module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 56,
"column": 4
} | {
"line": 57,
"column": 84
} | {
"line": 57,
"column": 85
} | [
{
"pp": "E : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommMonoid E\ninst✝¹ : Module ℝ E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : IsGaussian μ\nthis : (Measure.map (⇑0) μ) Set.univ = 1\n⊢ μ Set.univ = 1",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"E : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommMonoid E\ninst✝¹ : Module ℝ E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : IsGaussian μ\nthis : (Measure.map (⇑0) μ) Set.univ = 1\n⊢ μ Set.univ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 288,
"column": 8
} | {
"line": 288,
"column": 24
} | {
"line": 288,
"column": 25
} | [
{
"pp": "case e'_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\ninst✝ : IsZeroOrProbabilityMeasure μ\nht : 0 < t\nhc : ∫ (x : Ω), X x ∂μ = 0\nhs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)\nhu : UniqueDiffOn ℝ (Set.Icc 0 t)\nx✝ : ℝ\n⊢ 0 = deriv (cgf X μ) 0",
"ppTerm": "?e'_3",
... | [
"case e'_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\ninst✝ : IsZeroOrProbabilityMeasure μ\nht : 0 < t\nhc : ∫ (x : Ω), X x ∂μ = 0\nhs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)\nhu : UniqueDiffOn ℝ (Set.Icc 0 t)\nx✝ : ℝ\n⊢ 0 = deriv (cgf X μ) 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | {
"line": 339,
"column": 14
} | [
{
"pp": "μ μ' : Measure ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure μ'\nh : complexMGF id μ = complexMGF id μ'\n⊢ μ = μ'",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"μ μ' : Measure ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure μ'\nh : complexMGF id μ = complexMGF id μ'\n⊢ μ = μ'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 32
} | {
"line": 525,
"column": 33
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nx✝ : ℝ\n⊢ x✝ ∈ integrableExpSet id (gaussianReal μ v) ↔ x✝ ∈ Set.univ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"congrArg",
"Set.mem_univ._simp_1",
... | [
"μ : ℝ\nv : ℝ≥0\nx✝ : ℝ\n⊢ Integrable (fun ω ↦ rexp (x✝ * ω)) (gaussianReal μ v)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 55
} | {
"line": 552,
"column": 2
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ Var[fun x ↦ x; gaussianReal μ v] = ↑v",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.variance_eq_integral",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Measurable.aemeasurable",
"Real.instRCLike",
... | [
"μ : ℝ\nv : ℝ≥0\n⊢ ∫ (ω : ℝ), (ω - ∫ (x : ℝ), x ∂gaussianReal μ v) ^ 2 ∂gaussianReal μ v = ↑v"
] | rw [variance_eq_integral measurable_id'.aemeasurable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 661,
"column": 39
} | {
"line": 661,
"column": 59
} | {
"line": 662,
"column": 2
} | [
{
"pp": "case hf\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nm₁ m₂ : ℝ\nv₁ v₂ : ℝ≥0\nX Y : Ω → ℝ\nhXY : X ⟂ᵢ[P] Y\nhX : Measure.map X P = gaussianReal m₁ v₁\nhY : Measure.map Y P = gaussianReal m₂ v₂\n⊢ Measure.map X P ≠ 0",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Fals... | [] | simp [NeZero.ne, hX] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 21
} | [
{
"pp": "case h₁\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\np : ℝ≥0∞\n𝕜 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual 𝕜 E\nh_Lp : MemLp id p μ\nhp : ¬p = 0\nhp_top : ¬p = ∞\nh0 : 0 < p.toReal... | [
"case h₁\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\np : ℝ≥0∞\n𝕜 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual 𝕜 E\nh_Lp : MemLp id p μ\nhp : ¬p = 0\nhp_top : ¬p = ∞\nh0 : 0 < p.toReal\nthis : ∫⁻ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "case refine_1\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDu... | [
"case refine_1\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), cha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "case refine_2\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDu... | [
"case refine_2\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), cha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "case refine_3\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDu... | [
"case refine_3\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), cha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "case refine_4\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), char... | [
"case refine_4\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = ce... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "case refine_5\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), char... | [
"case refine_5\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = ce... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "case refine_6\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), char... | [
"case refine_6\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = ce... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 15
} | {
"line": 189,
"column": 16
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : CompleteSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsGaussian μ\np : ℝ≥0∞\nhp : p ≠ ∞\nthis : MemLp (fun x ↦ ‖x‖ ^ 2) (p / 2) μ\n⊢ MemLp (fun x ↦ ‖... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : CompleteSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsGaussian μ\np : ℝ≥0∞\nhp : p ≠ ∞\nthis : MemLp (fun x ↦ ‖x‖ ^ 2) (p / 2) μ\n⊢ MemLp (fun x ↦ ‖x‖ ^ 2) (p /... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 162,
