module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 596,
"column": 4
} | {
"line": 597,
"column": 44
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i))... | rw [Measure.measure_univ_eq_zero] at hμ
exact hμ.symm ▸ unifIntegrable_zero_meas | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 596,
"column": 4
} | {
"line": 597,
"column": 44
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i))... | rw [Measure.measure_univ_eq_zero] at hμ
exact hμ.symm ▸ unifIntegrable_zero_meas | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 201,
"column": 32
} | {
"line": 201,
"column": 49
} | [
{
"pp": "case h.e'_3.h.e'_6.h.e'_6.h.e'_2.h.h.e'_4\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : IsFiniteMeasure μ\ng : α → ℝ\nhint : Integrable g μ\nℱ : ι → MeasurableSpace α\nhℱ : ∀ (i : ι), ℱ i ≤ m0\nA : MeasurableSpace α := m0\nhmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ... | ENNReal.rpow_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 202,
"column": 28
} | {
"line": 202,
"column": 45
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : IsFiniteMeasure μ\ng : α → ℝ\nhint : Integrable g μ\nℱ : ι → MeasurableSpace α\nhℱ : ∀ (i : ι), ℱ i ≤ m0\nA : MeasurableSpace α := m0\nhmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}\nhg : MemLp g 1 μ\nε : ℝ... | ENNReal.rpow_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 8
} | {
"line": 264,
"column": 42
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν F\n⊢ inst✝⁴ =... | BorelSpace.measurable_eq (α := Ω), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1166,
"column": 8
} | {
"line": 1166,
"column": 37
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid N\ninst✝² : TopologicalSpace N\nv₁ v₂ : VectorMeasure α M\nw : VectorMeasure α N\ninst✝¹ : T2Space N\ninst✝ : ContinuousAdd M\nu : Set α\nhmu : Measur... | rcases ht hx with (hxu | hxv) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 793,
"column": 6
} | {
"line": 794,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x... | · rw [Set.indicator_of_mem, zero_add]
simpa using hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 896,
"column": 4
} | {
"line": 896,
"column": 41
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nhf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nε : ℝ\nh... | refine ⟨δ, hδ₁, fun n s hs hle => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 59
} | [
{
"pp": "case h\nΩ : Type u_1\ninst✝ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ns : Set Ω\n_s_mble : MeasurableSet s\n⊢ μ s = (μ.mass • μ.normalize.toFiniteMeasure) s",
"usedConstants": [
"MeasureTheory.FiniteMeasure.mass",
"MeasureTheory.FiniteMeasure",
"Eq.mpr",
"NN... | rw [μ.self_eq_mass_mul_normalize s, smul_apply, smul_eq_mul,
ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 462,
"column": 2
} | {
"line": 465,
"column": 72
} | [
{
"pp": "case hfg\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nle_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)\nω : Ω\n⊢ (fun a ↦ ↑(f a)) ω ≤ (fun a ↦ ↑(g a) + edist f g) ω",
"usedConstants": [
"Iff.m... | have le' : f ω ≤ g ω + nndist f g := by
calc f ω
_ ≤ g ω + nndist (f ω) (g ω) := NNReal.le_add_nndist (f ω) (g ω)
_ ≤ g ω + nndist f g := (add_le_add_iff_left (g ω)).mpr (le_dist ω) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 538,
"column": 2
} | {
"line": 538,
"column": 33
} | [
{
"pp": "case neg\nΩ : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμs_lim :\n ∀ (f : Ω →ᵇ ℝ≥0),\n Tendsto (fun i ↦ ((ProbabilityMeasure.toFiniteMeasure ∘ fun i ↦... | exact tendsto_mul.comp lim_pair | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 33
} | [
{
"pp": "Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμs_lim : Tendsto μs F (𝓝 μ)\nnonzero : μ ≠ 0\nf : Ω →ᵇ ℝ≥0\nlim_mass : Tendsto (fun i ↦ (μs i).mass) F (𝓝 ... | exact tendsto_mul.comp lim_pair | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 429,
"column": 46
} | {
"line": 429,
"column": 54
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → Set (Set Ω)\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : iIndepSets s κ μ\ni j : ι\nhij : i ≠ j\nt₁ t₂ : Set Ω\nht₁ : t₁ ∈ s i\nht₂ : t₂ ∈ s j\nhf_m : ∀ x ∈ {i, j}, (if x = i then t₁ else t₂) ∈ s x\... | h_indep' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 561,
"column": 44
