module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Sublocale | {
"line": 207,
"column": 48
} | {
"line": 207,
"column": 79
} | {
"line": 209,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nm n : Nucleus X\n⊢ m.toSublocale ≤ n.toSublocale ↔ n ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Sublocale.instPartialOrder",
"CompleteLattice.toLattice",
"congrArg",
"Nucleus",
"PartialOrder.toPreorder",
... | [] | simp [← SetLike.coe_subset_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Sublocale | {
"line": 207,
"column": 48
} | {
"line": 207,
"column": 79
} | {
"line": 209,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nm n : Nucleus X\n⊢ m.toSublocale ≤ n.toSublocale ↔ n ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Sublocale.instPartialOrder",
"CompleteLattice.toLattice",
"congrArg",
"Nucleus",
"PartialOrder.toPreorder",
... | [] | simp [← SetLike.coe_subset_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.UpperLower.LocallyFinite | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 23
} | {
"line": 33,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderBot α\nhs : s.Finite\n⊢ (↑(lowerClosure s)).Finite",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Preorder.toLE",
"Set.Finite",
"Membership.mem",
"id"... | [
"α : Type u_1\ninst✝¹ : Preorder α\ns : Set α\ninst✝ : LocallyFiniteOrderBot α\nhs : s.Finite\n⊢ (⋃ a ∈ s, Iic a).Finite"
] | rw [coe_lowerClosure] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Process.Predictable | {
"line": 235,
"column": 6
} | {
"line": 238,
"column": 44
} | {
"line": 239,
"column": 4
} | [
{
"pp": "case refine_2.hs\nΩ : Type u_1\nm✝ : MeasurableSpace Ω\nE : Type u_3\ninst✝ : TopologicalSpace E\n𝓕 : Filtration ℕ m✝\nu : ℕ → Ω → E\nh₀ : StronglyMeasurable (u 0)\nh : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))\nthis : MeasurableSpace (ℕ × Ω) := ⋯\nX : ℕ → ℕ × Ω → E := ⋯\nY : ℕ → ℕ × Ω → E := ⋯\nm : ℕ... | [] | refine Set.Finite.biUnion' (by aesop) (fun n hn ↦ ?_)
rcases n with rfl | n
· exact @(h₀.approx m).finite_range
· exact @((h n).approx m).finite_range | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.Predictable | {
"line": 235,
"column": 6
} | {
"line": 238,
"column": 44
} | {
"line": 239,
"column": 4
} | [
{
"pp": "case refine_2.hs\nΩ : Type u_1\nm✝ : MeasurableSpace Ω\nE : Type u_3\ninst✝ : TopologicalSpace E\n𝓕 : Filtration ℕ m✝\nu : ℕ → Ω → E\nh₀ : StronglyMeasurable (u 0)\nh : ∀ (n : ℕ), StronglyMeasurable (u (n + 1))\nthis : MeasurableSpace (ℕ × Ω) := ⋯\nX : ℕ → ℕ × Ω → E := ⋯\nY : ℕ → ℕ × Ω → E := ⋯\nm : ℕ... | [] | refine Set.Finite.biUnion' (by aesop) (fun n hn ↦ ?_)
rcases n with rfl | n
· exact @(h₀.approx m).finite_range
· exact @((h n).approx m).finite_range | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 34
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case h\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\ni : ℕ\na✝¹ : i ∈ ↑(Finset.range N)\nj : ℕ\na✝ : j ∈ ↑(Finset.range N)\nhij : i ≠ j\n⊢ Function.onFun Disjoint (fun k ↦ Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)) i j",
"ppTerm": "?h",
"assigned": t... | [
"case h\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\ni : ℕ\na✝¹ : i ∈ ↑(Finset.range N)\nj : ℕ\na✝ : j ∈ ↑(Finset.range N)\nhij : i ≠ j\n⊢ min (upperCrossingTime a b f N (i + 1) ω) (upperCrossingTime a b f N (j + 1) ω) ≤\n max (lowerCrossingTime a b f N i ω) (lowerCrossingTime a b f N j ω)"
] | simp only [Set.Ico_disjoint_Ico] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Process.HittingTime | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 25
} | {
"line": 321,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\nn m : ι\nE F : Set β\nhEF : E ⊆ F\nω : Ω\n⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ F then sInf (Set.Icc n m ∩ {i | u i ω ∈ F}) else m) ≤\n if ∃ j ∈ Set.Icc n m, u j ω ∈ E then sInf (Set.Icc n m ∩ {i | u i ω ... | [
"case pos\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\nn m : ι\nE F : Set β\nhEF : E ⊆ F\nω : Ω\nhF : ∃ j ∈ Set.Icc n m, u j ω ∈ F\nhE : ∃ j ∈ Set.Icc n m, u j ω ∈ E\n⊢ sInf (Set.Icc n m ∩ {i | u i ω ∈ F}) ≤ sInf (Set.Icc n m ∩ {i | u i ω ∈ E})",
"case neg\... | split_ifs with hF hE hE | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 72
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\na b : ℝ\nN : ℕ\nf : ℕ → Ω → ℝ\nω : Ω\nhab : a < b\nhN : ¬N = 0\nn : ℕ\nhn : upperCrossingTime a b f N n ω < N\nhnN : ¬n ≤ N\n⊢ False",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"not_le",
"Preorder.toLT",
"PartialOrder.toPreorder",
... | [] | exact hn.ne (upperCrossingTime_eq_of_bound_le hab (not_le.1 hnN).le) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.HittingTime | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 25
} | {
"line": 334,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\nn : ι\nE F : Set β\nhEF : E ⊆ F\nω : Ω\n⊢ (if ∃ j, n ≤ j ∧ u j ω ∈ F then ↑(sInf {i | n ≤ i ∧ u i ω ∈ F}) else ⊤) ≤\n if ∃ j, n ≤ j ∧ u j ω ∈ E then ↑(sInf {i | n ≤ i ∧ u i ω ∈ E}) else ⊤",
"ppTe... | [
"case pos\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\nn : ι\nE F : Set β\nhEF : E ⊆ F\nω : Ω\nhF : ∃ j, n ≤ j ∧ u j ω ∈ F\nhE : ∃ j, n ≤ j ∧ u j ω ∈ E\n⊢ ↑(sInf {j | n ≤ j ∧ u j ω ∈ F}) ≤ ↑(sInf {j | n ≤ j ∧ u j ω ∈ E})",
"case neg\nΩ : Type u_1\nβ : Type ... | split_ifs with hF hE hE | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Probability.Martingale.Convergence | {
"line": 177,
"column": 10
} | {
"line": 178,
"column": 29
} | {
"line": 179,
"column": 6
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\na b : ℝ\nf : ℕ → Ω → ℝ\nR : ℝ≥0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R\nhab : a < b\nthis : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (... | [] | rw [posPart_eq_zero.2 hnonneg.le]
exact norm_nonneg _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Convergence | {
"line": 177,
"column": 10
} | {
"line": 178,
"column": 29
} | {
"line": 179,
"column": 6
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\na b : ℝ\nf : ℕ → Ω → ℝ\nR : ℝ≥0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R\nhab : a < b\nthis : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (... | [] | rw [posPart_eq_zero.2 hnonneg.le]
exact norm_nonneg _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.HittingTime | {
"line": 409,
"column": 2
} | {
"line": 412,
"column": 62
} | {
"line": 414,
"column": 0
} | [
{
"pp": "case inr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn n' : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nhi : i < ... | [] | · have h_set_eq_Union : {ω | hittingBtwn u s n n' ω ≤ i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s := by
