module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 38
} | {
"line": 762,
"column": 39
} | [
{
"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",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Set.Ici",
"Re... | [
"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 (Ici x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 18
} | {
"line": 109,
"column": 2
} | [
{
"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 : ∀ ... | [
"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 : ∀ (i : ι), ∫⁻ ... | rw [funext this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 140,
"column": 2
} | {
"line": 154,
"column": 93
} | {
"line": 156,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : ℕ → Set α\nf : α → E\nhsm : ∀ (i : ℕ), MeasurableSet (s i)\nh_anti : Antitone s\nhfi : IntegrableOn f (s 0) μ\n⊢ Tendsto (fun i ↦ ∫ (a : α) in s i, f a ∂μ) atTop (𝓝 (∫ (a ... | [] | let bound : α → ℝ := indicator (s 0) fun a => ‖f a‖
have h_int_eq : (fun i => ∫ a in s i, f a ∂μ) = fun i => ∫ a, (s i).indicator f a ∂μ :=
funext fun i => (integral_indicator (hsm i)).symm
rw [h_int_eq]
rw [← integral_indicator (MeasurableSet.iInter hsm)]
refine tendsto_integral_of_dominated_convergence bo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 140,
"column": 2
} | {
"line": 154,
"column": 93
} | {
"line": 156,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : ℕ → Set α\nf : α → E\nhsm : ∀ (i : ℕ), MeasurableSet (s i)\nh_anti : Antitone s\nhfi : IntegrableOn f (s 0) μ\n⊢ Tendsto (fun i ↦ ∫ (a : α) in s i, f a ∂μ) atTop (𝓝 (∫ (a ... | [] | let bound : α → ℝ := indicator (s 0) fun a => ‖f a‖
have h_int_eq : (fun i => ∫ a in s i, f a ∂μ) = fun i => ∫ a, (s i).indicator f a ∂μ :=
funext fun i => (integral_indicator (hsm i)).symm
rw [h_int_eq]
rw [← integral_indicator (MeasurableSet.iInter hsm)]
refine tendsto_integral_of_dominated_convergence bo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 293,
"column": 6
} | {
"line": 293,
"column": 48
} | {
"line": 293,
"column": 49
} | [
{
"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 =ᵐ[μ.withDensity fun x ↦ ↑(f x)] v\n... | [
"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 =ᵐ[μ.withDensity fun x ↦ ↑(f x)] v\na✝ : Integra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 195,
"column": 59
} | {
"line": 195,
"column": 84
} | {
"line": 195,
"column": 85
} | [
{
"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... | [
"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, b]]\nhl : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.Prod | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 13
} | {
"line": 38,
"column": 14
} | [
{
"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... | [
"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\ninst✝ : Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 235,
"column": 4
} | {
"line": 236,
"column": 11
} | {
"line": 236,
"column": 12
} | [
{
"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 : ↥{... | [
"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 : ↥{ carrier := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.TangentCone.Prod | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 13
} | {
"line": 48,
"column": 14
} | [
{
"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... | [
"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\ninst✝ : Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 48
} | {
"line": 156,
"column": 0
} | [
{
"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✝ : NullSingletonClass μ\nhab : a ≤ b\nha : ‖f a‖ₑ ≠ ∞\nhb : ‖f b‖ₑ ≠ ∞\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ",
"ppTerm": "?m.29",
... | [] | 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": 153,
"column": 2
} | {
"line": 154,
"column": 48
} | {
"line": 156,
"column": 0
} | [
{
"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✝ : NullSingletonClass μ\nhab : a ≤ b\nha : ‖f a‖ₑ ≠ ∞\nhb : ‖f b‖ₑ ≠ ∞\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ",
"ppTerm": "?m.29",
... | [] | 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": 153,
"column": 2
} | {
"line": 154,
"column": 48
} | {
"line": 156,
"column": 0
} | [
{
"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✝ : NullSingletonClass μ\nhab : a ≤ b\nha : ‖f a‖ₑ ≠ ∞\nhb : ‖f b‖ₑ ≠ ∞\n⊢ IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ",
"ppTerm": "?m.29",
... | [] | 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.IntervalIntegral.Basic | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 35
} | {
"line": 391,
"column": 36
} | [
{
"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",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"M... | [
"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⁻¹) μ a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 417,
