module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.Binomial | {
"line": 536,
"column": 2
} | {
"line": 536,
"column": 13
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nF : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\ninst✝³ : BinomialRing R\ninst✝² : BinomialRing S\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\na : R\nn : ℕ\n⊢ f (choose a n) = choose (f a) n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Polynomial | {
"line": 33,
"column": 23
} | {
"line": 33,
"column": 51
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CommSemiring A\nz : E\ns : Set E\ninst✝² : NormedRing B\ninst✝¹ : NormedAlgebra 𝕜 B\ninst✝ : Algebra A B\nf : E → B\nhf : Anal... | apply analyticWithinAt_const | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Analytic.RadiusLiminf | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 15
} | [
{
"pp": "case h\n𝕜 : 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\np : FormalMultilinearSeries 𝕜 E F\nthis : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.RadiusLiminf | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 34
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\n⊢ p.radius⁻¹ = limsup (fun n ↦ ↑(‖p n‖₊ ^ (1 / ↑n))) atTop",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.MetricSeparated | {
"line": 120,
"column": 50
} | {
"line": 120,
"column": 61
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\ns t : Set X\nh : AreSeparated s t\nr : ℝ≥0∞\nr0 : r ≠ 0\nhr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y\nx : X\nhx1 : x ∈ s\nhx2 : x ∈ t\n⊢ r = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.MetricSeparated | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 32
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nι : Type u_3\nI : Set ι\nhI : I.Finite\ns : Set X\nt : ι → Set X\n⊢ AreSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, AreSeparated s (t i)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Metric.AreSeparated.comm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Cover | {
"line": 82,
"column": 38
} | {
"line": 82,
"column": 49
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nε : ℝ≥0\ns : Set X\nf : X → Y\nhf : Isometry f\nC : Set X\nh : IsCover ε s C\n⊢ IsCover ε (f '' s) (f '' C)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Cover | {
"line": 103,
"column": 32
} | {
"line": 103,
"column": 79
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nε : ℝ≥0\ns N : Set X\nhN : Maximal (fun N ↦ N ⊆ s ∧ IsSeparated (↑ε) N) N\n⊢ Maximal (fun N ↦ N ⊆ s ∧ SetRel.IsSeparated {(x, y) | edist x y ≤ ↑ε} N) N",
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"ENNReal.ofNNReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 106,
"column": 26
} | {
"line": 106,
"column": 49
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nD : DeloneSet X\ncarrier : Set X\npackingRadius coveringRadius : ℝ≥0\nh_carrier : carrier = D.carrier\nh_packing : packingRadius = D.packingRadius\nh_covering : coveringRadius = D.coveringRadius\n⊢ 0 < packingRadius",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 38
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nD : DeloneSet X\ncarrier : Set X\npackingRadius coveringRadius : ℝ≥0\nh_carrier : carrier = D.carrier\nh_packing : packingRadius = D.packingRadius\nh_covering : coveringRadius = D.coveringRadius\n⊢ IsSeparated (↑packingRadius) c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 110,
"column": 27
} | {
"line": 110,
"column": 51
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nD : DeloneSet X\ncarrier : Set X\npackingRadius coveringRadius : ℝ≥0\nh_carrier : carrier = D.carrier\nh_packing : packingRadius = D.packingRadius\nh_covering : coveringRadius = D.coveringRadius\n⊢ 0 < coveringRadius",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nD : DeloneSet X\ncarrier : Set X\npackingRadius coveringRadius : ℝ≥0\nh_carrier : carrier = D.carrier\nh_packing : packingRadius = D.packingRadius\nh_covering : coveringRadius = D.coveringRadius\n⊢ IsCover coveringRadius Set.uni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 28
} | [
{
"pp": "X : Type u_1\ninst✝ : MetricSpace X\nD : DeloneSet X\nx y : X\nhx : x ∈ D\nhy : y ∈ D\nhne : x ≠ y\n⊢ ENNReal.ofReal ↑D.packingRadius < ENNReal.ofReal (dist x y)",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.ofReal",
