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.Probability.Process.Stopping | {
"line": 747,
"column": 80
} | {
"line": 747,
"column": 94
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\n⊢ MeasurableSet (Set.univ ∩ {ω | τ ω ≤ π ω})",
... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 755,
"column": 6
} | {
"line": 755,
"column": 33
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\nthis : MeasurableSet {ω | τ ω ≤ π ω}\n⊢ Measura... | measurableSet_min_iff hτ hπ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 773,
"column": 6
} | {
"line": 773,
"column": 33
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\nh : MeasurableSet {ω | τ ω = π ω}\n⊢ Measurable... | measurableSet_min_iff hτ hπ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.BorelCantelli | {
"line": 77,
"column": 98
} | {
"line": 78,
"column": 41
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ns : ℕ → Set Ω\nhsm : ∀ (n : ℕ), MeasurableSet (s n)\nhs : iIndepSet s μ\nhs' : ∑' (n : ℕ), μ (s n) = ∞\nthis✝ : IsProbabilityMeasure μ\nthis :\n {ω |\n Tendsto (fun n ↦ ∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator 1 | ↑(filtrationOfSet hsm) k]... | by
rw [measure_congr this, measure_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 217,
"column": 6
} | {
"line": 219,
"column": 46
} | [
{
"pp": "case hfs\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ IntegrableOn (stoppedValue f fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω))\n {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ ... | · exact Integrable.integrableOn (hsub.integrable_stoppedValue
(hsub.stronglyAdapted.adapted.isStoppingTime_hittingBtwn measurableSet_Ici)
(fun ω ↦ mod_cast hittingBtwn_le ω)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 220,
"column": 6
} | {
"line": 222,
"column": 46
} | [
{
"pp": "case hft\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ IntegrableOn (stoppedValue f fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω))\n {ω | ((range (n + 1)).sup' ⋯ fun ... | · exact Integrable.integrableOn (hsub.integrable_stoppedValue
(hsub.stronglyAdapted.adapted.isStoppingTime_hittingBtwn measurableSet_Ici)
(fun ω ↦ mod_cast hittingBtwn_le ω)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | {
"line": 294,
"column": 61
} | {
"line": 303,
"column": 47
} | [
{
"pp": "α : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nr : ℚ\n⊢ stieltjesFunctionAux f a ↑r = f a r",
"usedConstants": [
"ProbabilityTheory.IsMeasurableRatCDF.iInf_rat_gt_eq",
"Eq.mpr",
"Real",
"Set.Ioi",
"Preorder.toLT",
"iInf"... | by
rw [← hf.iInf_rat_gt_eq a r, IsMeasurableRatCDF.stieltjesFunctionAux]
refine Equiv.iInf_congr ?_ ?_
· exact
{ toFun := fun t ↦ ⟨t.1, mod_cast t.2⟩
invFun := fun t ↦ ⟨t.1, mod_cast t.2⟩
left_inv := fun t ↦ by simp only [Subtype.coe_eta]
right_inv := fun t ↦ by simp only [Subtype.co... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 147,
"column": 6
} | {
"line": 148,
"column": 41
} | [
{
"pp": "case e_f.h.hfin\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : ¬(ν a).restrict s = 0\nb : β\n⊢ ∃ i, (s... | obtain ⟨q, hq⟩ := exists_rat_gt x
exact ⟨⟨q, hq⟩, measure_ne_top _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 147,
"column": 6
} | {
"line": 148,
"column": 41
} | [
{
"pp": "case e_f.h.hfin\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : ¬(ν a).restrict s = 0\nb : β\n⊢ ∃ i, (s... | obtain ⟨q, hq⟩ := exists_rat_gt x
exact ⟨⟨q, hq⟩, measure_ne_top _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 154,
"column": 2
} | {
"line": 156,
"column": 28
} | [
{
"pp": "case neg.hf_int\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : ¬(ν a).restrict s = 0\nh :\n ∫⁻ (b : β... | · intro b
rw [setLIntegral_stieltjesOfMeasurableRat_rat hf a _ hs]
exact measure_ne_top _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Process.Stopping | {
"line": 1078,
"column": 4
} | {
"line": 1082,
"column": 24
} | [
{
"pp": "case h.inr\nΩ : Type u_1\nι : Type u_3\ninst✝² : Nonempty ι\nτ : Ω → WithTop ι\nE : Type u_4\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nω : Ω\ni : ι\nhi : ↑i = τ ω\nh : ↑i < ↑n\nhbdd : ↑i ∈ WithTop.some '' ↑s\n⊢ u (↑i).untopA ω = {a | ↑n ≤ τ a}.indicator (u n)... | · simp only [untopD_coe]
rw [Set.indicator_of_notMem, zero_add, Set.indicator_of_mem] <;> rw [Set.mem_setOf]
· exact hi.symm
· rw [← hi]
exact not_le.2 h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.Basic | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 21
} | [
{
"pp": "case h\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nhμ : μ ≠ 0\nh_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ\nthis : ∫ (x : Ω), rexp (t * X x) ∂μ = ∫ (x : Ω) in Set.univ, rexp (t * X x) ∂μ\nx : Ω\n⊢ 0 x ≤ rexp (t * X x)",
"usedConstants": [
"Eq.mpr",
"Real.i... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Basic | {
"line": 367,
"column": 2
} | {
"line": 376,
"column": 36
