module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 72
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫ (x : α), (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd ↑r).real univ",
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr",
"InnerProductSpace.toNo... | rw [← setIntegral_univ, setIntegral_preCDF_fst ρ _ MeasurableSet.univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 72
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫ (x : α), (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd ↑r).real univ",
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr",
"InnerProductSpace.toNo... | rw [← setIntegral_univ, setIntegral_preCDF_fst ρ _ MeasurableSet.univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Stopping | {
"line": 706,
"column": 2
} | {
"line": 706,
"column": 78
} | [
{
"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\ns : Set Ω\nhs : MeasurableSet s ∧ ∀ (i : ι), Me... | · exact ((hτ.min hπ).min_const i).measurable_of_le fun _ => min_le_right _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Process.Stopping | {
"line": 764,
"column": 2
} | {
"line": 764,
"column": 74
} | [
{
"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\nthis : {ω | τ ω = π ω} = {ω | τ ω ≤ π ω} ∩ {ω |... | refine MeasurableSet.inter (measurableSet_stopping_time_le_min hτ hπ) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.Basic | {
"line": 212,
"column": 47
} | {
"line": 212,
"column": 75
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nhμ : NeZero μ\nhX : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ rexp (log (mgf X μ t)) = mgf X μ t",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congrArg",
"id",
"Real.exp",
... | exp_log (mgf_pos' hμ.out hX) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Basic | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 14
} | [
{
"pp": "case e_f.h\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt α : ℝ\nx : Ω\n⊢ rexp (t * (α + X x)) = rexp (t * α) * rexp (t * X x)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.h... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Basic | {
"line": 296,
"column": 30
} | {
"line": 296,
"column": 38
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_indep : X ⟂ᵢ[μ] Y\nhX : AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) μ\nhY : AEStronglyMeasurable (fun ω ↦ rexp (t * Y ω)) μ\n⊢ ∫ (x : Ω), rexp (t * (X x + Y x)) ∂μ = (∫ (x : Ω), rexp (t * X x) ∂μ) * ∫ (x : Ω), rexp (t * Y x) ∂μ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.Basic | {
"line": 321,
"column": 25
} | {
"line": 321,
"column": 33
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_int_X : AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) μ\nh_int_Y : AEStronglyMeasurable (fun ω ↦ rexp (t * Y ω)) μ\n⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * (X ω + Y ω))) μ",
"usedConstants": [
"Distrib.leftDistribClas... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.Basic | {
"line": 343,
"column": 25
} | {
"line": 343,
"column": 33
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_indep : X ⟂ᵢ[μ] Y\nh_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ\nh_int_Y : Integrable (fun ω ↦ rexp (t * Y ω)) μ\n⊢ Integrable (fun ω ↦ rexp (t * (X ω + Y ω))) μ",
"usedConstants": [
"Distrib.leftDistribClass",
... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 515,
"column": 4
} | {
"line": 516,
"column": 45
} | [
{
"pp": "case basic\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 → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nt✝ t : Set ℝ\nht : t ∈ range Iic\n⊢ ∫⁻ (b : β) in ... | obtain ⟨q, rfl⟩ := ht
exact setLIntegral_toKernel_Iic hf a _ hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 515,
"column": 4
} | {
"line": 516,
"column": 45
} | [
{
"pp": "case basic\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 → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nt✝ t : Set ℝ\nht : t ∈ range Iic\n⊢ ∫⁻ (b : β) in ... | obtain ⟨q, rfl⟩ := ht
exact setLIntegral_toKernel_Iic hf a _ hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 40
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\nhX : AEMeasurable X μ\nl u : ℝ\nhlu : z.re ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\nt : ℝ := min (z.re - l) (u - z.re) / 2\nh_pos : 0 < min (z.re... | · exact .of_forall fun z ↦ by fun_prop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 169,
"column": 53
} | {
"line": 169,
"column": 63
} | [
{
"pp": "case refine_2.inr\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\nh_int_add : Integrable (fun a ↦ rexp ((v + t) * X a) + rexp ((v - t) * X a)) μ\nω : Ω\nh_nonpos ... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 605,