"column": 12
} | {
"line": 162,
"column": 23
} | {
"line": 162,
"column": 24
} | [
{
"pp": "case refine_2\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDu... | [
"case refine_2\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), cha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 162,
"column": 12
} | {
"line": 162,
"column": 23
} | {
"line": 162,
"column": 24
} | [
{
"pp": "case refine_3\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDu... | [
"case refine_3\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), cha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 162,
"column": 12
} | {
"line": 162,
"column": 23
} | {
"line": 162,
"column": 24
} | [
{
"pp": "case refine_5\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), char... | [
"case refine_5\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = ce... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 162,
"column": 12
} | {
"line": 162,
"column": 23
} | {
"line": 162,
"column": 24
} | [
{
"pp": "case refine_6\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), char... | [
"case refine_6\nE : 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\nx✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = ce... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 236,
"column": 8
} | {
"line": 236,
"column": 67
} | {
"line": 236,
"column": 67
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nμ : Measure E\ninst✝² : IsGaussian μ\ninst✝¹ : CompleteSpace E\ninst✝ : SecondCountableTopology E\nh : ∀ (x : E), μ ≠ Measure.dirac x\nx : E\nL : StrongDual ℝ E\nhL : Var[⇑L; μ] ≠ 0... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nμ : Measure E\ninst✝² : IsGaussian μ\ninst✝¹ : CompleteSpace E\ninst✝ : SecondCountableTopology E\nh : ∀ (x : E), μ ≠ Measure.dirac x\nx : E\nL : StrongDual ℝ E\nhL : Var[⇑L; μ] ≠ 0\nhL_zero : ... | Measure.map_apply (by fun_prop) (measurableSet_singleton _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 178,
"column": 19
} | {
"line": 178,
"column": 34
} | {
"line": 178,
"column": 35
} | [
{
"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... | [
"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 : E), charF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 178,
"column": 58
} | {
"line": 178,
"column": 73
} | {
"line": 178,
"column": 74
} | [
{
"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... | [
"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 : E), charF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily | {
"line": 68,
"column": 32
} | {
"line": 73,
"column": 42
} | {
"line": 75,
"column": 0
} | [
{
"pp": "I : Finset ℝ≥0\n⊢ (covMatrix I).PosSemidef",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.IsAddHaarMeasure.toIsFiniteMeasureOnCompacts",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"ConditionallyCompleteLinearOrder.toCompactIc... | [] | by
have : covMatrix I = .of fun s t ↦ volume.real ((Icc 0 s.1.1) ∩ (Icc 0 t.1.1)) := by
ext; simp [Icc_inter_Icc]
rw [this]
exact posSemidef_matrix_measure_inter (fun _ ↦ measurableSet_Icc)
(fun _ ↦ isCompact_Icc.measure_ne_top) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Fernique | {
"line": 408,
"column": 8
} | {
"line": 408,
"column": 19
} | {
"line": 408,
"column": 20
} | [
{
"pp": "case hab\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 μ) =... | [
"case hab\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 μ) = μ.prod μ\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic | {
"line": 171,
"column": 66
} | {
"line": 171,
"column": 77
} | {
"line": 171,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nX : Ω → E\ninst✝ : NormedSpace ℝ E\nhX : HasGaussianLaw X P\n⊢ HasGaussianLaw (-X) P",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": ... | [
"Ω : Type u_1\nE : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nX : Ω → E\ninst✝ : NormedSpace ℝ E\nhX : HasGaussianLaw X P\n⊢ HasGaussianLaw (-X) P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.FiniteDimensionalLaws | {
"line": 64,
"column": 97
} | {
"line": 73,
"column": 47
} | {
"line": 75,
"column": 0
} | [
{
"pp": "T : 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\n⊢ Measure.map (fun ω x ↦ X x ω) P = Measure.map... | [] | by
refine ⟨fun h I ↦ ?_, fun h ↦ ?_⟩
· have hX' : P.map (fun ω ↦ I.restrict (X · ω)) = (P.map (fun ω ↦ (X · ω))).map I.restrict := by
rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) hX, Function.comp_def]
have hY' : P.map (fun ω ↦ I.restrict (Y · ω)) = (P.map (fun ω ↦ (Y · ω))).map I.restrict := by... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 149,
"column": 10
} | {
"line": 149,
"column": 21
} | {
"line": 149,
"column": 22
} | [
{
"pp": "case hm\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nX : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nh1 : IsGaussianProcess X P\nh2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0\nh3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s\nI : Finset ℝ≥0\nthis : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)\ni : ↥I\n⊢ ∫ (x :... | [
