module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Real | {
"line": 476,
"column": 2
} | {
"line": 476,
"column": 20
} | {
"line": 477,
"column": 2
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nhs : MeasurableSet s\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ.real u < μ.real s + μ.real t\nhu : μ u ≠ ∞\n⊢ (s ∩ t).Nonempty",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real",
"MeasureTheory.Measure",
... | [
"α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nhs : MeasurableSet s\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ.real u < μ.real t + μ.real s\nhu : μ u ≠ ∞\n⊢ (s ∩ t).Nonempty"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.IndicatorFunction | {
"line": 117,
"column": 34
} | {
"line": 117,
"column": 51
} | {
"line": 117,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : One β\nl : Filter α\nf : α → β\ns : Set α\nhf : f =ᶠ[l] 1\n⊢ s.mulIndicator 1 =ᶠ[l] 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mulIndicator",
"Filter.EventuallyEq",
"id",
"Pi.... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : One β\nl : Filter α\nf : α → β\ns : Set α\nhf : f =ᶠ[l] 1\n⊢ 1 =ᶠ[l] 1"
] | mulIndicator_one' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 42
} | {
"line": 183,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\nhf : MemLp f p μ\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\n⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ",
"ppTerm": "?m.27",
"assigned": true,
"usedCons... | [] | rw [← memLp_one_iff_integrable]
exact hf.enorm_rpow hp_ne_zero hp_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 42
} | {
"line": 183,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\nhf : MemLp f p μ\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\n⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ",
"ppTerm": "?m.27",
"assigned": true,
"usedCons... | [] | rw [← memLp_one_iff_integrable]
exact hf.enorm_rpow hp_ne_zero hp_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 52
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_5\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\ninst✝ : PseudoMetrizableSpace ε\nι : Type u_8\nm : MeasurableSpace α\nf : α → ε\nμ : ι → Measure α\ns : Finset ι\n⊢ Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i)",
"ppTerm": "?m.27",
"assigned": tru... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 52
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_5\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\ninst✝ : PseudoMetrizableSpace ε\nι : Type u_8\nm : MeasurableSpace α\nf : α → ε\nμ : ι → Measure α\ns : Finset ι\n⊢ Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i)",
"ppTerm": "?m.27",
"assigned": tru... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 52
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_5\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\ninst✝ : PseudoMetrizableSpace ε\nι : Type u_8\nm : MeasurableSpace α\nf : α → ε\nμ : ι → Measure α\ns : Finset ι\n⊢ Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i)",
"ppTerm": "?m.27",
"assigned": tru... | [] | induction s using Finset.induction_on <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 506,
"column": 2
} | {
"line": 508,
"column": 67
} | {
"line": 510,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nh_meas : AEStronglyMeasurable f μ\nhf : 0 ≤ᵐ[μ] f\nhg : 0 ≤ᵐ[μ] g\nh_int : Integrable (f + g) μ\n⊢ Integrable f μ",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Norm.norm",
"Re... | [] | refine h_int.mono' h_meas ?_
filter_upwards [hf, hg] with a haf hag
exact (Real.norm_of_nonneg haf).symm ▸ le_add_of_nonneg_right hag | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 506,
"column": 2
} | {
"line": 508,
"column": 67
} | {
"line": 510,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nh_meas : AEStronglyMeasurable f μ\nhf : 0 ≤ᵐ[μ] f\nhg : 0 ≤ᵐ[μ] g\nh_int : Integrable (f + g) μ\n⊢ Integrable f μ",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Norm.norm",
"Re... | [] | refine h_int.mono' h_meas ?_
filter_upwards [hf, hg] with a haf hag
exact (Real.norm_of_nonneg haf).symm ▸ le_add_of_nonneg_right hag | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 17
} | {
"line": 506,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set α\nhs : s ∈ l ∧ IntegrableOn (f ∘ e) s μ\n⊢ ∃ s ∈ map e l, Integrabl... | · exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 67
} | {
"line": 58,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace E\ninst✝² : Countable ι\nf : ι → X → E\nhE : Nonempty E\ninst✝¹ : IsCompletelyMetrizableSpace E\nhf : ∀ (i : ι), StronglyMeasurable (f i)\ninst✝ : ⊥.IsCountablyGenerated\n⊢ StronglyMeasurable fun x... | [
"case inr\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace E\ninst✝² : Countable ι\nl : Filter ι\ninst✝¹ : l.IsCountablyGenerated\nf : ι → X → E\nhE : Nonempty E\ninst✝ : IsCompletelyMetrizableSpace E\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nhl : l.NeBot\n⊢ Strongly... | · simpa [limUnder, Filter.map_bot] using stronglyMeasurable_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 512,
"column": 2
} | {
"line": 514,
"column": 50
} | {
"line": 516,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : UniformSpace E\ninst✝⁴ : T2Space E\ninst✝³ : IsUniformAddGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n⊢ CompleteSpace E",
"p... | [] | set e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm
have : IsUniformEmbedding e.toEquiv.symm := e.symm.isUniformEmbedding
exact (completeSpace_congr this).1 inferInstance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 512,
"column": 2
} | {
"line": 514,
"column": 50
} | {
"line": 516,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : UniformSpace E\ninst✝⁴ : T2Space E\ninst✝³ : IsUniformAddGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n⊢ CompleteSpace E",
"p... | [] | set e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm
have : IsUniformEmbedding e.toEquiv.symm := e.symm.isUniformEmbedding
exact (completeSpace_congr this).1 inferInstance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 277,
"column": 2
} | {
"line": 280,
"column": 43
} | {
"line": 281,
"column": 2
} | [
{
"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 ≠ ∞\nf : α →ₛ E\nhf : MemLp (⇑f) p μ\ny : E\nhy_ne : y ≠ 0\nh_fin : (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f).FinMeasSupp μ\n⊢ μ (⇑f ⁻¹' {y}) < ∞",
"ppTerm": "?m.47",... | [
"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 ≠ ∞\nf : α →ₛ E\nhf : MemLp (⇑f) p μ\ny : E\nhy_ne : y ≠ 0\nh_fin : (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f).FinMeasSupp μ\nhf_fin : f.FinMeasSupp μ\n⊢ μ (⇑f ⁻¹' {y}) < ∞"
] | have hf_fin : f.FinMeasSupp μ := by
have {b : E} : (fun x ↦ ‖x‖ₑ ^ p.toReal) b = 0 ↔ b = 0 := by
simp [rpow_eq_zero_iff_of_pos (toReal_pos hp_pos hp_ne_top)]
rwa [FinMeasSupp.map_iff this] at h_fin | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 602,
"column": 2
} | {
"line": 602,
"column": 17
} | {
"line": 603,
"column": 2
} | [
{
"pp": "case neg\nα : 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\nhs_empty : s.Nonempty\nhs_un... | [
"case neg\nα : 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\nhs_empty : s.Nonempty\nhs_univ : s ≠ Set... | rw [sum_insert] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 195,