"column": 14
} | {
"line": 605,
"column": 82
} | [
{
"pp": "case e_f.h\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 → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set (β × ℝ)\nf' : ℕ → Set (β × ℝ)\nhf_disj : Pairwise (Disjoint on f')\nhf_meas... | measure_iUnion (h_disj x) fun i ↦ measurable_prodMk_left (hf_meas i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 31
} | [
{
"pp": "case refine_2.refine_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v x : ℝ\nh_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nh_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\nh_nonneg : 0 ≤ x\nhx : x < |t|\np : ℝ\nhp : 0 ≤ p\nht : t ≠ 0\nhX : AEMeasurable X μ\nh_le ... | refine (h_le ω).trans_eq ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 26
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nc : ℝ\n⊢ Measure.map (fun x ↦ x / c) (gaussianReal μ v) = gaussianReal (μ / c) (v / ⟨c ^ 2, ⋯⟩)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"MeasureTheory.Measure... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.CentralLimitTheorem | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 100
} | [
{
"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\nmX : ∀ (n : ℕ), AEMeasurable (X n) P\n⊢ charFun (Measure.map (fun ω ↦ (√↑n)⁻¹ * ∑ k ∈ Finset.range n, X k ω) P) t =\n charFun (Measure.map (X 0)... | rw [charFun_map_mul_comp, (hindep.restrict _).charFun_map_fun_finset_sum_eq_prod (fun _ _↦ mX _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Disintegration.StandardBorel | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 19
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nΩ : Type u_5\ninst✝² : Nonempty Ω\ninst✝¹ : MeasurableSpace Ω\ninst✝ : StandardBorelSpace Ω\nη : Kernel α ℝ\na : α\n⊢ Measure.comap (embeddingReal Ω)\n (if (η a) (range (embeddingReal Ω))ᶜ = 0 then η a else Measure.dirac (Exists.choose ⋯)) =\n if (η a) (ran... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 95,
"column": 45
} | {
"line": 95,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nX' : α → β\nY' : α → Ω\nhY : Y =ᶠ[ae μ] Y'\nhX : X =ᶠ[ae μ] X'\na : α... | by rw [ha, hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 17
} | [
{
"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\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nthis : IsFiniteKernel κ\nS : F... | intro u v huv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 17
} | [
{
"pp": "case hn\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\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nthis : IsFiniteKernel... | intro u v huv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Kernel.Integral | {
"line": 112,
"column": 29
} | {
"line": 112,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\n⊢ (∫ (b : β), g b ∂if a ∈ s then κ a... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Integral | {
"line": 117,
"column": 29
} | {
"line": 117,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\nt : Set β\n⊢ (∫ (b : β) in t, g b ∂i... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 658,
"column": 2
} | {
"line": 682,
"column": 14
} | [
{
"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\n⊢ ∀ᵐ (x : γ) ∂κ.fst a, κ.densityProcess κ.fst n a x univ = 1",
"usedConstants": [
"_pr... | rw [ae_iff]
have : {x | ¬ densityProcess κ (fst κ) n a x univ = 1}
⊆ {x | fst κ a (countablePartitionSet n x) = 0} := by
intro x hx
simp only [mem_setOf_eq] at hx ⊢
rw [densityProcess_fst_univ] at hx
simpa using hx
refine measure_mono_null this ?_
have : {x | fst κ a (countablePartitionSet n... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 658,
"column": 2
} | {
"line": 682,
"column": 14
} | [
{
"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\n⊢ ∀ᵐ (x : γ) ∂κ.fst a, κ.densityProcess κ.fst n a x univ = 1",
"usedConstants": [
"_pr... | rw [ae_iff]
have : {x | ¬ densityProcess κ (fst κ) n a x univ = 1}
⊆ {x | fst κ a (countablePartitionSet n x) = 0} := by
intro x hx
simp only [mem_setOf_eq] at hx ⊢
rw [densityProcess_fst_univ] at hx
simpa using hx
refine measure_mono_null this ?_
have : {x | fst κ a (countablePartitionSet n... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.CondVar | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 85
} | [
{
"pp": "Ω : Type u_1\nm₀ m : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhm : m ≤ m₀\ninst✝ : IsProbabilityMeasure μ\nhX : MemLp X 2 μ\n⊢ ∫ (x : Ω), (μ[X ^ 2 | m] - μ[X | m] ^ 2) x ∂μ + (∫ (x : Ω), (μ[X | m] ^ 2) x ∂μ - (∫ (x : Ω), μ[X | m] x ∂μ) ^ 2) =\n ∫ (x : Ω), (X ^ 2) x ∂μ - ∫ (x : Ω), (μ[X | m] ^ 2)... | rw [integral_sub' integrable_condExp, integral_condExp hm, integral_condExp hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.ProductMeasure | {