"case hm\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nX : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nh1 : IsGaussianProcess X P\nh2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0\nh3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s\nI : Finset ℝ≥0\nthis : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)\ni : ↥I\n⊢ ∫ (x : Ω), X (↑i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 34
} | {
"line": 133,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : (ν a).restrict s = 0\nq : ℚ\nhq : x < ↑q\nthis : (κ a) (... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : (ν a).restrict s = 0\nq : ℚ\nhq : x < ↑q\nthis : (κ a) (s ×ˢ Iic ↑q)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 96,
"column": 4
} | {
"line": 103,
"column": 45
} | {
"line": 104,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nh_empty : ρ (s ×ˢ ∅) = 0\nh_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))\n⊢ Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑r)) atBot (𝓝 (ρ (⋂ i, s ×ˢ Iic ↑i)))",
... | [] | have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ⋂ r : ℚ, s ×ˢ Iic (r : ℝ) := by
ext1 x
push _ ∈ _
refine ⟨fun h i ↦ ⟨(h i).1, ?_⟩, fun h i ↦ ⟨(h i).1, ?_⟩⟩ <;> have h' := h (-i)
· rw [neg_neg] at h'; exact h'.2
· exact h'.2
rw [h_inter_eq] at h_neg
exact tendsto_comp_neg_atTop_iff.mp h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 96,
"column": 4
} | {
"line": 103,
"column": 45
} | {
"line": 104,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nh_empty : ρ (s ×ˢ ∅) = 0\nh_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))\n⊢ Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑r)) atBot (𝓝 (ρ (⋂ i, s ×ˢ Iic ↑i)))",
... | [] | have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ⋂ r : ℚ, s ×ˢ Iic (r : ℝ) := by
ext1 x
push _ ∈ _
refine ⟨fun h i ↦ ⟨(h i).1, ?_⟩, fun h i ↦ ⟨(h i).1, ?_⟩⟩ <;> have h' := h (-i)
· rw [neg_neg] at h'; exact h'.2
· exact h'.2
rw [h_inter_eq] at h_neg
exact tendsto_comp_neg_atTop_iff.mp h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 15
} | {
"line": 214,
"column": 16
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nt : ℝ≥0\n⊢ HasLaw ((-B) t) (gaussianReal 0 t) P",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Real",
"Pi.instNeg",
"Real.instZero",
"Prob... | [
"case refine_1\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nt : ℝ≥0\n⊢ HasLaw (-B t) (gaussianReal 0 t) P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 54
} | {
"line": 282,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nr : ℚ\na : α\nha : ↑(condCDF ρ a) ↑r = (preCDF ρ r a).toReal\nha_le_one : ∀ (r : ℚ), preCDF ρ r a ≤ 1\n⊢ preCDF ρ r a ≠ ∞",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.... | [] | exact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 37
} | {
"line": 328,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝ : IsFiniteKernel ν\na : α\nq : ℚ\nt : β\nhbdd_below : ∀ (q : ℚ), BddBelow (range fun r ↦ f (a, t) ↑r)\nh_nonneg : ∀ (q : ℚ), 0 ≤ f... | [] | · exact le_ciInf fun r ↦ h_nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 15
} | {
"line": 256,
"column": 16
} | [
{
"pp": "case refine_2.hX\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\nth... | [
"case refine_2.hX\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✝² : IsPro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 257,
"column": 4
} | {
"line": 257,
"column": 15
} | {
"line": 257,
"column": 16
} | [
{
"pp": "case refine_2.hY\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\nth... | [
"case refine_2.hY\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✝² : IsPro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.CentralLimitTheorem | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 87
} | {
"line": 90,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nP : Measure Ω\nP' : Measure Ω'\nX : ℕ → Ω → ℝ\nY : Ω' → ℝ\ninst✝¹ : IsProbabilityMeasure P\ninst✝ : IsProbabilityMeasure P'\nhY : HasLaw Y (gaussianReal 0 1) P'\nh0 : ∫ (x : Ω), X 0 x ∂P = 0\nh1 : ∫ (x : Ω), (X 0 ^ 2) x ∂P =... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nP : Measure Ω\nP' : Measure Ω'\nX : ℕ → Ω → ℝ\nY : Ω' → ℝ\ninst✝¹ : IsProbabilityMeasure P\ninst✝ : IsProbabilityMeasure P'\nhY : HasLaw Y (gaussianReal 0 1) P'\nh0 : ∫ (x : Ω), X 0 x ∂P = 0\nh1 : ∫ (x : Ω), (X 0 ^ 2) x ∂P = 1\nhindep :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 426,
"column": 43
} | {
"line": 426,
"column": 65
} | {
"line": 426,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\ninst✝ : IsFiniteKernel κ\nf : α × β → StieltjesFunction ℝ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nx : ℝ\n⊢ ENNReal.ofReal (∫ (x_1 : β) in s, ↑(f (a, x_1)) x ∂ν ... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\ninst✝ : IsFiniteKernel κ\nf : α × β → StieltjesFunction ℝ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nx : ℝ\n⊢ ENNReal.ofReal ((κ a).real (s ×ˢ Iic x)) = (κ a) (s ×ˢ Iic x)"
] | hf.setIntegral a hs x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Basic | {
"line": 65,
"column": 75
} | {
"line": 66,
"column": 86
} | {
"line": 68,
"column": 0
} | [
{
