module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.MeanInequalities | {
"line": 1033,
"column": 2
} | {
"line": 1033,
"column": 28
} | {
"line": 1034,
"column": 2
} | [
{
"pp": "ι : Type u\np q : ℝ\nhpq : p.HolderConjugate q\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhf_sum : HasSum (fun a ↦ f a ^ p) (A ^ p)\nhg_sum : HasSum (fun a ↦ g a ^ q) (B ^ q)\nC : ℝ≥0\nhC : C ≤ A * B\nH : HasSum (fun i ↦ f i * g i) C\n⊢ ∃ C, 0 ≤ C ∧ C ≤ ↑A * ↑B ∧ HasSum (fun i ↦ ↑(f i) * ↑(g i)) C",
"ppTerm": "?m.... | [
"ι : Type u\np q : ℝ\nhpq : p.HolderConjugate q\nf g : ι → ℝ≥0\nA B : ℝ≥0\nhf_sum : HasSum (fun a ↦ f a ^ p) (A ^ p)\nhg_sum : HasSum (fun a ↦ g a ^ q) (B ^ q)\nC : ℝ≥0\nhC : C ≤ A * B\nH : HasSum (fun i ↦ f i * g i) C\n⊢ HasSum (fun i ↦ ↑(f i) * ↑(g i)) ↑C"
] | refine ⟨C, C.prop, hC, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 371,
"column": 2
} | {
"line": 376,
"column": 61
} | {
"line": 378,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsFiniteMeasure μ\nf : ↥(Lp E p μ)\nC : ℝ≥0\nhfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C\n⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | by_cases hμ : μ = 0
· simp [hμ, nnnorm_def]
rw [← ENNReal.coe_le_coe, nnnorm_def, ENNReal.coe_toNNReal (eLpNorm_ne_top _)]
refine (eLpNorm_le_of_ae_nnnorm_bound hfC).trans_eq ?_
rw [← coe_measureUnivNNReal μ, ← ENNReal.coe_rpow_of_ne_zero (measureUnivNNReal_pos hμ).ne',
ENNReal.coe_mul, mul_comm, ENNReal.sm... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 371,
"column": 2
} | {
"line": 376,
"column": 61
} | {
"line": 378,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : IsFiniteMeasure μ\nf : ↥(Lp E p μ)\nC : ℝ≥0\nhfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C\n⊢ ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | by_cases hμ : μ = 0
· simp [hμ, nnnorm_def]
rw [← ENNReal.coe_le_coe, nnnorm_def, ENNReal.coe_toNNReal (eLpNorm_ne_top _)]
refine (eLpNorm_le_of_ae_nnnorm_bound hfC).trans_eq ?_
rw [← coe_measureUnivNNReal μ, ← ENNReal.coe_rpow_of_ne_zero (measureUnivNNReal_pos hμ).ne',
ENNReal.coe_mul, mul_comm, ENNReal.sm... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 396,
"column": 10
} | {
"line": 396,
"column": 34
} | {
"line": 397,
"column": 10
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nhp : Fact (1 ≤ p)\nf g : ↥(Lp E p μ)\n⊢ ‖f + g‖ₑ ≤ ‖f‖ₑ + ‖g‖ₑ",
"ppTerm": "?m.81",
"assigned": true,
"usedCo... | [
"α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nhp : Fact (1 ≤ p)\nf g : ↥(Lp E p μ)\n⊢ eLpNorm (↑↑(f + g)) p μ ≤ eLpNorm (↑↑f) p μ + eLpNorm (↑↑g) p μ"
] | simp only [Lp.enorm_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 906,
"column": 2
} | {
"line": 906,
"column": 26
} | {
"line": 907,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nν : Measure α\nc : ℝ≥0∞\nhc : c ≠ ∞\nh : μ ≤ c • ν\ninst✝ : Fact (1 ≤ p)\nf : ↥(Lp E p ν)\n⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ₑ ≤ c ^ (1 / p).toReal * ‖f‖ₑ",
"ppTerm"... | [
"α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nν : Measure α\nc : ℝ≥0∞\nhc : c ≠ ∞\nh : μ ≤ c • ν\ninst✝ : Fact (1 ≤ p)\nf : ↥(Lp E p ν)\n⊢ eLpNorm (↑↑((LpToLpOfMeasureLeSMulₗ hc h) f)) p μ ≤ c ^ (1 / p).toReal * eLpNorm (↑↑f) p ... | simp only [Lp.enorm_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 86,
"column": 44
} | {
"line": 86,
"column": 77
} | {
"line": 86,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : R\nhx : x ∈ nonunits R\nh₁ : x ∈ Metric.ball 1 1\n⊢ ‖1 - x‖ < 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"NormedRing.toRing",
"congrArg",
"AddGroupWi... | [] | by rwa [mem_ball_iff_norm'] at h₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 219,
"column": 63
} | {
"line": 219,
"column": 85
} | {
"line": 221,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_2\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_8\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedAddCommGroup G\ninst✝¹⁰ : NontriviallyNormedField 𝕜₁\ninst✝⁹ : NormedSpace 𝕜₁ E\ninst✝⁸ : NontriviallyNormedField 𝕜₂\ninst✝⁷ : Nor... | [] | by simp [← coe_nnnorm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 224,
"column": 62
} | {
"line": 224,
"column": 84
} | {
"line": 226,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_2\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_8\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedAddCommGroup G\ninst✝¹⁰ : NontriviallyNormedField 𝕜₁\ninst✝⁹ : NormedSpace 𝕜₁ E\ninst✝⁸ : NontriviallyNormedField 𝕜₂\ninst✝⁷ : Nor... | [] | by simp [← coe_nnnorm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 73,
"column": 65
} | {
"line": 75,
"column": 80
} | {
"line": 77,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf T : M →L[R] M\nhf : IsIdempotentElem f\n⊢ (↑f).ker ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = f ∘SL T",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
... | [] | by
simpa [← toLinearMap_comp] using
LinearMap.IsIdempotentElem.ker_mem_invtSubmodule_iff (T := T) hf.toLinearMap | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Maps.Strict.Basic | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 81
} | {
"line": 125,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nf_quot : IsQuotientMap f\n⊢ IsStrictMap g ↔ IsStrictMap (g ∘ f)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Topology.IsQ... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nf_quot : IsQuotientMap f\nΦ : ↑(range (g ∘ f)) ≃ₜ ↑(range g) := Homeomorph.setCongr ⋯\n⊢ IsStrictMap g ↔ IsStrictMap (g ∘ f)"