"line": 176,
"column": 4
} | {
"line": 188,
"column": 19
} | [
{
"pp": "case inl\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\nhμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)\na b : ℕ\nhab : a < b\n⊢ (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a b).map (restrict₂ ⋯) =\n const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun... | refine Nat.le_induction ?_ (fun n hn hind ↦ ?_) b (Nat.succ_le_of_lt hab) <;> ext1 x₀
· rw [partialTraj_succ_self, ← map_comp_right, map_apply, prod_apply, map_apply, const_apply,
const_apply, Measure.map_piSingleton, restrict₂_comp_IicProdIoc, Measure.map_snd_prod,
measure_univ, one_smul]
all... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProductMeasure | {
"line": 176,
"column": 4
} | {
"line": 188,
"column": 19
} | [
{
"pp": "case inl\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\nhμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)\na b : ℕ\nhab : a < b\n⊢ (partialTraj (fun n ↦ const ((i : ↥(Iic n)) → X ↑i) (μ (n + 1))) a b).map (restrict₂ ⋯) =\n const ((i : ↥(Iic a)) → X ↑i) (Measure.pi fun... | refine Nat.le_induction ?_ (fun n hn hind ↦ ?_) b (Nat.succ_le_of_lt hab) <;> ext1 x₀
· rw [partialTraj_succ_self, ← map_comp_right, map_apply, prod_apply, map_apply, const_apply,
const_apply, Measure.map_piSingleton, restrict₂_comp_IicProdIoc, Measure.map_snd_prod,
measure_univ, one_smul]
all... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProductMeasure | {
"line": 417,
"column": 2
} | {
"line": 420,
"column": 50
} | [
{
"pp": "case hν\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 : Set ι\ns : Finset ↑I\nt : (i : ↑I) → Set (X ↑i)\nht : ∀ (i : ↑I), MeasurableSet (t i)\n⊢ (map I.restrict (infinitePi μ)) ((↑s).pi t) = ∏ i ∈ s, (μ ... | classical
rw [map_apply (by fun_prop), restrict_preimage, infinitePi_pi _ (by measurability)]
· simp
· exact .pi s.countable_toSet (by measurability) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Probability.ProductMeasure | {
"line": 451,
"column": 54
} | {
"line": 454,
"column": 17
} | [
{
"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)\ninst✝ : Countable ι\nt : (i : ι) → Set (X i)\nmt : ∀ (i : ι), MeasurableSet (t i)\n⊢ (infinitePi μ) (Set.univ.pi t) = ∏' (i : ι), (μ i) (t i)",
"usedConstan... | by
rw [infinitePi_pi_of_countable, tprod_univ (f := fun i ↦ μ i (t i))]
· simpa [Set.countable_univ_iff]
· measurability | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 653,
"column": 8
} | {
"line": 653,
"column": 29
} | [
{
"pp": "case h.e'_7\nX : ℕ → Type u_1\ninst✝³ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝² : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ℕ\nx₀ : (i : ↥(Iic a)) → X ↑i\nf : ((n : ℕ) → X n) → E\n... | traj_map_updateFinset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 58,
"column": 33
} | {
"line": 58,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsGaussian μ\nL : StrongDual ℝ E\n⊢ cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2) =\n c... | covarianceBilinDual_self_eq_variance, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 112,
"column": 11
} | {
"line": 114,
"column": 23
} | [
{
"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 : IsGaussian μ\ninst✝ : SecondCountableTopology E\nL : StrongDual ℝ E\n⊢ ∫ (x : E), L x ∂μ + ∫ (y : E), L y ∂Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ = 0... | by
rw [integral_map (by fun_prop) (by fun_prop)]
simp [integral_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nL₁ L₂ : StrongDual ℝ E\nh : MemLp id 2 μ\n⊢ ∫ (x : E), ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖ ∂μ = ‖L₁‖ * ‖L₂‖ * ∫ (x : E), ‖x‖ ^ 2 ∂μ",
"usedConstants": [
"Norm.norm... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 176,
"column": 58
} | {
"line": 176,
"column": 85
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : E →L[ℝ] E →L[ℝ] ℝ\nhf : f.toBilinForm.IsPosSemidef\nh : ∀ (t... | simpa [g] using hf.nonneg _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 176,
"column": 58
} | {
"line": 176,
"column": 85
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : E →L[ℝ] E →L[ℝ] ℝ\nhf : f.toBilinForm.IsPosSemidef\nh : ∀ (t... | simpa [g] using hf.nonneg _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 176,
"column": 58
} | {
"line": 176,
"column": 85
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : E →L[ℝ] E →L[ℝ] ℝ\nhf : f.toBilinForm.IsPosSemidef\nh : ∀ (t... | simpa [g] using hf.nonneg _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Fernique | {
"line": 458,
"column": 2
} | {
"line": 459,
"column": 44
} | [
{
"pp": "case neg.h₂.h\nE : 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... | refine (le_trans ?_ (lintegral_closedBall_diff_exp_logRatio_mul_sq_le h_rot