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nh_meas : ∀ (i : ι), AEMeasurable (X i) μ\ns : Finset ι\nthis : IsProbabilityMeasure μ\n⊢ mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t",
"usedConstants": [
"AEMeasurable.aestrong... | classical
induction s using Finset.induction_on with
| empty => simp
| insert i s hi_notin_s h_rec =>
have h_int' : ∀ i : ι, AEStronglyMeasurable (fun ω : Ω => exp (t * X i ω)) μ := fun i =>
((h_meas i).const_mul t).exp.aestronglyMeasurable
rw [sum_insert hi_notin_s,
IndepFun.mgf_add (h_indep.... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Probability.Moments.Basic | {
"line": 388,
"column": 11
} | {
"line": 388,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nh_meas : ∀ (i : ι), AEMeasurable (X i) μ\ns : Finset ι\nh_int : ∀ i ∈ s, Integrable (fun ω ↦ rexp (t * X i ω)) μ\nthis : IsProbabilityMeasure μ\n⊢ cgf (∑ i ∈ s, X i) μ t = ∑ i ∈ s, cgf (X i)... | cgf | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 48
} | [
{
"pp": "case h\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nY : Ω' → ℝ\nμ' : Measure Ω'\nhXY : mgf X μ = mgf Y μ'\nhμμ' : μ = 0 ↔ μ' = 0\nt : ℝ\n⊢ t ∈ integrableExpSet X μ ↔ t ∈ integrableExpSet Y μ'",
"usedConstants": [
"Real",
"Membe... | simp only [integrableExpSet, Set.mem_setOf_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 228,
"column": 56
} | {
"line": 234,
"column": 64
} | [
{
"pp": "x t p : ℝ\nhp : 0 ≤ p\nht : 0 < t\nhp_zero : ¬p = 0\nh_x_le : ∀ (c : ℝ), 0 < c → x ≤ c⁻¹ * rexp (c * x)\nh_neg_x_le : ∀ (c : ℝ), 0 < c → -x ≤ c⁻¹ * rexp (-c * x)\nh_abs_le : ∀ (c : ℝ), 0 < c → |x| ≤ c⁻¹ * max (rexp (c * x)) (rexp (-c * x))\n⊢ ((t / p)⁻¹ * max (rexp (t / p * x)) (rexp (-t / p * x))) ^ p... | by
rw [mul_rpow (by positivity) (by positivity)]
congr
· simp
· rw [rpow_max (by positivity) (by positivity) hp, ← exp_mul, ← exp_mul]
ring_nf
congr <;> rw [mul_assoc, mul_inv_cancel₀ hp_zero, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 88
} | [
{
"pp": "case insert.hX\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ns✝ : Finset ι\nE : Type u_3\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nX : ι → Ω → E\ninst✝ : NormedSpace ℝ E\nthis : IsProbabilityMeasure P\ni ... | exact hX.precomp (g := fun x : s ↦ ⟨x.1, mem_insert_of_mem x.2⟩) (fun _ ↦ by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 88
} | [
{
"pp": "case insert.hX\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ns✝ : Finset ι\nE : Type u_3\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nX : ι → Ω → E\ninst✝ : NormedSpace ℝ E\nthis : IsProbabilityMeasure P\ni ... | exact hX.precomp (g := fun x : s ↦ ⟨x.1, mem_insert_of_mem x.2⟩) (fun _ ↦ by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 88
} | [
{
"pp": "case insert.hX\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ns✝ : Finset ι\nE : Type u_3\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nX : ι → Ω → E\ninst✝ : NormedSpace ℝ E\nthis : IsProbabilityMeasure P\ni ... | exact hX.precomp (g := fun x : s ↦ ⟨x.1, mem_insert_of_mem x.2⟩) (fun _ ↦ by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 60
} | [
{
"pp": "l u v : ℝ\nhv : v ∈ Set.Ioo l u\n⊢ v + min (v - l) (u - v) / 2 ∈ Set.Ioo l u",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"sub_pos._simp_1",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Lattic... | have h_pos : 0 < (v - l) ⊓ (u - v) := by simp [hv.1, hv.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 60
} | [
{
"pp": "l u v : ℝ\nhv : v ∈ Set.Ioo l u\n⊢ v - min (v - l) (u - v) / 2 ∈ Set.Ioo l u",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"sub_pos._simp_1",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Lattic... | have h_pos : 0 < (v - l) ⊓ (u - v) := by simp [hv.1, hv.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 83,
"column": 8
} | {
"line": 83,
"column": 19
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nκ : Kernel α β\ninst✝ : SFinite μ\nhκ : ¬IsSFiniteKernel κ\n⊢ μ ⊗ₘ κ = ⇑(Kernel.id ∥ₖ κ) ∘ₘ ⇑(Kernel.copy α) ∘ₘ μ",
"usedConstants": [
"False",
"MeasureTheory.Measure",
"eq_false",... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.CentralLimitTheorem | {
"line": 47,
"column": 50
} | {
"line": 51,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : ℕ → Ω → ℝ\nhindep : iIndepFun X P\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P\nn : ℕ\nt : ℝ\n⊢ charFun (Measure.map (fun ω ↦ (√↑n)⁻¹ * ∑ k ∈ Finset.range n, X k ω) P) t =\n charFun (Measure.map (X 0) P) ((√↑n)⁻¹ * t) ^ n",
"usedConst... | by
have mX n := (hident n).aemeasurable_fst
rw [charFun_map_mul_comp, (hindep.restrict _).charFun_map_fun_finsetSum_eq_prod (fun _ _ ↦ mX _)]
· simp [fun i ↦ (hident i).map_eq]
· exact Finset.aemeasurable_fun_sum _ fun _ _ ↦ mX _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Process.PartitionFiltration | {
"line": 92,
"column": 56
} | {
"line": 92,
"column": 75
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nt : ℕ → Set α\nht : ∀ (n : ℕ), MeasurableSet (t n)\n| generateFrom (⋃ n, memPartition t n)",
