module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 65,
"column": 76
} | {
"line": 74,
"column": 67
} | {
"line": 76,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\n⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)",
"ppTerm": "?m.... | [] | by
rcases eq_or_ne p 0 with (rfl | hp)
· simp
rcases lt_or_ge p 1 with (h'p | h'p)
· simp only [eLpNorm_eq_eLpNorm' hp (h'p.trans ENNReal.one_lt_top).ne]
convert! eLpNorm'_add_le_of_le_one hf ENNReal.toReal_nonneg _
· have : p ∈ Set.Ioo (0 : ℝ≥0∞) 1 := ⟨hp.bot_lt, h'p⟩
simp only [LpAddConst, if_po... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalities | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 81
} | {
"line": 299,
"column": 0
} | [
{
"pp": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 < w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\n⊢ ∑ i ∈ s, w i * z i ≤ ∏ i ∈ s, z i ^ w i ↔ ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partia... | [] | exact geom_mean_le_arith_mean_weighted s w z (hw · · |>.le) hw' hz |>.ge_iff_eq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 50
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\np q : ℝ\ninst✝ : IsFiniteMeasure μ\nhf : AEStronglyMeasurable f μ\nhfq_lt_top : eLpNorm' f q μ < ∞\nhp_nonneg : 0 ≤ p\nhpq : p ≤ q\nhp_pos : 0 < p\n⊢ eLpNorm' ... | [
"case inr\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\np q : ℝ\ninst✝ : IsFiniteMeasure μ\nhf : AEStronglyMeasurable f μ\nhfq_lt_top : eLpNorm' f q μ < ∞\nhp_nonneg : 0 ≤ p\nhpq : p ≤ q\nhp_pos : 0 < p\nhq_pos : 0 < q\n⊢ eLpNo... | have hq_pos : 0 < q := lt_of_lt_of_le hp_pos hpq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 59,
"column": 2
} | {
"line": 63,
"column": 52
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : SeminormedAddGroup E\nι : Type u_3\ninst✝¹ : Nonempty ι\ninst✝ : LinearOrder ι\nf : ι → α → E\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))\n⊢ eLpNorm f_lim ∞ μ = essSup (fun x ↦ liminf (fun... | [] | rw [eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]
refine essSup_congr_ae (h_lim.mono fun x hx => ?_)
dsimp only
apply (Tendsto.liminf_eq ..).symm
exact (continuous_enorm.tendsto (f_lim x)).comp hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 59,
"column": 2
} | {
"line": 63,
"column": 52
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : SeminormedAddGroup E\nι : Type u_3\ninst✝¹ : Nonempty ι\ninst✝ : LinearOrder ι\nf : ι → α → E\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))\n⊢ eLpNorm f_lim ∞ μ = essSup (fun x ↦ liminf (fun... | [] | rw [eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]
refine essSup_congr_ae (h_lim.mono fun x hx => ?_)
dsimp only
apply (Tendsto.liminf_eq ..).symm
exact (continuous_enorm.tendsto (f_lim x)).comp hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 84
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α →ₘ[μ] E\n⊢ f ∈ Lp E p μ ↔ MemLp (↑f) p μ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"congrArg",
"M... | [] | simp [mem_Lp_iff_eLpNorm_lt_top, MemLp, f.stronglyMeasurable.aestronglyMeasurable] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 84
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α →ₘ[μ] E\n⊢ f ∈ Lp E p μ ↔ MemLp (↑f) p μ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"congrArg",
"M... | [] | simp [mem_Lp_iff_eLpNorm_lt_top, MemLp, f.stronglyMeasurable.aestronglyMeasurable] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 84
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α →ₘ[μ] E\n⊢ f ∈ Lp E p μ ↔ MemLp (↑f) p μ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"congrArg",
"M... | [] | simp [mem_Lp_iff_eLpNorm_lt_top, MemLp, f.stronglyMeasurable.aestronglyMeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 952,
"column": 4
} | {
"line": 952,
"column": 58
} | {
"line": 954,
"column": 0
} | [
{
"pp": "case hx\nι : Type u\ns : Finset ι\np : ℝ\nhp : 1 ≤ p\nw f : ι → ℝ\nhw : ∀ (i : ι), 0 ≤ w i\nhf : ∀ (i : ι), 0 ≤ f i\n⊢ 0 ≤ ∑ i ∈ s, w i",
"ppTerm": "?hx✝",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Finset",
"Membership.mem",
... | [] | exact sum_nonneg fun i _ ↦ by have := hw i; positivity | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.MeanInequalities | {
"line": 952,
"column": 4
} | {
"line": 952,
"column": 58
} | {
"line": 954,
"column": 0
} | [
{
"pp": "case hx\nι : Type u\ns : Finset ι\np : ℝ\nhp : 1 ≤ p\nw f : ι → ℝ\nhw : ∀ (i : ι), 0 ≤ w i\nhf : ∀ (i : ι), 0 ≤ f i\n⊢ 0 ≤ ∑ i ∈ s, w i",
"ppTerm": "?hx✝",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Finset",
"Membership.mem",
... | [] | exact sum_nonneg fun i _ ↦ by have := hw i; positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 952,
"column": 4
} | {
"line": 952,
"column": 58
} | {
"line": 954,
"column": 0
} | [
{
"pp": "case hx\nι : Type u\ns : Finset ι\np : ℝ\nhp : 1 ≤ p\nw f : ι → ℝ\nhw : ∀ (i : ι), 0 ≤ w i\nhf : ∀ (i : ι), 0 ≤ f i\n⊢ 0 ≤ ∑ i ∈ s, w i",
"ppTerm": "?hx✝",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Finset",
"Membership.mem",
... | [] | exact sum_nonneg fun i _ ↦ by have := hw i; positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 55
} | {
"line": 335,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PseudoEMetricSpace E\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nhfg : TendstoInMeasure μ f u g\nms : ℕ → ι\nhms1 : Tendsto ms atTop u\nhms2 : TendstoInMeasure μ (fun ... | [
"α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PseudoEMetricSpace E\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nhfg : TendstoInMeasure μ f u g\nms : ℕ → ι\nhms1 : Tendsto ms atTop u\nhms2 : TendstoInMeasure μ (fun n ↦ f (ms n)... | obtain ⟨ns, hns1, hns2⟩ := hms2.exists_seq_tendsto_ae | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 969,
