module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 426,
"column": 43
} | {
"line": 426,
"column": 65
} | [
{
"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 ... | hf.setIntegral a hs x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 40
} | [
{
"pp": "Ω : 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\n⊢ HasDerivAt (fun z ↦ ∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ n * cexp (z * ↑(X ω))) x ... | let t := ((z.re - l) ⊓ (u - z.re)) / 2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"pp": "case h.e'_6.h\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nh : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) μ\nt : ℝ\n⊢ t ∈ integrableExpSet X μ ↔ t ∈ Set.univ",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 126,
"column": 2
} | {
"line": 129,
"column": 36
} | [
{
"pp": "case refine_1\nΩ✝ : Type u_1\nm✝ : MeasurableSpace Ω✝\nX✝ : Ω✝ → ℝ\nμ✝ : Measure Ω✝\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt₁ : ℝ\nht₁ : t₁ ∈ integrableExpSet X μ\nt₂ : ℝ\nht₂ : t₂ ∈ integrableExpSet X μ\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh_le : t₁ ≤ t₂\n⊢ t₁ ≤ ... | · simp only [smul_eq_mul]
calc t₁
_ = a * t₁ + b * t₁ := by rw [← add_mul, hab, one_mul]
_ ≤ a * t₁ + b * t₂ := by gcongr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 154,
"column": 25
} | {
"line": 154,
"column": 36
} | [
{
"pp": "Ω : 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\n⊢ Integrable (fun a ↦ rexp ((v - t) * X a)) μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 36
} | [
{
"pp": "case neg\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\n⊢ deriv (deriv (cgf X μ)) v = (∫ (x : Ω), (fun ω ↦ X ω ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2",
"usedConstants": [
"Real",
"IsOp... | have h_mem : ∀ᶠ y in 𝓝 v, y ∈ interior (integrableExpSet X μ) :=
isOpen_interior.eventually_mem h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 15
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\nh : Integrable (fun ω ↦ rexp (t * |X ω| + 0 * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp (t * |X ω|)) μ",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 15
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp ((0 + t) * X ω)) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 15
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 192,
"column": 52
} | {
"line": 192,
"column": 63
} | [
{
"pp": "Ω : 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\nht_nonpos : t ≤ 0\n⊢ Integrable (fun ω ↦ rexp ((v - -t) * X ω)) μ",
"usedConstants": [
"AddGroup.toSubtra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 203,
"column": 48
} | {
"line": 203,
"column": 59
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\nht_nonpos : t ≤ 0\n⊢ Integrable (fun ω ↦ rexp (- -t * X ω)) μ",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 213,
"column": 65
} | {
"line": 213,
"column": 76
} | [
{
"pp": "x t p : ℝ\nhp : 0 ≤ p\nht : 0 < t\nhp_zero : ¬p = 0\nh_x_le : ∀ (c : ℝ), 0 < c → x ≤ c⁻¹ * rexp (c * x)\nc : ℝ\nhc : 0 < c\n⊢ -x ≤ c⁻¹ * rexp (-c * x)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 229,
"column": 4
} | {
"line": 234,
"column": 64
} | [
{
"pp": "x t p : ℝ\nhp : 0 ≤ p\nht : 0 < t\nhp_zero : ¬p = 0\nh_x_le : ∀ (c : ℝ), 0 < c → x ≤ c⁻¹ * rexp (c * x)\nh_neg_x_le : ∀ (c : ℝ), 0 < c → -x ≤ c⁻¹ * rexp (-c * x)\nh_abs_le : ∀ (c : ℝ), 0 < c → |x| ≤ c⁻¹ * max (rexp (c * x)) (rexp (-c * x))\n⊢ ((t / p)⁻¹ * max (rexp (t / p * x)) (rexp (-t / p * x))) ^ p... | rw [mul_rpow (by positivity) (by positivity)]
congr
· simp
· rw [rpow_max (by positivity) (by positivity) hp, ← exp_mul, ← exp_mul]
ring_nf
congr <;> rw [mul_assoc, mul_inv_cancel₀ hp_zero, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 229,
"column": 4
} | {
"line": 234,
"column": 64
} | [
{
"pp": "x t p : ℝ\nhp : 0 ≤ p\nht : 0 < t\nhp_zero : ¬p = 0\nh_x_le : ∀ (c : ℝ), 0 < c → x ≤ c⁻¹ * rexp (c * x)\nh_neg_x_le : ∀ (c : ℝ), 0 < c → -x ≤ c⁻¹ * rexp (-c * x)\nh_abs_le : ∀ (c : ℝ), 0 < c → |x| ≤ c⁻¹ * max (rexp (c * x)) (rexp (-c * x))\n⊢ ((t / p)⁻¹ * max (rexp (t / p * x)) (rexp (-t / p * x))) ^ p... | rw [mul_rpow (by positivity) (by positivity)]
congr
· simp
· rw [rpow_max (by positivity) (by positivity) hp, ← exp_mul, ← exp_mul]
ring_nf
congr <;> rw [mul_assoc, mul_inv_cancel₀ hp_zero, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 288,
"column": 8
} | {
"line": 288,
"column": 24
} | [
{
"pp": "case h.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",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 17
} | [
{
"pp": "case neg.refine_1.right\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nY : Ω' → ℝ\nμ' : Measure Ω'\nhXY : mgf X μ = mgf Y μ'\nhμμ' : μ = 0 ↔ μ' = 0\nt : ℝ\nht : t ∈ interior (integrableExpSet Y μ')\nhX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | [
{
"pp": "μ μ' : Measure ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure μ'\nh : complexMGF id μ = complexMGF id μ'\n⊢ μ = μ'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 13