"usedConstants": [
"congrArg",
"iSup",
"CompleteLattice.toConditionallyCompleteLattice",
"memPartition",
"MeasurableSpace.generateFrom",
... | ← iSup_generateFrom | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Probability.Process.PartitionFiltration | {
"line": 143,
"column": 62
} | {
"line": 143,
"column": 81
} | [
{
"pp": "α : Type u_2\nm : MeasurableSpace α\ninst✝ : CountablyGenerated α\n| generateFrom (⋃ n, countablePartition α n)",
"usedConstants": [
"congrArg",
"iSup",
"CompleteLattice.toConditionallyCompleteLattice",
"MeasurableSpace.generateFrom",
"MeasurableSpace",
"Nat",
... | ← iSup_generateFrom | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 51
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\nm n : ι\ni : ↥(Iic n)\nh : ↑i ≤ m\n⊢ Measurable fun c ↦ IicProdIoc m n c i",
"usedConstants": [
"... | · simpa [IicProdIoc, h] using measurable_fst.eval | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 301,
"column": 2
} | {
"line": 306,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns s' : Set β\nh : s ⊆ s'\n⊢ κ.densityProcess ν n a x s ≤ κ.densityProcess ν n a x s... | unfold densityProcess
obtain h₀ | h₀ := eq_or_ne (ν a (countablePartitionSet n x)) 0
· simp [h₀]
· gcongr
simp only [ne_eq, ENNReal.div_eq_top, h₀, and_false, false_or, not_and, not_not]
exact eq_top_mono (meas_countablePartitionSet_le_of_fst_le hκν n a x s') | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 301,
"column": 2
} | {
"line": 306,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns s' : Set β\nh : s ⊆ s'\n⊢ κ.densityProcess ν n a x s ≤ κ.densityProcess ν n a x s... | unfold densityProcess
obtain h₀ | h₀ := eq_or_ne (ν a (countablePartitionSet n x)) 0
· simp [h₀]
· gcongr
simp only [ne_eq, ENNReal.div_eq_top, h₀, and_false, false_or, not_and, not_not]
exact eq_top_mono (meas_countablePartitionSet_le_of_fst_le hκν n a x s') | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 355,
"column": 2
} | {
"line": 360,
"column": 35
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ ... | have : Tendsto (fun m ↦ κ a (countablePartitionSet n x ×ˢ seq m)) atTop
(𝓝 ((κ a) (⋂ n_1, countablePartitionSet n x ×ˢ seq n_1))) := by
apply tendsto_measure_iInter_atTop
· measurability
· exact fun _ _ h ↦ prod_mono_right <| hseq h
· exact ⟨0, measure_ne_top _ _⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 199,
"column": 44
} | {
"line": 204,
"column": 69
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nZ : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nκ : Kernel X Y\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (X × Y) Z\ninst✝ : IsSFiniteKernel η\n⊢ deterministic Prod.fst ⋯ ×ₖ η ∘ₖ (Kernel.id ×ₖ κ) = Kernel.id ×ₖ (η ∘ₖ (Kernel.id ×ₖ κ))",
... | by
ext x s ms
simp_rw [comp_apply' _ _ _ ms, lintegral_id_prod (Kernel.measurable_coe _ ms),
deterministic_prod_apply' _ _ _ ms, id_prod_apply' _ _ ms,
comp_apply' _ _ _ (measurable_prodMk_left ms),
lintegral_id_prod (η.measurable_coe (measurable_prodMk_left ms))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 610,
"column": 73
} | {
"line": 622,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\na : α\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ i, seq i = ∅\nhs... | by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
simp_rw [integral_density hκν a (hseq_meas _)]
rw [← ENNReal.toReal_zero]
have h_cont := ENNReal.continuousAt_toReal ENNReal.zero_ne_top
refine h_cont.tendsto.comp ?_
have h : Tendsto (fun m ↦ κ a (univ ×ˢ seq... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 721,
"column": 2
} | {
"line": 723,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, seq i = univ\n⊢ ∀ᵐ (x : γ) ∂κ.fst a, Te... | filter_upwards [densityProcess_fst_univ_ae κ n a] with x hx
rw [← hx]
exact tendsto_densityProcess_fst_atTop_univ_of_monotone κ n a x seq hseq hseq_iUnion | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 721,
"column": 2
} | {
"line": 723,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, seq i = univ\n⊢ ∀ᵐ (x : γ) ∂κ.fst a, Te... | filter_upwards [densityProcess_fst_univ_ae κ n a] with x hx
rw [← hx]
exact tendsto_densityProcess_fst_atTop_univ_of_monotone κ n a x seq hseq hseq_iUnion | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 47
} | [
{
"pp": "ι : Type u_1\ns : Set ι\np : ↑I\ninst✝ : Countable ι\nu : Finset ι\nhsu : s ⊆ ↑u\n⊢ ∏' (i : ι), ((if i ∈ u ↔ i ∈ s then ↑(toNNReal p) else 0) + if i ∈ s then 0 else ↑(toNNReal (σ p))) =\n ∏ i ∈ u, if i ∈ s then ↑(toNNReal p) else ↑(toNNReal (σ p))",
"usedConstants": [
"Eq.mpr",
"ENNR... | rw [tprod_eq_prod, Finset.prod_congr rfl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case pos\nι : Type u_1\np : ↑I\ninst✝ : Countable ι\ns : Set (Set ι)\nhs : MeasurableSet s\nh : ∅ ∈ s\nthis : {t | t ∈ s ∧ t ⊆ ∅} = {∅}\n⊢ setBer(∅, p) {s_1 | s_1 ∈ s ∧ s_1 ⊆ ∅} = (dirac ∅) s",
"usedConstants": [
"subset_refl._simp_1",
"MulOne.toOne",
"ENNReal.ofNNReal",