"column": 20
} | {
"line": 417,
"column": 36
} | {
"line": 417,
"column": 37
} | [
{
"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",
"ppTerm": "?m.61",
"assigned": false,
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 29
} | {
"line": 424,
"column": 30
} | [
{
"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)",
... | [
"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 (c * x)) volume (a / c) (b / c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 435,
"column": 2
} | {
"line": 435,
"column": 13
} | {
"line": 436,
"column": 4
} | [
{
"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... | [
"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 : StronglyMeasurableAtFilter f lb' μ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 13
} | {
"line": 455,
"column": 4
} | [
{
"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... | [
"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 : StronglyMeasurableAtFilter f la' μ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 515,
"column": 2
} | {
"line": 516,
"column": 9
} | {
"line": 516,
"column": 10
} | [
{
"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... | [
"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\nhmeas_a : Stron... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 36
} | {
"line": 641,
"column": 4
} | [
{
"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... | [
"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 x) (-c) a"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 447,
"column": 14
} | {
"line": 447,
"column": 25
} | {
"line": 447,
"column": 26
} | [
{
"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)",
"ppTerm": "?m.42",
"assigned": ... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 36
} | {
"line": 650,
"column": 37
} | [
{
"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",
"ppTerm... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 450,
"column": 4
} | {
"line": 450,
"column": 15
} | {
"line": 450,
"column": 16
} | [
{
"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",
... | [
"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": 455,
"column": 2
} | {
"line": 455,
"column": 24
} | {
"line": 455,
"column": 25
} | [
{
"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)",
"ppTerm": "?m.35",
"assigned": fals... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 46
} | {
"line": 485,
"column": 47
} | [
{
"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)",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 491,
"column": 14
} | {
"line": 491,
"column": 25
} | {
"line": 491,
"column": 26
} | [
{
"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",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 823,
"column": 2
} | {
"line": 823,
"column": 31
} | {
"line": 823,
"column": 32
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 828,
"column": 2
} | {
"line": 828,
"column": 35
} | {
"line": 828,
"column": 36
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"DivInvMonoi... | [
"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⁻¹ ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) * r⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 428,
"column": 8
} | {
"line": 428,
"column": 34
} | {
"line": 428,
"column": 35
} | [
{
"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 :... | [
"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 : X) in 𝓝 x₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 935,
"column": 2
} | {
"line": 935,
"column": 31
} | {
"line": 935,
"column": 32
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSem... | [
"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⁻¹ • ∫ (x : ℝ) in a * c..b * c, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 896,
"column": 2
} | {
"line": 896,
"column": 61
} | {
"line": 897,
"column": 2
} | [
{
"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... | [
"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 : MeasurableSpace α\... | 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": 964,
"column": 2
} | {
"line": 964,
"column": 31
} | {
"line": 964,
"column": 32
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"cong... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 989,
"column": 2
} | {
"line": 989,
"column": 44
} | {
"line": 989,
"column": 45
} | [
{
"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",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instH... | [
"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⁻¹ * x + d) = c • ∫ (x : ℝ) in c⁻¹ * a + d..c⁻¹ * b + d, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 999,
"column": 2
} | {
"line": 999,
"column": 44
} | {
"line": 999,
"column": 45