"column": 46
} | {
"line": 969,
"column": 77
} | {
"line": 971,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nf : ↥(Lp ℝ p μ)\na : α\nh : ↑↑(negPart f) a = max (-↑↑f a) 0\n⊢ ↑↑(negPart f) a = -min (↑↑f a) 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Lattice.t... | [] | rw [h, ← max_neg_neg, neg_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 969,
"column": 46
} | {
"line": 969,
"column": 77
} | {
"line": 971,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nf : ↥(Lp ℝ p μ)\na : α\nh : ↑↑(negPart f) a = max (-↑↑f a) 0\n⊢ ↑↑(negPart f) a = -min (↑↑f a) 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Lattice.t... | [] | rw [h, ← max_neg_neg, neg_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 969,
"column": 46
} | {
"line": 969,
"column": 77
} | {
"line": 971,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nf : ↥(Lp ℝ p μ)\na : α\nh : ↑↑(negPart f) a = max (-↑↑f a) 0\n⊢ ↑↑(negPart f) a = -min (↑↑f a) 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Lattice.t... | [] | rw [h, ← max_neg_neg, neg_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 50,
"column": 40
} | {
"line": 57,
"column": 48
} | {
"line": 57,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : Rˣ\nt : R\nh : ‖t‖ < ‖↑x⁻¹‖⁻¹\n⊢ ‖-(↑x⁻¹ * t)‖ < 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",... | [] | by
nontriviality R using zero_lt_one
have hpos : 0 < ‖(↑x⁻¹ : R)‖ := Units.norm_pos x⁻¹
calc
‖-(↑x⁻¹ * t)‖ = ‖↑x⁻¹ * t‖ := by rw [norm_neg]
_ ≤ ‖(↑x⁻¹ : R)‖ * ‖t‖ := norm_mul_le (x⁻¹).1 _
_ < ‖(↑x⁻¹ : R)‖ * ‖(↑x⁻¹ : R)‖⁻¹ := by nlinarith only [h, hpos]
_ = 1 := mul_inv_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Span | {
"line": 108,
"column": 64
} | {
"line": 108,
"column": 69
} | {
"line": 108,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nx : E\nhx : ‖x‖ = 1\n⊢ x ≠ 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real.partialOrder... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Normed.Module.Span | {
"line": 108,
"column": 64
} | {
"line": 108,
"column": 69
} | {
"line": 108,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nx : E\nhx : ‖x‖ = 1\n⊢ x ≠ 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real.partialOrder... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Span | {
"line": 108,
"column": 64
} | {
"line": 108,
"column": 69
} | {
"line": 108,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nx : E\nhx : ‖x‖ = 1\n⊢ x ≠ 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real.partialOrder... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Span | {
"line": 114,
"column": 45
} | {
"line": 114,
"column": 50
} | {
"line": 114,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nx : E\nhx : ‖x‖ = 1\nr : 𝕜\n⊢ r • x ∈ 𝕜 ∙ x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"Sub... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Normed.Module.Span | {
"line": 114,
"column": 45
} | {
"line": 114,
"column": 50
} | {
"line": 114,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nx : E\nhx : ‖x‖ = 1\nr : 𝕜\n⊢ r • x ∈ 𝕜 ∙ x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"Sub... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Span | {
"line": 114,
"column": 45
} | {
"line": 114,
"column": 50
} | {
"line": 114,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nx : E\nhx : ‖x‖ = 1\nr : 𝕜\n⊢ r • x ∈ 𝕜 ∙ x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
"Sub... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 548,
"column": 4
} | {
"line": 549,
"column": 41
} | {
"line": 550,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\ne : E ≃L[𝕜] F\nO : (E →L[𝕜] F) → E →L[𝕜] E := ⋯\nh_O : Continuous ... | [] | rintro ⟨e', rfl⟩
exact ⟨(e'.trans e.symm).toUnit, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 548,
"column": 4
} | {
"line": 549,
"column": 41
} | {
"line": 550,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\ne : E ≃L[𝕜] F\nO : (E →L[𝕜] F) → E →L[𝕜] E := ⋯\nh_O : Continuous ... | [] | rintro ⟨e', rfl⟩
exact ⟨(e'.trans e.symm).toUnit, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Complement | {
"line": 115,
"column": 27
} | {
"line": 117,
"column": 51
} | {
"line": 119,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nM : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np q : Submodule R M\ninst✝ : ContinuousSub M\nh : IsTopCompl p q\n⊢ Continuous[inst✝³, inst✝³] ⇑(q.projection p ⋯)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": ... | [] | by
rw [projection_eq_id_sub_projection h.isCompl]
exact continuous_id.sub h.continuous_projection | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Complement | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 38
} | {
"line": 324,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nM : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np q : Submodule R M\ninst✝ : IsTopologicalAddGroup M\nh : IsTopCompl p q\nhq : IsClosed[inst✝³] ↑q\nthis✝ : IsClosed[instTopologicalSpaceSubtype] {0}\nthis : T1Space ↥p\n⊢ T3Space ↥p"... | [
"R : Type u_1\ninst✝⁴ : Ring R\nM : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np q : Submodule R M\ninst✝ : IsTopologicalAddGroup M\nh : IsTopCompl p q\nhq : IsClosed[inst✝³] ↑q\nthis✝ : IsClosed[instTopologicalSpaceSubtype] {0}\nthis : T1Space ↥p\n⊢ T0Space ↥p"
] | RegularSpace.t3Space_iff_t0Space | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.IndicatorFunction | {
"line": 68,
"column": 87
} | {
"line": 69,
"column": 57
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : One β\ns : ι → Set α\nhs : Monotone s\nf : α → β\na : α\n⊢ (fun i ↦ (s i).mulIndicator f a) =ᶠ[atTop] fun x ↦ (⋃ i, s i).mulIndicator f a",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Classical.propDec... | [] | by