] | set Φ : range (g ∘ f) ≃ₜ range g := .setCongr <| f_quot.surjective.range_comp g | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Measure.Real | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 86
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns₁ s₂ : Set α\nh : μ.real s₂ = 0\nh' : μ s₂ ≠ ∞\nH : μ s₁ < ∞\n⊢ μ.real (s₁ \\ s₂) = μ.real s₁",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"PartialOrder.toPreorder",
"Preor... | [] | · exact measureReal_sdiff_null' (measureReal_mono_null inter_subset_right h h') H.ne | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSpace.Indicator | {
"line": 131,
"column": 43
} | {
"line": 134,
"column": 62
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nc : E\nhμs_ne_zero : μ s ≠ 0\n⊢ ‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | by
rw [Lp.norm_def, eLpNorm_congr_ae indicatorConstLp_coeFn,
eLpNorm_indicator_const' hs hμs_ne_zero ENNReal.top_ne_zero, ENNReal.toReal_top,
_root_.div_zero, ENNReal.rpow_zero, mul_one, toReal_enorm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 443,
"column": 96
} | {
"line": 446,
"column": 51
} | {
"line": 448,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ns : 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\nh't : ∀ᵐ (x : α) ∂μ, x ∉ s → f x = 0\n⊢ Integrable f μ",
"ppTerm": "?m.29",
"assigned... | [] | by
rw [← integrableOn_univ]
apply hf.of_ae_sdiff_eq_zero nullMeasurableSet_univ
filter_upwards [h't] with x hx h'x using hx h'x.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 754,
"column": 6
} | {
"line": 754,
"column": 53
} | {
"line": 754,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetrizableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nh's : IsSep... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetrizableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nh's : IsSeparable s\nth... | aestronglyMeasurable_iff_aemeasurable_separable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 771,
"column": 4
} | {
"line": 771,
"column": 72
} | {
"line": 772,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nh : SecondCountableTopologyEither α β\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet... | [] | exact isSeparable_range <| continuousOn_iff_continuous_restrict.1 hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 782,
"column": 6
} | {
"line": 782,
"column": 53
} | {
"line": 782,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns t : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : IsCompact s\nht : MeasurableSet t\nhts : t ⊆ s\nthis✝¹ : Mea... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns t : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : IsCompact s\nht : MeasurableSet t\nhts : t ⊆ s\nthis✝¹ : MeasurableSpace... | aestronglyMeasurable_iff_aemeasurable_separable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 463,
"column": 4
} | {
"line": 463,
"column": 55
} | {
"line": 464,
"column": 4
} | [
{
"pp": "case refine_1\n𝕜 : 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 : ContinuousMult... | [
"case refine_1\n𝕜 : 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 �... | simp only [hasBasis_nhds_zero.mem_iff, Prod.exists] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 1241,
"column": 25
} | {
"line": 1241,
"column": 80
} | {
"line": 1241,
"column": 80
} | [
{
"pp": "G : ℕ → ℝ → ℝ\nf : ℝ → ℝ\nμ : Measure ℝ\nhGf : ∀ᵐ (x : ℝ) ∂μ, Tendsto (fun n ↦ G n x) atTop (𝓝 (f x))\nhG : ∀ (n : ℕ), AEMeasurable (fun x ↦ ‖G n x‖ₑ) μ\n⊢ ?m.51 =ᵐ[?m.50] ?m.52",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toSemi... | [] | filter_upwards [hGf] with x hx using hx.enorm.liminf_eq | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 1241,
"column": 25
} | {
"line": 1241,
"column": 80
} | {
"line": 1241,
"column": 80
} | [
{
"pp": "G : ℕ → ℝ → ℝ\nf : ℝ → ℝ\nμ : Measure ℝ\nhGf : ∀ᵐ (x : ℝ) ∂μ, Tendsto (fun n ↦ G n x) atTop (𝓝 (f x))\nhG : ∀ (n : ℕ), AEMeasurable (fun x ↦ ‖G n x‖ₑ) μ\n⊢ ?m.51 =ᵐ[?m.50] ?m.52",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toSemi... | [] | filter_upwards [hGf] with x hx using hx.enorm.liminf_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 1241,
"column": 25
} | {
"line": 1241,
"column": 80
} | {
"line": 1241,
"column": 80
} | [
{
"pp": "G : ℕ → ℝ → ℝ\nf : ℝ → ℝ\nμ : Measure ℝ\nhGf : ∀ᵐ (x : ℝ) ∂μ, Tendsto (fun n ↦ G n x) atTop (𝓝 (f x))\nhG : ∀ (n : ℕ), AEMeasurable (fun x ↦ ‖G n x‖ₑ) μ\n⊢ ?m.51 =ᵐ[?m.50] ?m.52",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toSemi... | [] | filter_upwards [hGf] with x hx using hx.enorm.liminf_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 53
} | {
"line": 147,
"column": 4
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nμ : Measure β\nfmeas : Measurable f\nhf : MemLp f p μ\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhi₀ : MemLp (fun x ↦ y₀) p μ\n... | [
"β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nμ : Measure β\nfmeas : Measurable f\nhf : MemLp f p μ\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhi₀ : MemLp (fun x ↦ y₀) p μ\nn : ℕ\nhf' :... | convert! norm_approxOn_y₀_le fmeas h₀ x n using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1071,
"column": 23
} | {
"line": 1071,
"column": 25
} | {
"line": 1072,
"column": 8
} | [
{
"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... | [
"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) → SeminormedAddCommGroup (... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 38
} | {