ext; simp [hittingBtwn_le_iff_of_lt _ hi]
simpa [h_set_eq_Union] using MeasurableSet.iUnion fun j =>
MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j) hs) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Convergence | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 13
} | {
"line": 373,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\ng : Ω → ℝ\nhg : Integrable g μ\nhgmeas : StronglyMeasurable g\nhle : ⨆ n, ↑ℱ n ≤ m0\nhunif : UniformIntegrable (fun n ↦ μ[g | ↑ℱ n]) 1 μ\nR : ℝ≥0\nhR : ∀ (i : ℕ), eLpNorm ((fun n ↦ μ[g | ↑ℱ n]) i) 1 μ ≤... | [] | rwa [heq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Probability.Process.Stopping | {
"line": 872,
"column": 90
} | {
"line": 873,
"column": 40
} | {
"line": 875,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : Nonempty ι\nu : ι → Ω → β\nτ σ : Ω → WithTop ι\ninst✝ : LinearOrder ι\nω : Ω\nhω : σ ω ≠ ⊤\n⊢ stoppedValue (stoppedProcess u τ) σ ω = stoppedValue u (fun ω ↦ min (σ ω) (τ ω)) ω",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
... | [] | by
simp [stoppedValue_stoppedProcess, hω] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 811,
"column": 6
} | {
"line": 811,
"column": 51
} | {
"line": 812,
"column": 6
} | [
{
"pp": "case pos\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : a < b\nthis : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)\n⊢ ENNReal.ofReal (b -... | [
"case pos\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : a < b\nthis : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)\n⊢ ∀ (i : ℕ),\n ENNReal.ofReal ... | simp_rw [this, ENNReal.mul_iSup, iSup_le_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.ConditionalExpectation | {
"line": 57,
"column": 16
} | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 37
} | [
{
"pp": "case pos.refine_2\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : ∀ (t1 t2 : Set Ω),... | [
"case pos.refine_2\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : ∀ (t1 t2 : Set Ω), MeasurableS... | ← hindp _ _ hmt hms, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 941,
"column": 2
} | {
"line": 941,
"column": 58
} | {
"line": 942,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁷ : Nonempty ι\nτ : Ω → WithTop ι\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : SecondCountableTopology ι\ninst✝¹ : BorelSpace ι\nf : Filtration ι m\ninst✝ : PseudoMetrizableSp... | [
"case refine_1\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁷ : Nonempty ι\nτ : Ω → WithTop ι\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : SecondCountableTopology ι\ninst✝¹ : BorelSpace ι\nf : Filtration ι m\ninst✝ : PseudoMetrizabl... | refine measurable_of_restrict_of_restrict_compl hs ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.Basic | {
"line": 232,
"column": 89
} | {
"line": 232,
"column": 96
} | {
"line": 232,
"column": 96
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\n⊢ ∫ (x : Ω), rexp (-(t * X x)) ∂μ = ∫ (x : Ω), rexp (-t * X x) ∂μ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"NonUnitalCommRing.toNonUnitalNonAss... | [] | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.Basic | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 20
} | {
"line": 273,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nY : Ω → ℝ\nhXY : X ≤ᵐ[μ] Y\nht : 0 ≤ t\nhtY : Integrable (fun ω ↦ rexp (t * Y ω)) μ\nhtX : ¬Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ 0 ≤ mgf Y μ t",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | exact mgf_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 9
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case e_f\nμ : Measure ℝ\nt x : ℝ\n⊢ cexp (↑t * I * ↑x) = cexp (↑t * ↑x * I)",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.mul_congr",... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Moments.Basic | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 20
} | {
"line": 283,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nY : Ω → ℝ\nhXY : X ≤ᵐ[μ] Y\nht : t ≤ 0\nhtX : Integrable (fun ω ↦ rexp (t * X ω)) μ\nhtY : ¬Integrable (fun ω ↦ rexp (t * Y ω)) μ\n⊢ 0 ≤ mgf X μ t",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | exact mgf_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.Basic | {
"line": 374,
"column": 6
} | {
"line": 375,
"column": 63
} | {
"line": 376,
"column": 6
} | [
{
"pp": "case insert\nΩ : 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\nthis : IsProbabilityMeasure μ\ni : ι\ns : Finset ι\nhi_notin_s : i ∉ s\nh_rec : mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t\nh_int'... | [
"case insert\nΩ : 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\nthis : IsProbabilityMeasure μ\ni : ι\ns : Finset ι\nhi_notin_s : i ∉ s\nh_rec : mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t\nh_int' : ∀ (i : ι)... | IndepFun.mgf_add (h_indep.indepFun_finsetSum_of_notMem₀ h_meas hi_notin_s).symm (h_int' i)
(aestronglyMeasurable_exp_mul_sum fun i _ => h_int' i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 13
} | {
"line": 234,
"column": 6
} | [
{
"pp": "case e_a\nx 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⊢ max (rexp (t / p * x * p)) (rexp (-t / p * x * p))... | [
"case e_a\nx 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⊢ max (rexp (t * p * p⁻¹ * x)) (rexp (-(t * p * p⁻¹ * x))) = max... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Process.Stopping | {
"line": 1378,
"column": 4
} | {
"line": 1378,
"column": 76
} | {
"line": 1379,
"column": 4
} | [
{
"pp": "case neg\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝¹ : Preorder ι\n𝒢 : Filtration ι m\nτ η : Ω → WithTop ι\ni : ι\ns : Set Ω\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhτ_st : IsStoppingTime 𝒢 τ\nhη_st : IsStoppingTime 𝒢 η\nhτ : ∀ (ω : Ω), ↑i ≤ τ ω\nhη : ∀ (ω : Ω), ↑i ≤ η ω\nhs : Measurabl... | [
"case neg\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝¹ : Preorder ι\n𝒢 : Filtration ι m\nτ η : Ω → WithTop ι\ni : ι\ns : Set Ω\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhτ_st : IsStoppingTime 𝒢 τ\nhη_st : IsStoppingTime 𝒢 η\nhτ : ∀ (ω : Ω), ↑i ≤ τ ω\nhη : ∀ (ω : Ω), ↑i ≤ η ω\nhs : MeasurableSet s\nn : ... | have hηn : ∀ ω, ¬η ω ≤ n := fun ω hηn => hin (mod_cast (hη ω).trans hηn) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 333,
"column": 2
} | {
"line": 337,
"column": 6
} | {
"line": 339,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\nn : ℕ\n⊢ Integrable (fun ω ↦ |X ω| ^ n * rexp (v * X ω)) μ",
"ppTerm": "?m.54",
"assigned": tru... | [] | convert!