"Me... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 12
} | [
{
"pp": "case neg\nι : Type u_1\np : ↑I\ninst✝ : Countable ι\ns : Set (Set ι)\nhs : MeasurableSet s\nh : ∅ ∉ s\nthis : {t | t ∈ s ∧ t ⊆ ∅} = ∅\n⊢ setBer(∅, p) ∅ = (dirac ∅) s",
"usedConstants": [
"False",
"MeasureTheory.Measure",
"eq_false",
"congrArg",
"Set.indicator",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.ProductMeasure | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 95
} | [
{
"pp": "case refine_1\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nI J : Finset ι\nhJI : J ⊆ I\ns : (i : ↥J) → Set (X ↑i)\nms : ∀ (i : ↥J), MeasurableSet (s i)\nx : ι\nhx : x ∈ I \\ J\n⊢ Function.extend Subtype.v... | rw [Function.extend_val_apply (mem_sdiff.1 hx).1, dif_neg (mem_sdiff.1 hx).2, measure_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 68
} | [
{
"pp": "V : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (SimpleGraph V)\n⊢ G(V, p) S =\n (infinitePi fun e ↦ (toNNReal p • dirac ¬e.IsDiag) + toNNReal (σ p) • dirac False) ((fun G e ↦ e ∈ G.edgeSet) '' S)",
"usedConstants": [
"False",
"instHSMul",
"MeasureTheory.Measure",
"Sim... | simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 68
} | [
{
"pp": "V : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (SimpleGraph V)\n⊢ G(V, p) S =\n (infinitePi fun e ↦ (toNNReal p • dirac ¬e.IsDiag) + toNNReal (σ p) • dirac False) ((fun G e ↦ e ∈ G.edgeSet) '' S)",
"usedConstants": [
"False",
"instHSMul",
"MeasureTheory.Measure",
"Sim... | simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 68
} | [
{
"pp": "V : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (SimpleGraph V)\n⊢ G(V, p) S =\n (infinitePi fun e ↦ (toNNReal p • dirac ¬e.IsDiag) + toNNReal (σ p) • dirac False) ((fun G e ↦ e ∈ G.edgeSet) '' S)",
"usedConstants": [
"False",
"instHSMul",
"MeasureTheory.Measure",
"Sim... | simp [binomialRandom_apply', setBernoulli_apply, ← Set.image_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 95
} | [
{
"pp": "case refine_1\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nI J : Finset ι\nhJI : J ⊆ I\ns : (i : ↥J) → Set (X ↑i)\nms : ∀ (i : ↥J), MeasurableSet (s i)\nx : ι\nhx : x ∈ I \\ J\n⊢ Function.extend Subtype.v... | rw [Function.extend_val_apply (mem_sdiff.1 hx).1, dif_neg (mem_sdiff.1 hx).2, measure_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 95
} | [
{
"pp": "case refine_1\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nI J : Finset ι\nhJI : J ⊆ I\ns : (i : ↥J) → Set (X ↑i)\nms : ∀ (i : ↥J), MeasurableSet (s i)\nx : ι\nhx : x ∈ I \\ J\n⊢ Function.extend Subtype.v... | rw [Function.extend_val_apply (mem_sdiff.1 hx).1, dif_neg (mem_sdiff.1 hx).2, measure_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 43
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_b... | let x_ : Π n, X n := Classical.ofNonempty | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Distributions.Gamma | {
"line": 112,
"column": 4
} | {
"line": 115,
"column": 93
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0\nrightSide :\n ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x =\n ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))\n⊢ ∫ (a_1 : ℝ) in Ici 0, r ^ a / Gamma a * a_1 ^ (a - 1) * rexp (-(r * a_1... | simp_rw [integral_Ici_eq_integral_Ioi, mul_assoc]
rw [integral_const_mul, integral_rpow_mul_exp_neg_mul_Ioi ha hr, div_mul_eq_mul_div,
← mul_assoc, mul_div_assoc, div_self (Gamma_pos_of_pos ha).ne', mul_one,
div_rpow zero_le_one hr.le, one_rpow, mul_one_div, div_self (rpow_pos_of_pos hr _).ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gamma | {
"line": 112,
"column": 4
} | {
"line": 115,
"column": 93
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0\nrightSide :\n ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x =\n ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))\n⊢ ∫ (a_1 : ℝ) in Ici 0, r ^ a / Gamma a * a_1 ^ (a - 1) * rexp (-(r * a_1... | simp_rw [integral_Ici_eq_integral_Ioi, mul_assoc]
rw [integral_const_mul, integral_rpow_mul_exp_neg_mul_Ioi ha hr, div_mul_eq_mul_div,
← mul_assoc, mul_div_assoc, div_self (Gamma_pos_of_pos ha).ne', mul_one,
div_rpow zero_le_one hr.le, one_rpow, mul_one_div, div_self (rpow_pos_of_pos hr _).ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gamma | {
"line": 112,
"column": 2
} | {
"line": 115,
"column": 93
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0\nrightSide :\n ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x =\n ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))\n⊢ ∫ (a_1 : ℝ) in Ici 0, r ^ a / Gamma a * a_1 ^ (a - 1) * rexp (-(r * a_1... | · simp_rw [integral_Ici_eq_integral_Ioi, mul_assoc]
rw [integral_const_mul, integral_rpow_mul_exp_neg_mul_Ioi ha hr, div_mul_eq_mul_div,
← mul_assoc, mul_div_assoc, div_self (Gamma_pos_of_pos ha).ne', mul_one,
div_rpow zero_le_one hr.le, one_rpow, mul_one_div, div_self (rpow_pos_of_pos hr _).ne'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.ProductMeasure | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 12