"line": 99,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhμs : μ s ≠ ∞\nhνs : ν s ≠ ∞\nx : F\n⊢ (μ + ν).real s • x = μ.real s • x + ν.real s • x",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.... | [] | rw [measureReal_add_apply, add_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 55
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ E\ng : E → ℝ≥0∞\nhf : Integrable (⇑f) μ\nhg0 : g 0 = 0\nht : ∀ (b : E), g b ≠ ∞\nhf' : f.FinMeasSupp μ\na : E\na0 : a = 0\n⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞",
"ppTerm": "?pos✝",
"assigned... | [] | rw [a0, hg0, zero_mul]; exact WithTop.zero_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 55
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ E\ng : E → ℝ≥0∞\nhf : Integrable (⇑f) μ\nhg0 : g 0 = 0\nht : ∀ (b : E), g b ≠ ∞\nhf' : f.FinMeasSupp μ\na : E\na0 : a = 0\n⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞",
"ppTerm": "?pos✝",
"assigned... | [] | rw [a0, hg0, zero_mul]; exact WithTop.zero_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 73
} | {
"line": 267,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ E\ng : E → ℝ≥0∞\nhf : Integrable (⇑f) μ\nhg0 : g 0 = 0\nht : ∀ (b : E), g b ≠ ∞\nhf' : f.FinMeasSupp μ\na : E\na0 : ¬a = 0\n⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞",
"ppTerm": "?neg✝",
"assigne... | [] | apply mul_ne_top (ht a) (hf'.meas_preimage_singleton_ne_zero a0).ne | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 73
} | {
"line": 267,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ E\ng : E → ℝ≥0∞\nhf : Integrable (⇑f) μ\nhg0 : g 0 = 0\nht : ∀ (b : E), g b ≠ ∞\nhf' : f.FinMeasSupp μ\na : E\na0 : ¬a = 0\n⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞",
"ppTerm": "?neg✝",
"assigne... | [] | apply mul_ne_top (ht a) (hf'.meas_preimage_singleton_ne_zero a0).ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 73
} | {
"line": 267,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ E\ng : E → ℝ≥0∞\nhf : Integrable (⇑f) μ\nhg0 : g 0 = 0\nht : ∀ (b : E), g b ≠ ∞\nhf' : f.FinMeasSupp μ\na : E\na0 : ¬a = 0\n⊢ g a * μ (⇑f ⁻¹' {a}) ≠ ∞",
"ppTerm": "?neg✝",
"assigne... | [] | apply mul_ne_top (ht a) (hf'.meas_preimage_singleton_ne_zero a0).ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 470,
"column": 4
} | {
"line": 471,
"column": 45
} | {
"line": 472,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ↥(α →₁ₛ[μ] ℝ)\neq : ∀ (a : α), (toSimpleFunc f).posPart a = max ((toSimpleFunc f) a) 0\n⊢ ∀ᵐ (a : α) ∂μ, (toSimpleFunc (posPart f)) a = max ((toSimpleFunc f) a) 0",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Measur... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ↥(α →₁ₛ[μ] ℝ)\neq : ∀ (a : α), (toSimpleFunc f).posPart a = max ((toSimpleFunc f) a) 0\na✝¹ : α\na✝ : (toSimpleFunc (posPart f)) a✝¹ = ↑↑↑(posPart f) a✝¹\nh₂ : ↑↑(Lp.posPart ↑f) a✝¹ = max (↑↑↑f a✝¹) 0\nh₃ : (toSimpleFunc f) a✝¹ = ↑↑↑f a✝¹\n⊢ (toSimpleFunc (po... | filter_upwards [toSimpleFunc_eq_toFun (posPart f), Lp.coeFn_posPart (f : α →₁[μ] ℝ),
toSimpleFunc_eq_toFun f] with _ _ h₂ h₃ | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 441,
"column": 2
} | {
"line": 442,
"column": 43
} | {
"line": 444,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nX : Type u_6\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nF : X → α → G\nbound : α → ℝ\ns : Set X\nhF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ\nh_bound : ∀ x... | [] | exact continuousOn_setToFun_of_dominated (dominatedFinMeasAdditive_weightedSMul μ)
hF_meas h_bound bound_integrable h_cont | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 40
} | {
"line": 51,
"column": 40
} | [
{
"pp": "α : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : SeminormedAddCommGroup W\ninst✝¹ : PseudoMetricSpace Q\ninst✝ : NormedAddTorsor W Q\nc : V\nx y : P\n⊢ dist (c +ᵥ x) (c +ᵥ y) = d... | [] | simp [NormedAddTorsor.dist_eq_norm'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 40
} | {
"line": 51,
"column": 40
} | [
{
"pp": "α : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : SeminormedAddCommGroup W\ninst✝¹ : PseudoMetricSpace Q\ninst✝ : NormedAddTorsor W Q\nc : V\nx y : P\n⊢ dist (c +ᵥ x) (c +ᵥ y) = d... | [] | simp [NormedAddTorsor.dist_eq_norm'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 40
} | {
"line": 51,
"column": 40
} | [
{
"pp": "α : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : PseudoMetricSpace P\ninst✝³ : NormedAddTorsor V P\ninst✝² : SeminormedAddCommGroup W\ninst✝¹ : PseudoMetricSpace Q\ninst✝ : NormedAddTorsor W Q\nc : V\nx y : P\n⊢ dist (c +ᵥ x) (c +ᵥ y) = d... | [] | simp [NormedAddTorsor.dist_eq_norm'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 53
} | {
"line": 71,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : x ∉ F\nd : ℝ := infDist x ↑F\nhFn : (↑F).Nonempty\nhdp : 0 < d\nr' : ℝ... | [
"𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nr : ℝ\nhr : r < 1\nx : E\nhx : x ∉ F\nd : ℝ := infDist x ↑F\nhFn : (↑F).Nonempty\nhdp : 0 < d\nr' : ℝ := max r 2⁻... | refine ⟨x - y₀, x_ne_y₀, fun y hy => le_of_lt ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.Matrix | {
"line": 149,
"column": 38
} | {
"line": 151,
"column": 10
} | {
"line": 153,
"column": 0
} | [
{
"pp": "X : Type u_1\nn : Type u_5\nR : Type u_8\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Fintype n\ninst✝³ : Mul R\ninst✝² : AddCommMonoid R\ninst✝¹ : ContinuousAdd R\ninst✝ : ContinuousMul R\nA B : X → n → R\nhA : Continuous A\nhB : Continuous B\n⊢ Continuous fun x ↦ A x ⬝ᵥ B x",
... | [] | by
dsimp only [dotProduct]
fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.Matrix | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 61
