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.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 118,
"column": 6
} | {
"line": 123,
"column": 29
} | [
{
"pp": "α : 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\nε : ℝ≥0∞\nε0 : ε ≠ 0\nf : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)\nh : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞\nhc : ¬c = 0\n⊢ μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 17
} | [
{
"pp": "α : 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\nε : ℝ≥0∞\nε0 : ε ≠ 0\nf : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)\nh : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞\nhc : ¬c = 0\nne_t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 144,
"column": 10
} | {
"line": 144,
"column": 21
} | [
{
"pp": "α : 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\nε : ℝ≥0∞\nε0 : ε ≠ 0\nf : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)\nh : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞\nhc : ¬c = 0\nne_t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 212,
"column": 8
} | {
"line": 212,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 219,
"column": 54
} | {
"line": 219,
"column": 72
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 660,
"column": 12
} | {
"line": 660,
"column": 72
} | [
{
"pp": "case hz\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 295,
"column": 10
} | {
"line": 295,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nfmeas : AEMeasurable f μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nδ : ℝ≥0\nδpos : 0 < δ\nhδε : δ < ε\nint_f_ne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.CompleteCodomain | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nι : Type u_4\ninst✝⁴ : Finite ι\nM : ι → Type u_5\ninst✝³ : (i : ι) → NormedAddCommGroup (M i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (M i)\ninst✝¹ : ∀ (i : ι), SeparatingDual 𝕜 (M i... | have : ∀ i, ∃ φ : StrongDual 𝕜 (M i), φ (m i) = 1 := fun i ↦ exists_eq_one (hm i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 736,
"column": 6
} | {
"line": 736,
"column": 30
} | [
{
"pp": "case h\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : ℝ → F\ninst✝ : CompleteSpace F\nthis✝¹ : MeasurableSpace F := ⋯\nthis✝ : BorelSpace F\nt : Set ℝ\nt_count : t.Countable\nht : Dense t\nx : ℝ\n⊢ Ioi x ∩ closure t ⊆ closure (Ioi x ∩ t)",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 321,
"column": 35
} | {
"line": 321,
"column": 94
} | [
{
"pp": "α : 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\nhc : ¬c = 0\n⊢ μ s < ∞",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 753,
"column": 2
} | {
"line": 753,
"column": 50
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : ℝ → F\ninst✝ : CompleteSpace F\n⊢ MeasurableSet {x | DifferentiableWithinAt ℝ f (Ioi x) x}",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.pa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 758,
"column": 2
} | {
"line": 758,
"column": 38
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nf : ℝ → F\ninst✝² : CompleteSpace F\ninst✝¹ : MeasurableSpace F\ninst✝ : BorelSpace F\n⊢ Measurable fun x ↦ derivWithin f (Ioi x) x",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Set.Ici",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 38
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : ℝ → F\ninst✝ : CompleteSpace F\n⊢ StronglyMeasurable fun x ↦ derivWithin f (Ioi x) x",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Set.Ici",
"Real.denselyNormedField",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 341,
"column": 10
} | {
"line": 341,
"column": 21
} | [
{
"pp": "α : 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\nhc : ¬c = 0\nμs_lt_top : μ s < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 373,
"column": 28
} | {
"line": 373,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nint_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑(fs x) ≤ ↑(f x)\nint_fs : ∫⁻ (x : α), ↑(f x) ∂μ < (SimpleFunc.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nint_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), fs x ≤ f x\nint_fs : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(fs x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 388,