integrable_rpow_abs_mul_exp_of_integrable_exp_mul ht ht_int_pos ht_int_neg
(by positivity : 0 ≤ (n : ℝ)) with
ω
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 333,
"column": 2
} | {
"line": 337,
"column": 6
} | {
"line": 339,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\nn : ℕ\n⊢ Integrable (fun ω ↦ |X ω| ^ n * rexp (v * X ω)) μ",
"ppTerm": "?m.54",
"assigned": tru... | [] | convert!
integrable_rpow_abs_mul_exp_of_integrable_exp_mul ht ht_int_pos ht_int_neg
(by positivity : 0 ≤ (n : ℝ)) with
ω
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 219,
"column": 4
} | {
"line": 223,
"column": 77
} | {
"line": 224,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)\nh_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u\nh_d_mgf :... | [] | rw [deriv_fun_div]
· rw [h_d_mgf.symm.differentiableAt_iff, ← iteratedDeriv_one]
exact differentiableAt_iteratedDeriv_mgf h 1
· exact differentiableAt_mgf h
· exact (mgf_pos' hμ (interior_subset (s := integrableExpSet X μ) h)).ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 219,
"column": 4
} | {
"line": 223,
"column": 77
} | {
"line": 224,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)\nh_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u\nh_d_mgf :... | [] | rw [deriv_fun_div]
· rw [h_d_mgf.symm.differentiableAt_iff, ← iteratedDeriv_one]
exact differentiableAt_iteratedDeriv_mgf h 1
· exact differentiableAt_mgf h
· exact (mgf_pos' hμ (interior_subset (s := integrableExpSet X μ) h)).ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 53
} | {
"line": 119,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : Subsingleton E\ninst✝ : IsProbabilityMeasure μ\ns : Set E\n⊢ μ s = (Measure.dirac 0) s",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [
"case h0\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : Subsingleton E\ninst✝ : IsProbabilityMeasure μ\ns : Set E\n⊢ μ ∅ = (Measure.dirac 0) ∅",
"case h1\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : Norme... | apply Subsingleton.set_cases (p := fun s ↦ μ s = _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 454,
"column": 26
} | {
"line": 454,
"column": 45
} | {
"line": 454,
"column": 46
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ ∫ (x : ℝ), (↑√(2 * (π * ↑v)))⁻¹ * (cexp (-(↑x - ↑μ) ^ 2 / (2 * ↑↑v)) * cexp (z * ↑x)) =\n (↑√(2 * (π * ↑v)))⁻¹ * ∫ (x : ℝ), cexp (-(2 * ↑↑v)⁻¹ * ↑x ^ 2 + (z + ↑μ / ↑↑v) * ↑x + -↑μ ^ 2 / (2 * ↑↑v))",
"ppTerm": "?m.440",
"assigned": true,
"usedConstant... | [
"μ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ (↑√(2 * (π * ↑v)))⁻¹ * ∫ (a : ℝ), cexp (-(↑a - ↑μ) ^ 2 / (2 * ↑↑v)) * cexp (z * ↑a) =\n (↑√(2 * (π * ↑v)))⁻¹ * ∫ (x : ℝ), cexp (-(2 * ↑↑v)⁻¹ * ↑x ^ 2 + (z + ↑μ / ↑↑v) * ↑x + -↑μ ^ 2 / (2 * ↑↑v))"
] | integral_const_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 468,
"column": 6
} | {
"line": 468,
"column": 13
} | {
"line": 469,
"column": 4
} | [
{
"pp": "case e_a\nμ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ (↑π * (2 * ↑↑v)) ^ 2⁻¹ = (2 * ↑π * ↑↑v) ^ 2⁻¹",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Eq.mpr",
"NonAssocSemiring.toAddC... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 9
} | {
"line": 492,
"column": 0
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nt : ℝ\n⊢ ↑t * I * ↑μ + -(↑↑v * ↑t ^ 2) / 2 = ↑t * ↑μ * I + -(↑↑v * ↑t ^ 2 / 2)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 40
} | {
"line": 558,
"column": 2
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ ∫ (ω : ℝ), ω ^ 2 ∂gaussianReal 0 v = iteratedDeriv 2 (mgf (fun x ↦ x) (gaussianReal 0 v)) 0",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.denselyNormedField",
"Real.instZe... | [] | rw [iteratedDeriv_mgf_zero] <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 40
} | {
"line": 558,
"column": 2
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ ∫ (ω : ℝ), ω ^ 2 ∂gaussianReal 0 v = iteratedDeriv 2 (mgf (fun x ↦ x) (gaussianReal 0 v)) 0",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.denselyNormedField",
"Real.instZe... | [] | rw [iteratedDeriv_mgf_zero] <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 40
} | {
"line": 558,
"column": 2
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ ∫ (ω : ℝ), ω ^ 2 ∂gaussianReal 0 v = iteratedDeriv 2 (mgf (fun x ↦ x) (gaussianReal 0 v)) 0",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.denselyNormedField",
"Real.instZe... | [] | rw [iteratedDeriv_mgf_zero] <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 27
} | {
"line": 177,
"column": 28
} | [
{
"pp": "case e_a.e_a\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\nu : ℝ\... | [
"case e_a.e_a\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\nu : ℝ\n⊢ ↑u * ∫ (a... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 9
} | {
"line": 655,
"column": 0
} | [
{
"pp": "m₁ m₂ : ℝ\nv₁ v₂ : ℝ≥0\nt : ℝ\n⊢ cexp (↑t * ↑m₁ * Complex.I - ↑↑v₁ * ↑t ^ 2 / 2 + (↑t * ↑m₂ * Complex.I - ↑↑v₂ * ↑t ^ 2 / 2)) =\n cexp (↑t * (↑m₁ + ↑m₂) * Complex.I - (↑↑v₁ + ↑↑v₂) * ↑t ^ 2 / 2)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.m... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 83
} | {
"line": 267,
"column": 4
} | [
{
"pp": "case e_a.hf\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ\n⊢ Integrable (fun ω ↦ X ω ^ 2 * rexp (v * X ω) - 2 * X ω * deriv (cgf X μ) v * rexp (v * ... | [] | exact (integrable_pow_mul_exp_of_mem_interior_integrableExpSet h 2).sub h_int | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 83
} | {
"line": 267,
"column": 4
} | [
{
"pp": "case e_a.hf\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ\n⊢ Integrable (fun ω ↦ X ω ^ 2 * rexp (v * X ω) - 2 * X ω * deriv (cgf X μ) v * rexp (v * ... | [] | exact (integrable_pow_mul_exp_of_mem_interior_integrableExpSet h 2).sub h_int | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 83
} | {
"line": 267,
"column": 4
} | [
{
"pp": "case e_a.hf\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ\n⊢ Integrable (fun ω ↦ X ω ^ 2 * rexp (v * X ω) - 2 * X ω * deriv (cgf X μ) v * rexp (v * ... | [] | exact (integrable_pow_mul_exp_of_mem_interior_integrableExpSet h 2).sub h_int | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 273,