} | [
{
"pp": "case mt\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nα : Type u_3\ne : α ≃ ι\ns : Finset ι\nt : (i : ι) → Set (X i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ i ∈ s.preimage ⇑e ⋯, MeasurableSet (t (e i))",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.ProductMeasure | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 12
} | [
{
"pp": "case mt\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nα : Type u_3\ne : α ≃ ι\ns : Finset ι\nt : (i : ι) → Set (X i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ i ∈ s.preimage ⇑e ⋯, MeasurableSet (t (e i))",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 12
} | [
{
"pp": "case mt\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nα : Type u_3\ne : α ≃ ι\ns : Finset ι\nt : (i : ι) → Set (X i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ i ∈ s.preimage ⇑e ⋯, MeasurableSet (t (e i))",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 507,
"column": 48
} | {
"line": 507,
"column": 56
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nα : Type u_3\ne : α ≃ ι\ns : Finset ι\nt : (i : ι) → Set (X i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ i ∈ ⇑e '' ↑(s.preimage ⇑e ⋯), MeasurableSet (t i)",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.ProductMeasure | {
"line": 507,
"column": 48
} | {
"line": 507,
"column": 56
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nα : Type u_3\ne : α ≃ ι\ns : Finset ι\nt : (i : ι) → Set (X i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ i ∈ ⇑e '' ↑(s.preimage ⇑e ⋯), MeasurableSet (t i)",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 507,
"column": 48
} | {
"line": 507,
"column": 56
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nα : Type u_3\ne : α ≃ ι\ns : Finset ι\nt : (i : ι) → Set (X i)\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ i ∈ ⇑e '' ↑(s.preimage ⇑e ⋯), MeasurableSet (t i)",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 535,
"column": 75
} | {
"line": 535,
"column": 83
} | [
{
"pp": "ι : Type u_3\nκ : ι → Type u_4\nX : (i : ι) → κ i → Type u_5\nmX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)\nμ : (i : ι) → (j : κ i) → Measure (X i j)\nhμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)\ns : Finset ((i : ι) × κ i)\nt : (i : (i : ι) × κ i) → Set (X i.fst i.snd)\nht : ∀ (i : (i ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.ProductMeasure | {
"line": 535,
"column": 75
} | {
"line": 535,
"column": 83
} | [
{
"pp": "ι : Type u_3\nκ : ι → Type u_4\nX : (i : ι) → κ i → Type u_5\nmX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)\nμ : (i : ι) → (j : κ i) → Measure (X i j)\nhμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)\ns : Finset ((i : ι) × κ i)\nt : (i : (i : ι) × κ i) → Set (X i.fst i.snd)\nht : ∀ (i : (i ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 535,
"column": 75
} | {
"line": 535,
"column": 83
} | [
{
"pp": "ι : Type u_3\nκ : ι → Type u_4\nX : (i : ι) → κ i → Type u_5\nmX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)\nμ : (i : ι) → (j : κ i) → Measure (X i j)\nhμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)\ns : Finset ((i : ι) × κ i)\nt : (i : (i : ι) × κ i) → Set (X i.fst i.snd)\nht : ∀ (i : (i ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 34
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : Module ℝ E\nmE : MeasurableSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid F\ninst✝² : Module ℝ F\nmF : MeasurableSpace F\ninst✝¹ : OpensMeasurableSpace F\nμ : Measure E\nL : E →L[ℝ] F\ninst✝ : IsGa... | IsGaussian.map_eq_gaussianReal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 36
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\np : ℝ≥0∞\n𝕜 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual 𝕜 E\nhp : ¬p = 0\nhp_top : p = ∞\nh_Lp : MemLp id ∞ μ\n⊢ (eLpNormEssSup ... | simp only [eLpNormEssSup, id_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 92,
"column": 6
} | {
"line": 93,
"column": 60
} | [
{
"pp": "case pos.hb\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\np : ℝ≥0∞\n𝕜 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual 𝕜 E\nhp : ¬p = 0\nhp_top : p = ∞\nh_Lp : MemLp id ∞ μ\n⊢ essSup (fun ... | rw [ENNReal.essSup_const_mul]
exact ENNReal.mul_ne_top (by simp) h_Lp.eLpNorm_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 92,
"column": 6
} | {
"line": 93,
"column": 60
} | [
{
"pp": "case pos.hb\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\np : ℝ≥0∞\n𝕜 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual 𝕜 E\nhp : ¬p = 0\nhp_top : p = ∞\nh_Lp : MemLp id ∞ μ\n⊢ essSup (fun ... | rw [ENNReal.essSup_const_mul]
exact ENNReal.mul_ne_top (by simp) h_Lp.eLpNorm_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.CovarianceBilin | {
"line": 127,
"column": 8
} | {
"line": 127,
"column": 35
} | [
{