"column": 48
} | {
"line": 388,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nint_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), fs x ≤ f x\nint_fs : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(fs x... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemiconti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 416,
"column": 8
} | {
"line": 416,
"column": 19
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 516,
"column": 43
} | {
"line": 516,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integrabl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 54
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 30
} | [
{
"pp": "case refine_5\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsUniformGroup.Order | {
"line": 41,
"column": 86
} | {
"line": 41,
"column": 96
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nβ : Type u_3\ninst✝⁶ : UniformSpace β\ninst✝⁵ : AddGroup β\ninst✝⁴ : IsUniformAddGroup β\ninst✝³ : PartialOrder β\ninst✝² : OrderTopology β\ninst✝¹ : AddLeftMono β\ninst✝ : AddRightMono β\nf : ι → α → β\ng : α → β\nK : Set α\np : Filter ι\nu v : β\nhuv : u < v\nhg : ∀ x ∈ K,... | simp [huv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.IsUniformGroup.Order | {
"line": 41,
"column": 86
} | {
"line": 41,
"column": 96
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nβ : Type u_3\ninst✝⁶ : UniformSpace β\ninst✝⁵ : AddGroup β\ninst✝⁴ : IsUniformAddGroup β\ninst✝³ : PartialOrder β\ninst✝² : OrderTopology β\ninst✝¹ : AddLeftMono β\ninst✝ : AddRightMono β\nf : ι → α → β\ng : α → β\nK : Set α\np : Filter ι\nu v : β\nhuv : u < v\nhg : ∀ x ∈ K,... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.IsUniformGroup.Order | {
"line": 41,
"column": 86
} | {
"line": 41,
"column": 96
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nβ : Type u_3\ninst✝⁶ : UniformSpace β\ninst✝⁵ : AddGroup β\ninst✝⁴ : IsUniformAddGroup β\ninst✝³ : PartialOrder β\ninst✝² : OrderTopology β\ninst✝¹ : AddLeftMono β\ninst✝ : AddRightMono β\nf : ι → α → β\ng : α → β\nK : Set α\np : Filter ι\nu v : β\nhuv : u < v\nhg : ∀ x ∈ K,... | simp [huv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Order | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | [
{
"pp": "case h\nα : Type u_1\nι : Type u_2\nβ : Type u_3\ninst✝⁶ : UniformSpace β\ninst✝⁵ : AddGroup β\ninst✝⁴ : IsUniformAddGroup β\ninst✝³ : PartialOrder β\ninst✝² : OrderTopology β\ninst✝¹ : AddLeftMono β\ninst✝ : AddRightMono β\nf : ι → α → β\ng : α → β\nK : Set α\np : Filter ι\nu v : β\nhuv : u < v\nhg : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 75
} | [
{
"pp": "case pos.refine_1.hfi\nX : Type u_1\nE : Type u_3\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → ℝ≥0\nf_meas : Measurable f\ng : X → E\nhE : CompleteSpace E\nhg : Integrable g (μ.withDensity fun x ↦ ↑(f x))\nc : E\ns : Set X\ns_meas : Measura... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 293,
"column": 6
} | {
"line": 293,
"column": 48
} | [
{
"pp": "X : Type u_1\nE : Type u_3\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → ℝ≥0\nf_meas : Measurable f\ng : X → E\nhE : CompleteSpace E\nhg : Integrable g (μ.withDensity fun x ↦ ↑(f x))\nu v : X → E\nhuv : u =ᶠ[ae (μ.withDensity fun x ↦ ↑(f x))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 896,
"column": 2
} | {
"line": 896,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : LocallyCompactSpace E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nα : Type u_4\ninst✝⁵ : TopologicalSpace α\nf : α → E → F\ninst✝⁴ : Measur... | have : IsComplete (univ : Set (E →L[𝕜] F)) := complete_univ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 48
} | [
{
"pp": "ε : Type u_3\ninst✝³ : TopologicalSpace ε\ninst✝² : ENormedAddMonoid ε\ninst✝¹ : PseudoMetrizableSpace ε\nf : ℝ → ε\na b : ℝ\nμ : Measure ℝ\ninst✝ : NoAtoms μ\nhab : a ≤ b\nha : ‖f a‖ₑ ≠ ∞\nhb : ‖f b‖ₑ ≠ ∞\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ",
"usedConstants": [
"Eq.mpr... | rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab ha,
integrableOn_Icc_iff_integrableOn_Ioo ha hb] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 48
} | [
{