"column": 10
} | {
"line": 273,
"column": 29
} | {
"line": 273,
"column": 30
} | [
{
"pp": "case e_a.e_a\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ\n⊢ deriv (cgf X μ) v ^ 2 * mgf X μ v = ∫ (a : Ω), deriv (cgf X μ) v ^ 2 * rexp (v * X a) ... | [
"case e_a.e_a\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ\n⊢ deriv (cgf X μ) v ^ 2 * mgf X μ v = deriv (cgf X μ) v ^ 2 * ∫ (a : Ω), rexp (v * X a) ∂μ"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 262,
"column": 2
} | {
"line": 265,
"column": 13
} | {
"line": 267,
"column": 0
} | [
{
"pp": "case e_a\nE : Type u_1\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : MeasurableSpace E\ninst✝⁷ : BorelSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : MeasurableSpace F\ninst✝³ : BorelSpace F\nμ : Measure E\ninst✝² : IsGaussian μ\ninst✝¹ : S... | [] | · field_simp
rw [variance_dual_prod' (IsGaussian.memLp_dual μ (L.comp (.inl ℝ E F)) 2 (by simp))
(IsGaussian.memLp_dual ν (L.comp (.inr ℝ E F)) 2 (by simp))]
norm_cast | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Gaussian.Multivariate | {
"line": 173,
"column": 67
} | {
"line": 177,
"column": 7
} | {
"line": 179,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nμ : EuclideanSpace ℝ ι\nS : Matrix ι ι ℝ\nhS : ¬S.PosSemidef\n⊢ multivariateGaussian μ S = Measure.dirac μ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CFC.sqrt.eq_1",
"cfcₙ",
"Eq.mpr",
"Pi.Functi... | [] | by
rw [multivariateGaussian, CFC.sqrt, cfcₙ_apply_of_not_predicate]
· simp
change ¬ (S - 0).PosSemidef
simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Gaussian.Multivariate | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 64
} | {
"line": 200,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nμ : EuclideanSpace ℝ ι\nS : Matrix ι ι ℝ\n⊢ Integrable (⇑(toEuclideanCLM (CFC.sqrt S))) (stdGaussian (EuclideanSpace ℝ ι))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"InnerProductSpace.... | [] | · exact IsGaussian.integrable_id.comp_measurable (by fun_prop) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 37
} | {
"line": 83,
"column": 2
} | [
{
"pp": "case convert_1\nT : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝² : Preorder T\ninst✝¹ : MeasurableSpace E\ninst✝ : Sub E\nh : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P\nn : ℕ\nt : Fin (... | [
"case convert_2\nT : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝² : Preorder T\ninst✝¹ : MeasurableSpace E\ninst✝ : Sub E\nh : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P\nn : ℕ\nt : Fin (n + 1) → T\n... | · exact fun a b hab ↦ ht (by grind) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.Process.Basic | {
"line": 211,
"column": 2
} | {
"line": 215,
"column": 16
} | {
"line": 216,
"column": 2
} | [
{
"pp": "S : Type u_1\nΩ : Type u_3\nmΩ : MeasurableSpace Ω\nα : Type u_4\nmα : MeasurableSpace α\nκ : Kernel α Ω\nP : Measure α\nT : S → Type u_5\n𝓧 : (i : S) → T i → Type u_6\ninst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)\nX X' : (i : S) → (j : T i) → Ω → 𝓧 i j\nh1 : iIndepFun (fun i ω j ↦ X i j ω)... | [
"S : Type u_1\nΩ : Type u_3\nmΩ : MeasurableSpace Ω\nα : Type u_4\nmα : MeasurableSpace α\nκ : Kernel α Ω\nP : Measure α\nT : S → Type u_5\n𝓧 : (i : S) → T i → Type u_6\ninst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)\nX X' : (i : S) → (j : T i) → Ω → 𝓧 i j\nh1 : iIndepFun (fun i ω j ↦ X i j ω) κ P\nh2 : ∀... | have h4' a (f : (i : S) → (j : T i) → Ω → 𝓧 i j) :
∏ i ∈ s, κ a ((fun i ω j ↦ f i j ω) i ⁻¹' g i) =
∏ i ∈ s, κ a ((fun i ω j ↦ f i j ω) i ⁻¹' (I i).restrict ⁻¹' u i) := by
refine Finset.prod_congr rfl fun i hi ↦ ?_
rw [hu i hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Distributions.Fernique | {
"line": 498,
"column": 6
} | {
"line": 498,
"column": 42
} | {
"line": 499,
"column": 4
} | [
{
"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\... | [] | exact h_neg.trans_le (norm_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 27
} | {
"line": 227,
"column": 28
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nc : ℝ≥0\nhc : c ≠ 0\nt : ℝ≥0\n⊢ ∫ (x : Ω), (√↑c)⁻¹ * B (c * t) x ∂P = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toN... | [
"case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nc : ℝ≥0\nhc : c ≠ 0\nt : ℝ≥0\n⊢ (√↑c)⁻¹ * ∫ (a : Ω), B (c * t) a ∂P = 0"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 243,
"column": 4
} | {
"line": 245,
"column": 59
} | {
"line": 246,
"column": 4
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nt₀ s t : ℝ≥0\nhst : s ≤ t\nthis : IsProbabilityMeasure P\n⊢ cov[fun ω ↦ B (t₀ + s) ω - B t₀ ω, fun ω ↦ B (t₀ + t) ω - B t₀ ω; P] = ↑s",
"ppTerm": "?refine_2",
"assigned": true,
"... | [
"case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nt₀ s t : ℝ≥0\nhst : s ≤ t\nthis : IsProbabilityMeasure P\n⊢ ↑(t₀ + s) - ↑t₀ - (↑t₀ - ↑t₀) = ↑s",
"case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownia... | rw [covariance_fun_sub_left, covariance_fun_sub_right, covariance_fun_sub_right,
hB.covariance_eval, hB.covariance_eval, hB.covariance_eval, hB.covariance_eval, ← add_min,
min_eq_left hst, min_eq_right, min_eq_left, min_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 282,
"column": 8
} | {
"line": 282,
"column": 27
} | {
"line": 282,
"column": 28
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nt : ℝ≥0\n⊢ ∫ (x : Ω), ↑t * B (1 / t) x ∂P = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Normed... | [
"case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nt : ℝ≥0\n⊢ ↑t * ∫ (a : Ω), B (1 / t) a ∂P = 0"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 65
} | {
"line": 200,
"column": 65
} | [
{
"pp": "Ω : 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 : ι), SecondCountab... | [
"Ω : 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 : ι), SecondCountableTopology (... | iIndepFun_iff_charFunDual_pi fun i ↦ hX.aemeasurable.eval i | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Unique | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 91