"column": 93
} | {
"line": 199,
"column": 33
} | {
"line": 201,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\nf : α → E\n⊢ ∫ (a : α), f a ∂μ = setToFun μ (weightedSMul μ) ⋯ f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"NormedCommRi... | [] | by
by_cases hE : CompleteSpace E
· simp only [integral, hE, ↓reduceDIte, L1.integral, setToFun]
rfl
· simp [integral, hE, setToFun] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 386,
"column": 2
} | {
"line": 387,
"column": 85
} | {
"line": 389,
"column": 0
} | [
{
"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\nhfi : AEStronglyMeasurable f μ\nF : ι → α → G\nl : Filter ι\nhFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ\nhF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ)... | [] | simp only [integral_eq_setToFun]
exact tendsto_setToFun_of_L1 (dominatedFinMeasAdditive_weightedSMul μ) f hfi hFi hF | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 386,
"column": 2
} | {
"line": 387,
"column": 85
} | {
"line": 389,
"column": 0
} | [
{
"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\nhfi : AEStronglyMeasurable f μ\nF : ι → α → G\nl : Filter ι\nhFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ\nhF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ)... | [] | simp only [integral_eq_setToFun]
exact tendsto_setToFun_of_L1 (dominatedFinMeasAdditive_weightedSMul μ) f hfi hFi hF | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 581,
"column": 2
} | {
"line": 582,
"column": 61
} | {
"line": 583,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : IsOrderedModule ℝ E\ninst✝ : OrderClosedTopology E\nf : α → E\nν : Measure α\nhle : μ ≤ ν\nhf : 0 ≤ᵐ[ν] f... | [
"case pos\nα : Type u_1\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : IsOrderedModule ℝ E\ninst✝ : OrderClosedTopology E\nf : α → E\nν : Measure α\nhle : μ ≤ ν\nhf : 0 ≤ᵐ[ν] f\nhfi : Inte... | obtain ⟨g, hg, hg_nonneg, hfg⟩ := hfi.1.exists_stronglyMeasurable_range_subset
isClosed_Ici.measurableSet (Set.nonempty_Ici (a := 0)) hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 907,
"column": 2
} | {
"line": 907,
"column": 31
} | {
"line": 908,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nH : Type u_6\ninst✝ : NormedAddCommGroup H\nf : α → H\nhf : Integrable f μ\n⊢ ‖Integrable.toL1 f hf‖ = ∫ (a : α), ‖f a‖ ∂μ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"co... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nH : Type u_6\ninst✝ : NormedAddCommGroup H\nf : α → H\nhf : Integrable f μ\n⊢ ∫ (a : α), ‖↑↑(Integrable.toL1 f hf) a‖ ∂μ = ∫ (a : α), ‖f a‖ ∂μ"
] | rw [L1.norm_eq_integral_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 87
} | {
"line": 137,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\nP : Type u_10\nP₂ : Type u_11\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : NormedSpace 𝕜 V\ninst✝⁵ : PseudoMetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : SeminormedAddCommGroup V₂\ninst✝² : NormedSpace 𝕜 V₂\ninst✝¹ : PseudoMe... | [] | rw [dist_eq_norm_vsub V₂, dist_eq_norm_vsub V, ← map_vsub, f.linearIsometry.norm_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 87
} | {
"line": 137,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\nP : Type u_10\nP₂ : Type u_11\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : NormedSpace 𝕜 V\ninst✝⁵ : PseudoMetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : SeminormedAddCommGroup V₂\ninst✝² : NormedSpace 𝕜 V₂\ninst✝¹ : PseudoMe... | [] | rw [dist_eq_norm_vsub V₂, dist_eq_norm_vsub V, ← map_vsub, f.linearIsometry.norm_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 87
} | {
"line": 137,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\nP : Type u_10\nP₂ : Type u_11\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : NormedSpace 𝕜 V\ninst✝⁵ : PseudoMetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : SeminormedAddCommGroup V₂\ninst✝² : NormedSpace 𝕜 V₂\ninst✝¹ : PseudoMe... | [] | rw [dist_eq_norm_vsub V₂, dist_eq_norm_vsub V, ← map_vsub, f.linearIsometry.norm_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1091,
"column": 4
} | {
"line": 1093,
"column": 22
} | {
"line": 1095,
"column": 0
} | [] | [] | ∫ x, f x ∂Measure.dirac a = ∫ _, f a ∂Measure.dirac a :=
integral_congr_ae <| ae_eq_dirac' hfm.measurable
_ = f a := by simp | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 802,
"column": 2
} | {
"line": 808,
"column": 53
} | {
"line": 810,
"column": 0
} | [
{
"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\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\nh : f =ᵐ[μ] g\n⊢ setToFun μ T ... | [] | by_cases hF : CompleteSpace F; swap
· simp [setToFun, hF]
by_cases hfi : Integrable f μ
· have hgi : Integrable g μ := hfi.congr h
rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h]
· have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
rw [s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 802,
"column": 2
} | {
"line": 808,
"column": 53
} | {
"line": 810,
"column": 0
} | [
{
"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\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\nh : f =ᵐ[μ] g\n⊢ setToFun μ T ... | [] | by_cases hF : CompleteSpace F; swap
· simp [setToFun, hF]
by_cases hfi : Integrable f μ
· have hgi : Integrable g μ := hfi.congr h
rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h]
· have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
rw [s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1013,
"column": 2
} | {
"line": 1013,
"column": 18
} | {
"line": 1014,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nG : Type u_5\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ μ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ∞\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\n⊢ ∀ (b : ↥(Lp G 1 μ)),\n ∀ ε > 0, ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a b < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑b... | [
"case neg\nα : Type u_1\nG : Type u_5\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ μ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ∞\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a f < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε... | intro f ε hε_pos | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 458,
"column": 2
} | {
"line": 463,
"column": 32
} | {
"line": 464,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LocallyCompactSpace X\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nι : Type u_3\nF : ι → C(X, E)\nhF : ∀ (K : Compacts X), Summable fun i ↦ ‖restrict (↑K) (F i)‖\nK : Compacts X\n⊢ ∃ f, Filter.Tendsto (fun i ↦ restrict (↑K) (... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LocallyCompactSpace X\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nι : Type u_3\nF : ι → C(X, E)\nhF : ∀ (K : Compacts X), Summable fun i ↦ ‖restrict (↑K) (F i)‖\nK : Compacts X\nA : ∀ (s : Finset ι), restrict (↑K) (∑ i ∈ s, F i) = ∑ i ∈... | have A : ∀ s : Finset ι, restrict K (∑ i ∈ s, F i) = ∑ i ∈ s, restrict K (F i) := by
intro s
ext1 x
-- TODO: there is a non-confluence problem in the lemmas here,
-- and `SetLike.coe_sort_coe` prevents `restrict_apply` from being used.