} | [
{
"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",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instH... | [
"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 + c⁻¹ * x) = c • ∫ (x : ℝ) in d + c⁻¹ * a..d + c⁻¹ * b, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1009,
"column": 2
} | {
"line": 1009,
"column": 35
} | {
"line": 1009,
"column": 36
} | [
{
"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",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ins... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 924,
"column": 2
} | {
"line": 924,
"column": 46
} | {
"line": 924,
"column": 47
} | [
{
"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... | [
"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 : MeasurableSpace �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 44
} | {
"line": 466,
"column": 45
} | [
{
"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✝ : NullSingletonClass 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)",
"ppTerm": "?m.58"... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\nh_int : IntervalIntegrable f μ b₁ b₂\nha : a ∈ [[b₁, b₂]]\nx✝¹ : ℝ\nx✝ : x✝¹ ∈ [[b₁, b₂]]\n⊢ IntegrableOn f (Ioc (min b₁ b₂) (max b₁ b₂)) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1031,
"column": 2
} | {
"line": 1031,
"column": 44
} | {
"line": 1031,
"column": 45
} | [
{
"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",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instH... | [
"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⁻¹ * x - d) = c • ∫ (x : ℝ) in c⁻¹ * a - d..c⁻¹ * b - d, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1041,
"column": 2
} | {
"line": 1041,
"column": 44
} | {
"line": 1041,
"column": 45
} | [
{
"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",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instH... | [
"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 - c⁻¹ * x) = c • ∫ (x : ℝ) in d - c⁻¹ * b..d - c⁻¹ * a, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1050,
"column": 2
} | {
"line": 1050,
"column": 35
} | {
"line": 1050,
"column": 36
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"cong... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1054,
"column": 2
} | {
"line": 1054,
"column": 47
} | {
"line": 1054,
"column": 48
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1058,
"column": 2
} | {
"line": 1058,
"column": 29
} | {
"line": 1058,
"column": 30
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1078,
"column": 2
} | {
"line": 1078,
"column": 36
} | {
"line": 1078,
"column": 37
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real",
"PartialOrder.toPreorder",
... | [
"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",
"case inr\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → E\nμ : Meas... | 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": 1079,
"column": 4
} | {
"line": 1079,
"column": 53
} | {
"line": 1080,
"column": 6
} | [
{
"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",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Se... | [
"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 Ioc a b, f x ∂μ = ∫ (x : ℝ) in Ioc a b, g x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1079,
"column": 4
} | {
"line": 1079,
"column": 53
} | {
"line": 1080,
"column": 6
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Se... | [
"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 Ioc b a, f x ∂μ = ∫ (x : ℝ) in Ioc b a, g x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 60
} | {
"line": 545,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NullSingletonClass μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologic... | [
"E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NullSingletonClass μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Fu... | 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.Analysis.SpecialFunctions.NonIntegrable | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 15
} | {
"line": 178,
"column": 16
} | [
{
"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]]\nx : ℝ\nhx : x ∈ {c}ᶜ\n⊢ HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x",
"ppTerm": "?m.83",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"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]]\nx : ℝ\nhx : x ∈ {c}ᶜ\n⊢ HasDerivAt (fun x ↦ Real.log (x - c)) (x - c)⁻¹ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 21
} | {
"line": 182,
"column": 4
} | [
{
"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)",
"ppTerm": "?m.154",
"assigned":... | [
"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[≠] (c - c))"
] | 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
} | {
"line": 223,
"column": 64
} | [
{