} | {
"line": 74,
"column": 2
} | [
{
"pp": "case h_basic\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝¹ : IsFiniteMeasure ρ\nκ : Kernel α ℝ\ninst✝ : IsFiniteKernel κ\nhκ : ρ = ρ.fst ⊗ₘ κ\nhuniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ\n⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ t ∈ ⋃ a, {Iic ↑a}, (κ x) t = (ρ.condKernel x) t",
... | [
"case h_compl\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝¹ : IsFiniteMeasure ρ\nκ : Kernel α ℝ\ninst✝ : IsFiniteKernel κ\nhκ : ρ = ρ.fst ⊗ₘ κ\nhuniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ\n⊢ ∀ᵐ (x : α) ∂ρ.fst, ∀ (t : Set ℝ), MeasurableSet t → (κ x) t = (ρ.condKernel x) t → (κ ... | · simp only [iUnion_singleton_eq_range, mem_range, forall_exists_index, forall_apply_eq_imp_iff]
exact ae_all_iff.2 fun q ↦ eq_condKernel_of_measure_eq_compProd' κ hκ measurableSet_Iic | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 641,
"column": 68
} | {
"line": 655,
"column": 30
} | {
"line": 657,
"column": 0
} | [
{
"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\nx : γ\n⊢ κ.densityProcess κ.fst n a x univ = if (κ.fst a) (countablePartitionSet n x) = 0 then 0... | [] | by
rw [densityProcess]
split_ifs with h
· simp only [h]
by_cases h' : κ a (countablePartitionSet n x ×ˢ univ) = 0
· simp [h']
· simp
· rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
have : countablePartitionSet n x ×ˢ univ = {p : γ × β | p.1 ∈ countablePartitionSet n x} := by
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 132,
"column": 35
} | {
"line": 132,
"column": 56
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case zero\nX : ℕ → Type u_1\na : ℕ\nx : (i : ↥(Iic a)) → X ↑i\nind : (n : ℕ) → ((i : ↥(Iic n)) → X ↑i) → X (n + 1)\nhj : 0 ∈ Iic a\n⊢ iterateInduction x ind ↑⟨0, hj⟩ = x ⟨0, hj⟩",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
... | [] | rw [iterateInduction] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 132,
"column": 35
} | {
"line": 132,
"column": 56
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case succ\nX : ℕ → Type u_1\na : ℕ\nx : (i : ↥(Iic a)) → X ↑i\nind : (n : ℕ) → ((i : ↥(Iic n)) → X ↑i) → X (n + 1)\nj : ℕ\nhj : j + 1 ∈ Iic a\n⊢ iterateInduction x ind ↑⟨j + 1, hj⟩ = x ⟨j + 1, hj⟩",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iterateInduct... | [
"case succ\nX : ℕ → Type u_1\na : ℕ\nx : (i : ↥(Iic a)) → X ↑i\nind : (n : ℕ) → ((i : ↥(Iic n)) → X ↑i) → X (n + 1)\nj : ℕ\nhj : j + 1 ∈ Iic a\n⊢ (if h : j + 1 ≤ a then x ⟨j + 1, ⋯⟩ else ind j fun i ↦ iterateInduction x ind ↑i) = x ⟨j + 1, hj⟩"
] | rw [iterateInduction] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 297,
"column": 6
} | {
"line": 299,
"column": 19
} | {
"line": 301,
"column": 2
} | [
{
"pp": "case inr.inr\nX : 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... | [] | rw [lmarginalPartialTraj_le _ _ (mf n), (hcte n).lmarginalPartialTraj_of_le _ (mf n),
(hcte n).lmarginalPartialTraj_of_le _ (mf n)]
all_goals lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 297,
"column": 6
} | {
"line": 299,
"column": 19
} | {
"line": 301,
"column": 2
} | [
{
"pp": "case inr.inr\nX : 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... | [] | rw [lmarginalPartialTraj_le _ _ (mf n), (hcte n).lmarginalPartialTraj_of_le _ (mf n),
(hcte n).lmarginalPartialTraj_of_le _ (mf n)]
all_goals lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 28
} | {
"line": 303,
"column": 2
} | [
{
"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)\nA : ℕ → Set ((i : ι) → X i)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\nthis : ∀ (i : ι), Nonempty (X i)\ns : ... | [
"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)\nA : ℕ → Set ((i : ι) → X i)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\nthis : ∀ (i : ι), Nonempty (X i)\ns : ℕ → Finset ι... | let T n := (g n) ⁻¹' (S n) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 391,
"column": 4
} | {
"line": 392,
"column": 57
} | {
"line": 393,
"column": 4
} | [
{
"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)\nA : ℕ → Set ((n : ℕ) → X n)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\np : ℕ\nx₀ : (i : ↥(Iic... | [
"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)\nA : ℕ → Set ((n : ℕ) → X n)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\np : ℕ\nx₀ : (i : ↥(Iic p)) → X ↑i\... | rw [← lma_inv k ((a n).max (a m)) n (le_max_left _ _),
← lma_inv k ((a n).max (a m)) m (le_max_right _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 254,
"column": 60
} | {
"line": 254,
"column": 69
} | {
"line": 255,
"column": 4
} | [
{
"pp": "X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nhp : (fun x ↦ x ⟨a + 1, ⋯⟩) ∘ _root_.IicProdIoc a (a + 1) = ⇑(piSingleton a).symm ∘ Prod.snd\n⊢ (Kernel.id ×ₖ (κ a).map ⇑(piSingleton a)).snd... | [
"X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nhp : (fun x ↦ x ⟨a + 1, ⋯⟩) ∘ _root_.IicProdIoc a (a + 1) = ⇑(piSingleton a).symm ∘ Prod.snd\n⊢ ((κ a).map ⇑(piSingleton a)).map ⇑(piSingleton a).symm = ... | snd_prod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 368,
"column": 96
} | {
"line": 377,
"column": 10
} | {
"line": 379,
"column": 0
} | [
{
"pp": "X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\na b c : ℕ\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\nhab : a ≤ b\nhbc : b ≤ c\nf : ((n : ℕ) → X n) → ℝ≥0∞\nhf : Measurable f\n⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b c f) = lmarginalPartialTraj κ a c f",
"ppTerm"... | [] | by
ext x₀
obtain rfl | hab := eq_or_lt_of_le hab <;> obtain rfl | hbc := eq_or_lt_of_le hbc
· rw [lmarginalPartialTraj_le κ le_rfl (measurable_lmarginalPartialTraj _ _ hf)]
· rw [lmarginalPartialTraj_le κ le_rfl (measurable_lmarginalPartialTraj _ _ hf)]
· rw [lmarginalPartialTraj_le κ le_rfl hf]
simp_rw [lm... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 474,
"column": 38
} | {
"line": 478,
"column": 31
} | {
"line": 480,
"column": 0
} | [
{
"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 : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ MeasurableSet {a | κ a ⟂ₘ η a}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants"... | [] | by