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ Measure.map (fun x ↦ -x) (gaussianReal μ v) = gaussianReal (-μ) v",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 49
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nc : ℝ\n⊢ Measure.map (fun x ↦ x * c⁻¹) (gaussianReal μ v) = gaussianReal (μ * c⁻¹) (v * (NNReal.mk (c ^ 2) ⋯)⁻¹)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"MeasureTheory.Measure",
... | convert! gaussianReal_map_mul_const c⁻¹ using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 15
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\nh : Integrable (fun ω ↦ |X ω| ^ p * rexp (0 * X ω)) μ\n⊢ Integrable (fun ω ↦ |X ω|... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 15
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 + t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 15
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 15
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\nh : Integrable (fun ω ↦ X ω ^ p * rexp (0 * X ω)) μ\n⊢ Integrable (fun ω ↦ X ω ^ p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 15
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 + t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 421,
"column": 6
} | {
"line": 426,
"column": 10
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ ∫ (x : ℝ), ↑(gaussianPDFReal μ v x) * cexp (z * ↑x) =\n ↑(√(2 * π * ↑v))⁻¹ * ∫ (x : ℝ), cexp (-↑↑(2 * v)⁻¹ * ↑x ^ 2 + (z + ↑μ / ↑↑v) * ↑x + -↑μ ^ 2 / (2 * ↑↑v))",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.... | unfold gaussianPDFReal
push_cast
simp_rw [mul_assoc, integral_const_mul, ← Complex.exp_add]
congr with x
congr 1
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 421,
"column": 6
} | {
"line": 426,
"column": 10
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ ∫ (x : ℝ), ↑(gaussianPDFReal μ v x) * cexp (z * ↑x) =\n ↑(√(2 * π * ↑v))⁻¹ * ∫ (x : ℝ), cexp (-↑↑(2 * v)⁻¹ * ↑x ^ 2 + (z + ↑μ / ↑↑v) * ↑x + -↑μ ^ 2 / (2 * ↑↑v))",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.... | unfold gaussianPDFReal
push_cast
simp_rw [mul_assoc, integral_const_mul, ← Complex.exp_add]
congr with x
congr 1
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 429,
"column": 38
} | {
"line": 429,
"column": 49
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ (-↑↑(2 * v)⁻¹).re < 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 15
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 32
} | [
{
"pp": "case h\nμ : ℝ\nv : ℝ≥0\nx✝ : ℝ\n⊢ x✝ ∈ integrableExpSet id (gaussianReal μ v) ↔ x✝ ∈ Set.univ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"PseudoMetricSpace... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 55
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ Var[fun x ↦ x; gaussianReal μ v] = ↑v",
"usedConstants": [
"ProbabilityTheory.variance_eq_integral",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Measurable.aemeasurable",
"Real.instRCLike",
"congrArg",
"Real.instSub",
... | 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.Moments.IntegrableExpMul | {
"line": 455,
"column": 6
} | {
"line": 455,
"column": 24
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv l u : ℝ\nhvlu : v ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\nt : ℝ := min (v - l) (u - v) / 2\nh_pos : 0 < min (v - l) (u - v)\nht : 0 < t\nhvt : v + t = 0\nh_eq : v = t\n⊢ t = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 629,
"column": 39
} | {
"line": 629,
"column": 59
} | [
{
"pp": "case hf.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",
"usedConstants": [
"False",
"Real",
"MeasureTheory.M... | simp [NeZero.ne, hX] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 556,
"column": 2
} | {
"line": 556,
"column": 13
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nh : 0 ∈ interior (integrableExpSet X μ)\n⊢ Integrable X μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 568,
"column": 4
} | {
"line": 568,
"column": 34
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhX : AEMeasurable X μ\nhz : z.re ∈ integrableExpSet X μ\n⊢ Integrable (fun a ↦ ‖cexp (z * ↑(X a))‖) μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Measu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 583,
"column": 2
} | {
"line": 584,
"column": 9
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\np : ℝ\nhp : 0 ≤ p\nhX : AEMeasurable X μ\n⊢ Integrable (fun a ↦ ‖↑(|X a| ^ p) * cexp (z * ↑(X a))‖) μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 610,
"column": 2
} | {
"line": 611,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\n⊢ Integrable (fun ω ↦ ↑(X ω) ^ n * cexp (z * ↑(X ω))) μ",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"... | convert! integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet hz (Nat.cast_nonneg n)
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 610,
"column": 2
} | {
"line": 611,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\n⊢ Integrable (fun ω ↦ ↑(X ω) ^ n * cexp (z * ↑(X ω))) μ",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"... | convert! integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet hz (Nat.cast_nonneg n)