} | {
"line": 561,
"column": 58
} | [
{
"pp": "case inr\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s κ μ\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ x ∈ p1, f1 x ∈ s ... | Set.univ_inter | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 10
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nι : Type u_8\nβ : ι → Type u_9\nm : (x : ι) → MeasurableSpace (β x)\nf : (i : ι) → Ω → β i\nh :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Independence.Basic | {
"line": 700,
"column": 4
} | {
"line": 701,
"column": 57
} | [
{
"pp": "case mpr\nΩ : Type u_1\nβ : Type u_6\nβ' : Type u_7\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nσf : SigmaFinite (Measure.map f μ)\nσg : SigmaFinite (Measure.map g μ)\nh₀ :\n ∀ {s : Set ... | intro h s t hs ht
rw [(h₀ hs ht).1, (h₀ hs ht).2, h, Measure.prod_prod] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Basic | {
"line": 700,
"column": 4
} | {
"line": 701,
"column": 57
} | [
{
"pp": "case mpr\nΩ : Type u_1\nβ : Type u_6\nβ' : Type u_7\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nσf : SigmaFinite (Measure.map f μ)\nσg : SigmaFinite (Measure.map g μ)\nh₀ :\n ∀ {s : Set ... | intro h s t hs ht
rw [(h₀ hs ht).1, (h₀ hs ht).2, h, Measure.prod_prod] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 703,
"column": 4
} | {
"line": 704,
"column": 8
} | [
{
"pp": "case mpr.inr.refine_2\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), ... | apply (add_le_add (hM M rfl.le).le (le_refl (ε / 2))).trans_eq
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 703,
"column": 4
} | {
"line": 704,
"column": 8
} | [
{
"pp": "case mpr.inr.refine_2\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), ... | apply (add_le_add (hM M rfl.le).le (le_refl (ε / 2))).trans_eq
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Covariance | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 84
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\ns : Finset ι\ninst✝ : IsFiniteMeasure μ\nι' : Type u_3\nY : ι' → Ω → ℝ\nt : Finset ι'\nhX : ∀ i ∈ s, MemLp (X i) 2 μ\nhY : ∀ i ∈ t, MemLp (Y i) 2 μ\n⊢ ∑ i ∈ s, cov[X i, ∑ j ∈ t, Y j; μ] = ∑ i ∈ s, ∑ j ∈ t, cov[X i, Y j; μ... | exact Finset.sum_congr rfl fun i hi ↦ by rw [covariance_sum_right' hY (hX i hi)] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.Covariance | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 84
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\ns : Finset ι\ninst✝ : IsFiniteMeasure μ\nι' : Type u_3\nY : ι' → Ω → ℝ\nt : Finset ι'\nhX : ∀ i ∈ s, MemLp (X i) 2 μ\nhY : ∀ i ∈ t, MemLp (Y i) 2 μ\n⊢ ∑ i ∈ s, cov[X i, ∑ j ∈ t, Y j; μ] = ∑ i ∈ s, ∑ j ∈ t, cov[X i, Y j; μ... | exact Finset.sum_congr rfl fun i hi ↦ by rw [covariance_sum_right' hY (hX i hi)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.Covariance | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 84
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\ns : Finset ι\ninst✝ : IsFiniteMeasure μ\nι' : Type u_3\nY : ι' → Ω → ℝ\nt : Finset ι'\nhX : ∀ i ∈ s, MemLp (X i) 2 μ\nhY : ∀ i ∈ t, MemLp (Y i) 2 μ\n⊢ ∑ i ∈ s, cov[X i, ∑ j ∈ t, Y j; μ] = ∑ i ∈ s, ∑ j ∈ t, cov[X i, Y j; μ... | exact Finset.sum_congr rfl fun i hi ↦ by rw [covariance_sum_right' hY (hX i hi)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 404,
"column": 33
} | {
"line": 404,
"column": 47
} | [
{
"pp": "case inr\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_meas : ∀ (i : ι), Measurable (f i)\nhμ : μ ≠ 0\nη : Ker... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 414,
"column": 27
} | {
"line": 414,
"column": 41
} | [
{
"pp": "case h.e_a\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_meas : ∀ (i : ι), Measurable (f i)\nhμ : μ ≠ 0\nη : K... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 125,
"column": 41
} | {
"line": 125,
"column": 66
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : AEStronglyMeasurable X μ\n⊢ ¬MemLp X 2 μ → eVar[X; μ] = ∞",
"usedConstants": [
"ProbabilityTheory.evariance_eq_top"
]
}
] | exact evariance_eq_top hX | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 488,
"column": 2
} | {
"line": 491,
"column": 90
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k l : ι\nhik : i ≠ k\nhil : i ≠ l... | let g (i j : ι) (v : Π x : ({i, j} : Finset ι), β x) : β i × β j :=
⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem <| mem_singleton_self _⟩⟩
have hg (i j : ι) : Measurable (g i j) := by fun_prop
exact (hf_indep.indepFun_finset {i, j} {k, l} (by aesop) hf_meas).comp (hg i j) (hg k l) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 488,
"column": 2
} | {
"line": 491,
"column": 90
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k l : ι\nhik : i ≠ k\nhil : i ≠ l... | let g (i j : ι) (v : Π x : ({i, j} : Finset ι), β x) : β i × β j :=
⟨v ⟨i, mem_insert_self _ _⟩, v ⟨j, mem_insert_of_mem <| mem_singleton_self _⟩⟩
have hg (i j : ι) : Measurable (g i j) := by fun_prop
exact (hf_indep.indepFun_finset {i, j} {k, l} (by aesop) hf_meas).comp (hg i j) (hg k l) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Variance | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\ninst✝ : MeasurableSingletonClass Ω\nx : Ω\n⊢ Var[X; Measure.dirac x] = 0",
"usedConstants": [
"ProbabilityTheory.variance_eq_integral",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.instZero",
"Real.in... | rw [variance_eq_integral] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Moments.Variance | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 21
} | [
{
"pp": "case h.e'_4.e_a\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nc : ℝ≥0\nhc : c ≠ 0\nA : ↑c ≠ 0\nB : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ\n⊢ eVar[X; μ] = (∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ) ^ 1",
"usedConstants": [
"Eq.... | ENNReal.rpow_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.FactorsThrough | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 12
} | [
{
"pp": "case lim.h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng✝ : X → Z\ninst✝² : Nonempty Z\ninst✝¹ : TopologicalSpace Z\ninst✝ : IsCompletelyMetrizableSpace Z\nmX : MeasurableSpace X := MeasurableSpace.comap f mY\ng : ℕ → X → Z\ni : X → Z\nhg : ∀ (n : ℕ), StronglyMeasurabl... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 31
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n... | obtain ⟨n, hn⟩ := hb.exists | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.LpSpace.DomAct.Basic | {
"line": 62,
"column": 9
} | {
"line": 62,
"column": 65
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nα : Type u_3\nE : Type u_4\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝² : SMul M α\ninst✝¹ : SMulInvariantMeasure M α μ\ninst✝ : MeasurableConstSMul M α\nc : M\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nb : E\n⊢ μ ((fun x ... | by rwa [SMulInvariantMeasure.measure_preimage_smul c hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\nf : ℕ → α → β\nhp' : p ≠ ∞\nhf : ∀ (n : ℕ), MemLp (f n) p μ\nhf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε\nε : ℝ≥0\nhε : 0 < ε\nN : ℕ\nhNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε\nF : Fin N ... | refine ⟨s, hμs, fun n => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 111,
"column": 20
} | {
"line": 111,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 111,
"column": 20
} | {
"line": 111,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 111,
"column": 20
} | {
"line": 111,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 130,
"column": 27
} | {
"line": 130,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 130,
"column": 27
} | {
"line": 130,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 130,
"column": 27
} | {
"line": 130,
"column": 35
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 322,
"column": 2
} | {
"line": 325,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\nf f' : ℕ → α → β\ng g' : α → β\nhff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n\nhgg' : g =ᶠ[ae μ] g'\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f' n x) atTop (𝓝 (... | have hff'' := eventually_countable_forall.mpr hff'
filter_upwards [hff'', hgg', hfg] with x hff'x hgg'x hfgx
apply Tendsto.congr hff'x
rw [← hgg'x]; exact hfgx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 322,
"column": 2
} | {
"line": 325,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\nf f' : ℕ → α → β\ng g' : α → β\nhff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n\nhgg' : g =ᶠ[ae μ] g'\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f' n x) atTop (𝓝 (... | have hff'' := eventually_countable_forall.mpr hff'