"pp": "ε : Type u_3\ninst✝³ : TopologicalSpace ε\ninst✝² : ENormedAddMonoid ε\ninst✝¹ : PseudoMetrizableSpace ε\nf : ℝ → ε\na b : ℝ\nμ : Measure ℝ\ninst✝ : NoAtoms μ\nhab : a ≤ b\nha : ‖f a‖ₑ ≠ ∞\nhb : ‖f b‖ₑ ≠ ∞\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ",
"usedConstants": [
"Eq.mpr... | rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab ha,
integrableOn_Icc_iff_integrableOn_Ioo ha hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 48
} | [
{
"pp": "ε : Type u_3\ninst✝³ : TopologicalSpace ε\ninst✝² : ENormedAddMonoid ε\ninst✝¹ : PseudoMetrizableSpace ε\nf : ℝ → ε\na b : ℝ\nμ : Measure ℝ\ninst✝ : NoAtoms μ\nhab : a ≤ b\nha : ‖f a‖ₑ ≠ ∞\nhb : ‖f b‖ₑ ≠ ∞\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ",
"usedConstants": [
"Eq.mpr... | rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab ha,
integrableOn_Icc_iff_integrableOn_Ioo ha hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 18
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_4\ninst✝ : Countable ι\nF : ι → α → E\nhF_int : ∀ (i : ι), Integrable (F i) μ\nhF_sum : Summable fun i ↦ ∫ (a : α), ‖F i a‖ ∂μ\nhE : CompleteSpace E\nthis : ∀ ... | rw [funext this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 924,
"column": 2
} | {
"line": 924,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nα : Type u_4\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : OpensMeasurableSpace α\ninst✝⁵ : CompleteSpace F\ninst✝⁴ : LocallyCompactSpace 𝕜\ninst✝³ : Measu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.Prod | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : ContinuousAdd E\ninst✝⁵ : ContinuousConstSMul 𝕜 E\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module 𝕜 F\ninst✝² : TopologicalSpace F\ninst✝¹ : ContinuousAdd F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.Prod | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : ContinuousAdd E\ninst✝⁵ : ContinuousConstSMul 𝕜 E\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module 𝕜 F\ninst✝² : TopologicalSpace F\ninst✝¹ : ContinuousAdd F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 35
} | [
{
"pp": "a b : ℝ\nμ : Measure ℝ\n𝕜 : Type u_8\nf : ℝ → 𝕜\ninst✝ : NormedDivisionRing 𝕜\nh : IntervalIntegrable f μ a b\nc : 𝕜\n⊢ IntervalIntegrable (fun x ↦ f x / c) μ a b",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"MeasureTheory.Measure",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 235,
"column": 59
} | {
"line": 235,
"column": 84
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nl : Filter ι\ninst✝¹ : l.IsCountablyGenerated\nF : ι → ℝ → E\ninst✝ : IsLocallyFiniteMeasure μ\nhF : ∀ᶠ (i : ι) in l, ContinuousOn (F i) [[a, b]]\nh_lim : TendstoUniformlyOn F f l [[a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 408,
"column": 20
} | {
"line": 408,
"column": 36
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nc : ℝ\nhc : c ≠ 0\nh✝ : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\nh : IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c)\n⊢ IntervalIntegrable f volume a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 29
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntervalIntegrable f volume a b\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (x * c)) volume (a / c) (b / c)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 275,
"column": 4
} | {
"line": 276,
"column": 11
} | [
{
"pp": "case pos.h_lim\nι : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : Countable ι\nf : ι → C(ℝ, E)\nhf_sum : Summable fun i ↦ ‖ContinuousMap.restrict (↑{ carrier := [[a, b]], isCompact' := ⋯ }) (f i)‖\nhE : CompleteSpace E\nx✝ : ℝ\nhx : x✝ ∈ Ι a b\nx : ↥{... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 438,
"column": 14
} | {
"line": 438,
"column": 25
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc : ℝ\nh : ‖f (min a b + c)‖ₑ ≠ ∞\nhf : IntervalIntegrable (fun x ↦ f (x + c)) volume a b\n⊢ IntervalIntegrable f volume (a + c) (b + c)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 15
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc : ℝ\nh : ‖f (min a b + c)‖ₑ ≠ ∞\nhf : IntervalIntegrable f volume (a + c) (b + c)\nthis : ‖f (min (a + c) (b + c))‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (x + c)) volume a b",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 24