simp [-SetLike.coe_sort_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 17
} | {
"line": 542,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : CompleteSpace 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\nS : Submodule 𝕜 E\n⊢ ProperSpace ↥S",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nontrivi... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : CompleteSpace 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\nS : Submodule 𝕜 E\na✝ : Nontrivial E\n⊢ ProperSpace ↥S"
] | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 643,
"column": 34
} | {
"line": 648,
"column": 36
} | {
"line": 650,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nk : ℕ\nr : F\nhr : ‖r‖ < 1\nu : ℕ → F\nhu : u =O[atTop] fun n ↦ ↑(n ^ k)\nr' : ℝ\nhrr' : ‖r‖ < r'\nh : r' < 1\n⊢ (fun n ↦ ‖↑n ^ k‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖r' ^ n‖",
"ppTerm": "?m.169",
"assigned": true,
... | [] | by
convert!
isBigO_norm_right.mpr
(isBigO_norm_left.mpr
(isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt k hrr').isBigO)
simp only [norm_pow, norm_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 31
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\nx y : α\nh : infEDist x E ≤ infEDist y E\nhle : 1 ≤ infEDist x E / ENNReal.ofReal δ\n⊢ 1 - infEDist y E / ENNReal.ofReal δ ≤ 1 - infEDist x E / ENNReal.ofReal δ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | · rw [tsub_eq_zero_of_le hle, tsub_eq_zero_of_le]
exact hle.trans (by gcongr) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Content | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 55
} | {
"line": 377,
"column": 2
} | [
{
"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\nK : Set G\nhK : IsCompact K\n⊢ μ.outerMeasure (closure K) < ∞",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"IsCom... | [] | exact μ.outerMeasure_lt_top_of_isCompact hK.closure | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 210,
"column": 65
} | {
"line": 210,
"column": 70
} | {
"line": 210,
"column": 70
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index (K... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ ... | ← h2s | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 51
} | {
"line": 217,
"column": 51
} | [
{
"pp": "case h.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card ... | [
"case h.w\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index (K₁.... | simp only [Finset.mem_filter, h1g₀, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 138,
"column": 94
} | {
"line": 141,
"column": 29
} | {
"line": 143,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : X → E\nμ : Measure X\nι : Type u_5\ninst✝ : Fintype ι\ns : ι → Set X\nhs : ∀ (i : ι), MeasurableSet (s i)\nh's : Pairwise (Disjoint on s)\nhf : ∀ (i : ι), IntegrableOn f (s i) μ\n⊢ ∫ (x : X)... | [] | by
convert! integral_biUnion_finset Finset.univ (fun i _ => hs i) _ fun i _ => hf i
· simp
· simp [pairwise_univ, h's] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 217,
"column": 45
} | {
"line": 217,
"column": 70
} | {
"line": 217,
"column": 70
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns✝ : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\ni : δ\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ny : X i\ns : Finset δ\nih : i ∉ s → ∀ (x : (i : δ) → X i), (∫⋯∫⁻_s, f... | [] | exact mem_insert_self i s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 241,
"column": 50
} | {
"line": 241,
"column": 55
} | {
"line": 241,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : (interior V).Nonempty\ng : G\ns : Finset G\nh1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index K V\n⊢ index ((fun h ↦ g * h) '' K) V ≤ index K V",
"ppT... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : (interior V).Nonempty\ng : G\ns : Finset G\nh1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index K V\n⊢ index ((fun h ↦ g * h) '' K) V ≤ s.card"
] | ← h2s | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 44
} | {
"line": 255,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nι : Type u_5\nt : Finset ι\ns : ι → Set X\nhs : ∀ i ∈ t, MeasurableSet (s i)\nhf : ∀ i ∈ t, μ (s i) ≠ ∞\n⊢ μ.real (⋃ i ∈ t, s i) = ∑ u ∈ t.powerset with u.Nonempty, (-1) ^ (#u + 1) * μ.real (⋂ i ∈ u, s i)",
"ppTerm": "?m.71",
"assigned": true... | [
"X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nι : Type u_5\nt : Finset ι\ns : ι → Set X\nhs : ∀ i ∈ t, MeasurableSet (s i)\nhf : ∀ i ∈ t, μ (s i) ≠ ∞\n⊢ ∫ (x : X) in ⋃ i ∈ t, s i, 1 ∂μ = ∑ x ∈ t.powerset with x.Nonempty, (-1) ^ (#x + 1) * ∫ (x : X) in ⋂ i ∈ x, s i, 1 ∂μ"
] | simp_rw [← setIntegral_one_eq_measureReal] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 19
} | {
"line": 393,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁\nthis : Continuous eval\n⊢ chaar K₀ K₁ ≤ chaar K₀ K₂",
"ppTerm": "?m.45",
"assigned": true,
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁\nthis : Continuous eval\n⊢ 0 ≤ chaar K₀ K₂ - chaar K₀ K₁"
] | rw [← sub_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 19
} | {
"line": 406,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)\nthis : Continuous eval\n⊢ chaar K₀ (K₁ ⊔ K₂) ≤ chaar K₀ K₁ + chaar K₀ K₂",
"ppTerm": "?m.46",
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)\nthis : Continuous eval\n⊢ 0 ≤ chaar K₀ K₁ + chaar K₀ K₂ - chaar K₀ (K₁ ⊔ K₂)"
] | rw [← sub_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.LeftRightLim | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 30
} | {
"line": 248,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\nb : β\nh : Tendsto f atTop (𝓝 b)\n⊢ Tendsto (rightLim f) atTop (𝓝 b)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants"... | [
"case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\nb : β\nh : Tendsto f atTop (𝓝 b)\nval✝ : OrderTop α\n⊢ Tendsto (rightLim f) atTop (𝓝 b)",
"case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴... | cases topOrderOrNoTopOrder α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.Order.LeftRightLim | {
"line": 378,
"column": 4
} | {
"line": 379,
"column": 70
} | {