"pp": "a : ℝ\n⊢ ¬IntegrableOn (fun x ↦ x⁻¹) (Ioi a) volume",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"Set.Ioi",
"MeasureTheory.Measure",
"Set.Ici",
"congrAr... | [
"a : ℝ\n⊢ ¬Integrable (fun x ↦ x⁻¹) (volume.restrict (Ici a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1131,
"column": 2
} | {
"line": 1131,
"column": 36
} | {
"line": 1132,
"column": 2
} | [
{
"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 -... | [
"case inl\nE : 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\nhab : a ≤ b\n⊢ ∫ (y : ℝ) in a..b, f... | 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.FundThmCalculus | {
"line": 1170,
"column": 2
} | {
"line": 1170,
"column": 34
} | {
"line": 1170,
"column": 35
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf f' : ℝ → E\nhab : a < b\nfa fb : E\nhderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhint : IntervalIntegrable f' volume a b\nha : Tendsto f (𝓝[>] a) (𝓝 fa)\nhb : Tendsto f (𝓝[<] b) (𝓝 fb)\nF : ... | [
"E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf f' : ℝ → E\nhab : a < b\nfa fb : E\nhderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhint : IntervalIntegrable f' volume a b\nha : Tendsto f (𝓝[>] a) (𝓝 fa)\nhb : Tendsto f (𝓝[<] b) (𝓝 fb)\nF : ℝ → E := upd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1192,
"column": 2
} | {
"line": 1192,
"column": 36
} | {
"line": 1193,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf : ℝ → E\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x\nhint : IntervalIntegrable (deriv f) volume a b\n⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a",
"ppTerm"... | [
"case inl\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf : ℝ → E\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x\nhint : IntervalIntegrable (deriv f) volume a b\nhab : a ≤ b\n⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a",
... | 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": 1202,
"column": 4
} | {
"line": 1202,
"column": 28
} | {
"line": 1202,
"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",
"ppTerm": "?m.126... | [
"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 Ici a, f x ∂μ"
] | Ico_union_Ici_eq_Ici hab | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1231,
"column": 2
} | {
"line": 1231,
"column": 36
} | {
"line": 1232,
"column": 2
} | [
{
"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",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Real",
"PartialOrder.toPreorder",
"i... | [
"case inl\nE : 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\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ",
"case inr\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nμ : Measure ℝ\na b... | 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.FundThmCalculus | {
"line": 1242,
"column": 4
} | {
"line": 1242,
"column": 82
} | {
"line": 1242,
"column": 83
} | [
{
"pp": "g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x\nhab : a < b\nmeas_g' : AEMeasurable g' (volume.restrict (Ioo a b))\nH : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊\nf : SimpleFunc ℝ ℝ... | [
"g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x\nhab : a < b\nmeas_g' : AEMeasurable g' (volume.restrict (Ioo a b))\nH : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊\nf : SimpleFunc ℝ ℝ≥0\nfle : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1294,
"column": 2
} | {
"line": 1294,
"column": 36
} | {
"line": 1294,
"column": 37
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᵐ[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 =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Set.Ioc",
... | [
"case inl\nf : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f\nhfi : IntervalIntegrable f μ a b\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0",
"case inr\nf : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f\nhfi : Inte... | 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.FundThmCalculus | {
"line": 1266,
"column": 2
} | {
"line": 1266,
"column": 36
} | {
"line": 1267,
"column": 2
} | [
{
"pp": "g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\n⊢ IntervalIntegrable g' volume a b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"IntervalIn... | [
"case inl\ng' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\nhab : a ≤ b\n⊢ IntervalIntegrable g' volume a b",
"case inr\ng' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo ... | 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.FundThmCalculus | {
"line": 1270,
"column": 2
} | {
"line": 1272,
"column": 56
} | {
"line": 1274,
"column": 0
} | [
{
"pp": "case inr\ng' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\nhab : b ≤ a\n⊢ IntervalIntegrable g' volume a b",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Is... | [] | · simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, IntervalIntegrable, Ioc_eq_empty_of_le,