} | {
"line": 219,
"column": 0
} | [
{
"pp": "X : Type u_1\nn : Type u_5\nR : Type u_8\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace R\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : IsTopologicalRing R\nA : X → Matrix n n R\nhA : Continuous A\nl : Equiv.Perm n\nx✝ : l ∈ Finset.univ\n⊢ Continuous fun x ↦ ∏ i, A ... | [] | exact continuous_finsetProd _ fun l _ => hA.matrix_elem _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 775,
"column": 2
} | {
"line": 776,
"column": 36
} | {
"line": 778,
"column": 0
} | [
{
"pp": "case neg\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\nf : α → E\nhF : CompleteSpace F\n... | [] | · rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
rwa [← integrable_neg_iff] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 56
} | {
"line": 421,
"column": 4
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g✝ : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_2\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng : α →ᵇ β\nh : δ →ᵇ β\n⊢ ∃ C, ∀ (x y : δ), dist (Fu... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : PseudoMetricSpace γ\nf✝ g✝ : α →ᵇ β\nx : α\nC : ℝ\nδ : Type u_2\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng : α →ᵇ β\nh : δ →ᵇ β\n⊢ Bornology.IsBounded (range ⇑g ∪ ⇑h '' ... | rw [← isBounded_range_iff, range_extend f.injective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 369,
"column": 13
} | {
"line": 369,
"column": 64
} | {
"line": 371,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\nR : Type u_1\ninst✝ : SeminormedRing R\nf : α →ᵇ R\nn : ℕ\n⊢ ⇑(npowRec (n + 1) f) = ⇑f ^ (n + 1)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"BoundedContinuousFunction.coe_mul",
"Ring.toNon... | [] | by rw [npowRec, pow_succ, coe_mul, coe_npowRec f n] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 82
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\n⊢ thickenedIndicatorAux δ E x ∈ {a | a ≠ ∞}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"lt_of_le_of_lt",
"PartialOrder.toPreorder",
"ENNReal.one_lt_top",
"thickenedIndi... | [] | exact (lt_of_le_of_lt (@thickenedIndicatorAux_le_one _ _ δ E x) one_lt_top).ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 166,
"column": 4
} | {
"line": 169,
"column": 82
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\n⊢ Continuous[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ (thickenedIndicatorAux δ E x).toNNReal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
... | [] | apply ContinuousOn.comp_continuous continuousOn_toNNReal
(continuous_thickenedIndicatorAux δ_pos E)
intro x
exact (lt_of_le_of_lt (@thickenedIndicatorAux_le_one _ _ δ E x) one_lt_top).ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 166,
"column": 4
} | {
"line": 169,
"column": 82
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\n⊢ Continuous[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ (thickenedIndicatorAux δ E x).toNNReal",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
... | [] | apply ContinuousOn.comp_continuous continuousOn_toNNReal
(continuous_thickenedIndicatorAux δ_pos E)
intro x
exact (lt_of_le_of_lt (@thickenedIndicatorAux_le_one _ _ δ E x) one_lt_top).ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 295,
"column": 4
} | {
"line": 296,
"column": 17
} | {
"line": 297,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : TopologicalSpace R\nf : StieltjesFunction R\na b : R\nthis : Nonempty R\na' b' : R\nh : Ioc a b \\ botSet ⊆ Ioc a' b'\nab : b ≤ a\n⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal",
"ppTerm": "?inl",
"assigned": true,
"usedConstants... | [] | rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))]
apply zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 295,
"column": 4
} | {
"line": 296,
"column": 17
} | {
"line": 297,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : TopologicalSpace R\nf : StieltjesFunction R\na b : R\nthis : Nonempty R\na' b' : R\nh : Ioc a b \\ botSet ⊆ Ioc a' b'\nab : b ≤ a\n⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal",
"ppTerm": "?inl",
"assigned": true,
"usedConstants... | [] | rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))]
apply zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 254,
"column": 2
} | {
"line": 257,
"column": 30
} | {
"line": 259,
"column": 0
} | [
{
"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... | [] | simp_rw [← setIntegral_one_eq_measureReal]
apply integral_biUnion_eq_sum_powerset hs
intro i hi
simpa using (hf i hi).lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 254,
"column": 2
} | {
"line": 257,
"column": 30
} | {
"line": 259,
"column": 0
} | [
{
"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... | [] | simp_rw [← setIntegral_one_eq_measureReal]
apply integral_biUnion_eq_sum_powerset hs
intro i hi
simpa using (hf i hi).lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 308,
"column": 65
} | {
"line": 312,
"column": 68
} | {
"line": 313,
"column": 2
} | [
{
"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), MeasurableSet (s i)\nh_anti : Antitone s\nhne : atTop.NeBot\nthis... | [] | by
convert! this.congr' <| (eventually_ge_atTop i₀).mono fun i hi ↦ ?_
· rw [← sdiff_iInter, setIntegral_sdiff _ hi₀ (iInter_subset _ _), sub_sub_cancel]
exact .iInter_of_antitone h_anti hsm
· rw [setIntegral_sdiff (hsm i) hi₀ (h_anti hi), sub_sub_cancel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 599,
"column": 2
} | {
"line": 601,
"column": 53