"pp": "α : Type u_1\nΩ : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nρ : Measure (α × Ω)\nρCond : Kernel α Ω\ninst✝ : ρ.IsCondKernel ρCond\nhρ : ρ ≠ 0\n⊢ IsSFiniteKernel ρCond",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.c... | [] | by
contrapose hρ; rwa [← ρ.disintegrate ρCond, Measure.compProd_of_not_isSFiniteKernel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 61
} | {
"line": 117,
"column": 6
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nn : ℕ\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun p ↦ (κ p.1) (↑p.2 ×ˢ s)",
"ppTerm": "?refine_... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nn : ℕ\ns : Set β\nhs : MeasurableSet s\nt : ↑(countablePartition γ n)\n⊢ Measurable fun x ↦ (κ (x, t).1) (↑(x, t).2 ×ˢ... | refine measurable_from_prod_countable_left (fun t ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 219,
"column": 63
} | {
"line": 219,
"column": 74
} | {
"line": 219,
"column": 75
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nhν : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nu : Set γ\nhu : u ∈ countableParti... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nhν : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nu : Set γ\nhu : u ∈ countablePartition γ n\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 234,
"column": 35
} | {
"line": 234,
"column": 46
} | {
"line": 234,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nhν : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nu : Set γ\nhu : u ∈ countableParti... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\nhν : IsFiniteKernel ν\nn : ℕ\na : α\ns : Set β\nhs : MeasurableSet s\nu : Set γ\nhu : u ∈ countablePartition γ n\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Basic | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\nκ : Kernel α (β × Ω)\nκCond✝ : Kernel (α × β) Ω\ninst✝ : Countable α\nκCond : α → Kernel β Ω\nh_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y\nx y : α\nhx : β\nhy : y ∈ measu... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\nκ : Kernel α (β × Ω)\nκCond✝ : Kernel (α × β) Ω\ninst✝ : Countable α\nκCond : α → Kernel β Ω\nh_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y\nx y : α\nhx : β\nhy : y ∈ measurableAtom x\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Integral | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 20
} | {
"line": 54,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : CountableOrCountablyGenerated α β\nκ : Kernel α (β × Ω)\ninst✝ : IsFiniteKernel κ\na : α\ns : Set β\nhs : MeasurableSet s\nt... | [
"α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : CountableOrCountablyGenerated α β\nκ : Kernel α (β × Ω)\ninst✝ : IsFiniteKernel κ\na : α\ns : Set β\nhs : MeasurableSet s\nt : Set Ω\nht... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Integral | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 20
} | {
"line": 145,
"column": 21
} | [
{
"pp": "β : Type u_1\nΩ : Type u_2\nmβ : MeasurableSpace β\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nρ : Measure (β × Ω)\ninst✝ : IsFiniteMeasure ρ\ns : Set β\nhs : MeasurableSet s\nt : Set Ω\nht : MeasurableSet t\nthis : ρ (s ×ˢ t) = (ρ.fst ⊗ₘ ρ.condKernel) (s ×ˢ t)\n⊢ ∫... | [
"β : Type u_1\nΩ : Type u_2\nmβ : MeasurableSpace β\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nρ : Measure (β × Ω)\ninst✝ : IsFiniteMeasure ρ\ns : Set β\nhs : MeasurableSet s\nt : Set Ω\nht : MeasurableSet t\nthis : ρ (s ×ˢ t) = (ρ.fst ⊗ₘ ρ.condKernel) (s ×ˢ t)\n⊢ ∫⁻ (b : β) in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 47
} | {
"line": 361,
"column": 48
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ ... | [
"case neg.refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ i, seq i = ∅... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 31
} | {
"line": 37,
"column": 32
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ↑i ≤ b\n⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i",
"ppTerm": "?pos✝",
"assigned": true,
"used... | [
"case pos\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ↑i ≤ b\n⊢ Measurable fun c_1 ↦ c_1.1 ⟨↑i, ⋯⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 31
} | {
"line": 38,
"column": 32
} | [
{
"pp": "case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ¬↑i ≤ b\n⊢ Measurable fun c_1 ↦ IocProdIoc a b c c_1 i",
"ppTerm": "?neg✝",
"assigned": true,
"use... | [
"case neg\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : LocallyFiniteOrder ι\ninst✝¹ : DecidableLE ι\nX : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b c : ι\ni : ↥(Ioc a c)\nh : ¬↑i ≤ b\n⊢ Measurable fun c_1 ↦ c_1.2 ⟨↑i, ⋯⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 31
} | {
"line": 87,
"column": 32
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\nm n : ι\ni : ↥(Iic n)\nh : ↑i ≤ m\n⊢ Measurable fun c ↦ IicProdIoc m n c i",
"ppTerm": "?pos✝",
"as... | [
"case pos\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\nm n : ι\ni : ↥(Iic n)\nh : ↑i ≤ m\n⊢ Measurable fun c ↦ c.1 ⟨↑i, ⋯⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 31
} | {
"line": 88,
"column": 32
} | [
{
"pp": "case neg\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\nm n : ι\ni : ↥(Iic n)\nh : ¬↑i ≤ m\n⊢ Measurable fun c ↦ IicProdIoc m n c i",
"ppTerm": "?neg✝",
"a... | [