integrableOn_empty, true_and] at hcont hderiv hpos ⊢
exact integrableOn_deriv_of_nonneg hcont hderiv hpos | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1265,
"column": 91
} | {
"line": 1272,
"column": 56
} | {
"line": 1274,
"column": 0
} | [
{
"pp": "g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\n⊢ IntervalIntegrable g' volume a b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toConti... | [] | by
rcases le_total a b with hab | hab
· simp only [uIcc_of_le, min_eq_left, max_eq_right, IntervalIntegrable, hab,
Ioc_eq_empty_of_le, integrableOn_empty, and_true] at hcont hderiv hpos ⊢
exact integrableOn_deriv_of_nonneg hcont hderiv hpos
· simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, Interv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 49,
"column": 2
} | {
"line": 50,
"column": 63
} | {
"line": 51,
"column": 2
} | [
{
"pp": "K : 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)!",
"ppTerm": "?m.93",
"assi... | [
"K : 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⊢ a * c ^ n / ↑(n - d)! < ε"
] | · 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
} | {
"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",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Int.instNeZeroOfNatOfNat",
"AddGroupWithOne.toAddGroup",
"Int.instLinearOrder",
"... | [] | 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
} | {
"line": 103,
"column": 35
} | [
{
"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)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 33
} | {
"line": 111,
"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)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 33
} | {
"line": 122,
"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 - 1))",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"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))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 34
} | {
"line": 133,
"column": 35
} | [
{
"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))",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"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))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1377,
"column": 2
} | {
"line": 1377,
"column": 39
} | {
"line": 1377,
"column": 40
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhf : 0 ≤ᵐ[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",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [
"f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhf : 0 ≤ᵐ[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 Ioc a b, f u ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1390,
"column": 2
} | {
"line": 1390,
"column": 39
} | {
"line": 1390,
"column": 40
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instRCLike",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1422,
"column": 2
} | {
"line": 1422,
"column": 39
} | {
"line": 1422,
"column": 40
} | [
{
"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 ≤ᵐ[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⊢ ∫ (u ... | [
"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 ≤ᵐ[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⊢ ∫ (u : ℝ) in Ioc ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1425,
"column": 2
} | {
"line": 1425,
"column": 39
} | {
"line": 1425,
"column": 40
} | [
{
"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 ≤ᵐ[μ] g\n⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"InnerP... | [
"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 ≤ᵐ[μ] g\n⊢ ∫ (u : ℝ) in Ioc a b, f u ∂μ ≤ ∫ (u : ℝ) in Ioc a b, g u ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 216,
"column": 12
} | {
"line": 216,
"column": 31
} | {
"line": 216,
"column": 32
} | [
{
"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... | [
"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 : ContinuousOn (uncurry f... | nhdsWithin_prod_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1430,
"column": 2
} | {
"line": 1430,
"column": 39
} | {
"line": 1430,
"column": 40
} | [
{
"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 : ∀ x ∈ Icc a b, f x ≤ g x\nH : ∀ x ∈ Ioc a b, f x ≤ g x := fun x hx ↦ h x (Ioc_subset_Icc_self hx)\n⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ",
"ppTerm": "?m.41",
... | [
"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 : ∀ x ∈ Icc a b, f x ≤ g x\nH : ∀ x ∈ Ioc a b, f x ≤ g x := fun x hx ↦ h x (Ioc_subset_Icc_self hx)\n⊢ ∫ (u : ℝ) in Ioc a b, f u ∂μ ≤ ∫ (u : ℝ) in Ioc a b, g u ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1462,
"column": 2
} | {
"line": 1462,
"column": 72
} | {
"line": 1463,
"column": 4