} | {
"line": 602,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b ≤ a\n⊢ f.measure ... | [
"case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\n⊢ f.measure (Ico a b) = ... | · simp only [hab, measure_empty, Ico_eq_empty, not_lt]
symm
simp [ENNReal.ofReal_eq_zero, f.mono.leftLim hab] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Content | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 41
} | {
"line": 322,
"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\n⊢ μ.innerContent { carrier := ↑U' ∩ U, is_open' := ⋯ } + μ.outerMeasure (↑U' \\ U) ≤ μ.outerMeasure ↑U'",
"ppTerm": "?m.41",
... | [
"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\n⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \\ U) ≤ μ.outerMeasure ↑U'"
] | simp only [innerContent, iSup_subtype'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Content | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 41
} | {
"line": 337,
"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✝ : 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✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\nL : Compacts G\nL' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }\nhL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U\nh... | simp only [innerContent, iSup_subtype'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 281,
"column": 6
} | {
"line": 281,
"column": 23
} | {
"line": 281,
"column": 23
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nh : IsFundamentalDomain G s μ\nt : Set α\nht : ∀ (g : G), g • t = t\nhts : μ (t ∩ s) = ... | [
"G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nh : IsFundamentalDomain G s μ\nt : Set α\nht : ∀ (g : G), g • t = t\nhts : μ (t ∩ s) = 0\n⊢ ∑' (g :... | measure_eq_tsum h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 41
} | {
"line": 261,
"column": 41
} | [
{
"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 : F\n⊢ ⟪x, 0⟫ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inner.inner",
"congrArg",
"CommSemiring.toSemir... | [
"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 : F\n⊢ (starRingEnd 𝕜) 0 = 0"
] | rw [← inner_conj_symm, inner_zero_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 440,
"column": 6
} | {
"line": 444,
"column": 66
} | {
"line": 445,
"column": 2
} | [
{
"pp": "case h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : Disjoint K₁.carrier K₂.carrier\nh₂ : IsClosed[inst✝¹] K₂.carrier\nU₁ U₂ : Set G\nh1U₁ : IsOpen[inst✝¹] U₁\nh1U₂ : IsOpen[inst✝¹] U₂\nh2U₁ : K₁.carrier ⊆ U₁... | [] | refine disjoint_of_subset ?_ ?_ hU
· refine Subset.trans (mul_subset_mul Subset.rfl ?_) h2L₁
exact Subset.trans (inv_subset.mpr h1U) inter_subset_left
· refine Subset.trans (mul_subset_mul Subset.rfl ?_) h2L₂
exact Subset.trans (inv_subset.mpr h1U) inter_subset_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 440,
"column": 6
} | {
"line": 444,
"column": 66
} | {
"line": 445,
"column": 2
} | [
{
"pp": "case h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : Disjoint K₁.carrier K₂.carrier\nh₂ : IsClosed[inst✝¹] K₂.carrier\nU₁ U₂ : Set G\nh1U₁ : IsOpen[inst✝¹] U₁\nh1U₂ : IsOpen[inst✝¹] U₂\nh2U₁ : K₁.carrier ⊆ U₁... | [] | refine disjoint_of_subset ?_ ?_ hU
· refine Subset.trans (mul_subset_mul Subset.rfl ?_) h2L₁
exact Subset.trans (inv_subset.mpr h1U) inter_subset_left
· refine Subset.trans (mul_subset_mul Subset.rfl ?_) h2L₂
exact Subset.trans (inv_subset.mpr h1U) inter_subset_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 494,
"column": 6
} | {
"line": 494,
"column": 34
} | {
"line": 494,
"column": 34
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ 1 ≤ (haarContent K₀).outerMeasure ↑K₀",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"TopologicalSpace.PositiveCompacts.instSe... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ 1 ≤ ⨅ U, ⨅ (hU : IsOpen[inst✝¹] U), ⨅ (_ : ↑K₀ ⊆ U), (haarContent K₀).innerContent { carrier := U, is_open' := hU }"
] | Content.outerMeasure_eq_iInf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 681,
"column": 51
} | {
"line": 681,
"column": 92
} | {
"line": 681,
"column": 92
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : SecondCountableTopology G\nK₀ : PositiveCompacts G\nμ : Measure G\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsMulLeftInvariant\nh : μ ↑K₀ = 1\n⊢ haarMeasure K... | [] | rw [haarMeasure_unique μ K₀, h, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 681,
"column": 51
} | {
"line": 681,
"column": 92
} | {
"line": 681,
"column": 92
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : SecondCountableTopology G\nK₀ : PositiveCompacts G\nμ : Measure G\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsMulLeftInvariant\nh : μ ↑K₀ = 1\n⊢ haarMeasure K... | [] | rw [haarMeasure_unique μ K₀, h, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 681,
"column": 51
} | {
"line": 681,
"column": 92
} | {
"line": 681,
"column": 92
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : SecondCountableTopology G\nK₀ : PositiveCompacts G\nμ : Measure G\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsMulLeftInvariant\nh : μ ↑K₀ = 1\n⊢ haarMeasure K... | [] | rw [haarMeasure_unique μ K₀, h, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 121,
"column": 62
} | {
"line": 123,
"column": 67
} | {
"line": 125,
"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\nv : ι → E\nhv : Orthonormal 𝕜 v\nl : ι → 𝕜\ns : Finset ι\ni : ι\nhi : i ∈ s\n⊢ ⟪v i, ∑ i ∈ s, l i • v i⟫ = l i",
"ppTerm": "?m.25",
"assigned": true,
"usedCons... | [] | by
classical