classical exact hs.piecewise_eventually_eq_iUnion f 1 a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSpace.Indicator | {
"line": 246,
"column": 2
} | {
"line": 247,
"column": 32
} | {
"line": 249,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) c",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.i... | [] | rw [← MemLp.toLp_const, indicatorConstLp]
simp only [Set.indicator_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Indicator | {
"line": 246,
"column": 2
} | {
"line": 247,
"column": 32
} | {
"line": 249,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) c",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.i... | [] | rw [← MemLp.toLp_const, indicatorConstLp]
simp only [Set.indicator_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 68
} | {
"line": 390,
"column": 2
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nh's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0\nu : ℕ → ℝ≥0∞\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞\nu_lim : Tendsto u atTop (𝓝... | [] | exact (hf.measure_enorm_ge_lt_top (u_pos n).1 (u_pos n).2.ne).ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 30
} | {
"line": 424,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ns t : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nht : NullMeasurableSet t μ\nh't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0\nu : Set α := {x | x ... | [
"α : Type u_1\nmα : MeasurableSpace α\ns t : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nht : NullMeasurableSet t μ\nh't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0\nu : Set α := ⋯\nhu : IntegrableO... | apply (A.union B).mono_set _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 567,
"column": 52
} | {
"line": 569,
"column": 17
} | {
"line": 571,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_8\ninst✝³ : NormedAddCommGroup β\ninst✝² : Lattice β\ninst✝¹ : HasSolidNorm β\ninst✝ : IsOrderedAddMonoid β\nf g : α → β\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ Integrable (f ⊔ g) μ",
"ppTerm": "?m.27",
"assigned": true,
"u... | [] | by
rw [← memLp_one_iff_integrable] at hf hg ⊢
exact hf.sup hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 846,
"column": 2
} | {
"line": 848,
"column": 59
} | {
"line": 850,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nhflt : ∀ᵐ (x : α) ∂μ, f x < ∞\ng : α → ℝ\n⊢ Integrable g (μ.withDensity f) ↔ Integrable (fun x ↦ g x * (f x).toReal) μ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommR... | [] | have : (fun x => g x * (f x).toReal) = fun x => (f x).toReal • g x := by simp [mul_comm]
rw [this]
exact integrable_withDensity_iff_integrable_smul' hf hflt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 846,
"column": 2
} | {
"line": 848,
"column": 59
} | {
"line": 850,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nhflt : ∀ᵐ (x : α) ∂μ, f x < ∞\ng : α → ℝ\n⊢ Integrable g (μ.withDensity f) ↔ Integrable (fun x ↦ g x * (f x).toReal) μ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommR... | [] | have : (fun x => g x * (f x).toReal) = fun x => (f x).toReal • g x := by simp [mul_comm]
rw [this]
exact integrable_withDensity_iff_integrable_smul' hf hflt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 33
} | {
"line": 202,
"column": 33
} | [
{
"pp": "𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nl : E →ₗ[𝕜] 𝕜\ns : Set E\nhs₁ : IsOpen[inst✝²] s\nhs₃ : ∀ x ∈ s, l x ≠ 0\nhl : Dense ↑l.ker\nx : E\nh... | [
"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nl : E →ₗ[𝕜] 𝕜\ns : Set E\nhs₁ : IsOpen[inst✝²] s\nhs₃ : ∀ x ∈ s, l x ≠ 0\nhl : Dense ↑l.ker\nx : E\nhx : x ∈ s\n⊢... | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 465,
"column": 90
} | {
"line": 466,
"column": 99
} | {
"line": 467,
"column": 4
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nA : ∀ (f : ContinuousMultilinearMap 𝕜 E... | [] | by
simpa [NormedSpace.isVonNBounded_closedBall, closedBall_mem_nhds, Set.subset_def, Set.MapsTo] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 880,
"column": 15
} | {
"line": 892,
"column": 51
} | {
"line": 893,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : Topolo... | [] | by
intro r u
ext1
filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 (Lp.coeFn_smul r u),
(memL1_smul_of_L1_withDensity f_meas (r • u)).coeFn_toLp,
Lp.coeFn_smul r ((memL1_smul_of_L1_withDensity f_meas u).toLp _),
(memL1_smul_of_L1_withDensity f_meas u).coeFn_toLp]
intro ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 229,
"column": 6
} | {
"line": 235,
"column": 94
} | {
"line": 236,
"column": 6
} | [
{
"pp": "𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\nn : ℕ\nIH :\n ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]\n [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]\n (ξ : Bas... | [
"𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\nn : ℕ\nIH :\n ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]\n [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]\n (ξ : Basis ι 𝕜 E) (... | have U : IsUniformEmbedding b.equivFun.symm.toEquiv := by
have : Fintype.card (Basis.ofVectorSpaceIndex 𝕜 s) = n := by
rw [← s_dim]
exact (finrank_eq_card_basis b).symm
have : Continuous b.equivFun := IH b inferInstance this
exact
b.equivFun.symm.isUniformEmbedding... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 865,
"column": 6
} | {
"line": 866,
"column": 60
} | {
"line": 868,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁰ : Fintype ι'\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → SeminormedAd... | [] | simpa using ((f x).mkContinuous_norm_le' _).trans_eq <| by
rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 865,
"column": 6
} | {
"line": 866,
"column": 60
} | {