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nhfi : Integrable f volume\n⊢ HasSum (fun n ↦ ∫ (x : ℝ) in 0..1, f (x + ↑n)) (∫ (x : ℝ), f x)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"Meas... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nhfi : Integrable f volume\n⊢ HasSum (fun n ↦ ∫ (x : ℝ) in ↑n..↑n + 1, f x) (∫ (x : ℝ), f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 207,
"column": 84
} | {
"line": 224,
"column": 81
} | {
"line": 226,
"column": 0
} | [
{
"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": 45,
"column": 2
} | {
"line": 45,
"column": 69
} | {
"line": 46,
"column": 2
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc t (t + T)) μ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"nullMeasurableSet_Ioc",
"instClosedIicTopology",
"Real",
"Real.lattice",
"instHasSo... | [
"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\nx : ℝ\n⊢ ∃! g, g +ᵥ x ∈ Ioc t (t + T)"
] | refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 69
} | {
"line": 54,
"column": 2
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\n⊢ IsAddFundamentalDomain (↥(zmultiples T).op) (Ioc t (t + T)) μ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"nullMeasurableSet_Ioc",
"instClosedIicTopology",
"Real",
"Real.lattice",
"instHa... | [
"T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\nx : ℝ\n⊢ ∃! g, g +ᵥ x ∈ Ioc t (t + T)"
] | refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 70
} | {
"line": 114,
"column": 71
} | [
{
"pp": "T : 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)",
"ppTerm": "?m.222",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",... | [
"T : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\n⊢ I ⊆ Icc (-(T / 2)) (T / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 565,
"column": 87
} | {
"line": 573,
"column": 100
} | {
"line": 574,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NullSingletonClass μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologic... | [] | by
gcongr
· exact Eventually.of_forall (fun x ↦ norm_nonneg _)
· exact (hf.uncurry_left _).norm.integrableOn_Icc
· apply uIoc_subset_uIcc.trans (uIcc_subset_Icc ?_ ⟨hs.1.le, hs.2.le⟩ )
simp [δpos.le]
· exact Eventually.of_forall (fun x ↦ norm_nonneg _)
· exact ((hf.uncurry_le... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 328,
"column": 4
} | {
"line": 329,
"column": 11
} | {
"line": 329,
"column": 12
} | [
{
"pp": "case neg.refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume\nH : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0\n⊢ AEStronglyMeasurable (fun θ ↦ f (circleMa... | [
"case neg.refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume\nH : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0\n⊢ AEStronglyMeasurable (fun θ ↦ f (circleMap c R θ)) (v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 353,
"column": 4
} | {
"line": 358,
"column": 37
} | {
"line": 359,
"column": 4
} | [
{
"pp": "case mp\nc : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\n⊢ (fun x ↦ (x - θ)⁻¹) =O[𝓝[≠] θ] fun θ_1 ↦ (circleMap 0 R θ_1 * I) • (circleMap c R θ_1 - circleMap c R θ) ^ n",
"ppTerm": "?mp",
"assigned": true,
... | [
"case mp\nc : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\n⊢ (fun x ↦ (x - θ)⁻¹) =O[𝓝[≠] θ] fun θ_1 ↦ (circleMap 0 R θ_1 * I) • (circleMap c R θ_1 - circleMap c R θ) ^ n"
] | have : ∀ᶠ θ' in 𝓝[≠] θ, f θ' ∈ ball (0 : ℂ) 1 \ {0} := by
suffices ∀ᶠ z in 𝓝[≠] circleMap c R θ, z - circleMap c R θ ∈ ball (0 : ℂ) 1 \ {0} from
((differentiable_circleMap c R θ).hasDerivAt.tendsto_nhdsNE
(deriv_circleMap_ne_zero hR)).eventually this
filter_upwards [self_mem_nhdsWithin, ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 366,
"column": 6
} | {
"line": 367,
"column": 35
} | {
"line": 367,
"column": 36
} | [
{
"pp": "c : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis✝ : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\nθ' : ℝ\nhθ' : f θ' ∈ ball 0 1 \\ {0}\nx : ℝ := ‖f θ'‖\nthis : x⁻¹ ≤ x ^ n\n⊢ ‖circleMap c R θ' - circleMap c R θ‖⁻... | [
"c : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis✝ : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\nθ' : ℝ\nhθ' : f θ' ∈ ball 0 1 \\ {0}\nx : ℝ := ‖f θ'‖\nthis : x⁻¹ ≤ x ^ n\n⊢ ‖circleMap c R θ' - circleMap c R θ‖⁻¹ ≤ ‖circleM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 368,
"column": 35
} | {
"line": 368,
"column": 60
} | {
"line": 368,
"column": 61
} | [
{
"pp": "c : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\nθ' : ℝ\nhθ' : f θ' ∈ ball 0 1 \\ {0}\nx : ℝ := ‖f θ'‖\n⊢ x ∈ Ioo 0 1",
"ppTerm": "?m.355",
"assigned": true... | [
"c : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\nθ' : ℝ\nhθ' : f θ' ∈ ball 0 1 \\ {0}\nx : ℝ := ‖f θ'‖\n⊢ ¬f θ' = 0 ∧ ‖f θ'‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 15