simp_rw [← withDensity_rnDeriv_eq_zero_iff_mutuallySingular,
withDensity_rnDeriv_eq_zero_iff_measure_eq_zero]
exact measurable_kernel_prodMk_left (measurableSet_mutuallySingularSet κ η).compl
(measurableSet_singleton 0) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 427,
"column": 9
} | {
"line": 427,
"column": 21
} | {
"line": 427,
"column": 22
} | [
{
"pp": "case inr.refine_2\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nb : ℕ\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na : ℕ\nf : ((n : ℕ) → X n) → ℝ≥0∞\nhf : DependsOn f ↑(Iic b)\nmf : Measurable f\nx y : (i : ℕ) → X i\nhxy : ∀ i ∈ ↑(Iic a... | [
"case inr.refine_2\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nb : ℕ\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na : ℕ\nf : ((n : ℕ) → X n) → ℝ≥0∞\nhf : DependsOn f ↑(Iic b)\nmf : Measurable f\nx y : (i : ℕ) → X i\nhxy : ∀ i ∈ ↑(Iic a), x i = y i... | ← coe_sdiff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 427,
"column": 22
} | {
"line": 427,
"column": 50
} | {
"line": 427,
"column": 50
} | [
{
"pp": "case inr.refine_2\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nb : ℕ\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na : ℕ\nf : ((n : ℕ) → X n) → ℝ≥0∞\nhf : DependsOn f ↑(Iic b)\nmf : Measurable f\nx y : (i : ℕ) → X i\nhxy : ∀ i ∈ ↑(Iic a... | [
"case inr.refine_2\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nb : ℕ\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)\na : ℕ\nf : ((n : ℕ) → X n) → ℝ≥0∞\nhf : DependsOn f ↑(Iic b)\nmf : Measurable f\nx y : (i : ℕ) → X i\nhxy : ∀ i ∈ ↑(Iic a), x i = y i... | Iic_sdiff_Ioc_self_of_le hab | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 104,
"column": 48
} | {
"line": 106,
"column": 43
} | {
"line": 108,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteKernel κ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : Nonempty Ω\n⊢ ⇑(Kernel.swap Ω 𝓧) ∘ₘ ⇑(Kernel.id ∥ₖ κ) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ = ⇑(κ ∥ₖ Kernel.id... | [] | by
rw [Measure.comp_assoc, Kernel.swap_parallelComp, Measure.comp_assoc, Kernel.comp_assoc,
Kernel.swap_copy, Measure.comp_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Posterior | {
"line": 192,
"column": 7
} | {
"line": 203,
"column": 68
} | {
"line": 205,
"column": 0
} | [] | [] | (Kernel.id ∥ₖ κ†μ) ∘ₘ (Kernel.id ∥ₖ η†(κ ∘ₘ μ)) ∘ₘ (Kernel.copy 𝓨) ∘ₘ η ∘ₘ κ ∘ₘ μ
_ = (Kernel.id ∥ₖ κ†μ) ∘ₘ (η ∥ₖ Kernel.id) ∘ₘ Kernel.copy 𝓧 ∘ₘ κ ∘ₘ μ := by
rw [parallelProd_posterior_comp_copy_comp]
_ = (η ∥ₖ Kernel.id) ∘ₘ (Kernel.id ∥ₖ κ†μ) ∘ₘ Kernel.copy 𝓧 ∘ₘ κ ∘ₘ μ := by
rw [Measure.comp_assoc, Kern... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Probability.Kernel.Deterministic | {
"line": 146,
"column": 2
} | {
"line": 178,
"column": 69
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case inr\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 :... | [] | · /- In Example 11.25 of [gritz2020], the case where `((η ∘ₖ κ) a) s = 1` is not explicitly
treated. We prove it here by using the fact that the hypothesis implies that
`((η ∘ₖ κ) a) sᶜ = 0`, and thus that the integral of `1 - (η b) s` over `κ a` is zero. -/
rw [h₁, one_mul]
have integral_le_kernel : ∫⁻... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 25
} | {
"line": 119,
"column": 25
} | [
{
"pp": "case h\nΘ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 →... | [
"case h\nΘ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻ (θ ... | lintegral_mul_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 32
} | {
"line": 281,
"column": 2
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω... | [] | exact rnDeriv_posterior h_ac | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Posterior | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 32
} | {
"line": 281,
"column": 2
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω... | [] | exact rnDeriv_posterior h_ac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Posterior | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 32
} | {
"line": 281,
"column": 2
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω... | [] | exact rnDeriv_posterior h_ac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gamma | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 27
} | {
"line": 113,
"column": 28
} | [
{
"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 Ioi 0, r ^ a / Gamma a * (a_1 ^ (a - 1) * rexp (-(r * a_... | [
"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⊢ r ^ a / Gamma a * ∫ (a_1 : ℝ) in Ioi 0, a_1 ^ (a - 1) * rexp (-(r * a_1)) = 1"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Exponential | {
"line": 144,
"column": 10
} | {
"line": 144,
"column": 39
} | {
"line": 144,
"column": 40
} | [
{
"pp": "r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\nthis : ∫ (a : ℝ) in uIoc 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0..x, r * rexp (-(r * a))\n⊢ ∫ (a : ℝ) in Icc 0 x, r * rexp (-(r * a)) ∂volume = (ENNReal.ofReal (1 - rexp (-(r * x)))).toReal",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
... | [
"r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\nthis : ∫ (a : ℝ) in uIoc 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0..x, r * rexp (-(r * a))\n⊢ ∫ (t : ℝ) in Ioc 0 x, r * rexp (-(r * t)) ∂volume = (ENNReal.ofReal (1 - rexp (-(r * x)))).toReal"
] | integral_Icc_eq_integral_Ioc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Pareto | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 71
} | {
"line": 50,
"column": 0
} | [
{
"pp": "t r x : ℝ\nhx : x < t\n⊢ paretoPDF t r x = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instZero",
"ENNReal.ofReal",
"con... | [] | simp only [paretoPDF_eq, if_neg (not_le.mpr hx), ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Distributions.Pareto | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 71
} | {
"line": 50,
"column": 0
} | [
{
"pp": "t r x : ℝ\nhx : x < t\n⊢ paretoPDF t r x = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instZero",