"line": 380,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh : ContinuousAt f x\nA : leftLim f x = f x\n⊢ ... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh : ContinuousAt f x\nA : leftLim f x = f x\nB : rightLim f... | have B : rightLim f x = f x :=
hf.continuousWithinAt_Ioi_iff_rightLim_eq.1 h.continuousWithinAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 10
} | {
"line": 261,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\ninst✝ : OrderTopology R\nf : StieltjesFunction R\nx : R\nhx : x ∈ {x | leftLim (↑f) x ≠ ↑f x}\nh'x : ContinuousAt (↑f) x\n⊢ False",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hx"
... | [
"R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\ninst✝ : OrderTopology R\nf : StieltjesFunction R\nx : R\nhx : x ∈ {x | leftLim (↑f) x ≠ ↑f x}\nh'x : ContinuousAt (↑f) x\n⊢ leftLim (↑f) x = ↑f x"
] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 40
} | {
"line": 530,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : MeasurableSpace G\ninst✝ : BorelSpace G\nK₀ : PositiveCompacts G\n⊢ (haarMeasure K₀).IsMulLeftInvariant",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Measu... | [
"G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : MeasurableSpace G\ninst✝ : BorelSpace G\nK₀ : PositiveCompacts G\n⊢ ∀ (g : G) (A : Set G), MeasurableSet A → (haarMeasure K₀) ((fun h ↦ g * h) ⁻¹' A) = (haarMeasure K₀) A"
] | rw [← forall_measure_preimage_mul_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 353,
"column": 8
} | {
"line": 353,
"column": 27
} | {
"line": 353,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nthis : discrim (normSq x) (2 * ‖⟪x, y⟫‖) (normSq y) ≤ 0\n⊢ ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫",
"ppTerm": "?m.60",
"assigned": true,
"usedCons... | [
"𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nthis : discrim (normSq x) (2 * ‖⟪x, y⟫‖) (normSq y) ≤ 0\n⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫"
] | norm_inner_symm y x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 784,
"column": 2
} | {
"line": 784,
"column": 49
} | {
"line": 785,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nc : ℝ\n⊢ (StieltjesFunction.const R c).measure = 0",
"ppTerm": "?... | [
"R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nc : ℝ\na b : R\nhab : a ≤ b\n⊢ (StieltjesFunction.const R c).measure (Icc a b) = ... | apply Measure.ext_of_Icc _ _ (fun a b hab ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 768,
"column": 4
} | {
"line": 771,
"column": 49
} | {
"line": 772,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh₀ : x ≠ 0\nh₀' : ¬↑‖x‖ = 0\nh : ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖\n⊢ (↑‖y‖ / ↑‖x‖) • x = y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Norm.norm",
"... | [] | have : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫ : 𝕜) • x :=
((norm_inner_eq_norm_tfae 𝕜 x y).out 0 1).1 (by simp [h])
rw [this.resolve_left h₀, h]
simp [norm_smul, mul_div_cancel_right₀ _ h₀'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 768,
"column": 4
} | {
"line": 771,
"column": 49
} | {
"line": 772,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh₀ : x ≠ 0\nh₀' : ¬↑‖x‖ = 0\nh : ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖\n⊢ (↑‖y‖ / ↑‖x‖) • x = y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Norm.norm",
"... | [] | have : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫ : 𝕜) • x :=
((norm_inner_eq_norm_tfae 𝕜 x y).out 0 1).1 (by simp [h])
rw [this.resolve_left h₀, h]
simp [norm_smul, mul_div_cancel_right₀ _ h₀'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 31
} | {
"line": 228,
"column": 2
} | [
{
"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\nv w : ι → E\nhv : Orthonormal 𝕜 v\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\n⊢ Orthonormal 𝕜 w",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"N... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhv : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\n⊢ ∀ (i j : ι), ⟪w i, w j⟫ = if i = j then 1 else 0"
] | rw [orthonormal_iff_ite] at * | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 208,
"column": 66
} | {
"line": 214,
"column": 48
} | {
"line": 216,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : WithLp p (α × β)\n⊢ edist f f = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"WithLp.prod_edist_eq_card",
"Iff.mpr",
"WithLp",
"GroupWithZero.toMonoid... | [] | by
rcases p.trichotomy with (rfl | rfl | h)
· classical
simp
· simp [prod_edist_eq_sup]
· simp [prod_edist_eq_add h, ENNReal.zero_rpow_of_pos h,
ENNReal.zero_rpow_of_pos (inv_pos.2 <| h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 395,
"column": 6
} | {
"line": 396,
"column": 40
} | {
"line": 396,
"column": 41
} | [
{
"pp": "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\nf g : WithLp p (α × β)\nh : 1 ≤ p.toReal\nthis : 0 < p.toReal\n⊢ (ENNReal.ofReal (dist f.fst g.fst) ^ p.toReal + ENNReal.ofReal (dist f.snd g.snd) ^ p.toReal) ^ p... | [
"case inr.hx_nonneg\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\nf g : WithLp p (α × β)\nh : 1 ≤ p.toReal\nthis : 0 < p.toReal\n⊢ 0 ≤ dist f.snd g.snd",
"case inr.hp_nonneg\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\n... | rw [← ENNReal.ofReal_rpow_of_nonneg, ENNReal.ofReal_add, ← ENNReal.ofReal_rpow_of_nonneg,
← ENNReal.ofReal_rpow_of_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 98,
"column": 2
} | {
"line": 102,
"column": 84
} | {
"line": 103,
"column": 2
} | [
{
"pp": "case left\nn : Type u\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\nR : Type u_1\nm : Type u_2\ninst✝³ : Semiring R\ninst✝² : StarRing R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nU₁ : Matrix n n R\nU₂ : Matrix m m R\nhU₁ : U₁ ∈ unitary (Matrix n n R)\nhU₂ : U₂ ∈ unitary (Matrix m m R)\ni✝ j✝ : n × m\n... | [
"case right\nn : Type u\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\nR : Type u_1\nm : Type u_2\ninst✝³ : Semiring R\ninst✝² : StarRing R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nU₁ : Matrix n n R\nU₂ : Matrix m m R\nhU₁ : U₁ ∈ unitary (Matrix n n R)\nhU₂ : U₂ ∈ unitary (Matrix m m R)\ni✝ j✝ : n × m\n⊢ ∑ x, U₁ i... | · simp_rw [mul_assoc _ (star U₁ _ _), ← Finset.univ_product_univ, Finset.sum_product]