"pp": "case pos.h.h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : CompleteSpace E\ninst✝ : IsProbabilityMeasure μ\nc : E\nh : MemLp id 2 μ\nx y : E\nh_Lp : MemLp id 2 (Measure.map (fun x ↦ c + x) μ)\n⊢ (... | covarianceBilin_apply h_Lp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Fernique | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : SecondCountableTopology E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\na : ℝ\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\... | · simp [ENNReal.toReal_pos_iff, tsub_pos_iff_lt, hc_lt, hc_one_sub_lt_top] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 111,
"column": 2
} | {
"line": 115,
"column": 35
} | [
{
"pp": "T : Type u_2\nΩ : Type u_3\nE : Type u_4\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SecondCountableTopology E\nhX : IsGaussianProcess X P\nn : ℕ\nt : Fin (n + 1) → T\n⊢ HasGau... | let L : ((univ.image t) → E) →L[ℝ] Fin n → E :=
{ toFun x i := x ⟨t i.succ, by simp⟩ - x ⟨t i.castSucc, by simp⟩
map_add' x y := by ext; simp; abel
map_smul' m x := by ext; simp; module }
exact (hX.hasGaussianLaw _).map L | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 111,
"column": 2
} | {
"line": 115,
"column": 35
} | [
{
"pp": "T : Type u_2\nΩ : Type u_3\nE : Type u_4\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SecondCountableTopology E\nhX : IsGaussianProcess X P\nn : ℕ\nt : Fin (n + 1) → T\n⊢ HasGau... | let L : ((univ.image t) → E) →L[ℝ] Fin n → E :=
{ toFun x i := x ⟨t i.succ, by simp⟩ - x ⟨t i.castSucc, by simp⟩
map_add' x y := by ext; simp; abel
map_smul' m x := by ext; simp; module }
exact (hX.hasGaussianLaw _).map L | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 135,
"column": 4
} | {
"line": 138,
"column": 55
} | [
{
"pp": "S : Type u_1\nT : Type u_2\nΩ : Type u_3\nE : Type u_4\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : MeasurableSpace E\ninst✝⁷ : BorelSpace E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : SecondCountableTopology E\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup F\ninst... | let K : (I.biUnion J → E) →L[ℝ] I → F :=
{ toFun x s := L s (fun t ↦ x ⟨t.1, mem_biUnion.2 ⟨s.1, s.2, t.2⟩⟩)
map_add' x y := by ext; simp [← Pi.add_def]
map_smul' c x := by ext; simp [← Pi.smul_def] } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 78
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : InnerProductSpace ℝ E\nhX... | simpa using h t₁ t₂ ht s₁ s₂ ((toDual ℝ E).symm L₁) ((toDual ℝ E).symm L₂) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 78
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : InnerProductSpace ℝ E\nhX... | simpa using h t₁ t₂ ht s₁ s₂ ((toDual ℝ E).symm L₁) ((toDual ℝ E).symm L₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 78
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : InnerProductSpace ℝ E\nhX... | simpa using h t₁ t₂ ht s₁ s₂ ((toDual ℝ E).symm L₁) ((toDual ℝ E).symm L₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 109,
"column": 59
} | {
"line": 110,
"column": 74
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nS : T → Type u_4\nX : (t : T) → S t → Ω → ℝ\nhX : IsGaussianProcess (fun p ω ↦ X p.fst p.snd ω) P\nmX : ∀ (t : T) (s : S t), AEMeasurable (X t s) P\nh : ∀ (t₁ t₂ : T), t₁ ≠ t₂ → ∀ (s₁ : S t₁) (s₂ : S t₂), cov[X t₁ s₁, X t₂ s₂; P] = 0\nx... | by
simp [covariance_mul_const_left, covariance_mul_const_right, h _ _ h'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 113,
"column": 84
} | {
"line": 113,
"column": 91
} | [
{
"pp": "α : Type u_1\np : PMF α\na : α\nh : p a = 1\na' : α\nha' : a' ∈ p.support\nha : a' ∉ {a}\nthis : 0 < ∑' (b : α), if b = a then 0 else p b\n⊢ (p a + ∑' (b : α), if b = a then 0 else p b) = (if True then p a else 0) + ∑' (b : α), if b = a then 0 else p b",
"usedConstants": [
"Eq.mpr",
"EN... | if_true | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Pareto | {
"line": 109,
"column": 4
} | {
"line": 110,
"column": 71
} | [
{
"pp": "case hf\nt r : ℝ\nht : 0 < t\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0\nrightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))\n⊢ 0 ≤ᶠ[ae (volume.restrict (Ici t))] fun x ↦ r * t ^ r * x ^ (-(r + 1))",
"usedConstant... | rw [EventuallyLE, ae_restrict_iff' measurableSet_Ici]
filter_upwards with x hx using by positivity [lt_of_lt_of_le ht hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Pareto | {
"line": 109,
"column": 4
} | {
"line": 110,
"column": 71
} | [
{
"pp": "case hf\nt r : ℝ\nht : 0 < t\nhr : 0 < r\nleftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0\nrightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))\n⊢ 0 ≤ᶠ[ae (volume.restrict (Ici t))] fun x ↦ r * t ^ r * x ^ (-(r + 1))",
"usedConstant... | rw [EventuallyLE, ae_restrict_iff' measurableSet_Ici]
filter_upwards with x hx using by positivity [lt_of_lt_of_le ht hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 28
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\n⊢ (bernoulli p h).support = {b | bif b then p ≠ 0 else p ≠ 1}",