"case neg\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\nm n : ι\ni : ↥(Iic n)\nh : ¬↑i ≤ m\n⊢ Measurable fun c ↦ c.2 ⟨↑i, ⋯⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 21
} | {
"line": 109,
"column": 22
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b : ι\nhab : a ≤ b\nx : ↥(Iic b)\nh : ↑x ≤ a\n⊢ Measurable fun c ↦\n { toFun := fun x i ↦ if h : ↑i ≤ ... | [
"case pos\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b : ι\nhab : a ≤ b\nx : ↥(Iic b)\nh : ↑x ≤ a\n⊢ Measurable fun c ↦ c.1 ⟨↑x, ⋯⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 21
} | {
"line": 110,
"column": 22
} | [
{
"pp": "case neg\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b : ι\nhab : a ≤ b\nx : ↥(Iic b)\nh : ¬↑x ≤ a\n⊢ Measurable fun c ↦\n { toFun := fun x i ↦ if h : ↑i ≤... | [
"case neg\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\na b : ι\nhab : a ≤ b\nx : ↥(Iic b)\nh : ¬↑x ≤ a\n⊢ Measurable fun c ↦ c.2 ⟨↑x, ⋯⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 600,
"column": 6
} | {
"line": 600,
"column": 17
} | {
"line": 600,
"column": 18
} | [
{
"pp": "case refine_1.refine_1\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\na : α\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUn... | [
"case refine_1.refine_1\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\na : α\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 42
} | {
"line": 622,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\na : α\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ i, seq i = ∅\nhs... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nhκν : κ.fst ≤ ν\ninst✝ : IsFiniteKernel ν\na : α\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ i, seq i = ∅\nhseq_meas : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 15
} | {
"line": 665,
"column": 16
} | [
{
"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\nhx : ¬(if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1) = 1\n⊢ (κ.fst a) (coun... | [
"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\nhx : ¬(if (κ.fst a) (countablePartitionSet n x) = 0 then 0 else 1) = 1\n⊢ (κ.fst a) (countablePartiti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 36
} | {
"line": 142,
"column": 36
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\nu : Set ι\nhu : u.Finite\np : ↑I\nk : ℕ\nthis : {s | s ⊆ u ∧ s.ncard ∈ {k}}.Finite\nh1 : ∀ s ∈ this.toFinset, setBer(u, p).real {s} = ↑p ^ k * (1 - ↑p) ^ (u.ncard - k)\n⊢ ↑(#this.toFinset) * (↑p ^ k * (1 - ↑p) ^ (u.ncard - k)) = ↑(u.ncard.choose k) * (↑p ^ k * (1 - ↑p... | [
"ι : Type u_1\ninst✝ : Countable ι\nu : Set ι\nhu : u.Finite\np : ↑I\nk : ℕ\nthis : {s | s ⊆ u ∧ s.ncard ∈ {k}}.Finite\nh1 : ∀ s ∈ this.toFinset, setBer(u, p).real {s} = ↑p ^ k * (1 - ↑p) ^ (u.ncard - k)\n⊢ ↑{s | s ⊆ u ∧ s.ncard ∈ {k}}.ncard * (↑p ^ k * (1 - ↑p) ^ (u.ncard - k)) =\n ↑(u.ncard.choose k) * (↑p ^ k... | ← Set.ncard_eq_toFinset_card _ _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 156,
"column": 4
} | {
"line": 157,
"column": 12
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\np : ↑I\ninst✝ : Countable ι\ns : Set (Set ι)\nhs : MeasurableSet s\nh : ∅ ∈ s\n⊢ setBer(∅, p) {s_1 | s_1 ∈ s ∧ s_1 ⊆ ∅} = (dirac ∅) s",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"ENNReal.ofNNReal",
"MeasureTheory.Measur... | [] | have : {t | t ∈ s ∧ t ⊆ ∅} = {∅} := by grind
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 156,
"column": 4
} | {
"line": 157,
"column": 12
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\np : ↑I\ninst✝ : Countable ι\ns : Set (Set ι)\nhs : MeasurableSet s\nh : ∅ ∈ s\n⊢ setBer(∅, p) {s_1 | s_1 ∈ s ∧ s_1 ⊆ ∅} = (dirac ∅) s",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"ENNReal.ofNNReal",
"MeasureTheory.Measur... | [] | have : {t | t ∈ s ∧ t ⊆ ∅} = {∅} := by grind
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 59,
"column": 27
} | {
"line": 59,
"column": 96
} | {
"line": 59,
"column": 97
} | [
{
"pp": "V : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (Sym2 V)\nhS : S ⊆ Sym2.diagSetᶜ\n⊢ (fromEdgeSet S).edgeSet = S",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.edgeSet_fromEdgeSet",
"congrArg",
"SimpleGraph.fromEdgeSet",
"sdi... | [
"V : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (Sym2 V)\nhS : S ⊆ Sym2.diagSetᶜ\n⊢ Disjoint S Sym2.diagSet"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 73
} | {
"line": 77,
"column": 74
} | [
{
"pp": "V : Type u_1\np : ↑I\ninst✝ : Countable V\ns : Set (Sym2 V)\nhs : s ⊆ Sym2.diagSetᶜ\n⊢ (fromEdgeSet s).edgeSet = s",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.edgeSet_fromEdgeSet",
"congrArg",
"SimpleGraph.fromEdgeSet",
"sdi... | [
"V : Type u_1\np : ↑I\ninst✝ : Countable V\ns : Set (Sym2 V)\nhs : s ⊆ Sym2.diagSetᶜ\n⊢ Disjoint s Sym2.diagSet"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 145,
"column": 51
} | {
"line": 145,
"column": 87
} | {
"line": 145,
"column": 87
} | [
{
"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\ns : (i : ↥(Iic b)) → Set (X ↑i)\nms : ∀ (i : ↥(Iic b)), MeasurableSet (s i)\n⊢ ∏ x ∈ Iic a ∪ Ioc a b, Function.extend Subtype.val (fun i ↦ (μ ... | [