"line": 868,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁰ : Fintype ι'\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → SeminormedAd... | [] | simpa using ((f x).mkContinuous_norm_le' _).trans_eq <| by
rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 865,
"column": 6
} | {
"line": 866,
"column": 60
} | {
"line": 868,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁰ : Fintype ι'\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → SeminormedAd... | [] | simpa using ((f x).mkContinuous_norm_le' _).trans_eq <| by
rw [max_mul_of_nonneg _ _ (norm_nonneg x), zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 268,
"column": 42
} | {
"line": 268,
"column": 65
} | {
"line": 268,
"column": 66
} | [
{
"pp": "α : Type u_1\nF : Type u_5\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup F\np : ℝ\nf : α →ₛ F\nμ : Measure α\nh_map : (fun x ↦ ‖f x‖ₑ ^ p) = ⇑(map (fun x ↦ ‖x‖ₑ ^ p) f)\n⊢ MeasureTheory.lintegral μ ⇑(map (fun x ↦ ‖x‖ₑ ^ p) f) ^ (1 / p) = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (⇑f ⁻¹' {y})) ^ (1 / p)",
... | [
"α : Type u_1\nF : Type u_5\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup F\np : ℝ\nf : α →ₛ F\nμ : Measure α\nh_map : (fun x ↦ ‖f x‖ₑ ^ p) = ⇑(map (fun x ↦ ‖x‖ₑ ^ p) f)\n⊢ (map (fun x ↦ ‖x‖ₑ ^ p) f).lintegral μ ^ (1 / p) = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (⇑f ⁻¹' {y})) ^ (1 / p)"
] | lintegral_eq_lintegral, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 347,
"column": 2
} | {
"line": 348,
"column": 54
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ∞\nc : E\nhc : c ≠ 0\ns : Set α\nhs : MeasurableSet s\nhcs : MemLp (⇑(piecewise s hs (const α c) (const α 0))) p μ\nthis : support ⇑(const α c) = Set.univ\n⊢ μ ... | [] | simpa only [memLp_iff_finMeasSupp hp_pos hp_ne_top, finMeasSupp_iff_support,
support_indicator, Set.inter_univ, this] using hcs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 15
} | {
"line": 331,
"column": 15
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\nm : MeasurableSpace α\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nT : Set α → F →L[ℝ] F'\nhT : T ∅ = 0\nf : α →ₛ F\ns : Finset F\nhs : {x ∈ f.range | ... | [
"α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\nm : MeasurableSpace α\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nT : Set α → F →L[ℝ] F'\nhT : T ∅ = 0\nf : α →ₛ F\ns : Finset F\nhs : {x ∈ f.range | x ≠ 0} ⊆ s\n... | rintro x - hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 15
} | {
"line": 214,
"column": 15
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nf : α →ₛ F\ns : Finset F\nhs : {x ∈ f.range | x ≠ 0} ⊆ s\n⊢ ∀ x ∈ s, x ∉ {x ∈ f.range | x ≠ 0} → μ.real (⇑f ⁻¹' {x}) • x = 0",
"ppTerm": "?... | [
"α : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nf : α →ₛ F\ns : Finset F\nhs : {x ∈ f.range | x ≠ 0} ⊆ s\nx : F\nhx : x ∉ {x ∈ f.range | x ≠ 0}\n⊢ μ.real (⇑f ⁻¹' {x}) • x = 0"
] | rintro x - hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 613,
"column": 76
} | {
"line": 615,
"column": 33
} | {
"line": 617,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\n⊢ Continuous fun f ↦ integral f",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | [] | by
simp only [integral]
exact L1.integralCLM.continuous | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 41
} | {
"line": 364,
"column": 2
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_6\nf : α → G\nhf : HasFiniteIntegral f μ\nl : Filter ι\ns : ι → Set α\nhs : Tendsto (⇑μ ∘ s) l (𝓝 0)\n⊢ Tendsto (fun i ↦ ∫ (x : α) in s i, f x ∂μ) l (𝓝 0)",
"ppTerm... | [
"α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_6\nf : α → G\nhf : HasFiniteIntegral f μ\nl : Filter ι\ns : ι → Set α\nhs : Tendsto (⇑μ ∘ s) l (𝓝 0)\n⊢ Tendsto (fun x ↦ ‖∫ (x : α) in s x, f x ∂μ‖) l (𝓝 0)"
] | rw [tendsto_zero_iff_norm_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 603,
"column": 2
} | {
"line": 624,
"column": 6
} | {
"line": 626,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\n⊢ setToSimpleFunc T (piecewise s hs (c... | [] | classical
obtain rfl | hs_empty := s.eq_empty_or_nonempty
· simp only [hT_empty, zero_apply, piecewise_empty, const_zero,
setToSimpleFunc_zero_apply]
simp_rw [setToSimpleFunc]
obtain rfl | hs_univ := eq_or_ne s univ
· haveI hα := hs_empty.to_type
simp [← Function.const_def]
rw [range_indicator hs ... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 603,
"column": 2
} | {
"line": 624,
"column": 6
} | {
"line": 626,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\n⊢ setToSimpleFunc T (piecewise s hs (c... | [] | classical
obtain rfl | hs_empty := s.eq_empty_or_nonempty
· simp only [hT_empty, zero_apply, piecewise_empty, const_zero,
setToSimpleFunc_zero_apply]
simp_rw [setToSimpleFunc]
obtain rfl | hs_univ := eq_or_ne s univ
· haveI hα := hs_empty.to_type
simp [← Function.const_def]
rw [range_indicator hs ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 603,
"column": 2
} | {
"line": 624,
"column": 6
} | {
"line": 626,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nT : Set α → F →L[ℝ] F'\nhT_empty : T ∅ = 0\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nx : F\n⊢ setToSimpleFunc T (piecewise s hs (c... | [] | classical
obtain rfl | hs_empty := s.eq_empty_or_nonempty
· simp only [hT_empty, zero_apply, piecewise_empty, const_zero,
setToSimpleFunc_zero_apply]
simp_rw [setToSimpleFunc]
obtain rfl | hs_univ := eq_or_ne s univ
· haveI hα := hs_empty.to_type
simp [← Function.const_def]
rw [range_indicator hs ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 686,
"column": 2
} | {
"line": 688,
"column": 68
} | {
"line": 690,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : ... | [] | simp_rw [← neg_convexOn_iff] at hf_conc ⊢
simpa only [Pi.neg_apply, integral_neg] using!