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntervalIntegrable f volume a b\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (c + x)) volume (a - c) (b - c)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 35
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntervalIntegrable f volume a b\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (x - c)) volume (a + c) (b + c)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 476,
"column": 2
} | {
"line": 476,
"column": 46
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nhf : IntervalIntegrable f volume a b\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (c - x)) volume (c - a) (c - b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 482,
"column": 14
} | {
"line": 482,
"column": 25
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nc : ℝ\nh✝ : ‖f (min a b)‖ₑ ≠ ∞\nh : IntervalIntegrable (fun x ↦ f (c - x)) volume (c - a) (c - b)\n⊢ IntervalIntegrable f volume a b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 15
} | [
{
"pp": "case h\nF : Type u_2\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x ↦ (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nx : ℝ\nhx : x ∈ {c}ᶜ\n⊢ HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 21
} | [
{
"pp": "F : Type u_2\ninst✝ : NormedAddCommGroup F\nf : ℝ → F\na b c : ℝ\nhf : (fun x ↦ (x - c)⁻¹) =O[𝓝[≠] c] f\nhne : a ≠ b\nhc : c ∈ [[a, b]]\nA : ∀ᶠ (x : ℝ) in 𝓝[≠] c, HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x\n⊢ Tendsto (fun x ↦ x - c) (𝓝[≠] c) (𝓝[≠] 0)",
"usedConstants": [
"Eq.mpr",
... | rw [← sub_self c] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 63
} | [
{
"pp": "a : ℝ\n⊢ ¬IntegrableOn (fun x ↦ x⁻¹) (Ioi a) volume",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"Set.Ioi",
"MeasureTheory.Measure",
"Set.Ici",
"congrArg",
"Real.instInv",
"MeasureTheor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 783,
"column": 2
} | {
"line": 783,
"column": 35
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf g : ℝ → E\nμ : Measure ℝ\nhf : IntervalIntegrable f μ a b\nhg : IntervalIntegrable g μ a b\n⊢ ∫ (x : ℝ) in a..b, f x - g x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in a..b, g x ∂μ",
"usedConstants": [
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 814,
"column": 2
} | {
"line": 814,
"column": 31
} | [
{
"pp": "a b : ℝ\nμ : Measure ℝ\n𝕜 : Type u_8\ninst✝ : RCLike 𝕜\nr : 𝕜\nf : ℝ → 𝕜\n⊢ ∫ (x : ℝ) in a..b, f x * r ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) * r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 35
} | [
{
"pp": "a b : ℝ\nμ : Measure ℝ\n𝕜 : Type u_8\ninst✝ : RCLike 𝕜\nr : 𝕜\nf : ℝ → 𝕜\n⊢ ∫ (x : ℝ) in a..b, f x / r ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) / r",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"DivInvMonoid.toInv",
"MeasureTheory.Measure",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 909,
"column": 2
} | {
"line": 909,
"column": 31
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\n⊢ ∫ (x : ℝ) in a..b, f (c * x) = c⁻¹ • ∫ (x : ℝ) in c * a..c * b, f x",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 919,
"column": 2
} | {
"line": 919,
"column": 28
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\n⊢ ∫ (x : ℝ) in a..b, f (x / c) = c • ∫ (x : ℝ) in a / c..b / c, f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 938,
"column": 2
} | {
"line": 938,
"column": 31
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (d + x) = ∫ (x : ℝ) in d + a..d + b, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.MeasureSpace.toMe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 963,
"column": 2
} | {
"line": 963,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (x / c + d) = c • ∫ (x : ℝ) in a / c + d..b / c + d, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 973,
"column": 2
} | {
"line": 973,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (d + x / c) = c • ∫ (x : ℝ) in d + a / c..d + b / c, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 983,
"column": 2
} | {
"line": 983,
"column": 35
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (c * x - d) = c⁻¹ • ∫ (x : ℝ) in c * a - d..c * b - d, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1005,
"column": 2
} | {