"congrA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 129,
"column": 19
} | {
"line": 129,
"column": 43
} | [
{
"pp": "X : Type u_1\ninst✝ : MetricSpace X\nD : DeloneSet X\nx y : X\nhy : y ∈ D.carrier\nhdist : (x, y) ∈ {(x, y) | edist x y ≤ ↑D.coveringRadius}\n⊢ dist x y ≤ ↑D.coveringRadius",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"PartialOrder.toPreorder"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Cover | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 49
} | [
{
"pp": "X : Type u_1\ninst✝¹ : PseudoMetricSpace X\nε : ℝ≥0\ns N : Set X\ninst✝ : ProperSpace X\nhN : IsClosed N\n⊢ IsCover ε (closure s) N ↔ IsCover ε s N",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Metric.isCover_iff_subset_cthickening",
"PseudoMetricSpace.toUniformSpace",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.IteratedFDeriv | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 19
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 17
} | [
{
"pp": "case convert_7.hz\na : ℂ\nthis : binomialSeries ℂ a = FormalMultilinearSeries.ofScalars ℂ fun n ↦ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 / ↑n !\nz : ℂ\nhz : z ∈ Metric.ball 0 1\n⊢ ‖(fun x ↦ x) z‖ < 1",
"usedConstants": [
"Norm.norm",
"Real",
"Real.instLT",
"Complex.instNorm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 15
} | [
{
"pp": "a : ℂ\nn : ℕ\nB : Set ℂ := ⋯\nthis : iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 = (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) 0\n⊢ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 = (descPochhammer ℤ n).smeval a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 21
} | [
{
"pp": "case h.succ.hz\na : ℂ\nB : Set ℂ := ⋯\nn : ℕ\nih :\n Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))\n B\nthis✝ :\n iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B = derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 15
} | [
{
"pp": "a : ℂ\nH :\n (binomialSeries ℂ (-a)).compContinuousLinearMap (-1) =\n FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (a + ↑n - 1) n\n⊢ HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ (-a)) (binomialSeries ℂ (-a)) (-0) 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 64
} | [
{
"pp": "a : ℂ\nH :\n (binomialSeries ℂ (-a)).compContinuousLinearMap (-1) =\n FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (a + ↑n - 1) n\nthis : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ (-a)) (binomialSeries ℂ (-a)) (-0) 1\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)\n (FormalMultiline... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 13
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x)) (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + 1))⁻¹) 0 ‖z‖ₑ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Algebra.to_sm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 26
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ 2)\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + 2))⁻¹ * (↑n + 1)) 0 ‖z‖ₑ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | [
{
"pp": "⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x)) (FormalMultilinearSeries.ofScalars ℂ 1) 0 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Algebra.to_smulCommClass",
"DivInvMonoid.toInv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 13
} | [
{
"pp": "⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ 2) (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑n + 1) 0 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Algebra.to_smulCommClass",
"DivInv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 227,
"column": 4
} | {
"line": 228,
"column": 11
} | [
{
"pp": "case refine_1\nw x : ℂ\nhw : w ≠ x\n⊢ HasFPowerSeriesOnBall (fun z ↦ 1 / (z - w) ^ 2)\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (w - x) ^ (-↑(i + 2))) x ‖w - x‖ₑ",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 38
} | [
{
"pp": "u : ℕ → EReal\na : EReal\n⊢ linearGrowthSup u ≤ a ↔ ∀ b > a, ∀ᶠ (n : ℕ) in atTop, u n ≤ b * ↑n",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"congrArg",
"Filter.isBo... | rw [linearGrowthSup, limsup_le_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Binomial | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 43
} | [
{