"usedConstants": [
"cond",
"Set.ext",
"setOf",
"PMF.support",
"NNReal",
"Ne",
"NNReal.instZero",
"PMF.bernoulli",
"Bool",
"One.toOfNat1",
"Zero.toOfN... | refine Set.ext fun b => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Independence.InfinitePi | {
"line": 51,
"column": 25
} | {
"line": 51,
"column": 61
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\n𝓧 : ι → Type u_3\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nX : (i : ι) → Ω → 𝓧 i\nmX : AEMeasurable (fun ω i ↦ X i ω) P\nh : ∀ (s : Finset ι), iIndepFun (s.restrict X) P\nthis✝ : IsProbabilityMeasure P\nx✝ : ∀ (i : ι), IsProbabilityMea... | AEMeasurable.map_map_of_aemeasurable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 100,
"column": 4
} | {
"line": 101,
"column": 82
} | [
{
"pp": "case neg\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nhκη : κ ≤ η\na : α\nx : γ\nhα : ¬Countable α\n⊢ 0 ≤ (κ.map fun a ↦ (a, ())).density η a x univ",
"usedConstants": [
"Unit.unit",
... | have := hαγ.countableOrCountablyGenerated.resolve_left hα
exact density_nonneg ((fst_map_id_prod _ measurable_const).trans_le hκη) _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 100,
"column": 4
} | {
"line": 101,
"column": 82
} | [
{
"pp": "case neg\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nhκη : κ ≤ η\na : α\nx : γ\nhα : ¬Countable α\n⊢ 0 ≤ (κ.map fun a ↦ (a, ())).density η a x univ",
"usedConstants": [
"Unit.unit",
... | have := hαγ.countableOrCountablyGenerated.resolve_left hα
exact density_nonneg ((fst_map_id_prod _ measurable_const).trans_le hκη) _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 119,
"column": 4
} | {
"line": 120,
"column": 80
} | [
{
"pp": "case pos\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\nhα : Countable α\n⊢ Measurable fun p ↦ ((∂κ p.1/∂η p.1) p.2).toReal",
"usedConstants": [
"MeasureTheory.Measure",
"MeasureTheo... | refine Measurable.ennreal_toReal <| measurable_from_prod_countable_right'
(fun a ↦ Measure.measurable_rnDeriv (κ a) (η a)) fun a a' c ha'_mem_a ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 213,
"column": 4
} | {
"line": 214,
"column": 59
} | [
{
"pp": "case e_z.e_a.e_f.h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ninst✝⁶ : Finite ι\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → MeasurableSpace (E i)\ninst✝³ : ∀ (i : ι), CompleteSpace (E i)\ninst✝² : ∀ (i : ι), BorelSpace (E i)\ninst✝¹ : ∀ (i... | rw [sum_eq_single_of_mem i (by grind) (fun j _ hij ↦ h i j hij.symm _ _),
covariance_self ((hX.eval i).map_fun _).aemeasurable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.WithDensity | {
"line": 207,
"column": 2
} | {
"line": 212,
"column": 25
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β → ℝ≥0∞\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nhf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞\nhf : Measurable (Function.uncurry f)\nfs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n\n⊢... | have h_le : ∀ a b n, ⌈(f a b).toReal⌉₊ ≤ n → f a b ≤ n := by
intro a b n hn
have : (f a b).toReal ≤ n := Nat.le_of_ceil_le hn
rw [← ENNReal.le_ofReal_iff_toReal_le (hf_ne_top a b) _] at this
· simpa
· exact n.cast_nonneg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 291,
"column": 41
} | {
"line": 291,
"column": 54
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\nx : γ\nhx : κ.rnDerivAux (κ + η) a x < 1\n⊢ 0 = 0 x",
"usedConstants": [
"Eq.mpr... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 363,
"column": 8
} | {
"line": 364,
"column": 28
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ\nthis :\n η.wi... | rw [ne_eq, sub_eq_zero]
exact (hs' x hx).ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 363,
"column": 8
} | {
"line": 364,
"column": 28
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ\nthis :\n η.wi... | rw [ne_eq, sub_eq_zero]
exact (hs' x hx).ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 493,
"column": 4
} | {
"line": 493,
"column": 67
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η ξ : Kernel α γ\nf : α → γ → ℝ≥0∞\ninst✝ : IsFiniteKernel η\nh : κ = η.withDensity f + ξ\nhf : Measurable (Function.uncurry f)\na : α\nhξ : ξ a ⟂ₘ η a\n⊢ κ a = ξ a + (η a).withDensity (f a)",
"usedConstants": [
"Eq... | rw [h, coe_add, Pi.add_apply, η.withDensity_apply hf, add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 501,
"column": 4
} | {
"line": 501,
"column": 67
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η ξ : Kernel α γ\nf : α → γ → ℝ≥0∞\ninst✝ : IsFiniteKernel η\nh : κ = η.withDensity f + ξ\nhf : Measurable (Function.uncurry f)\na : α\nhξ : ξ a ⟂ₘ η a\n⊢ κ a = ξ a + (η a).withDensity (f a)",
"usedConstants": [
"Eq... | rw [h, coe_add, Pi.add_apply, η.withDensity_apply hf, add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 204,
"column": 18
} | {
"line": 204,
"column": 26
} | [
{
"pp": "case pos\nΩ : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Finty... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 204,