filter_upwards [hff'', hgg', hfg] with x hff'x hgg'x hfgx
apply Tendsto.congr hff'x
rw [← hgg'x]; exact hfgx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 525,
"column": 6
} | {
"line": 525,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\na : E\ns : Set E\nω : E → E →L[𝕜] F\nhs : Convex ℝ s\nhω : ∀ᶠ (x : E) in... | eventually_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 214,
"column": 20
} | {
"line": 214,
"column": 29
} | [
{
"pp": "case pos\nf : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\nN : ℕ\nN_nonzero : 0 < N\nh : ℝ := (b - a) / ↑N\nak : ℕ → ℝ := ... | fpp_bound | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 177,
"column": 2
} | {
"line": 187,
"column": 28
} | [
{
"pp": "α : Type u\ns : Set (Set α)\ni✝¹ i✝ i : Ordinal.{v}\nIH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀\nhi : i ≤ ω_ 1\nA : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀\nB : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀\n⊢ #↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀",
"usedConstants": [
"le_max_right",
"E... | have C : #(⋃ j < i, generateMeasurableRec s j) ≤ max #s 2 ^ ℵ₀ := by
apply mk_biUnion_le_of_le_lift _ B _
· intro j hj
exact IH j hj (hj.trans_le hi).le
· rw [lift_power, lift_aleph0]
rw [← Ordinal.lift_le.{u}, lift_omega, Ordinal.lift_one, ← ord_aleph] at hi
have H := card_le_of_le_ord hi... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 136,
"column": 2
} | {
"line": 147,
"column": 65
} | [
{
"pp": "case h.refine_2\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\na ε : ℝ\nhε : 0 < ε\nN : ℕ\nhf : range ⇑f ⊆ Ioo a (a + ↑N * ε)\nb : ℝ := a + ↑N * ε\ny : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)\nhy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m\nE : F... | · -- The sets `E n` are pairwise disjoint.
intro m _ n _ hmn
apply Disjoint.preimage
simp_rw [mem_preimage, mem_Ioc, disjoint_left]
intro x hx
rw [mem_setOf_eq, and_assoc] at hx
simp_rw [mem_setOf_eq, not_and_or, not_lt, not_le, or_assoc]
rcases (by lia : m < n ∨ n < m) with hc | hc
· le... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 52,
"column": 60
} | {
"line": 52,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nhint : MemLp id (↑n) μ\nk : ℕ\nhk : ↑k ≤ ↑n\n⊢ ↑k ≤ ↑n",
"usedConstants": [
"ENN... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 52,
"column": 60
} | {
"line": 52,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nhint : MemLp id (↑n) μ\nk : ℕ\nhk : ↑k ≤ ↑n\n⊢ ↑k ≤ ↑n",
"usedConstants": [
"ENN... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 52,
"column": 60
} | {
"line": 52,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nhint : MemLp id (↑n) μ\nk : ℕ\nhk : ↑k ≤ ↑n\n⊢ ↑k ≤ ↑n",
"usedConstants": [
"ENN... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 62,
"column": 64
} | {
"line": 62,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ∞\nhint : ∀ (k : ℕ), MemLp id (↑k) μ\nk : ℕ\nhk : ↑k ≤ n\n⊢ ↑k ≤ ↑k",
"usedConstants": [... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 62,
"column": 64
} | {
"line": 62,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ∞\nhint : ∀ (k : ℕ), MemLp id (↑k) μ\nk : ℕ\nhk : ↑k ≤ n\n⊢ ↑k ≤ ↑k",
"usedConstants": [... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 62,
"column": 64
} | {
"line": 62,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ∞\nhint : ∀ (k : ℕ), MemLp id (↑k) μ\nk : ℕ\nhk : ↑k ≤ n\n⊢ ↑k ≤ ↑k",
"usedConstants": [... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 75,
"column": 62
} | {
"line": 75,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = (innerSL ℝ).toLinearMap₁₂\nk ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 75,
"column": 62
} | {
"line": 75,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = (innerSL ℝ).toLinearMap₁₂\nk ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 75,
"column": 62
} | {
"line": 75,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = (innerSL ℝ).toLinearMap₁₂\nk ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 120,
"column": 54
} | {
"line": 120,
"column": 62
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nhint : MemLp id (↑n) μ\nt : ℝ\nk : ℕ\nhkn : k ∈ Finset.range (n + 1)\n⊢ ↑k ≤ ↑n",
"usedConstants": [
"ENNReal.instIsOrderedRing",
"Finset.mem_range._simp_1",
"Nat.instOne",