"pp": "a : ℝ\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))\n 0 1\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)\n (FormalMultilinearSeries.ofScalars ℝ fun n... | rw [← Complex.ofRealCLM.map_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Binomial | {
"line": 274,
"column": 25
} | {
"line": 274,
"column": 54
} | [
{
"pp": "a : ℝ\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))\n (Complex.ofRealCLM 0) 1\nx : ℝ\nhx : x ∈ Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ)\n⊢ |x| < 1",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 43
} | [
{
"pp": "a : ℕ\nr : ℝ\nhr : r ≠ 0\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (↑r - x) ^ (a + 1))\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (↑r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)))\n 0 ‖↑r‖ₑ\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x) ^ (a + 1... | rw [← Complex.ofRealCLM.map_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Binomial | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 13
} | [
{
"pp": "r : ℝ\nhr : r ≠ 0\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x)) (FormalMultilinearSeries.ofScalars ℝ fun n ↦ (r ^ (n + 1))⁻¹) 0 ‖r‖ₑ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 26
} | [
{
"pp": "r : ℝ\nhr : r ≠ 0\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x) ^ 2)\n (FormalMultilinearSeries.ofScalars ℝ fun n ↦ (r ^ (n + 2))⁻¹ * (↑n + 1)) 0 ‖r‖ₑ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 13
} | [
{
"pp": "⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x)) (FormalMultilinearSeries.ofScalars ℝ 1) 0 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Algebra.to_smulCommClass",
"DivInvMon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 13
} | [
{
"pp": "⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ 2) (FormalMultilinearSeries.ofScalars ℝ fun n ↦ ↑n + 1) 0 1",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Algebra.to_smulCommClass",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 42
} | [
{
"pp": "α : Type u_1\ninst✝ : NormedAddCommGroup α\nf : ℕ → α\ng : ℕ → ℝ\nh : f =o[atTop] g\nhg : 0 ≤ g\nh'g : Tendsto (fun n ↦ ∑ i ∈ range n, g i) atTop atTop\nA : ∀ (i : ℕ), ‖g i‖ = g i\nB : ∀ (n : ℕ), ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i\nε : ℝ\nεpos : 0 < ε\n⊢ ∃ N, ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nl : Filter α\nk : α → β\nz : ℕ\n⊢ Tendsto (fun a ↦ k a ^ z * 0 a) l (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : CommSemiring β\ninst✝ : ContinuousAdd β\nhf : SuperpolynomialDecay l k f\nhg : SuperpolynomialDecay l k g\nz : ℕ\n⊢ Tendsto (fun a ↦ k a ^ z * (f + g) a) l (𝓝 0)",
"usedConstants": [
"Distrib.leftD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : CommSemiring β\ninst✝ : ContinuousMul β\nhf : SuperpolynomialDecay l k f\nhg : SuperpolynomialDecay l k g\nz : ℕ\n⊢ Tendsto (fun a ↦ k a ^ z * (f * g) a) l (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : CommSemiring β\ninst✝ : ContinuousMul β\nhf : SuperpolynomialDecay l k f\nc : β\nz : ℕ\n⊢ Tendsto (fun a ↦ k a ^ z * (fun n ↦ f n * c) a) l (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nhf : SuperpolynomialDecay l k f\nz : ℕ\ns : Set β\nhs : IsOpen[inst✝¹] s\nhs0 : 0 ∈ s\nx : α\nhx : x ∈ (fun a ↦ k a ^ (z + 1) * f a) ⁻¹' s\n⊢ x ∈ (fun a ↦ k a ^ z * (k * f) a) ⁻¹' s",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 109,
"column": 12
} | {
"line": 109,
"column": 48
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nhf : SuperpolynomialDecay l k f\n⊢ SuperpolynomialDecay l k (k ^ 0 * f)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 110,
"column": 17
} | {
"line": 110,
"column": 56
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : CommSemiring β\nhf : SuperpolynomialDecay l k f\nn : ℕ\nhn : SuperpolynomialDecay l k (k ^ n * f)\n⊢ SuperpolynomialDecay l k (k ^ (n + 1) * f)",
"usedConstants": [
"Eq.mpr",
"Semigrou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 119,
"column": 50
} | {
"line": 119,
"column": 71
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : CommSemiring β\ninst✝¹ : ContinuousAdd β\ninst✝ : ContinuousMul β\nhf : SuperpolynomialDecay l k f\np✝ p q : β[X]\nhp : SuperpolynomialDecay l k fun x ↦ eval (k x) p * f x\nhq : SuperpolynomialDecay l k fun x ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : CommSemiring β\ninst✝¹ : ContinuousAdd β\ninst✝ : ContinuousMul β\nhf : SuperpolynomialDecay l k f\np : β[X]\nn : ℕ\nc : β\n⊢ SuperpolynomialDecay l k fun x ↦ eval (k x) ((monomial n) c) * f x",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 182,