} | {
"line": 310,
"column": 16
} | [
{
"pp": "case e'_4\nE✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max... | [
"case e'_4\nE✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 58
} | {
"line": 561,
"column": 59
} | [
{
"pp": "case inr\nn : ℤ\nc w : ℂ\nR : ℝ\nhn : n < 0\nhw : w ∈ sphere c |R|\nh0 : R ≠ 0\n⊢ ¬CircleIntegrable (fun z ↦ (z - w) ^ n) c R",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"circleIntegrable_sub_zpow_iff._simp_1",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSe... | [
"case inr\nn : ℤ\nc w : ℂ\nR : ℝ\nhn : n < 0\nhw : w ∈ sphere c |R|\nh0 : R ≠ 0\n⊢ ‖w - c‖ = |R|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 350,
"column": 6
} | {
"line": 350,
"column": 63
} | {
"line": 351,
"column": 8
} | [
{
"pp": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x ... | [
"case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 74
} | {
"line": 658,
"column": 6
} | [
{
"pp": "case refine_4\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nw : ℂ\nhf : CircleIntegrable f c R\nhw : ‖w‖ < R\nhR : 0 < R\nhwR : ‖w‖ / R ∈ Ico 0 1\n⊢ IntervalIntegrable (fun t ↦ ∑' (n : ℕ), ‖f (circleMap c R t)‖ * (‖w‖ / R) ^ n) volume 0 (2 * π)",
"p... | [
"case refine_4\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nw : ℂ\nhf : CircleIntegrable f c R\nhw : ‖w‖ < R\nhR : 0 < R\nhwR : ‖w‖ / R ∈ Ico 0 1\n⊢ IntervalIntegrable (fun t ↦ ‖f (circleMap c R t)‖ * (1 - ‖w‖ / R)⁻¹) volume 0 (2 * π)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 662,
"column": 49
} | {
"line": 662,
"column": 76
} | {
"line": 662,
"column": 77
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nw : ℂ\nhf : CircleIntegrable f c R\nhw : ‖w‖ < R\nhR : 0 < R\nhwR : ‖w‖ / R ∈ Ico 0 1\nθ : ℝ\nx✝ : θ ∈ Ι 0 (2 * π)\n⊢ ‖w / (circleMap c R θ - c)‖ < 1",
"ppTerm": "?m.474",
"assigned": true,
"usedC... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nw : ℂ\nhf : CircleIntegrable f c R\nhw : ‖w‖ < R\nhR : 0 < R\nhwR : ‖w‖ / R ∈ Ico 0 1\nθ : ℝ\nx✝ : θ ∈ Ι 0 (2 * π)\n⊢ ‖w‖ / R < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 693,
"column": 66
} | {
"line": 693,
"column": 77
} | {
"line": 693,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ≥0\nhf : CircleIntegrable f c ↑R\nhR : 0 < R\ny✝ : ℂ\nhy : y✝ ∈ eball 0 ↑R\n⊢ ‖y✝‖ < ↑R",
"ppTerm": "?m.80",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ≥0\nhf : CircleIntegrable f c ↑R\nhR : 0 < R\ny✝ : ℂ\nhy : y✝ ∈ eball 0 ↑R\n⊢ ‖y✝‖ < ↑R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 705,
"column": 4
} | {
"line": 705,
"column": 59
} | {
"line": 706,
"column": 6
} | [
{
"pp": "c w : ℂ\nR : ℝ\nhw : w ∈ ball c R\nhR : 0 < R\nH : HasSum (fun n ↦ ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)⁻¹) (2 * ↑π * I)\nA : CircleIntegrable (fun x ↦ 1) c R\n⊢ HasSum (fun n ↦ ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)⁻¹) (∮ (z : ℂ) in C(c, R), (z - w)⁻¹)",
"ppTerm"... | [
"c w : ℂ\nR : ℝ\nhw : w ∈ ball c R\nhR : 0 < R\nH : HasSum (fun n ↦ ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)⁻¹) (2 * ↑π * I)\nA : CircleIntegrable (fun x ↦ 1) c R\n⊢ HasSum (fun n ↦ ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)⁻¹) (∮ (z : ℂ) in C(c, R), (z - w)⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 15
} | {
"line": 30,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nμ : Measure α\np : ℝ≥0∞\nf : α → ε\nhf : MemLp f p μ\nν : Measure β\ninst✝ : IsFiniteMeasure ν\nhf' : MemLp f p (ν Set.univ • μ)\n⊢ MemLp f p (Measure.map P... | [
"α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nμ : Measure α\np : ℝ≥0∞\nf : α → ε\nhf : MemLp f p μ\nν : Measure β\ninst✝ : IsFiniteMeasure ν\nhf' : MemLp f p (ν Set.univ • μ)\n⊢ MemLp f p (ν Set.univ • μ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 15
} | {
"line": 31,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nμ : Measure α\np : ℝ≥0∞\nf : α → ε\nhf : MemLp f p μ\nν : Measure β\ninst✝ : IsFiniteMeasure ν\nhf' : MemLp f p (ν Set.univ • μ)\n⊢ AEStronglyMeasurable f (... | [
"α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nμ : Measure α\np : ℝ≥0∞\nf : α → ε\nhf : MemLp f p μ\nν : Measure β\ninst✝ : IsFiniteMeasure ν\nhf' : MemLp f p (ν Set.univ • μ)\n⊢ AEStronglyMeasurable f (ν Set.univ •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 15
} | {
"line": 39,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nν : Measure β\np : ℝ≥0∞\nf : β → ε\nhf : MemLp f p ν\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : SFinite ν\nhf' : MemLp f p (μ Set.univ • ν)\n⊢ MemL... | [
"α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nν : Measure β\np : ℝ≥0∞\nf : β → ε\nhf : MemLp f p ν\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : SFinite ν\nhf' : MemLp f p (μ Set.univ • ν)\n⊢ MemLp f p (μ Set... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 15