"ENNReal.instAddCommMonoid",
"IsOrderedRing.toZ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 120,
"column": 54
} | {
"line": 120,
"column": 62
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nhint : MemLp id (↑n) μ\nt : ℝ\nk : ℕ\nhkn : k ∈ Finset.range (n + 1)\n⊢ ↑k ≤ ↑n",
"usedConstants": [
"ENNReal.instIsOrderedRing",
"Finset.mem_range._simp_1",
"Nat.instOne",
"ENNReal.instAddCommMonoid",
"IsOrderedRing.toZ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 120,
"column": 54
} | {
"line": 120,
"column": 62
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nhint : MemLp id (↑n) μ\nt : ℝ\nk : ℕ\nhkn : k ∈ Finset.range (n + 1)\n⊢ ↑k ≤ ↑n",
"usedConstants": [
"ENNReal.instIsOrderedRing",
"Finset.mem_range._simp_1",
"Nat.instOne",
"ENNReal.instAddCommMonoid",
"IsOrderedRing.toZ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 10
} | [
{
"pp": "case convert_7\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nX : Ω → ℝ\nhX : AEMeasurable X P\nh0 : ∫ (x : Ω), X x ∂P = 0\nh1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1\n⊢ ∫ (x : Ω), id (X x) ^ 2 ∂P ≠ 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 22
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I'), fs i x = ... | use I'.attach.sup fs | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 255,
"column": 61
} | {
"line": 255,
"column": 69
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T35Space X\nx y : X\nx_ne_y : x ≠ y\nf : X → ↑I\nf_cont : Continuous[inst✝¹, _] f\nf_zero : f x = 0\nf_one : EqOn f 1 {y}\n⊢ (fun x ↦ ↑(f x)) x ≠ (fun x ↦ ↑(f x)) y",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCom... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 255,
"column": 61
} | {
"line": 255,
"column": 69
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T35Space X\nx y : X\nx_ne_y : x ≠ y\nf : X → ↑I\nf_cont : Continuous[inst✝¹, _] f\nf_zero : f x = 0\nf_one : EqOn f 1 {y}\n⊢ (fun x ↦ ↑(f x)) x ≠ (fun x ↦ ↑(f x)) y",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCom... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 255,
"column": 61
} | {
"line": 255,
"column": 69
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T35Space X\nx y : X\nx_ne_y : x ≠ y\nf : X → ↑I\nf_cont : Continuous[inst✝¹, _] f\nf_zero : f x = 0\nf_one : EqOn f 1 {y}\n⊢ (fun x ↦ ↑(f x)) x ≠ (fun x ↦ ↑(f x)) y",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCom... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 262,
"column": 23
} | {
"line": 262,
"column": 31
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T35Space X\nx y : X\nx_ne_y : x ≠ y\nf : X → ↑I\nf_cont : Continuous[inst✝¹, _] f\nf_zero : f x = 0\nf_one : EqOn f 1 {y}\n⊢ f x ≠ f y",
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 262,
"column": 23
} | {
"line": 262,
"column": 31
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T35Space X\nx y : X\nx_ne_y : x ≠ y\nf : X → ↑I\nf_cont : Continuous[inst✝¹, _] f\nf_zero : f x = 0\nf_one : EqOn f 1 {y}\n⊢ f x ≠ f y",
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 262,
"column": 23
} | {
"line": 262,
"column": 31
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T35Space X\nx y : X\nx_ne_y : x ≠ y\nf : X → ↑I\nf_cont : Continuous[inst✝¹, _] f\nf_zero : f x = 0\nf_one : EqOn f 1 {y}\n⊢ f x ≠ f y",
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 142,
"column": 13
} | {
"line": 142,
"column": 15
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν κ : Measure Ω\nLPμν_finite : ¬levyProkhorovEDist μ ν = ∞\nLPνκ_finite : ¬levyProkhorovEDist ν κ = ∞\nε : ℝ≥0∞\nB : Set Ω\nε_pos : 0 < ε\nε_lt_top : ε < ∞\nB_mble : MeasurableSet B\nhalf_ε_pos : ... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 343,
"column": 10
} | {
"line": 343,
"column": 44
} | [
{
"pp": "case a.a.refine_1\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : BorelSpace Ω\nμ ν : LevyProkhorov (ProbabilityMeasure Ω)\nh : dist μ ν = 0\n⊢ inst✝³ = MeasurableSpace.generateFrom {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopo... | BorelSpace.measurable_eq (α := Ω), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 335,
"column": 6
} | {
"line": 335,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ⇑f ⊆ Ioo a b\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 467,
"column": 6
} | {
"line": 467,
"column": 33