"column": 45
} | {
"line": 182,
"column": 85
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Field β\ninst✝ : ContinuousMul β\nc : β\nhc0 : c ≠ 0\nh : SuperpolynomialDecay l k fun n ↦ f n * c\nx : α\n⊢ f x * c * c⁻¹ = f x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"MulOne.to... | by simp [mul_assoc, mul_inv_cancel₀ hc0] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 217,
"column": 39
} | {
"line": 217,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : ∀ (z : ℤ), Tendsto (fun a ↦ k a ^ z * f a) l (𝓝 0)\nn : ℕ\n⊢ Tendsto (fun a ↦ k a ^ n * f a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 15
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℕ\n⊢ Filter.map (fun a ↦ k a ^ ↑z * f a) l ≤ 𝓝 0"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 224,
"column": 37
} | {
"line": 224,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nh : SuperpolynomialDecay l k f\nz : ℤ\nhz : z < 0\nthis : Tendsto (fun a ↦ k a ^ z) l (𝓝 0)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nhf : SuperpolynomialDecay l k f\n⊢ SuperpolynomialDecay l k (k⁻¹ * f)",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\n⊢ SuperpolynomialDecay l k (f * k) ↔ SuperpolynomialDecay l k f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 62
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nn : ℕ\nhn : SuperpolynomialDecay l k (k ^ n * f) ↔ SuperpolynomialDecay l k f\n⊢ Supe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nn : ℕ\n⊢ SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 313,
"column": 10
} | {
"line": 313,
"column": 21
} | [
{
"pp": "α : 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\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\n⊢ Tendsto (fun x ↦ 1 / k x) l (𝓝 0)",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 15
} | [
{
"pp": "α : 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\nh : SuperpolynomialDecay l k f\nz : ℤ\nhk0 : ∀ᶠ (x : α) in l, k x ≠ 0\nthis : (fun x ↦ 1) =o[l] k\n⊢ f =o[l] fun x ↦ k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.VitaliFamily | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 45
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\nm0 : MeasurableSpace X\nμ : Measure X\nv : VitaliFamily μ\nι : Sort u_2\np : ι → Prop\ns : ι → Set X\nx : X\nh : (𝓝 x).HasBasis p s\n⊢ (v.filterAt x).HasBasis p fun i ↦ {t | t ∈ v.setsAt x ∧ t ⊆ s i}",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 42
} | [
{
"pp": "case inl\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 ∈ u,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 64
} | [
{
"pp": "α : 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_nnreal_μ : ∀ (... | have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 41
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν ξ : Measure α\ninst✝ : IsFiniteMeasure ν\nh_le : ν ≤ μ\nh : μ - ν ≤ ξ\n⊢ μ ≤ ξ + ν",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 100
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\n⊢ ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Mea... | rw [add_comm, singularPart_add_rnDeriv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 100
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\n⊢ ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Mea... | rw [add_comm, singularPart_add_rnDeriv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 100
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\n⊢ ν.withDensity (μ.rnDeriv ν) + μ.singularPart ν = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Mea... | rw [add_comm, singularPart_add_rnDeriv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 13
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν μ' : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : μ'.HaveLebesgueDecomposition ν\nthis : ∀ (b : Bool), (bif b then μ else μ').HaveLebesgueDecomposition ν\n⊢ (μ + μ').HaveLebesgueDecomposition ν",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 52
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhs : NullMeasurableSet s μ\nhs₀ : μ s ≠ 0\nhsc₀ : μ sᶜ ≠ 0\n⊢ ⨍⁻ (x : α), f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in sᶜ, f x ∂μ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 13