rw [Finset.sum_comm]
simp_rw [← Finset.sum_mul, ← Finset.mul_sum, ← Matrix.mul_apply, hU₁.1, Matrix.one_apply,
mul_boole, ite_mul, zero_mul, Finset.sum_ite_irrel, ← Matrix.mul_apply, hU₂.1,
Matrix.one_apply, Finset... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 1152,
"column": 4
} | {
"line": 1153,
"column": 85
} | {
"line": 1155,
"column": 0
} | [
{
"pp": "case inr.inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nα' : Type u_5\nβ' : Type u_6\nhp✝ : Fact (1 ≤ p)\ninst✝⁵ : PseudoEMetricSpace α\ninst✝⁴ : PseudoEMetricSpace β\ninst✝³ : PseudoEMetricSpace γ\ninst✝² : PseudoEMetricSpace α'\ninst✝¹ : PseudoEMetricSpace β'\ninst✝ : Unique ... | [] | · simp_rw [WithLp.prod_edist_eq_add hp, Unique.eq_default, edist_self,
ENNReal.zero_rpow_of_pos hp, add_zero, one_div, ENNReal.rpow_rpow_inv hp.ne'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 961,
"column": 27
} | {
"line": 961,
"column": 41
} | {
"line": 961,
"column": 41
} | [
{
"pp": "p✝ : ℝ≥0∞\n𝕜 : Type u_1\nι✝ : Type u_2\nα✝ : ι✝ → Type u_3\nβ : ι✝ → Type u_4\nhp : Fact (1 ≤ p✝)\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : Semiring 𝕜\ninst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)\ninst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)\ninst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)\ninst✝⁵ : (i : ι✝) → ... | [
"p✝ : ℝ≥0∞\n𝕜 : Type u_1\nι✝ : Type u_2\nα✝ : ι✝ → Type u_3\nβ : ι✝ → Type u_4\nhp : Fact (1 ≤ p✝)\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : Semiring 𝕜\ninst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)\ninst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)\ninst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)\ninst✝⁵ : (i : ι✝) → Module 𝕜 (β... | NNReal.coe_inj | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 406,
"column": 2
} | {
"line": 407,
"column": 92
} | {
"line": 409,
"column": 0
} | [
{
"pp": "case mk\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ns : ι → Set ℝ\nhs : ∀ (i : ι), MeasurableSet (s i)\nh2s : MeasurableSet (univ.pi s)\nx : ι → ℝ\nt_i t_j : ι\nt_hij : t_i ≠ t_j\nt_c : ℝ\nht : Measurable ⇑(toLin' { i := t_i, j := t_j, hij := t_hij, c := t_c }.toMatrix)\n⊢ ∫⁻ (xᵢ : ℝ),\n ... | [] | simp [transvection, single_mulVec, t_hij.symm, ← Function.update_add,
lintegral_add_right_eq_self fun xᵢ ↦ indicator (univ.pi s) 1 (Function.update x t_i xᵢ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 406,
"column": 2
} | {
"line": 407,
"column": 92
} | {
"line": 409,
"column": 0
} | [
{
"pp": "case mk\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ns : ι → Set ℝ\nhs : ∀ (i : ι), MeasurableSet (s i)\nh2s : MeasurableSet (univ.pi s)\nx : ι → ℝ\nt_i t_j : ι\nt_hij : t_i ≠ t_j\nt_c : ℝ\nht : Measurable ⇑(toLin' { i := t_i, j := t_j, hij := t_hij, c := t_c }.toMatrix)\n⊢ ∫⁻ (xᵢ : ℝ),\n ... | [] | simp [transvection, single_mulVec, t_hij.symm, ← Function.update_add,
lintegral_add_right_eq_self fun xᵢ ↦ indicator (univ.pi s) 1 (Function.update x t_i xᵢ)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 406,
"column": 2
} | {
"line": 407,
"column": 92
} | {
"line": 409,
"column": 0
} | [
{
"pp": "case mk\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ns : ι → Set ℝ\nhs : ∀ (i : ι), MeasurableSet (s i)\nh2s : MeasurableSet (univ.pi s)\nx : ι → ℝ\nt_i t_j : ι\nt_hij : t_i ≠ t_j\nt_c : ℝ\nht : Measurable ⇑(toLin' { i := t_i, j := t_j, hij := t_hij, c := t_c }.toMatrix)\n⊢ ∫⁻ (xᵢ : ℝ),\n ... | [] | simp [transvection, single_mulVec, t_hij.symm, ← Function.update_add,
lintegral_add_right_eq_self fun xᵢ ↦ indicator (univ.pi s) 1 (Function.update x t_i xᵢ)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 160,
"column": 6
} | {
"line": 161,
"column": 65
} | {
"line": 162,
"column": 2
} | [
{
"pp": "case mp.inr\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\na t : ι → ℝ\ni : ι\nht : 0 ≤ t i ∧ t i ≤ 1\nhai : 0 ≤ a i\n⊢ min 0 (a i) ≤ t i * a i ∧ t i * a i ≤ max 0 (a i)",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
... | [] | rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai]
exact ⟨mul_nonneg ht.1 hai, mul_le_of_le_one_left hai ht.2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 160,
"column": 6
} | {
"line": 161,
"column": 65
} | {
"line": 162,
"column": 2
} | [
{
"pp": "case mp.inr\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\na t : ι → ℝ\ni : ι\nht : 0 ≤ t i ∧ t i ≤ 1\nhai : 0 ≤ a i\n⊢ min 0 (a i) ≤ t i * a i ∧ t i * a i ≤ max 0 (a i)",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
... | [] | rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai]
exact ⟨mul_nonneg ht.1 hai, mul_le_of_le_one_left hai ht.2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 25
} | {
"line": 265,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\n⊢ (match ↑I with\n | some val => true\n | none => false) =\n true ↔\n (↑↑I).Nonempty",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real",
"WithBot.some",
"congrArg",
"BoxIntegral.Box.toSet",
"Option.isSome... | [] | simp [I.nonempty_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 258,
"column": 53
} | {
"line": 258,
"column": 67
} | {
"line": 258,
"column": 68
} | [
{
"pp": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁶ : NormedField K\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace K E\nb : Basis ι K E\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\ny : ↥(span ℤ (Set.range ⇑b))\nx : E\n⊢ y +ᵥ x ∈ fundamentalDomain ... | [
"E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁶ : NormedField K\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace K E\nb : Basis ι K E\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\ny : ↥(span ℤ (Set.range ⇑b))\nx : E\n⊢ y +ᵥ x ∈ fundamentalDomain b ↔ ↑y +ᵥ x ... | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 544,
"column": 6
} | {
"line": 544,
"column": 26
} | {
"line": 545,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhr : r ≠ 0\nx y : E\ns t : Set E\n⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) * μ s * (ENNReal.ofReal (|r|... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhr : r ≠ 0\nx y : E\ns t : Set E\n⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) * μ s * ((ENNReal.ofReal (|r| ^ finrank ... | rw [ENNReal.mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 638,