} | {
"line": 40,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nν : Measure β\np : ℝ≥0∞\nf : β → ε\nhf : MemLp f p ν\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : SFinite ν\nhf' : MemLp f p (μ Set.univ • ν)\n⊢ AESt... | [
"α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nν : Measure β\np : ℝ≥0∞\nf : β → ε\nhf : MemLp f p ν\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : SFinite ν\nhf' : MemLp f p (μ Set.univ • ν)\n⊢ AEStronglyMeasur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 653,
"column": 4
} | {
"line": 653,
"column": 20
} | {
"line": 653,
"column": 21
} | [
{
"pp": "case right\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\nha : a₀ ≤ a\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a₀ ∩ Ici a) a",
"ppTerm": "?right",
... | [
"case right\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\nha : a₀ ≤ a\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 10
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\nX : Type u_4\ninst✝¹ : TopologicalSpace X\nf : β → X\ninst✝ : SFinite ν\nhf : AEStronglyMeasurable (fun x ↦ f x.2) (μ.prod ν)\nhμ : μ ≠ 0\n⊢ AEStronglyMeasurable f ν",
"ppTerm": "?m.28"... | [] | have := NeZero.mk hμ
obtain ⟨y, hy⟩ := hf.prodMk_left.exists
exact hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 10
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\nX : Type u_4\ninst✝¹ : TopologicalSpace X\nf : β → X\ninst✝ : SFinite ν\nhf : AEStronglyMeasurable (fun x ↦ f x.2) (μ.prod ν)\nhμ : μ ≠ 0\n⊢ AEStronglyMeasurable f ν",
"ppTerm": "?m.28"... | [] | have := NeZero.mk hμ
obtain ⟨y, hy⟩ := hf.prodMk_left.exists
exact hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 678,
"column": 4
} | {
"line": 678,
"column": 20
} | {
"line": 678,
"column": 21
} | [
{
"pp": "case left\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ≤ a₀\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀ ∩ Iic a) a",
"ppTerm": "?left",
... | [
"case left\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NullSingletonClass μ\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ≤ a₀\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 191,
"column": 8
} | {
"line": 194,
"column": 35
} | {
"line": 195,
"column": 6
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x\nHd : ... | [] | calc
dist (f y₁) (f y₂) ≤ dist (f y₁) (f x) + dist (f y₂) (f x) := dist_triangle_right _ _ _
_ ≤ ε / 2 / 2 + ε / 2 / 2 := add_le_add (hδ₁ _ <| this hy₁) (hδ₁ _ <| this hy₂)
_ = ε / 2 := add_halves _ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 198,
"column": 8
} | {
"line": 198,
"column": 19
} | {
"line": 198,
"column": 20
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.... | [
"case refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x\nHd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 296,
"column": 39
} | {
"line": 296,
"column": 50
} | {
"line": 296,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\nR : Type u_4\ninst✝² : NormedRing R\ninst✝¹ : Module R E\ninst✝ : IsBoundedSMul R E\nf : α → R\ng : β → E\nhf : Integrable f μ\nhg : Integrable ... | [
"α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\nR : Type u_4\ninst✝² : NormedRing R\ninst✝¹ : Module R E\ninst✝ : IsBoundedSMul R E\nf : α → R\ng : β → E\nhf : Integrable f μ\nhg : Integrable g ν\n⊢ ∀ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 504,
"column": 9
} | {
"line": 504,
"column": 31
} | {
"line": 504,
"column": 32
} | [
{
"pp": "E : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nF : ℝ → ℝ → E\na b c d : ℝ\nh : IntegrableOn (uncurry F) (uIoc a b ×ˢ uIoc c d) volume\n⊢ Integrable (uncurry fun y x ↦ F x y) ((volume.restrict (uIoc c d)).prod (volume.restrict (uIoc a b)))",
"ppTerm": "?m.76",
"assigned": ... | [
"E : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nF : ℝ → ℝ → E\na b c d : ℝ\nh : IntegrableOn (uncurry F) (uIoc a b ×ˢ uIoc c d) volume\n⊢ Integrable ((uncurry fun y x ↦ F x y) ∘ Prod.swap) ((volume.restrict (uIoc a b)).prod (volume.restrict (uIoc c d)))"
] | ← integrable_swap_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 91
} | {
"line": 259,
"column": 92
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz w : ℂ\nHd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHi : IntegrableOn (fun z ↦ I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]]) volume\nx : ℂ\nhx : x ∈ Ioo (min z.re w.re) (max z... | [
"E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz w : ℂ\nHd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHi : IntegrableOn (fun z ↦ I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]]) volume\nx : ℂ\nhx : x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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