(hc'_gt.trans_le hc') ha_lt n)).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Process.FiniteDimensionalLaws | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 72
} | [
{
"pp": "case refine_3\nT : Type u_1\nΩ : Type u_2\n𝓧 : T → Type u_3\nmΩ : MeasurableSpace Ω\nmα : (t : T) → MeasurableSpace (𝓧 t)\nX Y : (t : T) → Ω → 𝓧 t\nP : Measure Ω\ninst✝ : IsFiniteMeasure P\nhX : AEMeasurable (fun ω x ↦ X x ω) P\nhY : AEMeasurable (fun ω x ↦ Y x ω) P\nh : IdentDistrib (fun ω x ↦ X x ... | exact (map_eq_iff_forall_finset_map_restrict_eq hX hY).mp h.map_eq I | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.FiniteDimensionalLaws | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 72
} | [
{
"pp": "case refine_3\nT : Type u_1\nΩ : Type u_2\n𝓧 : T → Type u_3\nmΩ : MeasurableSpace Ω\nmα : (t : T) → MeasurableSpace (𝓧 t)\nX Y : (t : T) → Ω → 𝓧 t\nP : Measure Ω\ninst✝ : IsFiniteMeasure P\nhX : AEMeasurable (fun ω x ↦ X x ω) P\nhY : AEMeasurable (fun ω x ↦ Y x ω) P\nh : IdentDistrib (fun ω x ↦ X x ... | exact (map_eq_iff_forall_finset_map_restrict_eq hX hY).mp h.map_eq I | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.FiniteDimensionalLaws | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 72
} | [
{
"pp": "case refine_3\nT : Type u_1\nΩ : Type u_2\n𝓧 : T → Type u_3\nmΩ : MeasurableSpace Ω\nmα : (t : T) → MeasurableSpace (𝓧 t)\nX Y : (t : T) → Ω → 𝓧 t\nP : Measure Ω\ninst✝ : IsFiniteMeasure P\nhX : AEMeasurable (fun ω x ↦ X x ω) P\nhY : AEMeasurable (fun ω x ↦ Y x ω) P\nh : IdentDistrib (fun ω x ↦ X x ... | exact (map_eq_iff_forall_finset_map_restrict_eq hX hY).mp h.map_eq I | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Process.Basic | {
"line": 61,
"column": 34
} | {
"line": 61,
"column": 37
} | [
{
"pp": "case h\nS : Type u_1\nΩ : Type u_3\nmΩ : MeasurableSpace Ω\nα : Type u_4\nmα : MeasurableSpace α\nκ : Kernel α Ω\nP : Measure α\n𝓧 : S → Type u_5\n𝓨 : Type u_6\ninst✝¹ : (i : S) → MeasurableSpace (𝓧 i)\ninst✝ : MeasurableSpace 𝓨\nX X' : (i : S) → Ω → 𝓧 i\nY : Ω → 𝓨\nh1 : IndepFun (fun ω i ↦ X i ω... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Process.Basic | {
"line": 228,
"column": 32
} | {
"line": 228,
"column": 35
} | [
{
"pp": "case h\nS : 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 ↦ ... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Distributions.Gaussian.Multivariate | {
"line": 116,
"column": 56
} | {
"line": 116,
"column": 80
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nL : StrongDual ℝ E\n⊢ L (∫ (x : E), x ∂stdGaussian E) = 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | integral_id_stdGaussian, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 165,
"column": 82
} | {
"line": 165,
"column": 99
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\np : PMF α\nf : α → PMF β\ns : Set β\na : α\nb : β\n⊢ (if b ∈ s then (f a) b else 0) = if b ∈ s then (f a) b else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PMF",
"PMF.instFunLike",
"Classical.propDecidable",
"Membership.mem",
... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 165,
"column": 82
} | {
"line": 165,
"column": 99
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\np : PMF α\nf : α → PMF β\ns : Set β\na : α\nb : β\n⊢ (if b ∈ s then (f a) b else 0) = if b ∈ s then (f a) b else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PMF",
"PMF.instFunLike",
"Classical.propDecidable",
"Membership.mem",
... | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 165,
"column": 82
} | {
"line": 165,
"column": 99
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\np : PMF α\nf : α → PMF β\ns : Set β\na : α\nb : β\n⊢ (if b ∈ s then (f a) b else 0) = if b ∈ s then (f a) b else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PMF",
"PMF.instFunLike",
"Classical.propDecidable",
"Membership.mem",
... | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 13
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\nh : ∃ a ∈ s, a ∈ p.support\na : α\n⊢ (p.filter s h) a = s.indicator (⇑p) a * (∑' (a' : α), s.indicator (⇑p) a')⁻¹",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"CommSemiring.toSemir... | filter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 70,
"column": 63
} | {
"line": 70,
"column": 76
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\neq0 : k % (b + 1) = k\neq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)\nthis : 1 - ↑x ≥ 0\n⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) =\n ↑x ^ (k % (b + 1)) * (1 - ↑x) ^ (↑(Fin.last b) - k % (b + 1)) * ↑(b.choose (k % (b + 1)))",
"usedConstants... | Fin.val_last, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 43