"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\ns : (i : ↥(Iic b)) → Set (X ↑i)\nms : ∀ (i : ↥(Iic b)), MeasurableSet (s i)\n⊢ (∏ x ≤ a, Function.extend Subtype.val (fun i ↦ (μ ↑i) (s i)) 1 x) *\n ... | prod_union (Iic_disjoint_Ioc le_rfl) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.WithDensity | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 24
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β → ℝ≥0∞\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nhf : Measurable (Function.uncurry f)\na : α\ng : β → ℝ≥0∞\nhg : Measurable g\n⊢ ∫⁻ (a_1 : β), (f a * g) a_1 ∂κ a = ∫⁻ (b : β), f a b * g b ∂κ a",
"ppTerm": "?m.61... | [] | simp_rw [Pi.mul_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 41
} | {
"line": 302,
"column": 42
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_b... | [
"X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nf : ℕ → ((n : ℕ) → X n) → ℝ≥0∞\na : ℕ → ℕ\nhcte : ∀ (n : ℕ), DependsOn (f n) ↑(Iic (a n))\nmf : ∀ (n : ℕ), Measurable (f n)\nbound : ℝ≥0∞\nfin_bound : bound... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 41
} | {
"line": 359,
"column": 42
} | [
{
"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\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 190,
"column": 35
} | {
"line": 190,
"column": 52
} | {
"line": 190,
"column": 53
} | [
{
"pp": "X : 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\nhba : b ≤ a\n⊢ IsEmpty ↥(Ioc a b)",
"ppTerm": "?m.523",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Fins... | [
"X : 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\nhba : b ≤ a\n⊢ IsEmpty { x // False }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 41
} | {
"line": 227,
"column": 42
} | [
{
"pp": "X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\nhμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)\nA : Set ((n : ℕ) → X n)\nhA : A ∈ measurableCylinders X\n⊢ ∃ s S, MeasurableSet S ∧ A = cylinder s S",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
... | [
"X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\nhμ : ∀ (n : ℕ), IsProbabilityMeasure (μ n)\nA : Set ((n : ℕ) → X n)\nhA : A ∈ measurableCylinders X\n⊢ ∃ s S, MeasurableSet S ∧ A = cylinder s S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 214,
"column": 79
} | {
"line": 215,
"column": 52
} | {
"line": 216,
"column": 2
} | [
{
"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\nthis :\n (((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal) a) {x | ... | [] | by
rwa [withDensity_one_sub_rnDerivAux κ η] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 96
} | {
"line": 257,
"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\na : α\nx : γ\n⊢ κ.rnDeriv η a x = ∞ ↔ x ∈ κ.mutuallySingularSetSlice η a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq... | [] | rw [rnDeriv_eq_top_iff, mutuallySingularSet, mutuallySingularSetSlice, mem_ofPred, mem_ofPred] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 96
} | {
"line": 257,
"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\na : α\nx : γ\n⊢ κ.rnDeriv η a x = ∞ ↔ x ∈ κ.mutuallySingularSetSlice η a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq... | [] | rw [rnDeriv_eq_top_iff, mutuallySingularSet, mutuallySingularSetSlice, mem_ofPred, mem_ofPred] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 96
} | {
"line": 257,
"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\na : α\nx : γ\n⊢ κ.rnDeriv η a x = ∞ ↔ x ∈ κ.mutuallySingularSetSlice η a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq... | [] | rw [rnDeriv_eq_top_iff, mutuallySingularSet, mutuallySingularSetSlice, mem_ofPred, mem_ofPred] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 426,
"column": 33
} | {
"line": 426,
"column": 78
} | {
"line": 426,
"column": 79
} | [
{
"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\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 363,
"column": 6
} | {
"line": 364,
"column": 28
} | {
"line": 365,
"column": 6
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ\nthis :\n η.wi... | [
"α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ\nthis :\n η.withDensity (κ... | · rw [ne_eq, sub_eq_zero]
exact (hs' x hx).ne' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 17
} | {
"line": 454,
"column": 18
} | [
{
"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\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Deterministic | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 26
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nh : IsDeterministic κ\na : α\n⊢ IsZeroOneMeasure (κ a)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"MeasureTheory.IsZeroOneMeasure.mk",
"Mea... | [
"case mp\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nh : IsDeterministic κ\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ (κ a) s = 0 ∨ (κ a) s = 1"
] | refine ⟨fun s hs ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Deterministic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 37
} | {
"line": 135,
"column": 2
} | [
{
"pp": "α : 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 : Measurabl... | [
"α : 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 : MeasurableSet t\n⊢ ∫⁻... | rw [comp_apply' _ _ _ (hs.prod ht)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 501,
"column": 6
} | {
"line": 501,
"column": 43
} | {
"line": 501,
"column": 44
} | [
{