"line": 1005,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (x / c - d) = c • ∫ (x : ℝ) in a / c - d..b / c - d, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1015,
"column": 2
} | {
"line": 1015,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b c : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (d - x / c) = c • ∫ (x : ℝ) in d - b / c..d - a / c, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"MeasureTheory.Measure",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1024,
"column": 2
} | {
"line": 1024,
"column": 35
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (x - d) = ∫ (x : ℝ) in a - d..b - d, f x",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congrArg",
"Real.instSub",
"AddMonoid... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1028,
"column": 2
} | {
"line": 1028,
"column": 47
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nd : ℝ\n⊢ ∫ (x : ℝ) in a..b, f (d - x) = ∫ (x : ℝ) in d - b..d - a, f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1032,
"column": 2
} | {
"line": 1032,
"column": 29
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\n⊢ ∫ (x : ℝ) in a..b, f (-x) = ∫ (x : ℝ) in -b..-a, f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1052,
"column": 2
} | {
"line": 1052,
"column": 36
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nh : EqOn f g [[a, b]]\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ",
"usedConstants": [
"Real",
"le_total",
"Real.linearOrder"
]
}
] | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1053,
"column": 4
} | {
"line": 1053,
"column": 53
} | [
{
"pp": "case inl\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nh : EqOn f g [[a, b]]\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1053,
"column": 4
} | {
"line": 1053,
"column": 53
} | [
{
"pp": "case inr\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nh : EqOn f g [[a, b]]\nhab : b ≤ a\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"NegZeroClass.toNeg",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 468,
"column": 8
} | {
"line": 468,
"column": 34
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : TopologicalSpace X\nμ : Measure ℝ\ninst✝ : FirstCountableTopology X\nF : X → ℝ → E\nbound : ℝ → ℝ\na b a₀ b₀ : ℝ\nx₀ : X\nhF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nh_bound : ∀ᶠ (x :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 13
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc : E\nlb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v : ι → ℝ\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : FTCFilter b lb lb'\nhab : IntervalIntegrable f μ a b\nhmeas : StronglyMeasurableAtFilt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 13
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc : E\nla la' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v : ι → ℝ\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : FTCFilter a la la'\nhab : IntervalIntegrable f μ a b\nhmeas : StronglyMeasurableAtFilt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 30
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : ℝ → E\nc : E\nl l' : Filter ℝ\nlt : Filter ι\na : ℝ\ninst✝ : FTCFilter a l l'\nhfm : StronglyMeasurableAtFilter f l' volume\nhf : Tendsto f (l' ⊓ ae volume) (𝓝 c)\nu v : ι → ℝ\nhu : Tends... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 517,
"column": 2
} | {
"line": 518,
"column": 9
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : ℝ → E\nca cb : E\nla la' lb lb' : Filter ℝ\nlt : Filter ι\na b : ℝ\nua ub va vb : ι → ℝ\ninst✝¹ : FTCFilter a la la'\ninst✝ : FTCFilter b lb lb'\nhab : IntervalIntegrable f volume a b\nhme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 57
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : ℝ → E\nc : E\nlb lb' : Filter ℝ\nlt : Filter ι\na b : ℝ\nu v : ι → ℝ\ninst✝ : FTCFilter b lb lb'\nhab : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f lb' volume\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 44
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\nh_int : IntervalIntegrable f μ b₁ b₂\nha : a ∈ [[b₁, b₂]]\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ [[b₁, b₂]]\n⊢ IntervalIntegrable f μ (min b₁ b₂) (max b₁ b₂)",
"usedConstants": [