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.Basic | {
"line": 65,
"column": 75
} | {
"line": 66,
"column": 86
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose₂",
"MeasureTheory.Measure",
... | by
contrapose hρ; rwa [← ρ.disintegrate ρCond, Measure.compProd_of_not_isSFiniteKernel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.CentralLimitTheorem | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 87
} | [
{
"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 =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Basic | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Integral | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 20
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 61
} | [
{
"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)",
"usedConstants": [
... | refine measurable_from_prod_countable_left (fun t ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Disintegration.Integral | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 20
} | [
{
"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⊢ ∫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 219,
"column": 63
} | {
"line": 219,
"column": 74
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 234,
"column": 35
} | {
"line": 234,
"column": 46
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 13
} | [
{
"pp": "α : Type u_1\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\nY : α → Ω\n⊢ ⇑(condDistrib Y Y μ) =ᶠ[ae (Measure.map Y μ)] ⇑Kernel.id",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 31
} | [
{
"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",
"usedConstants": [
"dite_cond_eq_true",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 31
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"of_eq_f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 31
} | [
{
"pp": "case pos\nι : Type u_1\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrder ι\ninst✝² : DecidableLE ι\nX : ι → Type u_2\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : (i : ι) → MeasurableSpace (X i)\nm n : ι\ni : ↥(Iic n)\nh : ↑i ≤ m\n⊢ Measurable fun c ↦ IicProdIoc m n c i",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 31
} | [
{
"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",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 21
} | [
{
"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 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Maps | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 21
} | [
{
"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 ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 47
} | [
{
"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 : ⋂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 600,
"column": 6
} | {
"line": 600,
"column": 17
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 42
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 128,
"column": 4
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case pos\nι : Type u_1\np : ↑I\ninst✝ : Countable ι\ns : Set (Set ι)\nhs : MeasurableSet s\nh : ∅ ∈ s\n⊢ setBer(∅, p) {s_1 | s_1 ∈ s ∧ s_1 ⊆ ∅} = (dirac ∅) s",
"usedConstants": [
"subset_refl._simp_1",
"MulOne.toOne",
"ENNReal.ofNNReal",
"MeasureTheory.Measure",
"HMul.... | have : {t | t ∈ s ∧ t ⊆ ∅} = {∅} := by grind
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 128,
"column": 4
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case pos\nι : Type u_1\np : ↑I\ninst✝ : Countable ι\ns : Set (Set ι)\nhs : MeasurableSet s\nh : ∅ ∈ s\n⊢ setBer(∅, p) {s_1 | s_1 ∈ s ∧ s_1 ⊆ ∅} = (dirac ∅) s",
"usedConstants": [
"subset_refl._simp_1",
"MulOne.toOne",
"ENNReal.ofNNReal",
"MeasureTheory.Measure",
"HMul.... | 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": 95
} | [
{
"pp": "V : Type u_1\np : ↑I\ninst✝ : Countable V\nS : Set (Sym2 V)\nhS : S ⊆ Sym2.diagSetᶜ\n⊢ (fromEdgeSet S).edgeSet = S",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.edgeSet_fromEdgeSet",
"congrArg",
"SimpleGraph.fromEdgeSet",
"sdiff_eq_left._simp_1",
"BooleanAlgebra.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 72
} | [
{
"pp": "case h\nV : Type u_1\np : ↑I\ninst✝ : Countable V\ns : Set (Sym2 V)\nhs : s ⊆ Sym2.diagSetᶜ\n⊢ (fromEdgeSet s).edgeSet = s",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.edgeSet_fromEdgeSet",
"congrArg",
"SimpleGraph.fromEdgeSet",
"sdiff_eq_left._simp_1",
"BooleanAl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 144,
"column": 51
} | {
"line": 144,
"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 ... | prod_union (Iic_disjoint_Ioc le_rfl) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 41
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 189,
"column": 35
} | {
"line": 189,
"column": 52
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Finset",
"Nat.instLocallyFiniteOrder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 117,
"column": 77
} | {
"line": 117,
"column": 88
} | [
{
"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, ∫⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 41