"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\nt : Set ((n : ℕ) → X n)\nht : t ∈ measurableCylinders X\n⊢ ∃ N S, MeasurableSet S ∧ t = cylinder (Iic N) S",
"ppTerm": "?m.93",
"... | [
"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\nt : Set ((n : ℕ) → X n)\nht : t ∈ measurableCylinders X\n⊢ ∃ N S, MeasurableSet S ∧ t = cylinder (Iic N) S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 55
} | {
"line": 507,
"column": 56
} | [
{
"pp": "case compl\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)\na : ℕ\nt : Set ((n : ℕ) → X n)\nmt : MeasurableSet t\nht : Measurable fun x₀ ↦ (trajFun κ a x₀) t\nthis : ∀ (x₀ : (i : ↥(Iic a)) → X... | [
"case compl\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)\na : ℕ\nt : Set ((n : ℕ) → X n)\nmt : MeasurableSet t\nht : Measurable fun x₀ ↦ (trajFun κ a x₀) t\nthis : ∀ (x₀ : (i : ↥(Iic a)) → X ↑i), IsProb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 40
} | {
"line": 508,
"column": 41
} | [
{
"pp": "case union\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)\na : ℕ\nf : ℕ → Set ((n : ℕ) → X n)\ndisf : Pairwise (Disjoint on f)\nmf : ∀ (i : ℕ), MeasurableSet (f i)\nhf : ∀ (i : ℕ), Measurable... | [
"case union\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)\na : ℕ\nf : ℕ → Set ((n : ℕ) → X n)\ndisf : Pairwise (Disjoint on f)\nmf : ∀ (i : ℕ), MeasurableSet (f i)\nhf : ∀ (i : ℕ), Measurable fun x₀ ↦ (t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 447,
"column": 6
} | {
"line": 447,
"column": 17
} | {
"line": 447,
"column": 18
} | [
{
"pp": "case ha.hm\nι✝ : Type u_1\nX✝ : ι✝ → Type u_2\nmX✝ : (i : ι✝) → MeasurableSpace (X✝ i)\nμ✝ : (i : ι✝) → Measure (X✝ i)\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)\ns : Set ι\nhs : Countable ↑s\nt : (i : ι... | [
"case ha.hm\nι✝ : Type u_1\nX✝ : ι✝ → Type u_2\nmX✝ : (i : ι✝) → MeasurableSpace (X✝ i)\nμ✝ : (i : ι✝) → Measure (X✝ i)\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)\ns : Set ι\nhs : Countable ↑s\nt : (i : ι) → Set (X i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 153,
"column": 28
} | {
"line": 153,
"column": 66
} | {
"line": 153,
"column": 67
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.id ∥ₖ Kernel.deterministic f hf ∘ₖ (Kern... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.id ∘ₖ Kernel.deterministic f hf ∥ₖ (Kernel.determini... | Kernel.parallelComp_comp_parallelComp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 155,
"column": 28
} | {
"line": 155,
"column": 62
} | {
"line": 155,
"column": 62
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.deterministic f hf ∥ₖ Kernel.determinist... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.copy 𝓧 ∘ₖ Kernel.deterministic f hf) ∘ₘ μ = ⇑(Kerne... | Kernel.parallelComp_self_comp_copy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 15
} | {
"line": 164,
"column": 16
} | [
{
"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\nν : Measure 𝓧\ninst✝ : SFinite ν\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν\nthis : (⇑κ ∘ₘ μ) ⊗ₘ... | [
"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\nν : Measure 𝓧\ninst✝ : SFinite ν\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν\nthis : (⇑κ ∘ₘ μ) ⊗ₘ κ†μ ≪ ν ⊗ₘ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | {
"line": 216,
"column": 16
} | [
{
"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... | [
"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 μ, (κ†μ) b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 117,
"column": 77
} | {
"line": 117,
"column": 88
} | {
"line": 117,
"column": 89
} | [
{
"pp": "Θ : 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 : 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 : Θ), ℓ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Binomial | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "Ω : Type u_2\nm : MeasurableSpace Ω\nP : Measure Ω\nn : ℕ\np : ↑I\nX : Ω → ℕ\nhX : HasLaw X Bin(n, p) P\ns : Set ℕ\nhs : s ⊆ Iio n\n⊢ s.ncard ≤ n",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_2\nm : MeasurableSpace Ω\nP : Measure Ω\nn : ℕ\np : ↑I\nX : Ω → ℕ\nhX : HasLaw X Bin(n, p) P\ns : Set ℕ\nhs : s ⊆ Iio n\n⊢ s.ncard ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Cauchy | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 31
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case pos\nx₀ : ℝ\nγ : ℝ≥0\nh : γ = 0\n⊢ Integrable 0 volume",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
"PseudoMetricSpace.toUniformSpace",
"Real.measureSpace",
"Real.normedAddCommGroup",
... | [] | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
... | [] | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
... | [] | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
... | [] | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 70
} | {
"line": 257,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s ∩ p.support = t ∩ p.support\n⊢ p.toMeasure s = p.toMeasure t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Mea... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s ∩ p.support = t ∩ p.support\n⊢ p.toOuterMeasure s = p.toOuterMeasure t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 335,
"column": 4
} | {
"line": 336,
"column": 53
} | {