integral_convexOn_of_integrand_ae hs hf_conc (hf_int · · |>.neg) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 686,
"column": 2
} | {
"line": 688,
"column": 68
} | {
"line": 690,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : ... | [] | simp_rw [← neg_convexOn_iff] at hf_conc ⊢
simpa only [Pi.neg_apply, integral_neg] using!
integral_convexOn_of_integrand_ae hs hf_conc (hf_int · · |>.neg) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 472,
"column": 2
} | {
"line": 476,
"column": 49
} | {
"line": 478,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : Domin... | [] | apply setToL1_unique hT (A := setToL1 hT') _ f
intro f
suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]
rw [setToL1_eq_setToL1SCLM]
exact (setToL1SCLM_congr_left' hT hT' h f).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 472,
"column": 2
} | {
"line": 476,
"column": 49
} | {
"line": 478,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : Domin... | [] | apply setToL1_unique hT (A := setToL1 hT') _ f
intro f
suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]
rw [setToL1_eq_setToL1SCLM]
exact (setToL1SCLM_congr_left' hT hT' h f).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 22
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\nf g : C(α, β)\nC : ℝ\ninst✝ : Nonempty α\n⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topolo... | [] | simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 22
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\nf g : C(α, β)\nC : ℝ\ninst✝ : Nonempty α\n⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topolo... | [] | simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 22
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\nf g : C(α, β)\nC : ℝ\ninst✝ : Nonempty α\n⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topolo... | [] | simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 997,
"column": 2
} | {
"line": 999,
"column": 51
} | {
"line": 1000,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nβ : Type u_7\nx✝ : MeasurableSpace β\nμ'... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nβ : Type u_7\nx✝ : MeasurableSpace β\nμ' : Measure β... | have B : setToFun μ T hT (f ∘ φ) = setToFun μ T hT (g ∘ φ) := by
apply setToFun_congr_ae
exact ae_of_ae_map hφ.aemeasurable hfm.ae_eq_mk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1079,
"column": 4
} | {
"line": 1079,
"column": 72
} | {
"line": 1080,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ∞\nhc' : c' ≠ ∞\nhμ_le : μ ≤ ... | [] | exact setToFun_congr_measure_of_integrable c' hc' hμ'_le hT hT' f hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1079,
"column": 4
} | {
"line": 1079,
"column": 72
} | {
"line": 1080,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ∞\nhc' : c' ≠ ∞\nhμ_le : μ ≤ ... | [] | exact setToFun_congr_measure_of_integrable c' hc' hμ'_le hT hT' f hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1079,
"column": 4
} | {
"line": 1079,
"column": 72
} | {
"line": 1080,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ∞\nhc' : c' ≠ ∞\nhμ_le : μ ≤ ... | [] | exact setToFun_congr_measure_of_integrable c' hc' hμ'_le hT hT' f hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 142,
"column": 55
} | {
"line": 142,
"column": 85
} | {
"line": 142,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\ninst✝ : OrderTopology R\nf : StieltjesFunction R\nx : R\n⊢ ContinuousWithinAt (↑f) (Ioi x) x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"ContinuousWithinAt"... | [
"R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\ninst✝ : OrderTopology R\nf : StieltjesFunction R\nx : R\n⊢ ContinuousWithinAt (↑f) (Ici x) x"
] | continuousWithinAt_Ioi_iff_Ici | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1274,
"column": 4
} | {
"line": 1274,
"column": 24
} | {
"line": 1275,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nfs : ℕ → α → E\nf : α → E\nbound : α → ℝ\nf... | [
"case e'_3\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nfs : ℕ → α → E\nf : α → E\nbound : α → ℝ\nfs... | convert! this with n | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1281,
"column": 2
} | {
"line": 1285,
"column": 57
} | {
"line": 1286,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nfs : ℕ → α → E\nf : α → E\nbound ... | [
"case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nfs : ℕ → α → E\nf : α → E\nbound : α → ℝ\nfs_... | have lintegral_norm_tendsto_zero :
Tendsto (fun n => ENNReal.toReal <| ∫⁻ a, ENNReal.ofReal ‖fs n a - f a‖ ∂μ) atTop (𝓝 0) :=
(tendsto_toReal zero_ne_top).comp
(tendsto_lintegral_norm_of_dominated_convergence fs_measurable
bound_integrable.hasFiniteIntegral h_bound h_lim) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1304,
"column": 50
} | {
"line": 1304,
"column": 76
} | {
"line": 1305,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl : Filter ι\ninst✝ : l.IsCo... | [] | rwa [tendsto_atTop'] at xl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 284,
"column": 6
} | {
"line": 284,
"column": 31
} | {
"line": 284,
"column": 32
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : X → E\nμ : Measure X\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set X\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\nh_mono : Monotone s\nhne : atTop.NeBot... | [