} | [
{
"pp": "case refine_2\np : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nhp_lt_half : ∀ᶠ (n : ℕ) in atTop, p n < 1 / 2\nhEq : (fun n ↦ (1 - p n) ^ (n - k)) =ᶠ[atTop] fun n ↦ (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹\nthis : Real.exp (-r) = Real.exp (-r) * (1 ^ k)⁻¹\n⊢ Tendsto (fun n ↦ ((1 - p n) ^... | refine Tendsto.inv₀ (.pow ?_ k) (by simp) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 72,
"column": 2
} | {
"line": 86,
"column": 77
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\n⊢ Tendsto (fun n ↦ ↑(n.choose k) * p n ^ k * (1 - p n) ^ (n - k)) atTop (𝓝 (Real.exp (-r) * r ^ k / ↑k.factorial))",
"usedConstants": [
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"... | rw [mul_div_assoc, mul_comm]
refine (tendsto_choose_mul_pow_atTop k hr).mul ?_
have hp_lt_half : ∀ᶠ n in atTop, p n < 1 / 2 :=
(tendsto_zero_of_tendsto_mul_atTop hr).eventually (Iio_mem_nhds (by norm_num))
have hEq : (fun n => (1 - p n) ^ (n - k)) =ᶠ[atTop]
(fun n => (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹) :... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 72,
"column": 2
} | {
"line": 86,
"column": 77
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\n⊢ Tendsto (fun n ↦ ↑(n.choose k) * p n ^ k * (1 - p n) ^ (n - k)) atTop (𝓝 (Real.exp (-r) * r ^ k / ↑k.factorial))",
"usedConstants": [
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"... | rw [mul_div_assoc, mul_comm]
refine (tendsto_choose_mul_pow_atTop k hr).mul ?_
have hp_lt_half : ∀ᶠ n in atTop, p n < 1 / 2 :=
(tendsto_zero_of_tendsto_mul_atTop hr).eventually (Iio_mem_nhds (by norm_num))
have hEq : (fun n => (1 - p n) ^ (n - k)) =ᶠ[atTop]
(fun n => (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹) :... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 135,
"column": 79
} | {
"line": 148,
"column": 8
} | [
{
"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\na : α\ns : Set γ\nhs : MeasurableSet s\n⊢ ∫⁻ (x : γ) in s, ENNReal.ofReal (κ.rnDerivAux (κ + η) a x... | by
have h_le : κ ≤ κ + η := le_add_of_nonneg_right bot_le
simp_rw [rnDerivAux]
split_ifs with hα
· have h_ac : κ a ≪ (κ + η) a := Measure.absolutelyContinuous_of_le (h_le a)
rw [← Measure.setLIntegral_rnDeriv h_ac]
refine setLIntegral_congr_fun_ae hs ?_
filter_upwards [Measure.rnDeriv_lt_top (κ a) (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 26
} | [
{
"pp": "case hf\nα : 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\nh_le : κ ≤ κ + η\n⊢ Measurable (Function.uncurry fun a x ↦ 1)",
"usedConstants": [
... | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 26
} | [
{
"pp": "case hf\nα : 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\nh_le : κ ≤ κ + η\n⊢ Measurable (Function.uncurry fun a x ↦ 1)",
"usedConstants": [
... | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 26
} | [
{
"pp": "case hf\nα : 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\nh_le : κ ≤ κ + η\n⊢ Measurable (Function.uncurry fun a x ↦ 1)",
"usedConstants": [
... | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 256,
"column": 78
} | {
"line": 256,
"column": 87
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ninst✝¹ : Finite ι\nκ : ι → Type u_4\ninst✝ : ∀ (i : ι), Finite (κ i)\nX : (i : ι) → κ i → Ω → ℝ\nhX : HasGaussianLaw (fun ω i j ↦ X i j ω) P\nh : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0\nthis✝... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 68
} | [
{
"pp": "case refine_2\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | exact Monotone.directed_le fun i j hij k hki => le_trans hki hij | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.ZeroOne | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 68
} | [
{
"pp": "case refine_2\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | exact Monotone.directed_le fun i j hij k hki => le_trans hki hij | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.ZeroOne | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 68
} | [
{
"pp": "case refine_2\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | exact Monotone.directed_le fun i j hij k hki => le_trans hki hij | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.ZeroOne | {
"line": 304,
"column": 2
} | {
"line": 309,
"column": 55
} | [
{
"pp": "case refine_1\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | · simp only [mem_atBot_sets, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty]
rintro t ⟨a, ha⟩
obtain ⟨b, hb⟩ : ∃ b, b < a := exists_lt a
refine ⟨b, fun c hc hct => ?_⟩
suffices ∀ i ∈ t, c < i from lt_irrefl c (this c hct)
exact fun i hi => hc.trans_lt (hb.trans_le (ha hi)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.ZeroOne | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 61