"column": 8
} | {
"line": 638,
"column": 22
} | {
"line": 638,
"column": 23
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt u ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt u : Set E\nh'u... | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 638,
"column": 39
} | {
"line": 638,
"column": 70
} | {
"line": 638,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt u ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt u : Set E\nh'u... | affinity_unitClosedBall rpos.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 482,
"column": 48
} | {
"line": 482,
"column": 74
} | {
"line": 482,
"column": 75
} | [
{
"pp": "case refine_1.refine_2\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodu... | [
"case refine_1.refine_2\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst... | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 166,
"column": 89
} | {
"line": 168,
"column": 42
} | {
"line": 169,
"column": 2
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\nx : ι → ℝ\nJ₁ : Box ι\nh₁ : J₁ ∈ π\nhx₁ : x ∈ Box.Icc J₁\nJ₂ : Box ι\nh₂ : J₂ ∈ π\nhx₂ : x ∈ Box.Icc J₂\nH : {i | J₁.lower i = x i} = {i | J₂.lower i = x i}\nthis : ∀ (i : ι), (Set.Ioc (J₁.lower i) (J₁.upper i) ∩ Set.Ioc (J₂.lower i) (J₂.upper i)).Nonempty\n... | [] | by
choose y hy₁ hy₂ using this
exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 683,
"column": 6
} | {
"line": 686,
"column": 29
} | {
"line": 687,
"column": 2
} | [
{
"pp": "case h2\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝... | [] | filter_upwards [self_mem_nhdsWithin]
intro r rpos
rw [mul_zero]
exact mul_pos Rpos rpos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 683,
"column": 6
} | {
"line": 686,
"column": 29
} | {
"line": 687,
"column": 2
} | [
{
"pp": "case h2\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝... | [] | filter_upwards [self_mem_nhdsWithin]
intro r rpos
rw [mul_zero]
exact mul_pos Rpos rpos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 563,
"column": 24
} | {
"line": 563,
"column": 50
} | {
"line": 563,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst✝ : Discrete... | [
"K : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\... | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 494,
"column": 66
} | {
"line": 503,
"column": 50
} | {
"line": 505,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nπ' : Prepartition I\n⊢ π' ≤ π.biUnion πi ↔ π' ≤ π ∧ ∀ J ∈ π, π'.restrict J ≤ πi J",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"WithBot.some",
"WithBot",... | [] | by
refine ⟨fun H => ⟨H.trans (π.biUnion_le πi), fun J hJ => ?_⟩, ?_⟩
· rw [← π.restrict_biUnion πi hJ]
exact restrict_mono H
· rintro ⟨H, Hi⟩ J' hJ'
rcases H hJ' with ⟨J, hJ, hle⟩
have : J' ∈ π'.restrict J :=
π'.mem_restrict.2 ⟨J', hJ', (inf_of_le_right <| WithBot.coe_le_coe.2 hle).symm⟩
rca... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction | {
"line": 56,
"column": 77
} | {
"line": 56,
"column": 98
} | {
"line": 58,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nI J : Box ι\n⊢ J ∈ splitCenter I ↔ ∃ s, I.splitCenterBox s = J",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Finset.univ",
"BoxIntegral.Prepartition",
"congrArg",
"Finset",
"BoxIntegral.Prepartition.splitCenter._pro... | [] | by simp [splitCenter] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction | {
"line": 90,
"column": 2
} | {
"line": 92,
"column": 12
} | {
"line": 94,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\np : Box ι → Prop\nI : Box ι\nH_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.l... | [] | refine subbox_induction_on' I (fun J hle hs => H_ind J hle fun J' h' => ?_) H_nhds
rcases mem_splitCenter.1 h' with ⟨s, rfl⟩
exact hs s | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction | {
"line": 90,
"column": 2
} | {
"line": 92,
"column": 12
} | {
"line": 94,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\np : Box ι → Prop\nI : Box ι\nH_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.l... | [] | refine subbox_induction_on' I (fun J hle hs => H_ind J hle fun J' h' => ?_) H_nhds
rcases mem_splitCenter.1 h' with ⟨s, rfl⟩
exact hs s | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 650,
"column": 2
} | {
"line": 651,
"column": 49
} | {
"line": 653,
"column": 0
} | [
{
"pp": "ι : Type u_3\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝⁴ : DiscreteTopology ↥L\ninst✝³ : IsZLattice ℝ L\ninst✝² : Finite ι\nb : Basis ι ℤ ↥L\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : Measure E\n... | [] | convert! ZSpan.isAddFundamentalDomain (b.ofZLatticeBasis ℝ) μ
all_goals exact (b.ofZLatticeBasis_span ℝ).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 650,
"column": 2
} | {
"line": 651,
"column": 49
} | {
"line": 653,
"column": 0
} | [
{
"pp": "ι : Type u_3\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝⁴ : DiscreteTopology ↥L\ninst✝³ : IsZLattice ℝ L\ninst✝² : Finite ι\nb : Basis ι ℤ ↥L\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : Measure E\n... | [] | convert! ZSpan.isAddFundamentalDomain (b.ofZLatticeBasis ℝ) μ
all_goals exact (b.ofZLatticeBasis_span ℝ).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 799,
"column": 2
} | {
"line": 799,
"column": 52
} | {
"line": 800,
"column": 2
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : TopologicalSpace F\nL : Submodule ℤ E\ninst✝¹ : DiscreteTopology ↥L\ninst✝ : IsZLattice ℝ L\nf : E → F\nhf : Continuous f\nhf' : ∀ (z w : E), w ∈ L → f (z + w) = f z\nthis : Fre... | [
"E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : TopologicalSpace F\nL : Submodule ℤ E\ninst✝¹ : DiscreteTopology ↥L\ninst✝ : IsZLattice ℝ L\nf : E → F\nhf : Continuous f\nhf' : ∀ (z w : E), w ∈ L → f (z + w) = f z\nthis : Free ℤ ↥L\nb : ... | refine le_antisymm ?_ (Set.image_subset_range _ _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 947,
"column": 4
} | {
"line": 947,
"column": 67
} | {
"line": 949,
"column": 0
} | [
{