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)\nν : LevyProkhorov (ProbabilityMeasure Ω)\nhμs : Tendsto μs atTop (𝓝 ν)\nP : ProbabilityMeasure Ω := ν.toMeasure\nPs : ℕ → ProbabilityMeasure Ω := toMea... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 94,
"column": 2
} | {
"line": 95,
"column": 85
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : IsTopologicalAddGroup A\ninst✝⁵ : LocallyCompactSpace A\ninst✝⁴ : ContinuousConstSMul G A\ng : G\ninst✝³ : MeasurableSpace A\ninst✝² : BorelSpace A\nμ : Measure A... | refine ENNReal.coe_injective ?_
rw [distribHaarChar_eq_div hs₀ hs, hμgs, ENNReal.mul_div_cancel_right] <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 94,
"column": 2
} | {
"line": 95,
"column": 85
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : IsTopologicalAddGroup A\ninst✝⁵ : LocallyCompactSpace A\ninst✝⁴ : ContinuousConstSMul G A\ng : G\ninst✝³ : MeasurableSpace A\ninst✝² : BorelSpace A\nμ : Measure A... | refine ENNReal.coe_injective ?_
rw [distribHaarChar_eq_div hs₀ hs, hμgs, ENNReal.mul_div_cancel_right] <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 72
} | [
{
"pp": "case neg\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsFiniteMeasure μ\nhr : 0 < r\nh_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)\ny : ℝ\nhy : ¬y = 0\n⊢ ∫ (x : ℝ) in -r..r, cexp (↑x * ↑y * I) =\n if r * y = 0 then 2 * ↑r else ↑y⁻¹ * ∫ (x : ℝ) in -... | simp only [mul_eq_zero, hr.ne', hy, or_self, ↓reduceIte, ofReal_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 110,
"column": 71
} | {
"line": 124,
"column": 75
} | [
{
"pp": "μ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ 2 * ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 2⁻¹ ∂μ ≤ 2 * ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 1 - sinc (2 * r⁻¹ * x) ∂μ",
"usedConstants": [
"Iff.mpr"... | by
gcongr (2 : ℝ) * ?_
refine setIntegral_mono_on ?_
((integrable_const _).sub (integrable_sinc_const_mul _)).integrableOn ?_ fun x hx ↦ ?_
· exact Integrable.integrableOn <| by fun_prop
· exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop)
· have hx_ne : 2 * r⁻¹ * x ≠ 0 := by
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 45
} | [
{
"pp": "case h\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\... | apply le_trans (hcontradiction (sub c)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 68,
"column": 2
} | {
"line": 70,
"column": 51
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (μ ... | have h_le n r (hr : 0 < r) : (μ n).real {x | r < |⟪z, x⟫|} ≤
2⁻¹ * r * ‖∫ t in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ :=
measureReal_abs_inner_gt_le_integral_charFun hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 61
} | [
{
"pp": "case h.refine_2\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fu... | · exact ENNReal.toReal_nonneg.trans hu.exists.choose_spec | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 92
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns₁ s₂ : Set X\nhs₁ : MeasurableSet s₁\nhs₂ : MeasurableSet s₂\nh : s₁ ⊆ s₂\nP : Finpartition ⟨s₁, hs₁⟩\nheq : ¬s₁ = s₂\nb : Subtype MeasurableSet := ⟨s₂ \\ s₁, ⋯⟩\n⊢ ∑ p ∈ P.parts, f ↑p ≤ preVariationFun f s₂",
"usedConstants": [
... | have hb : b ≠ ⊥ := fun hc => heq (h.antisymm (Set.diff_eq_empty.mp (congrArg (·.1) hc))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 51
} | [
{
"pp": "X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\n⊢ ∑' (i : ℕ), preVariationFun f (s i) ≤ preVariationFun f (⋃ i, s i)",
"usedConstants": [
"ENNReal.tsum_le_of_sum_range_le",
"Nat",
"Measu... | refine ENNReal.tsum_le_of_sum_range_le fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 14
} | [
{
"pp": "case mp.inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsZeroOneMeasure μ\ns : Set α\nh : μ s = 1\nh₀ : μ univ = 0\nthis : 1 ≤ μ univ\n⊢ μ univ = 1",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"False",
"MeasureTheory.Measure",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 12
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsZeroOneMeasure μ\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : μ s = 1\nhμt : μ t = 1\nh✝ : μ (s ∩ t) = 0\nthis : 1 - μ (s ∩ t) ≤ 0\n⊢ False",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"False"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 85,