"ENNReal.ofReal",
"con... | [] | simp only [paretoPDF_eq, if_neg (not_le.mpr hx), ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Pareto | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 71
} | {
"line": 50,
"column": 0
} | [
{
"pp": "t r x : ℝ\nhx : x < t\n⊢ paretoPDF t r x = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instZero",
"ENNReal.ofReal",
"con... | [] | simp only [paretoPDF_eq, if_neg (not_le.mpr hx), ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Pareto | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 57
} | {
"line": 106,
"column": 58
} | [
{
"pp": "t 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⊢ ∫ (t_1 : ℝ) in Ioi t, r * t ^ r * t_1 ^ (-(r + 1)) = 1",
"ppTerm": "?m.141",
"assigned": t... | [
"t 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⊢ r * t ^ r * ∫ (a : ℝ) in Ioi t, a ^ (-(r + 1)) = 1"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 13
} | {
"line": 155,
"column": 2
} | [
{
"pp": "case e_f\nr : ℝ≥0\nt : ℝ\na : ℕ\n⊢ cexp (-↑↑r) * ↑↑r ^ a / ↑a ! * cexp (↑a * ↑t * I) = cexp (-↑↑r) * (↑↑r ^ a * cexp (↑a * (↑t * I)) / ↑a !)",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 13
} | {
"line": 163,
"column": 0
} | [
{
"pp": "r : ℝ≥0\nt : ℝ\n⊢ cexp (-↑↑r + ↑↑r * cexp (↑t * I)) = cexp (↑↑r * (cexp (↑t * I) - 1))",
"ppTerm": "?m.327",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"N... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 267,
"column": 77
} | {
"line": 272,
"column": 34
} | {
"line": 274,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nq : PMF β\nf : (a : α) → a ∈ p.support → (b : β) → b ∈ q.support → PMF γ\n⊢ (p.bindOnSupport fun a ha ↦ q.bindOnSupport (f a ha)) =\n q.bindOnSupport fun b hb ↦ p.bindOnSupport fun a ha ↦ f a ha b hb",
"ppTerm": "?m.33",
"assigned": true,
... | [] | by
apply PMF.ext; rintro c
simp only [bindOnSupport_apply, ← tsum_dite_right,
ENNReal.tsum_mul_left.symm]
refine _root_.trans ENNReal.tsum_comm (tsum_congr fun b => tsum_congr fun a => ?_)
split_ifs with h1 h2 h2 <;> ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.HasCondDistrib | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 19
} | {
"line": 80,
"column": 19
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_3\n𝓩 : Type u_4\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nm𝓩 : MeasurableSpace 𝓩\nP : Measure Ω\nX : Ω → 𝓧\ninst✝¹ : SFinite P\nY : Ω → 𝓨 × 𝓩\nκ : Kernel 𝓧 (𝓨 × 𝓩)\ninst✝ : IsSFiniteKernel κ\nh : HasCondDistrib Y X κ P... | [
"Ω : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_3\n𝓩 : Type u_4\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nm𝓩 : MeasurableSpace 𝓩\nP : Measure Ω\nX : Ω → 𝓧\ninst✝¹ : SFinite P\nY : Ω → 𝓨 × 𝓩\nκ : Kernel 𝓧 (𝓨 × 𝓩)\ninst✝ : IsSFiniteKernel κ\nh : HasCondDistrib Y X κ P\n⊢ HasCondD... | Kernel.fst_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.InfinitePi | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 30
} | {
"line": 142,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : ι → Type u_4\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\ni j : ι\nhij : i ≠ j\n⊢ (map (fun ω ↦ ω i) (infinitePi P)).prod (map (fun ω ↦ ω j) (infinitePi P)) = (P i).prod (P j)",
"ppTerm": "?m.48",
"assigned"... | [] | simp [infinitePi_map_eval] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Independence.InfinitePi | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 30
} | {
"line": 142,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : ι → Type u_4\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\ni j : ι\nhij : i ≠ j\n⊢ (map (fun ω ↦ ω i) (infinitePi P)).prod (map (fun ω ↦ ω j) (infinitePi P)) = (P i).prod (P j)",
"ppTerm": "?m.48",
"assigned"... | [] | simp [infinitePi_map_eval] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.InfinitePi | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 30
} | {
"line": 142,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : ι → Type u_4\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\ni j : ι\nhij : i ≠ j\n⊢ (map (fun ω ↦ ω i) (infinitePi P)).prod (map (fun ω ↦ ω j) (infinitePi P)) = (P i).prod (P j)",
"ppTerm": "?m.48",
"assigned"... | [] | simp [infinitePi_map_eval] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.InfinitePi | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 150,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_4\nΩ : ι → Type u_5\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\nf : α → ι\nhf : Function.Injective f\n⊢ (infinitePi fun i ↦ map (fun ω ↦ ω (f i)) (infinitePi P)) = infinitePi fun i ↦ P (f i)",
"ppTerm": ... | [] | simp [infinitePi_map_eval] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Independence.InfinitePi | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 150,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_4\nΩ : ι → Type u_5\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\nf : α → ι\nhf : Function.Injective f\n⊢ (infinitePi fun i ↦ map (fun ω ↦ ω (f i)) (infinitePi P)) = infinitePi fun i ↦ P (f i)",
"ppTerm": ... | [] | simp [infinitePi_map_eval] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.InfinitePi | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 150,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_4\nΩ : ι → Type u_5\nmΩ : (i : ι) → MeasurableSpace (Ω i)\nP : (i : ι) → Measure (Ω i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (P i)\nf : α → ι\nhf : Function.Injective f\n⊢ (infinitePi fun i ↦ map (fun ω ↦ ω (f i)) (infinitePi P)) = infinitePi fun i ↦ P (f i)",
"ppTerm": ... | [] | simp [infinitePi_map_eval] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Uniform | {
"line": 133,
"column": 2
} | {
"line": 134,
"column": 29
} | {
"line": 135,
"column": 2
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝ : MeasurableSpace E\nμ : Measure E\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → E\ns : Set E\nhms : MeasurableSet s\nhu : IsUniform X s ℙ μ\nhnt : ¬μ s = ∞\nhns : ¬μ s = 0\nthis✝ : HasPDF X ℙ μ\nthis : IsProbabilityMeasure ℙ\n⊢ Measure.map X ℙ = μ.withDens... | [