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 57
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : ℝ → E\nc : E\nla la' : Filter ℝ\nlt : Filter ι\na b : ℝ\nu v : ι → ℝ\ninst✝ : FTCFilter a la la'\nhab : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f la' volume\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 642,
"column": 2
} | {
"line": 642,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nc : E\na b : ℝ\nhf : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f (𝓝 a) volume\nha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)\n⊢ HasStrictDerivAt (fun u ↦ ∫ (x : ℝ) in u..b, f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 652,
"column": 2
} | {
"line": 652,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\na b : ℝ\nhf : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f (𝓝 a) volume\nha : ContinuousAt f a\n⊢ HasStrictDerivAt (fun u ↦ ∫ (x : ℝ) in u..b, f x) (-f a) a",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 60
} | [
{
"pp": "case h\nE : 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[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalS... | rintro ⟨p, s⟩ ⟨hp : p ∈ v, hs : s ∈ Ioo (b₀ - δ) (b₀ + δ)⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1176,
"column": 4
} | {
"line": 1176,
"column": 28
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhf : IntegrableOn f (Ici a) μ\nhab : a ≤ b\nha : IntegrableOn f (Ici b) μ\nh : IntegrableOn f (Ico a b) μ\n⊢ ∫ (x : ℝ) in Ici a, f x ∂μ = ∫ (x : ℝ) in Ico a b ∪ Ici b, f x ∂μ",
"usedConstants": ... | Ico_union_Ici_eq_Ici hab | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 49,
"column": 2
} | {
"line": 50,
"column": 63
} | [
{
"pp": "case h\nK : Type u_1\ninst✝⁵ : Field K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsStrictOrderedRing K\ninst✝² : FloorSemiring K\ninst✝¹ : TopologicalSpace K\ninst✝ : OrderTopology K\na c : K\nd : ℕ\nε : K\nhε : ε < 0\nn : ℕ\nh : a * c ^ n / ε < ↑(n - d)!\n⊢ ε < a * c ^ n / ↑(n - d)!",
"usedConstants": [
... | · rw [div_lt_iff_of_neg hε] at h
rwa [lt_div_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 79,
"column": 40
} | {
"line": 79,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝³ : Ring α\ninst✝² : LinearOrder α\ninst✝¹ : FloorRing α\ninst✝ : IsStrictOrderedRing α\nb : ℤ\n⊢ b - 1 ≤ b",
"usedConstants": [
"Int.instNeZeroOfNatOfNat",
"AddGroupWithOne.toAddGroup",
"Int.instLinearOrder",
"instIsLeftCancelAddOfAddLeftReflectLE",
... | exact (sub_one_lt _).le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto floor (𝓝[≥] ↑n) (pure n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1205,
"column": 2
} | {
"line": 1205,
"column": 36
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nμ : Measure ℝ\na b : ℝ\nh : support f ⊆ Ioc a b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ",
"usedConstants": [
"Real",
"le_total",
"Real.linearOrder"
]
}
] | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 33
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto ceil (𝓝[≤] ↑n) (pure n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 33
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto floor (𝓝[<] ↑n) (pure (n - 1))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto ceil (𝓝[>] ↑n) (pure (n + 1))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1258,
"column": 2
} | {
"line": 1258,
"column": 36
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᶠ[ae (μ.restrict (Ioc a b ∪ Ioc b a))] f\nhfi : IntervalIntegrable f μ a b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᶠ[ae (μ.restrict (Ioc a b ∪ Ioc b a))] 0",
"usedConstants": [
"Real",
"le_total",
"Real.linearOrder"
]
}
] | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 216,
"column": 12
} | {
"line": 216,
"column": 31
} | [
{
"pp": "case inl.left.refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁷ : Ring α\ninst✝⁶ : LinearOrder α\ninst✝⁵ : FloorRing α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : β → α → γ\nh : ContinuousO... | nhdsWithin_prod_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 207,
"column": 84
} | {
"line": 224,
"column": 81
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁷ : Ring α\ninst✝⁶ : LinearOrder α\ninst✝⁵ : FloorRing α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : β → α → γ\nh : ContinuousOn (uncurry f) (univ ×ˢ I... | by