"X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : X → E\nμ : Measure X\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set X\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\nh_mono : Monotone s\nhne : atTop.NeBot\nthis✝ : Is... | mem_closedBall_iff_norm', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 396,
"column": 6
} | {
"line": 400,
"column": 20
} | {
"line": 401,
"column": 4
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0\nht : IntegrableOn f t μ\nH : IntegrableOn f (s ∪ t) μ\nf' : X → E := AEStronglyMeasurable.mk f ⋯\n⊢ ∫ (x : X) in ... | [] | apply
integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk)
filter_upwards [ht_eq,
ae_mono (Measure.restrict_mono subset_union_right le_rfl) H.1.ae_eq_mk] with x hx h'x
rw [← h'x, hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 396,
"column": 6
} | {
"line": 400,
"column": 20
} | {
"line": 401,
"column": 4
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0\nht : IntegrableOn f t μ\nH : IntegrableOn f (s ∪ t) μ\nf' : X → E := AEStronglyMeasurable.mk f ⋯\n⊢ ∫ (x : X) in ... | [] | apply
integral_union_eq_left_of_ae_aux _ H.1.stronglyMeasurable_mk (H.congr_fun_ae H.1.ae_eq_mk)
filter_upwards [ht_eq,
ae_mono (Measure.restrict_mono subset_union_right le_rfl) H.1.ae_eq_mk] with x hx h'x
rw [← h'x, hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 422,
"column": 10
} | {
"line": 422,
"column": 63
} | {
"line": 422,
"column": 64
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nhts : s ⊆ t\nh't : ∀ᵐ (x : X) ∂μ, x ∈ t \\ s → f x = 0\nhaux : StronglyMeasurable f\nh'aux : IntegrableOn f t μ\nk : Set X := f ⁻¹' {0}\nhk : MeasurableSet ... | [
"X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nhts : s ⊆ t\nh't : ∀ᵐ (x : X) ∂μ, x ∈ t \\ s → f x = 0\nhaux : StronglyMeasurable f\nh'aux : IntegrableOn f t μ\nk : Set X := f ⁻¹' {0}\nhk : MeasurableSet k\n⊢ 0 + ∫ (... | setIntegral_eq_zero_of_forall_eq_zero fun x hx => ?_, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1477,
"column": 4
} | {
"line": 1477,
"column": 70
} | {
"line": 1478,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountabl... | [] | filter_upwards [self_mem_nhdsWithin] with x hx using hfs_meas x hx | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1477,
"column": 4
} | {
"line": 1477,
"column": 70
} | {
"line": 1478,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountabl... | [] | filter_upwards [self_mem_nhdsWithin] with x hx using hfs_meas x hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1477,
"column": 4
} | {
"line": 1477,
"column": 70
} | {
"line": 1478,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountabl... | [] | filter_upwards [self_mem_nhdsWithin] with x hx using hfs_meas x hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 589,
"column": 80
} | {
"line": 592,
"column": 50
} | {
"line": 594,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns : Set X\nμ : Measure X\nC : ℝ\nhs : μ s < ∞\nhC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C\n⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * μ.real s",
"ppTerm": "?m.39",
"assigned": true,
"u... | [] | by
rw [← Measure.restrict_apply_univ] at *
haveI : IsFiniteMeasure (μ.restrict s) := ⟨hs⟩
simpa using norm_integral_le_of_norm_le_const hC | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Content | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 23
} | {
"line": 340,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U... | [
"G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U\n... | rw [ENNReal.add_iSup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Content | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 57
} | {
"line": 343,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U... | [
"G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U\n... | let M' : Compacts G := ⟨closure M, M.isCompact.closure⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Content | {
"line": 377,
"column": 40
} | {
"line": 377,
"column": 69
} | {
"line": 377,
"column": 69
} | [
{
"pp": "G : Type w\ninst✝³ : TopologicalSpace G\nμ : Content G\ninst✝² : R1Space G\nS : MeasurableSpace G\ninst✝¹ : BorelSpace G\ninst✝ : WeaklyLocallyCompactSpace G\nthis : IsFiniteMeasureOnCompacts μ.measure\nU : Set G\nhU : IsOpen U\nr : ℝ≥0∞\nhr : r < μ.outerMeasure U\n⊢ ∃ K ⊆ U, IsCompact K ∧ r < μ.measur... | [
"G : Type w\ninst✝³ : TopologicalSpace G\nμ : Content G\ninst✝² : R1Space G\nS : MeasurableSpace G\ninst✝¹ : BorelSpace G\ninst✝ : WeaklyLocallyCompactSpace G\nthis : IsFiniteMeasureOnCompacts μ.measure\nU : Set G\nhU : IsOpen U\nr : ℝ≥0∞\nhr : r < μ.innerContent { carrier := U, is_open' := hU }\n⊢ ∃ K ⊆ U, IsCompa... | μ.outerMeasure_of_isOpen U hU | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1097,
"column": 38
} | {
"line": 1097,
"column": 70
} | {
"line": 1097,
"column": 70
} | [
{
"pp": "X : Type u_6\nf : X → ℝ\nm0 : MeasurableSpace X\nμ : Measure X\ng : SimpleFunc X ℝ\nhf : Integrable f μ\ng₁ g₂ : SimpleFunc X ℝ\nx✝ : Disjoint (support ⇑g₁) (support ⇑g₂)\nh_int₁ : Integrable (⇑g₁ * f) μ\nh_int₂ : Integrable (⇑g₂ * f) μ\n⊢ ⇑g₁ * f + ⇑g₂ * f =ᵐ[μ] ⇑(g₁ + g₂) * f",
"ppTerm": "?m.53",... | [] | rw [SimpleFunc.coe_add, add_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1097,
"column": 38
} | {
"line": 1097,
"column": 70
} | {
"line": 1097,
"column": 70
} | [
{