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\n⊢ ⨍⁻ (x : α), f x ∂μ ≤ essSup f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 53
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 422,
"column": 2
} | {
"line": 422,
"column": 52
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → E\ns : Set α\nhs : NullMeasurableSet s μ\nhs₀ : μ s ≠ 0\nhsc₀ : μ sᶜ ≠ 0\nhfi : Integrable f μ\n⊢ ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 13
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ\nhf : IntegrableOn f s μ\nhf₀ : 0 ≤ᶠ[ae (μ.restrict s)] f\n⊢ ENNReal.ofReal (⨍ (x : α) in s, f x ∂μ) = (∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ) / μ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 27
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhf : AEMeasurable f (μ.restrict s)\nhf' : ∀ᵐ (x : α) ∂μ.restrict s, f x ≠ ∞\n⊢ (⨍⁻ (x : α) in s, f x ∂μ).toReal = ⨍ (x : α) in s, (f x).toReal ∂μ",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Mea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 518,
"column": 2
} | {
"line": 518,
"column": 37
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : IntegrableOn f s μ\n⊢ 0 < μ {x | x ∈ s ∧ ⨍ (a : α) in s, f a ∂μ ≤ f x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 540,
"column": 2
} | {
"line": 540,
"column": 13
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nhf : Integrable f μ\n⊢ 0 < μ {x | f x ≤ ⨍ (a : α), f a ∂μ}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 13
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nhf : Integrable f μ\n⊢ 0 < μ {x | ⨍ (a : α), f a ∂μ ≤ f x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 573,
"column": 2
} | {
"line": 573,
"column": 37
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nN : Set α\nf : α → ℝ\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nhf : Integrable f μ\nhN : μ N = 0\n⊢ ∃ x ∉ N, ⨍ (a : α), f a ∂μ ≤ f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 53
} | [
{
"pp": "case refine_2\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 40
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsProbabilityMeasure μ\nhf : Integrable f μ\n⊢ 0 < μ {x | f x ≤ ∫ (a : α), f a ∂μ}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 40
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsProbabilityMeasure μ\nhf : Integrable f μ\n⊢ 0 < μ {x | ∫ (a : α), f a ∂μ ≤ f x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 595,
"column": 2
} | {
"line": 595,
"column": 40
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsProbabilityMeasure μ\nhf : Integrable f μ\n⊢ ∃ x, f x ≤ ∫ (a : α), f a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 40
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsProbabilityMeasure μ\nhf : Integrable f μ\n⊢ ∃ x, ∫ (a : α), f a ∂μ ≤ f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 40
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nN : Set α\nf : α → ℝ\ninst✝ : IsProbabilityMeasure μ\nhf : Integrable f μ\nhN : μ N = 0\n⊢ ∃ x ∉ N, f x ≤ ∫ (a : α), f a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 40
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nN : Set α\nf : α → ℝ\ninst✝ : IsProbabilityMeasure μ\nhf : Integrable f μ\nhN : μ N = 0\n⊢ ∃ x ∉ N, ∫ (a : α), f a ∂μ ≤ f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 626,
"column": 4
} | {
"line": 626,
"column": 83
} | [
{
"pp": "case inl\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : AEMeasurable f (μ.restrict s)\nh : ∫⁻ (a : α) in s, f a ∂μ = ∞\n⊢ 0 < μ {x | x ∈ s ∧ f x ≤ ⨍⁻ (a : α) in s, f a ∂μ}",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 184,
"column": 20
} | {
"line": 184,
"column": 31
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nh : IsHahnDecomposition μ ν s\n⊢ μ.restrict sᶜᶜ ≤ ν.restrict sᶜᶜ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"compl_compl",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝ : SecondCountableTopology α\n⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"Filter.Eventually",
... | set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 395,
"column": 4
} | {
"line": 403,
"column": 58
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set ι\nC : ℝ≥0\nr : ι → ℝ\nc : ι → α\nB : ι → Set α\nhB : ∀ a ∈ t, B a ⊆ closedBall ... | have b'_notmem_w : b' ∉ w := by