"column": 36
} | {
"line": 85,
"column": 53
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsZeroOneMeasure μ\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : μ s = 1\nhμt : μ t = 1\nh✝ : μ (s ∩ t) = 0\n⊢ μ univ - μ s + μ tᶜ ≤ 0",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"MeasureTheory.Me... | measure_univ hμs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 14
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nh : μ univ = 0\n⊢ IsProbabilityMeasure μ",
"usedConstants": [
"Eq.mpr",
"False",
"MeasureTheory.Measure",
"congrArg",
"Set.univ"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 14
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nh : μ univ = 0\n⊢ IsProbabilityMeasure μ",
"usedConstants": [
"Eq.mpr",
"False",
"MeasureTheory.Measure",
"congrArg",
"Set.univ"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 14
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nh : μ univ = 0\n⊢ IsProbabilityMeasure μ",
"usedConstants": [
"Eq.mpr",
"False",
"MeasureTheory.Measure",
"congrArg",
"Set.univ"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 14
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 14
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 14
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 16
} | [
{
"pp": "case neg.inr\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 16
} | [
{
"pp": "case neg.inr\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 16
} | [
{
"pp": "case neg.inr\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 14
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ → Set α :=... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis✝ : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 12
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsZeroOneMeasure μ\ninst✝¹ : StandardBorelSpace α\ninst✝ : NeZero μ\nthis✝ : IsProbabilityMeasure μ\nA : ℕ → Set α\nhAm : ∀ (n : ℕ), MeasurableSet (A n)\nhAsep : ∀ x ∈ univ, ∀ y ∈ univ, (∀ (n : ℕ), x ∈ A n ↔ y ∈ A n) → x = y\nB : ℕ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 12
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx : ι → ℝ\nf : (δ : ℝ) → 0 < δ → ι → ℝ\nhf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ\nhf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ\nH : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (c... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 12
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx : ι → ℝ\nf : (δ : ℝ) → 0 < δ → ι → ℝ\nhf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ\nhf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ\nH : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (c... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 12
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx : ι → ℝ\nf : (δ : ℝ) → 0 < δ → ι → ℝ\nhf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ\nhf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior sᶜ\nH : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (c... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 24
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nhs : IsUpperSet s\nx : ι → ℝ\nhx : x ∈ frontier s\nh : x ∈ closure s\nx✝ : ℝ\n⊢ x ∈ frontier sᶜ",
"usedConstants": [
"Eq.mpr",
"frontier_compl",
"frontier",
"Real",
"Pi.topologicalSpace",
"congrArg",
... | rwa [frontier_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 44
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁹ : (i : ι) → TopologicalSpace (X i)\ninst✝⁸ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁷ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁶ : (j : κ) → TopologicalSpac... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 24
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nhs : IsLowerSet s\nx : ι → ℝ\nhx : x ∈ frontier s\nh : x ∈ closure s\nx✝ : ℝ\n⊢ x ∈ frontier sᶜ",
"usedConstants": [
"Eq.mpr",
"frontier_compl",
"frontier",
"Real",
"Pi.topologicalSpace",
"congrArg",
... | rwa [frontier_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 75
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nP : Finset (Set X)\nhP₁ : ∀ t ∈ P, t ⊆ s\nhP₂ : (↑P).PairwiseDisjoint id\nQ : Finpartition (P.sup id) := Finpartit... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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