change Continuous (uncurry f ∘ Prod.map id fract)
rw [continuous_iff_continuousAt]
rintro ⟨s, t⟩
rcases em (∃ n : ℤ, t = n) with (⟨n, rfl⟩ | ht)
· rw [ContinuousAt, nhds_prod_eq, ← nhdsLT_sup_nhdsGE (n : α), prod_sup, tendsto_sup]
constructor
· refine (((h (s, 1) ⟨trivial, zero_le_one, le_rfl⟩).ten... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 69
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc t (t + T)) μ",
"usedConstants": [
"Set.Ioc",
"nullMeasurableSet_Ioc",
"instClosedIicTopology",
"Real",
"Real.lattice",
"instHasSolidNormReal",
"PartialOrder.toPreorder"... | refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 31
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\nx : ℝ\nthis : Bijective (codRestrict (fun n ↦ n • T) ↑(zmultiples T) ⋯)\n⊢ ∃! x_1, codRestrict (fun n ↦ n • T) ↑(zmultiples T) ⋯ x_1 +ᵥ x ∈ Ioc t (t + T)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 69
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\n⊢ IsAddFundamentalDomain (↥(zmultiples T).op) (Ioc t (t + T)) μ",
"usedConstants": [
"Set.Ioc",
"nullMeasurableSet_Ioc",
"instClosedIicTopology",
"Real",
"Real.lattice",
"instHasSolidNormReal",
"PartialOrder.toPreord... | refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 13
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\nx : ℝ\nthis : Bijective (codRestrict (fun n ↦ n • T) ↑(zmultiples T) ⋯)\n⊢ ∃! x_1, (⇑(zmultiples T).equivOp ∘ codRestrict (fun n ↦ n • T) ↑(zmultiples T) ⋯) x_1 +ᵥ x ∈ Ioc t (t + T)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Set.Ioc",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1328,
"column": 4
} | {
"line": 1329,
"column": 11
} | [
{
"pp": "case hle\nf g : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhgc : ContinuousOn g (Icc a b)\nhle : ∀ x ∈ Ioc a b, f x ≤ g x\nhlt : ∃ c ∈ Icc a b, f c < g c\n⊢ f ≤ᶠ[ae (volume.restrict (Ioc a b))] g",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Set.Ioc",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 70
} | [
{
"pp": "case hs\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := ⋯\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\n⊢ I ⊆ Metric.closedBall 0 (|T| / 2)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.lattice",
"Real.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1341,
"column": 2
} | {
"line": 1341,
"column": 39
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhf : 0 ≤ᶠ[ae (μ.restrict (Icc a b))] f\nH : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), 0 x ≤ f x := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf\n⊢ 0 ≤ ∫ (u : ℝ) in a..b, f u ∂μ",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"InnerP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1354,
"column": 2
} | {
"line": 1354,
"column": 39
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\n⊢ |∫ (x : ℝ) in a..b, f x ∂μ| ≤ ∫ (x : ℝ) in a..b, |f x| ∂μ",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instRCLike",
"abs",
"intervalIntegral",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1386,
"column": 2
} | {
"line": 1386,
"column": 39
} | [
{
"pp": "f g : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhf : IntervalIntegrable f μ a b\nhg : IntervalIntegrable g μ a b\nh : f ≤ᶠ[ae (μ.restrict (Icc a b))] g\nH : ∀ᵐ (x : ℝ) ∂μ.restrict (Ioc a b), f x ≤ g x :=\n Eventually.filter_mono (ae_mono (Measure.restrict_mono Ioc_subset_Icc_self (le_refl μ))) h\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1389,
"column": 2
} | {
"line": 1389,
"column": 39
} | [
{
"pp": "f g : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhf : IntervalIntegrable f μ a b\nhg : IntervalIntegrable g μ a b\nh : f ≤ᶠ[ae μ] g\n⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"InnerProductSpace.toNormedSpace",
"Real.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1133,
"column": 2
} | {
"line": 1133,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf f' : ℝ → E\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhint : IntervalIntegrable f' volume a b\n⊢ ∫ (y : ℝ) in a..b, f' y = f b -... | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
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