"pp": "X : Type u_6\nf : X → ℝ\nm0 : MeasurableSpace X\nμ : Measure X\ng : SimpleFunc X ℝ\nhf : Integrable f μ\ng₁ g₂ : SimpleFunc X ℝ\nx✝ : Disjoint (support ⇑g₁) (support ⇑g₂)\nh_int₁ : Integrable (⇑g₁ * f) μ\nh_int₂ : Integrable (⇑g₂ * f) μ\n⊢ ⇑g₁ * f + ⇑g₂ * f =ᵐ[μ] ⇑(g₁ + g₂) * f",
"ppTerm": "?m.53",... | [] | rw [SimpleFunc.coe_add, add_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1097,
"column": 38
} | {
"line": 1097,
"column": 70
} | {
"line": 1097,
"column": 70
} | [
{
"pp": "X : Type u_6\nf : X → ℝ\nm0 : MeasurableSpace X\nμ : Measure X\ng : SimpleFunc X ℝ\nhf : Integrable f μ\ng₁ g₂ : SimpleFunc X ℝ\nx✝ : Disjoint (support ⇑g₁) (support ⇑g₂)\nh_int₁ : Integrable (⇑g₁ * f) μ\nh_int₂ : Integrable (⇑g₂ * f) μ\n⊢ ⇑g₁ * f + ⇑g₂ * f =ᵐ[μ] ⇑(g₁ + g₂) * f",
"ppTerm": "?m.53",... | [] | rw [SimpleFunc.coe_add, add_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 15
} | {
"line": 274,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : (interior U).Nonempty\nK : Set G\nh1K : IsCompact K\nh2K : (interior K).Nonempty\n⊢ 0 < prehaar (↑K₀) U { carrier := K, isCompact' := h1K }",
"ppTerm": "?m.21",
"a... | [
"case ha\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : (interior U).Nonempty\nK : Set G\nh1K : IsCompact K\nh2K : (interior K).Nonempty\n⊢ 0 < ↑(index (↑{ carrier := K, isCompact' := h1K }) U)",
"case hb\nG : Type u_1\ninst✝² :... | apply div_pos | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 831,
"column": 2
} | {
"line": 831,
"column": 56
} | {
"line": 832,
"column": 2
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\nν : Measure α\ninst✝³ : SMulInvariantMeasure G α ν\ninst✝² : Countable G\ninst✝¹ : MeasurableConstSMul G α\ni : SigmaFinite ν\ni' : HasFundamentalDomain G α ν\nμ : Measure (Quotient α_mod_G)\ninst✝ : Quoti... | [
"case refine_1\nG : Type u_1\nα : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\nν : Measure α\ninst✝³ : SMulInvariantMeasure G α ν\ninst✝² : Countable G\ninst✝¹ : MeasurableConstSMul G α\ni : SigmaFinite ν\ni' : HasFundamentalDomain G α ν\nμ : Measure (Quotient α_mod_G)\ninst✝ : Qu... | refine ⟨⟨fun n ↦ π '' (A n), by simp, fun n ↦ ?_, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 575,
"column": 6
} | {
"line": 575,
"column": 81
} | {
"line": 576,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\ncd : PreInnerProductSpace.Core 𝕜 F\nthis : NormedSpace 𝕜 F := Core.toNormedSpace\nx : F\nh₁ : ‖x‖ ^ 2 = √(re ⟪x, x⟫_𝕜) ^ 2\n⊢ ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜",
"ppTerm": "?m.57",
"assigned": t... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\ncd : PreInnerProductSpace.Core 𝕜 F\nthis : NormedSpace 𝕜 F := Core.toNormedSpace\nx : F\nh₁ : ‖x‖ ^ 2 = √(re ⟪x, x⟫_𝕜) ^ 2\nh₂ : 0 ≤ re ⟪x, x⟫_𝕜\n⊢ ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜"
] | have h₂ : 0 ≤ re (cd.inner x x) := InnerProductSpace.Core.inner_self_nonneg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 596,
"column": 6
} | {
"line": 596,
"column": 81
} | {
"line": 597,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : AddCommGroup F\nhF : Module 𝕜 F\ninst✝² : TopologicalSpace F\ninst✝¹ : IsTopologicalAddGroup F\ninst✝ : ContinuousConstSMul 𝕜 F\ncd : Core 𝕜 F\nh : ContinuousAt (fun v ↦ ⟪v, v⟫_𝕜) 0\nh' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫_𝕜 < 1}... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : AddCommGroup F\nhF : Module 𝕜 F\ninst✝² : TopologicalSpace F\ninst✝¹ : IsTopologicalAddGroup F\ninst✝ : ContinuousConstSMul 𝕜 F\ncd : Core 𝕜 F\nh : ContinuousAt (fun v ↦ ⟪v, v⟫_𝕜) 0\nh' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫_𝕜 < 1}\nthis✝ : No... | have h₂ : 0 ≤ re (cd.inner x x) := InnerProductSpace.Core.inner_self_nonneg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 99
} | {
"line": 58,
"column": 2
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nne : K.Nonempty\nh₁ : IsComplete K\nh₂ : Convex ℝ K\nu : F\nδ : ℝ := ⨅ w, ‖u - ↑w‖\nthis : Nonempty ↑K := Set.Nonempty.to_subtype ne\nzero_le_δ : 0 ≤ δ\nδ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖\nδ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖\n... | [] | exact tendsto_of_tendsto_of_tendsto_of_le_of_le h h' (fun x => δ_le _) fun x => le_of_lt (hw _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 700,
"column": 39
} | {
"line": 700,
"column": 58
} | {
"line": 700,
"column": 59
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : F\nr : ℝ\nhx : x ≠ 0\nhr : r < 0\n⊢ r * (‖x‖ * ‖x‖) / -(r * (‖x‖ * ‖x‖)) = -1",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Semigroup.toMul",
"Real",
... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : F\nr : ℝ\nhx : x ≠ 0\nhr : r < 0\n⊢ -(r * (‖x‖ * ‖x‖) / (r * (‖x‖ * ‖x‖))) = -1"
] | div_neg_eq_neg_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 194,
"column": 2
} | {
"line": 197,
"column": 18
} | {
"line": 199,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nι' : Type u_5\nv : ι → E\nhv : Orthonormal 𝕜 v\nf : ι' → ι\nhf : Function.Injective f\n⊢ Orthonormal 𝕜 (v ∘ f)",
"ppTerm": "?m.20",
"assigned": true,
"usedCons... | [] | rw [orthonormal_iff_ite] at hv ⊢
intro i j
convert! hv (f i) (f j) using 1
simp [hf.eq_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 194,
"column": 2
} | {
"line": 197,
"column": 18
} | {