simp [inner_sum, inner_smul_right, orthonormal_iff_ite.mp hv, hi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 92
} | {
"line": 293,
"column": 4
} | [
{
"pp": "case refine_1\nι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : TopologicalSpace N\ninst✝⁵ : Module R N\ninst✝⁴ : AddCommMonoid O\ninst✝³ : Topologi... | [
"case refine_2\nι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : TopologicalSpace N\ninst✝⁵ : Module R N\ninst✝⁴ : AddCommMonoid O\ninst✝³ : TopologicalSpace O\n... | · exact subset_closure <| Submodule.mem_iSup_of_mem _ <| Submodule.mem_iSup_of_mem ha hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 55
} | {
"line": 228,
"column": 8
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nins... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ninst✝ : Complet... | have : s₁ ⊓ s₂ ⊆ s₁ := Finset.inter_subset_left | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 208,
"column": 4
} | {
"line": 210,
"column": 10
} | {
"line": 211,
"column": 4
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I = 0\n⊢ (starRingEnd 𝕜) ⟪T y, x⟫ =\n (⟪x, T x⟫ + (starRingEnd 𝕜) ⟪T y, x⟫ + (⟪T y, x⟫ + ⟪T y, y⟫) -\n (⟪x, T x⟫ - (... | [
"case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I = 0\n⊢ ↑(re ⟪T y, x⟫) = ⟪T y, x⟫"
] | suffices (re ⟪T y, x⟫ : 𝕜) = ⟪T y, x⟫ by
rw [conj_eq_iff_re.mpr this]
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 52
} | {
"line": 80,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 ↥K\n⊢ ↑(LinearEquiv.det K.reflection.toLinearEquiv) = ↑((-1) ^ finrank 𝕜 ↥Kᗮ)",
"ppTerm": "?m.45",
"assigned": true,
"usedConsta... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 ↥K\n⊢ LinearMap.det ↑K.reflection.toLinearEquiv = ↑(-1) ^ finrank 𝕜 ↥Kᗮ"
] | rw [LinearEquiv.coe_det, Units.val_pow_eq_pow_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 329,
"column": 79
} | {
"line": 331,
"column": 81
} | {
"line": 333,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\n⊢ ‖K.orthogonalProjectionOnto‖ ≤ 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Inne... | [] | by
refine K.orthogonalProjectionOnto.opNorm_le_bound zero_le_one ?_
simp [orthogonalProjectionOnto, projectionOntoL, norm_projection_orthogonal_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 585,
"column": 18
} | {
"line": 585,
"column": 58
} | {
"line": 587,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nu : ↥K\nv : E\n⊢ ⟪K.starProjection v, ↑u⟫ + ⟪v - K.starProjection v, ↑u⟫ = ⟪v, ↑u⟫",
"ppTerm": "?m.88",
"assigned": true,
"... | [] | by rw [← inner_add_left, add_sub_cancel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 45
} | {
"line": 288,
"column": 4
} | [
{
"pp": "case add\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(V i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nx✝ x y : E\nhx... | [
"case add\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(V i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nx✝ x y : E\nhx✝ : x ∈ ⨆ i,... | simp_rw [map_add, Finset.sum_add_distrib] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 46
} | {
"line": 405,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nf : E →L[𝕜] F\nhf : Injective ⇑f\nK : ℝ≥0\nhf' : Antili... | [] | exact hf'.isClosed_range f.uniformContinuous | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 629,
"column": 17
} | {
"line": 629,
"column": 74
} | {
"line": 630,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Normed... | [] | simpa [SetLike.ext'_iff] using! h.2.denseRange.closure_eq | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 629,
"column": 17
} | {
"line": 629,
"column": 74
} | {
"line": 630,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Normed... | [] | simpa [SetLike.ext'_iff] using! h.2.denseRange.closure_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 629,
"column": 17
} | {
"line": 629,
"column": 74
} | {
"line": 630,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Normed... | [] | simpa [SetLike.ext'_iff] using! h.2.denseRange.closure_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 442,
"column": 66
} | {
"line": 444,
"column": 75
} | {
"line": 446,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp p (α × β)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\ncancel : p.toReal * (1 / p.toReal) = 1\n⊢ (edist x.ofLp y.ofLp ^ p.toReal + edist x.ofLp y.ofLp ... | [] | by
simp only [← two_mul, ENNReal.mul_rpow_of_nonneg _ _ nonneg, ← ENNReal.rpow_mul, cancel,
ENNReal.rpow_one, ENNReal.coe_rpow_of_nonneg _ nonneg, coe_ofNat] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 760,
"column": 2
} | {
"line": 760,
"column": 62
} | {
"line": 762,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nf : WithLp ∞ (α × β)\n⊢ ‖f.ofLp‖₊ = ‖f‖₊",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"WithLp",
"Eq.mpr",
"Prod.seminormedAddGroup",
"WithLp.ofLp_fst",
... | [] | rw [prod_nnnorm_eq_sup, Prod.nnnorm_def, ofLp_fst, ofLp_snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 760,
"column": 2
} | {
"line": 760,
"column": 62
} | {
"line": 762,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nf : WithLp ∞ (α × β)\n⊢ ‖f.ofLp‖₊ = ‖f‖₊",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"WithLp",
"Eq.mpr",
"Prod.seminormedAddGroup",
"WithLp.ofLp_fst",
... | [] | rw [prod_nnnorm_eq_sup, Prod.nnnorm_def, ofLp_fst, ofLp_snd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 760,
"column": 2
} | {
"line": 760,
"column": 62
} | {
"line": 762,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nf : WithLp ∞ (α × β)\n⊢ ‖f.ofLp‖₊ = ‖f‖₊",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"WithLp",