"line": 336,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\n⊢ p.toMeasure Set.univ = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicato... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\n⊢ ∑' (x : α), p x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gamma | {
"line": 146,
"column": 81
} | {
"line": 149,
"column": 48
} | {
"line": 151,
"column": 0
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\nx : ℝ\n⊢ ↑(cdf (gammaMeasure a r)) x = (∫⁻ (x : ℝ) in Iic x, gammaPDF a r x).toReal",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"instClosedIicTopology",
"Real",
"Measu... | [] | by
have : IsProbabilityMeasure (gammaMeasure a r) := isProbabilityMeasure_gammaMeasure ha hr
simp only [gammaPDF, cdf_eq_real]
simp [gammaMeasure, gammaPDF, measureReal_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑n * p n * (1 / ↑n)) =ᶠ[atTop] p\n⊢ Tendsto p atTop (𝓝 0)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ → ℝ\nr : ℝ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑n * p n * (1 / ↑n)) =ᶠ[atTop] p\n⊢ Tendsto p atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 30
} | {
"line": 64,
"column": 31
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑(n.choose k) * p n ^ k) ~[atTop] fun n ↦ (↑n * p n) ^ k / ↑k.factorial\n⊢ Tendsto (fun n ↦ (↑n * p n) ^ k / ↑k.factorial) atTop (𝓝 (r ^ k / ↑k.factorial))",
"ppTerm": "?m.155",
"assigned": true,
"usedCo... | [
"p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑(n.choose k) * p n ^ k) ~[atTop] fun n ↦ (↑n * p n) ^ k / ↑k.factorial\n⊢ Tendsto (fun n ↦ (↑n * p n) ^ k * (↑k.factorial)⁻¹) atTop (𝓝 (r ^ k * (↑k.factorial)⁻¹))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "case refine_1\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 ↦ ↑n * -p n) a... | [
"case refine_1\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 x ↦ ↑x * p x) atTop (𝓝 r)"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 76
} | {
"line": 142,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\ng : β → PMF γ\nb : γ\n⊢ ((p.bind f).bind g) b = (p.bind fun a ↦ (f a).bind g) b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"ENNReal.tsum_mul_left",
"HMul.h... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\ng : β → PMF γ\nb : γ\n⊢ ∑' (a : β) (i : α), p i * ((f i) a * (g a) b) = ∑' (a : α) (i : β), p a * ((f a) i * (g i) b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 147,
"column": 4
} | {
"line": 148,
"column": 76
} | {
"line": 148,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nq : PMF β\nf : α → β → PMF γ\nb : γ\n⊢ (p.bind fun a ↦ q.bind (f a)) b = (q.bind fun b ↦ p.bind fun a ↦ f a b) b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.tsum_mul_left",
"HMul.hMul",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nq : PMF β\nf : α → β → PMF γ\nb : γ\n⊢ ∑' (a : α) (i : β), p a * (q i * (f a i) b) = ∑' (a : β) (i : α), p i * (q a * (f i a) b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 16
} | [
{
"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) atT... | [
"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) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 62
} | {
"line": 103,
"column": 4
} | [
{
"pp": "k : ℕ\nr : ℝ≥0\np : ℕ → ↑unitInterval\nhr : Tendsto (fun n ↦ ↑n * ↑(p n)) atTop (𝓝 ↑r)\nt1 : Tendsto (fun n ↦ ENNReal.ofReal (↑(n.choose k) * ↑(p n) ^ k * (1 - ↑(p n)) ^ (n - k))) atTop (𝓝 (Po(r) {k}))\n⊢ (fun n ↦ ENNReal.ofReal (↑(n.choose k) * ↑(p n) ^ k * (1 - ↑(p n)) ^ (n - k))) =ᶠ[atTop] fun n ↦... | [
"k : ℕ\nr : ℝ≥0\np : ℕ → ↑unitInterval\nhr : Tendsto (fun n ↦ ↑n * ↑(p n)) atTop (𝓝 ↑r)\nt1 : Tendsto (fun n ↦ ENNReal.ofReal (↑(n.choose k) * ↑(p n) ^ k * (1 - ↑(p n)) ^ (n - k))) atTop (𝓝 (Po(r) {k}))\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → ENNReal.ofReal (↑(b.choose k) * ↑(p b) ^ k * (1 - ↑(p b)) ^ (b - k)) = Bin(b, p b) {... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 29
} | {
"line": 124,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nq : PMF (α → β)\np : PMF α\nb : β\nf : α → β\na : α\n⊢ (q f * if b = f a then p a else 0) = if b = f a then q f * p a else 0",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nq : PMF (α → β)\np : PMF α\nb : β\nf : α → β\na : α\n⊢ (q f * if b = f a then p a else 0) = if b = f a then q f * p a else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 175,
"column": 22
} | {
"line": 175,
"column": 51
} | {
"line": 175,
"column": 52
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\ns : Finset α\nh : ∑ a ∈ s, f a = 1\nh' : ∀ a ∉ s, f a = 0\na : α\n⊢ a ∈ (ofFinset f s h h').support ↔ a ∈ ↑s ∩ Function.support f",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Function.mem_suppor... | [
"α : Type u_1\nf : α → ℝ≥0∞\ns : Finset α\nh : ∑ a ∈ s, f a = 1\nh' : ∀ a ∉ s, f a = 0\na : α\n⊢ ¬f a = 0 → a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 268,
"column": 36
} | {
"line": 268,
"column": 47
} | {
"line": 268,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns : Set α\nh : ∃ a ∈ s, a ∈ p.support\n⊢ tsum (s.indicator ⇑p) ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"P... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns : Set α\nh : ∃ a ∈ s, a ∈ p.support\n⊢ ∃ x ∈ s, ¬p x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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