"E : Type u_1\ninst✝ : MeasurableSpace E\nμ : Measure E\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → E\ns : Set E\nhms : MeasurableSet s\nhu : IsUniform X s ℙ μ\nhnt : ¬μ s = ∞\nhns : ¬μ s = 0\nthis✝ : HasPDF X ℙ μ\nthis : IsProbabilityMeasure ℙ\n⊢ AEMeasurable (s.indicator ((μ s)⁻¹ • 1)) μ"
] | · rw [hu, withDensity_indicator hms, withDensity_smul _ measurable_one, withDensity_one,
ProbabilityTheory.cond] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Uniform | {
"line": 171,
"column": 2
} | {
"line": 177,
"column": 47
} | {
"line": 179,
"column": 0
} | [
{
"pp": "Ω : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhuX : IsUniform X s ℙ volume\n⊢ ∫ (x : Ω), X x ∂ℙ = (volume s)⁻¹.toReal * ∫ (x : ℝ) in s, x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Re... | [] | rw [← smul_eq_mul, ← integral_smul_measure]
dsimp only [IsUniform, ProbabilityTheory.cond] at huX
rw [← huX]
by_cases hX : AEMeasurable X ℙ
· exact (integral_map hX aestronglyMeasurable_id).symm
· rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
rwa [aestronglyMeas... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Uniform | {
"line": 171,
"column": 2
} | {
"line": 177,
"column": 47
} | {
"line": 179,
"column": 0
} | [
{
"pp": "Ω : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ns : Set ℝ\nhuX : IsUniform X s ℙ volume\n⊢ ∫ (x : Ω), X x ∂ℙ = (volume s)⁻¹.toReal * ∫ (x : ℝ) in s, x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Re... | [] | rw [← smul_eq_mul, ← integral_smul_measure]
dsimp only [IsUniform, ProbabilityTheory.cond] at huX
rw [← huX]
by_cases hX : AEMeasurable X ℙ
· exact (integral_map hX aestronglyMeasurable_id).symm
· rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
rwa [aestronglyMeas... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 15
} | {
"line": 189,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n ∀ (s : Finset ι) (f : ι → Set Ω),\n (∀ i ∈ s, f i ∈ π i) →\n ∀ i ∈ s, (fun ω... | [
"Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n ∀ (s : Finset ι) (f : ι → Set Ω),\n (∀ i ∈ s, f i ∈ π i) →\n ∀ i ∈ s, (fun ω ↦ (((condEx... | intro s f H | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 88
} | {
"line": 142,
"column": 4
} | [
{
"pp": "X✝ Y✝ : SFinKer\nκ : X✝ ⟶ Y✝\na✝ : X✝.carrier × PUnit.{u + 1}\ns✝ : Set Y✝.carrier\nhs : MeasurableSet s✝\nthis : IsSFiniteKernel κ.hom\n⊢ ∫⁻ (b : Y✝.carrier), (Kernel.id a✝.2) (Prod.mk b ⁻¹' Prod.fst ⁻¹' s✝) ∂κ.hom a✝.1 = (κ.hom a✝.1) s✝",
"ppTerm": "?m.1052",
"assigned": true,
"usedConsta... | [
"X✝ Y✝ : SFinKer\nκ : X✝ ⟶ Y✝\na✝ : X✝.carrier × PUnit.{u + 1}\ns✝ : Set Y✝.carrier\nhs : MeasurableSet s✝\nthis : IsSFiniteKernel κ.hom\n⊢ ∫⁻ (b : Y✝.carrier), (Prod.mk b ⁻¹' Prod.fst ⁻¹' s✝).indicator 1 a✝.2 ∂κ.hom a✝.1 = (κ.hom a✝.1) s✝"
] | simp only [Kernel.id_apply, MeasurableSpace.measurableSet_top, Measure.dirac_apply'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 82
} | {
"line": 151,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁹ : (t : T) → TopologicalSpace (F t)\ninst✝⁸ : (t : T) → MeasurableSpace (F t)\ninst✝⁷ : ∀ (t : T), BorelSpace (F t)\ninst✝⁶ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁵ : TopologicalSpace G\ninst... | [] | convert! h f (∏ t, (g t).compContinuous ⟨Function.eval t, by fun_prop⟩) <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Representation | {
"line": 86,
"column": 4
} | {
"line": 89,
"column": 35
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case neg\nX : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\nIic_to_Icc : Iic x = Icc 0 x\nκ_in_I : (κ a).real (Icc 0 x) ∈ I\nξ : ↑I\nhξ : f a ξ ≤ x\nhx... | [
"case neg\nX : Type u_1\nmX : MeasurableSpace X\nκ : Kernel X ↑I\ninst✝ : IsMarkovKernel κ\nf : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}\nmeasurable_f : Measurable (uncurry f)\na : X\nx : ↑I\nIic_to_Icc : Iic x = Icc 0 x\nκ_in_I : (κ a).real (Icc 0 x) ∈ I\nξ : ↑I\nhξ : f a ξ ≤ x\nhx : ¬x = 1\ng... | let nebot : NeBot (𝓝[>] x) := by
refine nhdsGT_neBot_of_exists_gt ?_
use 1
exact lt_of_le_of_ne x.2.2 hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 98,
"column": 12
} | {
"line": 98,
"column": 42
} | {
"line": 98,
"column": 42
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [
"Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝² : SigmaFini... | ← @Measure.restrict_univ Ω _ μ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Conditional | {
"line": 841,
"column": 4
} | {
"line": 843,
"column": 21
} | {
"line": 844,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [
"Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\ninst✝¹ : St... | rw [Measure.bind_apply ((hk_meas hs).prod (ht.prod hu)) (by fun_prop),
Measure.bind_apply (hs.prod (ht.prod hu)) (by fun_prop), lintegral_map ?_ (by fun_prop),
lintegral_trim] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 23
} | {
"line": 219,
"column": 24
} | [
{
"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... | [
"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✝ : IsProbab... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 219,
"column": 24
} | {
"line": 219,
"column": 43
} | {
"line": 219,
"column": 44
} | [
{
"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... | [
"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✝ : IsProbab... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 219,
"column": 44
} | {
"line": 219,
"column": 63
} | {
"line": 219,
"column": 64
} | [
{
"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... | [
"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✝ : IsProbab... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 673,
"column": 2
} | {
"line": 673,
"column": 55
} | {
"line": 675,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nΩ' : Type u_2\nmΩ' : MeasurableSpace Ω'\nμ' : Measure Ω'\nY : Ω' → ℝ\nhX : HasSubgaussianMGF id c (Measure.map X μ)\nhXY : IdentDistrib X Y μ μ'\n⊢ HasSubgaussianMGF Y c μ'",
"ppTerm": "?m.35",
"assigned": true,
"usedC... | [] | rwa [← id_map_iff hXY.aemeasurable_snd, ← hXY.map_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
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