"Eq.mpr",
"Prod.seminormedAddGroup",
"WithLp.ofLp_fst",
... | [] | rw [prod_nnnorm_eq_sup, Prod.nnnorm_def, ofLp_fst, ofLp_snd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 823,
"column": 15
} | {
"line": 825,
"column": 32
} | {
"line": 827,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx y : WithLp 2 (α × β)\n⊢ ↑(nndist x y) = ↑(NNReal.sqrt (nndist x.fst y.fst ^ 2 + nndist x.snd y.snd ^ 2))",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"WithLp",
"Eq.mpr... | [] | by
push_cast
exact prod_dist_eq_of_L2 _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 207,
"column": 92
} | {
"line": 212,
"column": 65
} | {
"line": 214,
"column": 0
} | [
{
"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\nm n : E\n⊢ fract b m = fract b n ↔ -m + n ∈ span ℤ (Set.range ⇑b)"... | [] | by
classical
rw [eq_comm, Basis.ext_elem_iff b]
simp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,
sub_eq_neg_add, map_add, map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,
Pi.neg_apply, ← eq_intCast (algebraMap ℤ K) _, Set.mem_range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 63
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case intro.refine_1\nE : 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✝ : Finite ι\nx : E\nval✝ : Fintype ι\n⊢ (fun v ↦ v +ᵥ x ∈ f... | [] | exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 63
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case intro.refine_1\nE : 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✝ : Finite ι\nx : E\nval✝ : Fintype ι\n⊢ (fun v ↦ v +ᵥ x ∈ f... | [] | exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 63
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case intro.refine_1\nE : 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✝ : Finite ι\nx : E\nval✝ : Fintype ι\n⊢ (fun v ↦ v +ᵥ x ∈ f... | [] | exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 12
} | {
"line": 252,
"column": 13
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁸ : RCLike 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝³ : NormedAddCommGroup F'\ninst✝² : InnerProductSpace ℝ F'\ninst... | [
"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁸ : RCLike 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝³ : NormedAddCommGroup F'\ninst✝² : InnerProductSpace ℝ F'\ninst✝¹ : Fintype... | intro v₀ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 658,
"column": 56
} | {
"line": 658,
"column": 88
} | {
"line": 659,
"column": 4
} | [
{
"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 ... | [] | simp only [div_eq_mul_inv]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 658,
"column": 56
} | {
"line": 658,
"column": 88
} | {
"line": 659,
"column": 4
} | [
{
"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 ... | [] | simp only [div_eq_mul_inv]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 426,
"column": 2
} | {
"line": 428,
"column": 8
} | {
"line": 430,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι\n⊢ ⇑{ repr := e } = fun i ↦ e.symm (EuclideanSpace.single i 1)",
"ppTerm": "?m.50",
"assign... | [] | dsimp only [DFunLike.coe]
funext
congr! | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 426,
"column": 2
} | {
"line": 428,
"column": 8
} | {
"line": 430,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι\n⊢ ⇑{ repr := e } = fun i ↦ e.symm (EuclideanSpace.single i 1)",
"ppTerm": "?m.50",
"assign... | [] | dsimp only [DFunLike.coe]
funext
congr! | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 555,
"column": 4
} | {
"line": 555,
"column": 31
} | {
"line": 556,
"column": 4
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\np : Box ι → Prop\n⊢ (↑π.boxes ∪ ↑(π.filter p).boxes).Pairwise (Disjoint on Box.toSet)",
"ppTerm": "?m.423",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition.filter",
"Function.onFun",
... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\np : Box ι → Prop\n⊢ ↑π.boxes ∪ ↑(π.filter p).boxes = ↑π.boxes"
] | convert! π.pairwiseDisjoint | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 21
} | {
"line": 130,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ninst✝ : DecidableEq ι\ni : ι\nx : ℝ\nh : x ∈ Ioo (I.lower i) (I.upper i)\nh' : ∀ (j : ι), update I.lower i x j < I.upper j\n⊢ I.splitUpper i x = ↑{ lower := update I.lower i x, upper := I.upper, lower_lt_upper := h' }",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | simp +unfoldPartialApp only [splitUpper, mk'_eq_coe, max_eq_left h.1.le,
update, and_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 21
} | {
"line": 130,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ninst✝ : DecidableEq ι\ni : ι\nx : ℝ\nh : x ∈ Ioo (I.lower i) (I.upper i)\nh' : ∀ (j : ι), update I.lower i x j < I.upper j\n⊢ I.splitUpper i x = ↑{ lower := update I.lower i x, upper := I.upper, lower_lt_upper := h' }",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | simp +unfoldPartialApp only [splitUpper, mk'_eq_coe, max_eq_left h.1.le,
update, and_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 21
} | {
"line": 130,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ninst✝ : DecidableEq ι\ni : ι\nx : ℝ\nh : x ∈ Ioo (I.lower i) (I.upper i)\nh' : ∀ (j : ι), update I.lower i x j < I.upper j\n⊢ I.splitUpper i x = ↑{ lower := update I.lower i x, upper := I.upper, lower_lt_upper := h' }",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | simp +unfoldPartialApp only [splitUpper, mk'_eq_coe, max_eq_left h.1.le,