"pp": "case e'_3\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι'\na : OrthonormalBasis ι' 𝕜 E\nb : OrthonormalBasis ι 𝕜 E\ni j : ι\n⊢ 1 i j = ⟪b i, b ... | [] | rw [orthonormal_iff_ite.mp b.orthonormal i j, Matrix.one_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 947,
"column": 4
} | {
"line": 947,
"column": 67
} | {
"line": 949,
"column": 0
} | [
{
"pp": "case e'_3\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι'\na : OrthonormalBasis ι' 𝕜 E\nb : OrthonormalBasis ι 𝕜 E\ni j : ι\n⊢ 1 i j = ⟪b i, b ... | [] | rw [orthonormal_iff_ite.mp b.orthonormal i j, Matrix.one_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 947,
"column": 4
} | {
"line": 947,
"column": 67
} | {
"line": 949,
"column": 0
} | [
{
"pp": "case e'_3\nι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι'\na : OrthonormalBasis ι' 𝕜 E\nb : OrthonormalBasis ι 𝕜 E\ni j : ι\n⊢ 1 i j = ⟪b i, b ... | [] | rw [orthonormal_iff_ite.mp b.orthonormal i j, Matrix.one_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 64
} | {
"line": 321,
"column": 0
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nhB : hasIntegralVertices B\nx : ι → ℝ\nhx : x ∈ B\n⊢ ∃ ν ∈ admissibleIndex n B, box n ν = box n (index n x)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"BoxInteg... | [] | exact ⟨index n x, mem_admissibleIndex_of_mem_box n hB hx, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1192,
"column": 2
} | {
"line": 1192,
"column": 59
} | {
"line": 1193,
"column": 2
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\n... | [
"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fin... | haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 425,
"column": 2
} | {
"line": 452,
"column": 66
} | {
"line": 454,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nhF : Continuous F\nhs₁ : Bornology.IsBounded s\nhs₂ : MeasurableSet s\nhs₃ : volume (frontier s) = 0\n⊢ Tendsto (fun n ↦ (∑' (x : ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))), F ↑x) / ↑n ^ card ι) atTop\n (𝓝 (∫ (x : ι →... | [] | obtain ⟨B, hB, hs₀⟩ := le_hasIntegralVertices_of_isBounded hs₁
refine Metric.tendsto_atTop.mpr fun ε hε ↦ ?_
have h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖Set.indicator s F x‖ ≤ C := by
obtain ⟨C₀, h₀⟩ := (Box.isCompact_Icc B).exists_bound_of_continuousOn hF.continuousOn
refine ⟨max 0 C₀, fun x hx ↦ ?_⟩
rw [Set.indi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 425,
"column": 2
} | {
"line": 452,
"column": 66
} | {
"line": 454,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nhF : Continuous F\nhs₁ : Bornology.IsBounded s\nhs₂ : MeasurableSet s\nhs₃ : volume (frontier s) = 0\n⊢ Tendsto (fun n ↦ (∑' (x : ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))), F ↑x) / ↑n ^ card ι) atTop\n (𝓝 (∫ (x : ι →... | [] | obtain ⟨B, hB, hs₀⟩ := le_hasIntegralVertices_of_isBounded hs₁
refine Metric.tendsto_atTop.mpr fun ε hε ↦ ?_
have h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖Set.indicator s F x‖ ≤ C := by
obtain ⟨C₀, h₀⟩ := (Box.isCompact_Icc B).exists_bound_of_continuousOn hF.continuousOn
refine ⟨max 0 C₀, fun x hx ↦ ?_⟩
rw [Set.indi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Orientation | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 26
} | {
"line": 96,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\n⊢ reindex R M (Equiv.refl ι) = Equiv.refl (Orientation R M ι)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\n⊢ Ray.map (AlternatingMap.domDomCongrₗ R (Equiv.refl ι)) = Equiv.refl (Orientation R M ι)"
] | Orientation.reindex, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 84,
"column": 50
} | {
"line": 108,
"column": 36
} | {
"line": 110,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\na b : ι\nh₀ : a ≠ b\n⊢ ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫_𝕜 = 0",
"ppTe... | [] | by
suffices ∀ a b : ι, a < b → ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 by
rcases h₀.lt_or_gt with ha | hb
· exact this _ _ ha
· rw [inner_eq_zero_symm]
exact this _ _ hb
clear h₀ a b
intro a b h₀
revert a
apply wellFounded_lt.induction b
intro b ih a h₀
simp only [gramSchmidt_def 𝕜... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Orientation | {
"line": 239,
"column": 46
} | {
"line": 244,
"column": 29
} | {
"line": 246,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne : Basis ι R M\nx : Orientation R M ι\n⊢ x = e.orientation ∨ x = -e.orientation",
"ppTerm"... | [] | by
induction x using Module.Ray.ind with | h x hx =>
rw [← x.map_basis_ne_zero_iff e] at hx
rwa [Basis.orientation, ray_eq_iff, neg_rayOfNeZero, ray_eq_iff, x.eq_smul_basis_det e,
sameRay_neg_smul_left_iff_of_ne e.det_ne_zero hx, sameRay_smul_left_iff_of_ne e.det_ne_zero hx,
lt_or_lt_iff_ne, ne_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 41
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.car... | [] | · simp [gramSchmidtOrthonormalBasis_apply h hi, gramSchmidtNormed, inner_smul_left,
gramSchmidt_inv_triangular 𝕜 f hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 14
} | {
"line": 189,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ Nat.casesAuxOn (motive := fun a ↦ 0 = a → E [⋀^Fin 0]→ₗ[ℝ] ℝ) 0\n (fun h ↦\n if hp :\n -rayOfNeZero ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯ =\n rayOfNe... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ ¬-rayOfNeZero ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯ =\n rayOfNeZero ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯"
] | apply if_neg | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.ProdL2 | {
"line": 100,
"column": 79
} | {
"line": 101,
"column": 82
} | {
"line": 103,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nx : E\n⊢ K.orthogonalDecomposition x = WithLp.toLp 2 (K.orthogonalProjectionOnto x, Kᗮ.orthogonalProjectionOnto x)",
"ppTerm": "?m.... | [] | by
simp [orthogonalDecomposition, orthogonalProjectionOnto_apply_eq_projectionOnto] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 88
} | {
"line": 216,
"column": 2
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nb : OrthonormalBasis (Fin (n + 1)) ℝ E\nhb : b.toBasis.orientation ≠ o\n⊢ o.volumeForm = -b.toBasis.det",