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 41
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.TwoValued | {
"line": 46,
"column": 12
} | {
"line": 46,
"column": 70
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\n⊢ ∀ᵐ (ω : Ω) ∂μ, 1 - X ω = 0 ∨ 1 - X ω = 1",
"usedConstants": [
"MeasureTheory.ae",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real",
"Measu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Binomial | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 13
} | [
{
"pp": "case h\nΩ : 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 ⊆ Set.Iio n\n⊢ s.ncard ≤ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Cauchy | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 31
} | [
{
"pp": "case pos\nx₀ : ℝ\nγ : ℝ≥0\nh : γ = 0\n⊢ Integrable 0 volume",
"usedConstants": [
"Real",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
"PseudoMetricSpace.toUniformSpace",
"Real.measureSpace",
"Real.normedAddCommGroup",
"MeasureTheory.integrable_zero",
"Me... | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Distributions.TwoValued | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 12
} | [
{
"pp": "case inr\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsZeroOrProbabilityMeasure μ\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhμ : IsProbabilityMeasure μ\n⊢ Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 426,
"column": 33
} | {
"line": 426,
"column": 78
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProductMeasure | {
"line": 444,
"column": 6
} | {
"line": 444,
"column": 17
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gamma | {
"line": 146,
"column": 81
} | {
"line": 149,
"column": 48
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"instClosedIicTopology",
"Real",
"MeasureTheory.Measure",
"ProbabilityTheory.g... | 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.Kernel.IonescuTulcea.Traj | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 17
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 501,
"column": 6
} | {
"line": 501,
"column": 43
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nt : Set ((n : ℕ) → X n)\nht : t ∈ measurableCylinders X\n⊢ ∃ N S, MeasurableSet S ∧ t = cylinder (Iic N) S",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 55
} | [
{
"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 : (fun t ht ↦ Measurable fun x₀ ↦ (trajFun κ a x₀) t) t mt\nthis : ∀ (x₀ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 40
} | [
{
"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), (fun t ht ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 89
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 20
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Fernique | {
"line": 403,
"column": 8
} | {
"line": 403,
"column": 19
} | [
{
"pp": "case h.a.hbc.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.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 160,
"column": 12
} | {
"line": 160,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 160,
"column": 12
} | {
"line": 160,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 160,
"column": 12
} | {
"line": 160,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 160,
"column": 12
} | {
"line": 160,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 160,
"column": 12
} | {
"line": 160,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 160,
"column": 12
} | {
"line": 160,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 161,
"column": 12
} | {
"line": 161,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 15
} | [
{
"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 ↦ ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic | {
"line": 165,
"column": 66
} | {
"line": 165,
"column": 77
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 176,
"column": 19
} | {
"line": 176,
"column": 34
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 176,
"column": 58
} | {
"line": 176,
"column": 73
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 236,
"column": 8
} | {
"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... | Measure.map_apply (by fun_prop) (measurableSet_singleton _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.FiniteDimensionalLaws | {
"line": 64,
"column": 97
} | {
"line": 73,
"column": 47
} | [
{
"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.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 15
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : InnerProductSpace ℝ E\nhX... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
"PMF.instFunLike",
"Set.indicato... | 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
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
"PMF.instFunLike",
"Set.indicato... | 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
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
"PMF.instFunLike",
"Set.indicato... | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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