
exact le_add_of_nonneg_left (variationOnFromTo.nonneg_of_le f s ac) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1050,
"column": 2
} | {
"line": 1053,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\nb : α\nbs : b ∈ s\n⊢ AntitoneOn (fun a ↦ variationOnFromTo f s a b) s",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instAddMonoid",
"con... | rintro a as c cs ac
dsimp only
rw [← variationOnFromTo.add hf as cs bs]
exact le_add_of_nonneg_left (variationOnFromTo.nonneg_of_le f s ac) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1066,
"column": 6
} | {
"line": 1066,
"column": 49
} | [
{
"pp": "case hx\nα : Type u_1\ninst✝ : LinearOrder α\nf : α → ℝ\ns : Set α\nhf : LocallyBoundedVariationOn f s\na : α\nas : a ∈ s\nb : α\nbs : b ∈ s\nc : α\ncs : c ∈ s\nbc : b ≤ c\n⊢ b ∈ s ∩ Icc b c",
"usedConstants": [
"le_rfl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membersh... | exacts [⟨bs, le_rfl, bc⟩, ⟨cs, bc, le_rfl⟩] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 335,
"column": 2
} | {
"line": 354,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nx : MvPolynomial Bool R := ⋯\ny : MvPolynomial Bool R := ⋯\npderiv_true_x : (pderiv true) x = 1\npderiv_true_y : (pderiv true) y = 0\ne : Bool → R[X] := ⋯\n⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν =\n (MvPolynomial.aeval e) ((pd... | · -- We first prepare a tedious rewrite:
have w : ∀ k : ℕ, (k * (k - 1)) • bernsteinPolynomial R n k =
(n.choose k : R[X]) * ((1 - Polynomial.X) ^ (n - k) *
((k : R[X]) * ((↑(k - 1) : R[X]) * Polynomial.X ^ (k - 1 - 1)))) * Polynomial.X ^ 2 := by
rintro (_ | _ | k)
· simp
· simp
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 257,
"column": 2
} | {
"line": 258,
"column": 68
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | have hg : ContinuousOn (Function.extend Subtype.val f g) (σₙ R a) :=
continuousOn_iff_continuous_restrict.mpr <| h ▸ map_continuous f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 255,
"column": 2
} | {
"line": 264,
"column": 8
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | have h : f = (σₙ R a).restrict (Function.extend Subtype.val f g) := by
ext; simp
have hg : ContinuousOn (Function.extend Subtype.val f g) (σₙ R a) :=
continuousOn_iff_continuous_restrict.mpr <| h ▸ map_continuous f
have hg0 : (Function.extend Subtype.val f g) 0 = 0 := by
rw [← quasispectrum.coe_zero (R ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 255,
"column": 2
} | {
"line": 264,
"column": 8
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | have h : f = (σₙ R a).restrict (Function.extend Subtype.val f g) := by
ext; simp
have hg : ContinuousOn (Function.extend Subtype.val f g) (σₙ R a) :=
continuousOn_iff_continuous_restrict.mpr <| h ▸ map_continuous f
have hg0 : (Function.extend Subtype.val f g) 0 = 0 := by
rw [← quasispectrum.coe_zero (R ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 332,
"column": 4
} | {
"line": 335,
"column": 37
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A... | rw [cfcₙ_apply f a (h.2.1.congr hfg) (hfg (quasispectrum.zero_mem R a) ▸ h.2.2) h.1,
cfcₙ_apply g a h.2.1 h.2.2 h.1]
congr 3
exact Set.restrict_eq_iff.mpr hfg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 332,
"column": 4
} | {
"line": 335,
"column": 37
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A... | rw [cfcₙ_apply f a (h.2.1.congr hfg) (hfg (quasispectrum.zero_mem R a) ▸ h.2.2) h.1,
cfcₙ_apply g a h.2.1 h.2.2 h.1]
congr 3
exact Set.restrict_eq_iff.mpr hfg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 849,
"column": 34
} | {
"line": 852,
"column": 45
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : Semifield R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : Algebra R A\ninst✝ : ContinuousFunctionalCalculus R A p\n... | by
have h_cpct : CompactSpace (spectrum R a) := inferInstance
simp only [← isCompact_iff_compactSpace, quasispectrum_eq_spectrum_union_zero] at h_cpct ⊢
exact h_cpct |>.union isCompact_singleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.UnitizationL1 | {
"line": 114,
"column": 44
} | {
"line": 114,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : SMulCommClass 𝕜 A A\nx y : WithLp 1 (Unitization 𝕜 A)\n⊢ ‖(x * y).ofLp.toProd.1‖ + ‖(x * y).ofLp.toProd.2‖ ≤\n ‖x.ofLp.toProd.1‖ * (‖y.ofLp.toProd... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 382,
"column": 4
} | {
"line": 383,
"column": 18
} | [
{
"pp": "case h\n𝕜 : Type u_1\nX : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : TopologicalSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\nx₁ x₂ : X\nhx : x₁ ≠ x₂\nf : C(X, 𝕜)\nhfA : f ∈ ↑A.toSubalgebra\nhf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂\nF : C(X, 𝕜) := f - const X (f x₂)\na✝ : X\n⊢ (f x₂ ... | simp only [smul_apply, one_apply, smul_eq_mul, mul_one,
const_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 71
} | [
{
"pp": "F : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\nB : Type u_5\np : A → Prop\nq : B → Prop\ninst✝²⁹ : CommSemiring R\ninst✝²⁸ : Nontrivial R\ninst✝²⁷ : StarRing R\ninst✝²⁶ : MetricSpace R\ninst✝²⁵ : IsTopologicalSemiring R\ninst✝²⁴ : ContinuousStar R\ninst✝²³ : CommRing S\ninst✝²² : Algebra R S\n... | have hf' : ContinuousOn f (quasispectrum R (ψ a)) := hf.mono h_spec | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 265,
"column": 53
} | {
"line": 267,
"column": 77
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → Set β\nx : α\nγ : Type u_5\ninst✝ : TopologicalSpace γ\ni : γ → β\nhf : UpperHemicontinuousAt f x\nhi : IsInducing i\nh_cl : IsClosed (range i)\n⊢ UpperHemicontinuousAt (fun x ↦ i ⁻¹' f x) x",
"usedConstan... | by
simpa [upperHemicontinuousWithinAt_univ_iff] using
hf.upperHemicontinuousWithinAt (s := Set.univ) |>.isInducing_comp hi h_cl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 201,
"column": 45
} | {
"line": 211,
"column": 70
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\np : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Fact (map (algebraMap F E) p).Splits\n⊢ Function.Injective ⇑(galActionHom p E)",
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"instHSMul",
... | by
rw [injective_iff_map_eq_one]
intro ϕ hϕ
ext (x hx)
have key := Equiv.Perm.ext_iff.mp hϕ (rootsEquivRoots p E ⟨x, hx⟩)
change
rootsEquivRoots p E (ϕ • (rootsEquivRoots p E).symm (rootsEquivRoots p E ⟨x, hx⟩)) =
rootsEquivRoots p E ⟨x, hx⟩
at key
rw [Equiv.symm_apply_apply] at key
exact Su... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h.e'_4\n𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nn : ℕ\nk : 𝕜\nhk : k ∈ σ a\npow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))\nnnnorm_pow_le : ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊\nhn : 0 < ↑(n + 1)\n⊢ ... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
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