intro b'w
have b'k : B b' ⊆ k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w
have : (ball x (R x) \ k ∩ k).Nonempty := by
apply ab.mono (inter_subset_inter _ b'k)
refine ((hB _ hat).trans ?_).trans hd
gcongr
exact ad.... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 57
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : ν.HaveLebesgueDecomposition μ\nr : ℝ≥0∞\nhr : r ≠ ∞\nh : (r.toNNReal • ν).rnDeriv μ =ᶠ[ae μ] r.toNNReal • ν.rnDeriv μ\n⊢ (r • ν).rnDeriv μ =ᶠ[ae μ] r • ν.rnDeriv μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 20
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhμ : μ ≠ 0\nhint : ∫⁻ (a : α), f a ∂μ ≠ ∞\n⊢ 0 < μ {x | ⨍⁻ (a : α), f a ∂μ ≤ f x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 13
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nhf : AEMeasurable f μ\n⊢ 0 < μ {x | f x ≤ ⨍⁻ (a : α), f a ∂μ}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 728,
"column": 2
} | {
"line": 728,
"column": 42
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhf : AEMeasurable f μ\n⊢ 0 < μ {x | f x ≤ ∫⁻ (a : α), f a ∂μ}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 734,
"column": 2
} | {
"line": 734,
"column": 42
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhint : ∫⁻ (a : α), f a ∂μ ≠ ∞\n⊢ 0 < μ {x | ∫⁻ (a : α), f a ∂μ ≤ f x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 42
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhf : AEMeasurable f μ\n⊢ ∃ x, f x ≤ ∫⁻ (a : α), f a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 743,
"column": 2
} | {
"line": 743,
"column": 42
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhint : ∫⁻ (a : α), f a ∂μ ≠ ∞\n⊢ ∃ x, ∫⁻ (a : α), f a ∂μ ≤ f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 750,
"column": 2
} | {
"line": 750,
"column": 42
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nN : Set α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhf : AEMeasurable f μ\nhN : μ N = 0\n⊢ ∃ x ∉ N, f x ≤ ∫⁻ (a : α), f a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 42
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nN : Set α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhint : ∫⁻ (a : α), f a ∂μ ≠ ∞\nhN : μ N = 0\n⊢ ∃ x ∉ N, ∫⁻ (a : α), f a ∂μ ≤ f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 726,
"column": 59
} | {
"line": 726,
"column": 70
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)\nhf₃ : ∀ (n : ℕ) (t : Set α), Measur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 789,
"column": 10
} | {
"line": 789,
"column": 59
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : CompleteSpace E\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int :... | ← integrableOn_iff_integrable_of_support_subset A | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 753,
"column": 6
} | {
"line": 753,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)\nhf₃ : ∀ (n : ℕ) (t : Set ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Average | {
"line": 819,
"column": 21
} | {
"line": 819,
"column": 32
} | [
{
"pp": "case h.refine_2.hbc\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : CompleteSpace E\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.DensityTheorem | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 42
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\ninst✝³ : IsUnifLocDoublingMeasure μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nK : ℝ\nx : α\nι : Type u_2\nl : Filter ι\nw : ι → α\nδ : ι → ℝ\nxmem : ∀ᶠ (j : ι) in l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 442,
"column": 25
} | {
"line": 442,
"column": 41
} | [
{
"pp": "α : 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 : ℝ≥0\ns : Set α\nh : s ⊆ {x | v.limRatioMe... | by simp [(hρ A)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Monotone | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 17
} | [
{
"pp": "f : 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.rnDeriv volu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 1006,
"column": 2
} | {
"line": 1006,
"column": 57
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhr : r ≠ ∞\nh : (r.toNNReal • ν).rnDeriv μ =ᶠ[ae μ] r.toNNReal • ν.rnDeriv μ\n⊢ (r • ν).rnDeriv μ =ᶠ[ae μ] r • ν.rnDeriv μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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