"line": 199,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nι' : Type u_5\nv : ι → E\nhv : Orthonormal 𝕜 v\nf : ι' → ι\nhf : Function.Injective f\n⊢ Orthonormal 𝕜 (v ∘ f)",
"ppTerm": "?m.20",
"assigned": true,
"usedCons... | [] | rw [orthonormal_iff_ite] at hv ⊢
intro i j
convert! hv (f i) (f j) using 1
simp [hf.eq_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 98,
"column": 2
} | {
"line": 100,
"column": 55
} | {
"line": 102,
"column": 0
} | [
{
"pp": "V : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nS T : V →ₗ[ℂ] V\n⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Inn... | [] | rw [← sub_eq_zero, ← inner_map_self_eq_zero]
refine forall_congr' fun x => ?_
rw [LinearMap.sub_apply, inner_sub_left, sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 98,
"column": 2
} | {
"line": 100,
"column": 55
} | {
"line": 102,
"column": 0
} | [
{
"pp": "V : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nS T : V →ₗ[ℂ] V\n⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Inn... | [] | rw [← sub_eq_zero, ← inner_map_self_eq_zero]
refine forall_congr' fun x => ?_
rw [LinearMap.sub_apply, inner_sub_left, sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 39
} | {
"line": 55,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nx : E\nw : ↥K := K.orthogonalProjectionOnto x\nv : E := x ... | [
"case e'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nx : E\nw : ↥K := K.orthogonalProjectionOnto x\nv : E := x -... | convert norm_sub_eq_norm_add this | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 75,
"column": 38
} | {
"line": 75,
"column": 59
} | {
"line": 75,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\n⊢ K₁ᗮ ≤ K₀ᗮᗮ ↔ K₁ᗮ ≤ K₀",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\n⊢ K₁ᗮ ≤ K₀ ↔ K₁ᗮ ≤ K₀"
] | orthogonal_orthogonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 79,
"column": 38
} | {
"line": 79,
"column": 59
} | {
"line": 79,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\n⊢ K₁ᗮᗮ ≤ K₀ᗮ ↔ K₁ ≤ K₀ᗮ",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\n⊢ K₁ ≤ K₀ᗮ ↔ K₁ ≤ K₀ᗮ"
] | orthogonal_orthogonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 48
} | {
"line": 119,
"column": 4
} | [
{
"pp": "case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →... | [
"case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\ninst✝... | rw [mem_ball, dist_eq_norm, sub_zero] at hx₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 38
} | {
"line": 174,
"column": 6
} | [
{
"pp": "case pos\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F... | [
"case pos\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\nh... | obtain ⟨V, hV₁, hV₂⟩ := IH φ hn' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 314,
"column": 2
} | {
"line": 318,
"column": 71
} | {
"line": 320,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜' F\nσ' : 𝕜' →+* 𝕜\ninst✝⁵ : RingHomI... | [] | rw [continuous_def]
intro s hs
rw [← e.image_eq_preimage_symm]
rw [← e.coe_coe] at h ⊢
exact ContinuousLinearMap.isOpenMap (σ := σ) ⟨_, h⟩ e.surjective s hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 314,
"column": 2
} | {
"line": 318,
"column": 71
} | {
"line": 320,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜' F\nσ' : 𝕜' →+* 𝕜\ninst✝⁵ : RingHomI... | [] | rw [continuous_def]
intro s hs
rw [← e.image_eq_preimage_symm]
rw [← e.coe_coe] at h ⊢
exact ContinuousLinearMap.isOpenMap (σ := σ) ⟨_, h⟩ e.surjective s hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 487,
"column": 2
} | {
"line": 489,
"column": 5
} | {
"line": 491,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\n⊢ spectrum 𝕜 f = spectrum 𝕜 ↑f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
... | [] | ext μ
rw [spectrum.mem_iff, spectrum.mem_iff, ContinuousLinearMap.isUnit_iff_isUnit_toLinearMap]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 487,
"column": 2
} | {
"line": 489,
"column": 5
} | {
"line": 491,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\n⊢ spectrum 𝕜 f = spectrum 𝕜 ↑f",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
... | [] | ext μ
rw [spectrum.mem_iff, spectrum.mem_iff, ContinuousLinearMap.isUnit_iff_isUnit_toLinearMap]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 383,
"column": 6
} | {
"line": 383,
"column": 39
} | {
"line": 384,
"column": 6
} | [
{
"pp": "p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf g : WithLp p (α × β)\n⊢ 0 ≤ dist f g",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"WithLp",
"Real.instLE",
"Real",
"Real.... | [
"case inl\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nhp : Fact (1 ≤ ∞)\nf g : WithLp ∞ (α × β)\n⊢ 0 ≤ dist f g",
"case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\... | rcases p.dichotomy with (rfl | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Matrix.Reflection | {
"line": 104,
"column": 6
} | {
"line": 109,
"column": 11
} | {
"line": 111,
"column": 0
} | [
{
"pp": "α : Type u_1\nm n : ℕ\nA : Matrix (Fin m) (Fin (n + 1)) α\ni : Fin (n + 1)\nj : Fin m\n⊢ A.transposeᵣ i j = Aᵀ i j",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"FinVec.map",
"Matrix.submatrix",
"Equiv.instEquivLike"... | [] | simp_rw [transposeᵣ, transposeᵣ_eq]
refine i.cases ?_ fun i => ?_
· dsimp
rw [FinVec.map_eq, Function.comp_apply]
· simp only [of_apply, Matrix.cons_val_succ]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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