} | [
{
"pp": "case refine_2\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | exact Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.ZeroOne | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 61
} | [
{
"pp": "case refine_2\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | exact Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.ZeroOne | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 61
} | [
{
"pp": "case refine_2\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | exact Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 324,
"column": 21
} | {
"line": 324,
"column": 30
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝⁴ : Complete... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 356,
"column": 2
} | {
"line": 357,
"column": 56
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_4\nκ : Type u_5\ninst✝¹ : Finite ι\ninst✝ : Finite κ\nX : ι → Ω → ℝ\nY : κ → Ω → ℝ\nhXY : HasGaussianLaw (fun ω ↦ (fun i ↦ X i ω, fun j ↦ Y j ω)) P\nh : ∀ (i : ι) (j : κ), cov[X i, Y j; P] = 0\nthis✝¹ : IsProbabilityMeasure ... | · exact hXY.map_equiv (.prodCongr (PiLp.continuousLinearEquiv 2 ℝ (fun _ ↦ ℝ)).symm
(PiLp.continuousLinearEquiv 2 ℝ (fun _ ↦ ℝ)).symm) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 364,
"column": 66
} | {
"line": 364,
"column": 75
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_4\nκ : Type u_5\ninst✝¹ : Finite ι\ninst✝ : Finite κ\nX : ι → Ω → ℝ\nY : κ → Ω → ℝ\nhXY : HasGaussianLaw (fun ω ↦ (fun i ↦ X i ω, fun j ↦ Y j ω)) P\nh : ∀ (i : ι) (j : κ), cov[X i, Y j; P] = 0\nthis✝¹ : IsProbabilityMeasure ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Conditional | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\ns t : Set Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ CondIndepSet m' hm' s t μ ↔ CondIndep m' (generateFrom {s}) (generateFrom {t}) hm' μ",
"usedConstants": [
"ProbabilityTheory.Kernel.Indep",
"Me... | simp only [CondIndepSet, CondIndep, Kernel.IndepSet] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Independence.Conditional | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\ns t : Set Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ CondIndepSet m' hm' s t μ ↔ CondIndep m' (generateFrom {s}) (generateFrom {t}) hm' μ",
"usedConstants": [
"ProbabilityTheory.Kernel.Indep",
"Me... | simp only [CondIndepSet, CondIndep, Kernel.IndepSet] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Conditional | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\ns t : Set Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ CondIndepSet m' hm' s t μ ↔ CondIndep m' (generateFrom {s}) (generateFrom {t}) hm' μ",
"usedConstants": [
"ProbabilityTheory.Kernel.Indep",
"Me... | simp only [CondIndepSet, CondIndep, Kernel.IndepSet] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 736,
"column": 6
} | {
"line": 736,
"column": 46
} | [
{
"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\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nX : Ω → β\nY : Ω → β'\nhX : Measurable X\nhY : Measurable Y\ns : Set β\nhs : MeasurableSet s\nt... | show (fun ω : Ω ↦ (1 : ℝ)) = 1 from rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 213,
"column": 2
} | {
"line": 221,
"column": 7
} | [
{
"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 (b : 𝓧) ∂⇑κ ∘ₘ... | suffices μ ⊗ₘ κ ≪ μ.prod (κ ∘ₘ μ) by
rw [← Measure.compProd_const] at this
simpa using this.kernel_of_compProd
suffices (κ ∘ₘ μ) ⊗ₘ κ†μ ≪ (κ ∘ₘ μ).prod μ by
rw [← swap_compProd_posterior, ← Measure.prod_swap, Measure.swap_comp]
exact this.map measurable_swap
rw [← Measure.compProd_const]
refine Me... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Posterior | {
"line": 213,
"column": 2
} | {
"line": 221,
"column": 7
} | [
{
"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 (b : 𝓧) ∂⇑κ ∘ₘ... | suffices μ ⊗ₘ κ ≪ μ.prod (κ ∘ₘ μ) by
rw [← Measure.compProd_const] at this
simpa using this.kernel_of_compProd
suffices (κ ∘ₘ μ) ⊗ₘ κ†μ ≪ (κ ∘ₘ μ).prod μ by
rw [← swap_compProd_posterior, ← Measure.prod_swap, Measure.swap_comp]
exact this.map measurable_swap
rw [← Measure.compProd_const]
refine Me... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 63,
"column": 20
} | {
"line": 63,
"column": 44
} | [
{
"pp": "W✝ X✝ Y✝ Z✝ : SFinKer\nκ : W✝.Hom X✝\nη : X✝.Hom Y✝\nξ : Y✝.Hom Z✝\n⊢ { hom := ξ.hom ∘ₖ { hom := η.hom ∘ₖ κ.hom, property := ⋯ }.hom, property := ⋯ } =\n { hom := { hom := ξ.hom ∘ₖ η.hom, property := ⋯ }.hom ∘ₖ κ.hom, property := ⋯ }",
"usedConstants": [
"SFinKer.instIsSFiniteKernelCarrier... | simp [Kernel.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 63,
"column": 20