update, and_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 594,
"column": 4
} | {
"line": 595,
"column": 41
} | {
"line": 596,
"column": 4
} | [
{
"pp": "case refine_1\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\ni... | [
"case refine_1\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✝ : Discr... | obtain ⟨n, -, m, -, h_ne, h_eq⟩ := Set.Infinite.exists_ne_map_eq_of_mapsTo
Set.infinite_univ h_mapsto h_finite | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 79
} | {
"line": 145,
"column": 6
} | [
{
"pp": "case insert\nι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lowe... | [
"case insert\nι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.uppe... | rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Oscillation | {
"line": 64,
"column": 31
} | {
"line": 64,
"column": 49
} | {
"line": 64,
"column": 49
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : TopologicalSpace E\nf : E → F\nD : Set E\nx : E\nhf : ContinuousWithinAt f D x\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ 0 < ε / 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"instHDiv",
... | [] | simp [ne_of_gt hε] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Oscillation | {
"line": 64,
"column": 31
} | {
"line": 64,
"column": 49
} | {
"line": 64,
"column": 49
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : TopologicalSpace E\nf : E → F\nD : Set E\nx : E\nhf : ContinuousWithinAt f D x\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ 0 < ε / 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"instHDiv",
... | [] | simp [ne_of_gt hε] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Oscillation | {
"line": 64,
"column": 31
} | {
"line": 64,
"column": 49
} | {
"line": 64,
"column": 49
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : TopologicalSpace E\nf : E → F\nD : Set E\nx : E\nhf : ContinuousWithinAt f D x\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ 0 < ε / 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"instHDiv",
... | [] | simp [ne_of_gt hε] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 76,
"column": 23
} | {
"line": 76,
"column": 34
} | {
"line": 76,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : OrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\n⊢ f ⊥ - ∑ i ∈ Iio ⊥, (𝕜 ∙ gramSchmidt 𝕜 f i).starProjection (f... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : OrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\n⊢ f ⊥ - ∑ i ∈ Ico ⊥ ⊥, (𝕜 ∙ gramSchmidt 𝕜 f i).starProjection (f ⊥) = f ⊥"... | Iio_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 28
} | {
"line": 268,
"column": 29
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nν : ι → ℤ\nhν : ν ∈ admissibleIndex n B\nhI : box n ν ∈ (prepartition n B).boxes\n⊢ box n (index n ((prepartition n B).tag (box n ν))) = box n ν",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nν : ι → ℤ\nhν : ν ∈ admissibleIndex n B\nhI : box n ν ∈ (prepartition n B).boxes\n⊢ box n (index n (tag n ν)) = box n ν"
] | prepartition_tag n hν, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 270,
"column": 97
} | {
"line": 271,
"column": 58
} | {
"line": 273,
"column": 0
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny : F\nhf : HasIntegral I l f vol y\n⊢ HasIntegral I l (-... | [] | by
simpa only [HasIntegral, ← integralSum_neg] using hf.neg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 290,
"column": 44
} | {
"line": 290,
"column": 93
} | {
"line": 292,
"column": 0
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nh : HasIntegral I l f vol y\nh' : HasIntegral... | [] | by simpa only [sub_eq_add_neg] using h.add h'.neg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 79,
"column": 2
} | {
"line": 81,
"column": 9
} | {
"line": 82,
"column": 2
} | [
{
"pp": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne f : OrthonormalBasis ι ℝ E\n⊢ e.toBasis.det = f.toBasis.det → e.toBasis.orientation = f.toBasis.orientation",
"ppTerm": "?mp",
"assigned": true,
"... | [
"case mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ne f : OrthonormalBasis ι ℝ E\n⊢ e.toBasis.orientation = f.toBasis.orientation → e.toBasis.det = f.toBasis.det"
] | · intro h
dsimp [Basis.orientation]
congr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 56
} | {
"line": 199,
"column": 2
} | [
{
"pp": "U : Type u_4\nV : Type u_5\ninst✝⁹ : NormedAddCommGroup U\ninst✝⁸ : InnerProductSpace ℝ U\ninst✝⁷ : MeasurableSpace U\ninst✝⁶ : BorelSpace U\ninst✝⁵ : FiniteDimensional ℝ U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : FiniteD... | [
"U : Type u_4\nV : Type u_5\ninst✝⁹ : NormedAddCommGroup U\ninst✝⁸ : InnerProductSpace ℝ U\ninst✝⁷ : MeasurableSpace U\ninst✝⁶ : BorelSpace U\ninst✝⁵ : FiniteDimensional ℝ U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : FiniteDimensional ℝ... | refine (measurePreserving_sumPiEquivProdPi _).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.PSeries | {
"line": 248,
"column": 6
} | {
"line": 248,
"column": 26
} | {
"line": 248,
"column": 27
} | [
{
"pp": "f : ℕ → ℝ\nh_nonneg : 0 ≤ᶠ[Filter.atTop] f\nh_mono : ∀ᶠ (k : ℕ) in Filter.atTop, f (k + 1) ≤ f k\n⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable f",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"... | [
"f : ℕ → ℝ\nh_nonneg : ∀ᶠ (x : ℕ) in Filter.atTop, 0 x ≤ f x\nh_mono : ∀ᶠ (k : ℕ) in Filter.atTop, f (k + 1) ≤ f k\n⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable f"
] | Filter.EventuallyLE, | Lean.Elab.Tactic.evalRewriteSeq | null |
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