"ppTerm": "?succ",
"assigne... | [
"case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nb : OrthonormalBasis (Fin (n + 1)) ℝ E\nhb : b.toBasis.orientation ≠ o\ne : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o\n⊢... | let e : OrthonormalBasis (Fin n.succ) ℝ E := o.finOrthonormalBasis n.succ_pos Fact.out | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 88
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\n⊢ (-o).volumeForm = -o.volumeForm",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpac... | [
"case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\ne : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o\n⊢ (-o).volumeForm = -o.volumeForm"
] | let e : OrthonormalBasis (Fin n.succ) ℝ E := o.finOrthonormalBasis n.succ_pos Fact.out | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 88
} | {
"line": 298,
"column": 2
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\nφ : E ≃ₗᵢ[ℝ] F\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\ninst✝ : Fact (finrank ℝ F = n + 1)\nx : Fin (n ... | [
"case succ\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\nφ : E ≃ₗᵢ[ℝ] F\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\ninst✝ : Fact (finrank ℝ F = n + 1)\nx : Fin (n + 1) → F\ne ... | let e : OrthonormalBasis (Fin n.succ) ℝ E := o.finOrthonormalBasis n.succ_pos Fact.out | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.PSeries | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 29
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nf : ℕ → M\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ ∑ k ∈ range (2 ^ n), f k ≤ f 0 + ∑ k ∈ range n, 2 ^ k • f (2 ^ k)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"... | [] | grw [← le_sum_condensed' hf n, ← sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ,
sum_range_zero, zero_add] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Analysis.PSeries | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 29
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nf : ℕ → M\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ ∑ k ∈ range (2 ^ n), f k ≤ f 0 + ∑ k ∈ range n, 2 ^ k • f (2 ^ k)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"... | [] | grw [← le_sum_condensed' hf n, ← sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ,
sum_range_zero, zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.PSeries | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 29
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedAddMonoid M\nf : ℕ → M\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ ∑ k ∈ range (2 ^ n), f k ≤ f 0 + ∑ k ∈ range n, 2 ^ k • f (2 ^ k)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"... | [] | grw [← le_sum_condensed' hf n, ← sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ,
sum_range_zero, zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.PSeries | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 82
} | {
"line": 154,
"column": 2
} | [
{
"pp": "case h₂\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nhu : StrictMono u\nn : ℕ\n⊢ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) ≤ ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k)",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"co... | [
"case h₂\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nhu : StrictMono u\nn : ℕ\nthis : ∀ (k : ℕ), ↑(u (k + 1)) - ↑(u k) = ↑(u (k + 1) - u k)\n⊢ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) ≤ ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k)"
] | have (k : ℕ) : (u (k + 1) - u k : ℝ≥0∞) = (u (k + 1) - (u k : ℕ) : ℕ) := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 176,
"column": 2
} | {
"line": 177,
"column": 71
} | {
"line": 178,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\nL₁ L₂ : Submodule ℤ E\ninst✝³ : DiscreteTopology ↥L₁\ninst✝² : IsZLattice ℝ L₁\ninst✝¹ : DiscreteTopology ↥L₂\ninst✝ : IsZLattice ℝ L₂\nh : L₁ ... | [
"E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\nL₁ L₂ : Submodule ℤ E\ninst✝³ : DiscreteTopology ↥L₁\ninst✝² : IsZLattice ℝ L₁\ninst✝¹ : DiscreteTopology ↥L₂\ninst✝ : IsZLattice ℝ L₂\nh : L₁ ≤ L₂\nf : (F... | have hf : MeasurePreserving f := (stdOrthonormalBasis ℝ E).measurePreserving_repr_symm.comp
(EuclideanSpace.volume_preserving_symm_measurableEquiv_toLp _).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.PSeries | {
"line": 325,
"column": 59
} | {
"line": 326,
"column": 68
} | {
"line": 328,
"column": 0
} | [
{
"pp": "p : ℕ\n⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real",
"Real.summable_nat_rpow_inv._simp_1",
"FloorRing.toFloorSemiring",
"Real.instZeroLEOneClass",
"cong... | [] | by
simp only [← rpow_natCast, summable_nat_rpow_inv, Nat.one_lt_cast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 51
} | {
"line": 242,
"column": 4
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nL : Submodule ℤ E\ninst✝⁵ : DiscreteTopology ↥L\ninst✝⁴ : IsZLattice ℝ L\nι : Type u_2\ninst✝³ : Fintype ι\nb : Basis ι ℤ ↥L\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\ns : Set E... | [
"case refine_2\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nL : Submodule ℤ E\ninst✝⁵ : DiscreteTopology ↥L\ninst✝⁴ : IsZLattice ℝ L\nι : Type u_2\ninst✝³ : Fintype ι\nb : Basis ι ℤ ↥L\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\ns : Set E\nhs₁ : IsVo... | rw [← NormedSpace.isVonNBounded_iff ℝ] at hs₁ ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 263,
"column": 89
} | {
"line": 293,
"column": 87
} | {
"line": 295,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nL : Submodule ℤ E\ninst✝⁶ : DiscreteTopology ↥L\ninst✝⁵ : IsZLattice ℝ L\nι : Type u_2\ninst✝⁴ : Fintype ι\nb : Basis ι ℤ ↥L\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : Nonempty ι\nX :... | [] | by
refine Tendsto.congr' ?_ <| (tendsto_card_div_pow_atTop_volume'
((b.ofZLatticeBasis ℝ).equivFun '' {x ∈ X | F x ≤ 1}) ?_ ?_ h₄ fun x y hx hy ↦ ?_).comp
(tendsto_rpow_atTop <| inv_pos.mpr
(Nat.cast_pos.mpr card_pos) : Tendsto (fun x ↦ x ^ (card ι : ℝ)⁻¹) atTop atTop)
· filter_upwards [even... | [anonymous] | Lean.Parser.Term.byTactic |
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