} | {
"line": 63,
"column": 44
} | [
{
"pp": "W✝ X✝ Y✝ Z✝ : SFinKer\nκ : W✝.Hom X✝\nη : X✝.Hom Y✝\nξ : Y✝.Hom Z✝\n⊢ { hom := ξ.hom ∘ₖ { hom := η.hom ∘ₖ κ.hom, property := ⋯ }.hom, property := ⋯ } =\n { hom := { hom := ξ.hom ∘ₖ η.hom, property := ⋯ }.hom ∘ₖ κ.hom, property := ⋯ }",
"usedConstants": [
"SFinKer.instIsSFiniteKernelCarrier... | simp [Kernel.comp_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 63,
"column": 20
} | {
"line": 63,
"column": 44
} | [
{
"pp": "W✝ X✝ Y✝ Z✝ : SFinKer\nκ : W✝.Hom X✝\nη : X✝.Hom Y✝\nξ : Y✝.Hom Z✝\n⊢ { hom := ξ.hom ∘ₖ { hom := η.hom ∘ₖ κ.hom, property := ⋯ }.hom, property := ⋯ } =\n { hom := { hom := ξ.hom ∘ₖ η.hom, property := ⋯ }.hom ∘ₖ κ.hom, property := ⋯ }",
"usedConstants": [
"SFinKer.instIsSFiniteKernelCarrier... | simp [Kernel.comp_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 909,
"column": 36
} | {
"line": 909,
"column": 71
} | [
{
"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\... | by rw [h1_symm, h1, h2_symm, h2, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Proper | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 38
} | [
{
"pp": "X : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nB : Set X\nf : X → ℝ≥0∞\nx₀ : X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nhf : Measurable f\nhB : MeasurableSet B\n⊢ ∫⁻ (x : X), B.indicator 1 x * f x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f x ∂π x₀",
"usedConstants": [
"MeasureTheory.Measure... | refine hf.ennreal_induction ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RepresentationTheory.Action | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 48
} | [
{
"pp": "case h.h.h\nG : Type v\ninst✝¹ : Monoid G\nX Y Z : Action (Type w) G\nk : Type u\ninst✝ : CommSemiring k\nf : X ⟶ Y\na✝ : (X ⊗ Z).V\n⊢ (((rTensor (linearize k G Z) (linearizeMap f)).comp (δ X Z)).toLinearMap ∘ₗ Finsupp.lsingle a✝) 1 =\n (((δ Y Z).comp (linearizeMap (f ▷ Z))).toLinearMap ∘ₗ Finsupp.l... | simp [linearizeMap_single _, δ_apply_single _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Action | {
"line": 200,
"column": 7
} | {
"line": 200,
"column": 53
} | [
{
"pp": "case h.h.h\nG : Type v\ninst✝¹ : Monoid G\nX Y Z : Action (Type w) G\nk : Type u\ninst✝ : CommSemiring k\nf : X ⟶ Y\na✝ : (Z ⊗ X).V\n⊢ (((lTensor (linearize k G Z) (linearizeMap f)).comp (δ Z X)).toLinearMap ∘ₗ Finsupp.lsingle a✝) 1 =\n (((δ Z Y).comp (linearizeMap (Z ◁ f))).toLinearMap ∘ₗ Finsupp.l... | simp [linearizeMap_single _, δ_apply_single _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Action | {
"line": 222,
"column": 7
} | {
"line": 222,
"column": 53
} | [
{
"pp": "case h.h.h\nG : Type v\ninst✝¹ : Monoid G\nk : Type u\ninst✝ : CommSemiring k\nX : Action (Type u) G\na✝ : X.V\n⊢ ((↑(rid k (linearize k G X)).symm).toLinearMap ∘ₗ Finsupp.lsingle a✝) 1 =\n ((((lTensor (linearize k G X) (η k G)).comp (δ X (𝟙_ (Action (Type u) G)))).comp\n (linearizeMap (... | simp [linearizeMap_single _, δ_apply_single _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.SubGaussian | {
"line": 420,
"column": 15
} | {
"line": 420,
"column": 23
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (X ω + Y ω))) (⇑κ ∘ₘ ν)",
... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 432,
"column": 17
} | {
"line": 432,
"column": 25
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 435,
"column": 8
} | {
"line": 435,
"column": 48
} | [
{
"pp": "case hf_nonneg\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := ⋯\nq : ℝ≥0 := ⋯\nω' : Ω'\nhmX : ∀ (t : ℝ),... | · exact ae_of_all _ fun _ ↦ exp_nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.SubGaussian | {
"line": 436,
"column": 8
} | {
"line": 436,
"column": 48
} | [
{
"pp": "case hg_nonneg\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := ⋯\nq : ℝ≥0 := ⋯\nω' : Ω'\nhmX : ∀ (t : ℝ),... | · exact ae_of_all _ fun _ ↦ exp_nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.SubGaussian | {
"line": 475,
"column": 11
} | {
"line": 475,
"column": 19
} | [
{
"pp": "case pos.inr\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubga... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 502,
"column": 18
} | {
"line": 502,
"column": 26
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 502,
"column": 4
} | {
"line": 502,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | simp_rw [mgf, mul_add, exp_add] at h_int_mul ⊢ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.StrongLaw | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 16
} | [
{
"pp": "case refine_3\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\n⊢ Integrable (fun x ↦ |f x|) μ",
"usedConstants": [
"Real",
"Real.lattice",
"instHasSolidNormReal",
"Real.normedAddCommGroup",