"column": 18
} | {
"line": 204,
"column": 26
} | [
{
"pp": "case neg\nΩ : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Finty... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 209,
"column": 18
} | {
"line": 209,
"column": 26
} | [
{
"pp": "case pos\nΩ : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Finty... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 209,
"column": 18
} | {
"line": 209,
"column": 26
} | [
{
"pp": "case neg\nΩ : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Finty... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Independence.Process.HasIndepIncrements.IsGaussianProcess | {
"line": 96,
"column": 24
} | {
"line": 96,
"column": 44
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : LinearOrder T\nR : Type u_4\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid E\ninst✝² : Module R E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousAdd E\nI : Finset T\nm : R\nx : Fin #I → E\n⊢ (fun i ↦ ∑ j ∈ Iic ((I.... | ext; simp [smul_sum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Process.HasIndepIncrements.IsGaussianProcess | {
"line": 96,
"column": 24
} | {
"line": 96,
"column": 44
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : LinearOrder T\nR : Type u_4\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid E\ninst✝² : Module R E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousAdd E\nI : Finset T\nm : R\nx : Fin #I → E\n⊢ (fun i ↦ ∑ j ∈ Iic ((I.... | ext; simp [smul_sum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 218,
"column": 34
} | {
"line": 218,
"column": 57
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | integral_indicator₀ mA, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 218,
"column": 58
} | {
"line": 218,
"column": 81
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | integral_indicator₀ mA, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | integral_indicator₀ mA, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Deterministic | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 17
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nh : IsDeterministic κ\na : α\ns : Set β\nhs : MeasurableSet s\nthis : (κ a) s * (κ a) s = (κ a) s\nhκ : (κ a) s = 0\n⊢ (κ a) s = 0 ∨ (κ a) s = 1",
"usedConstants": [
... | · simp [hκ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.Conditional | {
"line": 665,
"column": 2
} | {
"line": 669,
"column": 21
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ CondIndepFun m' hm' f g μ ↔\n ... | rw [condIndepFun_iff _ _ _ _ hf hg]
refine ⟨fun h s t hs ht ↦ ?_, fun h s t ↦ ?_⟩
· exact h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
· rintro ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
exact h s t hs ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Conditional | {
"line": 665,
"column": 2
} | {
"line": 669,
"column": 21
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nhf : Measurable f\nhg : Measurable g\n⊢ CondIndepFun m' hm' f g μ ↔\n ... | rw [condIndepFun_iff _ _ _ _ hf hg]
refine ⟨fun h s t hs ht ↦ ?_, fun h s t ↦ ?_⟩
· exact h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
· rintro ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
exact h s t hs ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Deterministic | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝³ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : IsMarkovKernel κ\ninst✝¹ : IsMarkovKernel η\ninst✝ : IsDeterministic (η ∘ₖ κ)\na : α\ns : Set γ\nt : Set β\nhs : MeasurableSet s\nht :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Kernel.Deterministic | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝³ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : IsMarkovKernel κ\ninst✝¹ : IsMarkovKernel η\ninst✝ : IsDeterministic (η ∘ₖ κ)\na : α\ns : Set γ\nt : Set β\nhs : MeasurableSet s\nht :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Deterministic | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝³ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : IsMarkovKernel κ\ninst✝¹ : IsMarkovKernel η\ninst✝ : IsDeterministic (η ∘ₖ κ)\na : α\ns : Set γ\nt : Set β\nhs : MeasurableSet s\nht :... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Deterministic | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝³ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : IsMarkovKernel κ\ninst✝¹ : IsMarkovKernel η\ninst✝ : IsDeterministic (η ∘ₖ κ)\na : α\ns : Set γ\nt : Set β\nhs : MeasurableSet s\nht :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Kernel.Deterministic | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝³ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : IsMarkovKernel κ\ninst✝¹ : IsMarkovKernel η\ninst✝ : IsDeterministic (η ∘ₖ κ)\na : α\ns : Set γ\nt : Set β\nhs : MeasurableSet s\nht :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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