"MeasureTheory.Integrable.abs",
"Real... | · exact hf.abs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.StrongLaw | {
"line": 251,
"column": 10
} | {
"line": 253,
"column": 24
} | [
{
"pp": "case h.h\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := ⋯\nthis : IsProbabilityMeasure ρ\ni : ℕ\na✝ : i ∈ range N\nI : ↑i ≤ ↑(i + 1)\n⊢ ∀ x ∈ Set.Ioc ↑i ↑(i + 1), (↑i + 1) * 1 ≤ x + 1",
... | intro x hx
simp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx
simp [hx.1.le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.StrongLaw | {
"line": 251,
"column": 10
} | {
"line": 253,
"column": 24
} | [
{
"pp": "case h.h\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := ⋯\nthis : IsProbabilityMeasure ρ\ni : ℕ\na✝ : i ∈ range N\nI : ↑i ≤ ↑(i + 1)\n⊢ ∀ x ∈ Set.Ioc ↑i ↑(i + 1), (↑i + 1) * 1 ≤ x + 1",
... | intro x hx
simp only [Nat.cast_add, Nat.cast_one, Set.mem_Ioc] at hx
simp [hx.1.le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 730,
"column": 13
} | {
"line": 730,
"column": 21
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (X ω + Y ω))) μ",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Norme... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.StrongLaw | {
"line": 303,
"column": 8
} | {
"line": 304,
"column": 70
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK i : ℕ\nx✝ : i ∈ range K\n⊢ ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N} ⊆ {ω | X ω ∈ Set.Ioi ↑i}",
"usedConstants": [
"Set.Ioc",
"Real",
"Set.Ioi",
"Preorder.toLT",
... | simp +contextual only [Set.mem_Ioc, Set.mem_Ioi,
Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.StrongLaw | {
"line": 303,
"column": 8
} | {
"line": 304,
"column": 70
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK i : ℕ\nx✝ : i ∈ range K\n⊢ ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N} ⊆ {ω | X ω ∈ Set.Ioi ↑i}",
"usedConstants": [
"Set.Ioc",
"Real",
"Set.Ioi",
"Preorder.toLT",
... | simp +contextual only [Set.mem_Ioc, Set.mem_Ioi,
Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.StrongLaw | {
"line": 303,
"column": 8
} | {
"line": 304,
"column": 70
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK i : ℕ\nx✝ : i ∈ range K\n⊢ ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N} ⊆ {ω | X ω ∈ Set.Ioi ↑i}",
"usedConstants": [
"Set.Ioc",
"Real",
"Set.Ioi",
"Preorder.toLT",
... | simp +contextual only [Set.mem_Ioc, Set.mem_Ioi,
Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 756,
"column": 2
} | {
"line": 766,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\nh_meas : ∀ (i : ι), AEMeasurable (X i) μ\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ",
"usedCon... | have : IsProbabilityMeasure μ := h_indep.isProbabilityMeasure
classical
induction s using Finset.induction_on with
| empty => simp
| insert i s his h =>
simp_rw [← Finset.sum_apply, Finset.sum_insert his, Pi.add_apply, Finset.sum_apply]
have h_indep' := (h_indep.indepFun_finset_sum_of_notMem₀ h_meas his... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 756,
"column": 2
} | {
"line": 766,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\nh_meas : ∀ (i : ι), AEMeasurable (X i) μ\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ",
"usedCon... | have : IsProbabilityMeasure μ := h_indep.isProbabilityMeasure
classical
induction s using Finset.induction_on with
| empty => simp
| insert i s his h =>
simp_rw [← Finset.sum_apply, Finset.sum_insert his, Pi.add_apply, Finset.sum_apply]
have h_indep' := (h_indep.indepFun_finset_sum_of_notMem₀ h_meas his... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.StrongLaw | {
"line": 459,
"column": 8
} | {
"line": 459,
"column": 60
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | exact div_nonneg (variance_nonneg _ _) (sq_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Character | {
"line": 58,
"column": 52
} | {
"line": 58,
"column": 98
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type u\ninst✝ : Monoid G\nV : FDRep k G\ng h : G\n⊢ V.character (h * g) = V.character (g * h)",
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike",
"Semiring.toModule",
"HMul.hMul",
"Mon... | simp only [trace_mul_comm, character, map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Character | {
"line": 58,
"column": 52
} | {
"line": 58,
"column": 98
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type u\ninst✝ : Monoid G\nV : FDRep k G\ng h : G\n⊢ V.character (h * g) = V.character (g * h)",
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike",
"Semiring.toModule",
"HMul.hMul",
"Mon... | simp only [trace_mul_comm, character, map_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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