mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case refine_4
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
⊢ |X ω| ^ (n + 1) * rexp (ε * ↑(X ω)).re ≤ |X ω| ^ (n + 1) * rexp (z.re * X ω + t / 2 * |X ω|)

Tactic:
]

Goal state after:
case refine_4
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
⊢ |X ω| ^ (n + 1) * rexp (ε * ↑(X ω)).re ≤ |X ω| ^ (n + 1) * rexp (z.re * X ω + t / 2 * |X ω|)
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
x y : ℝ
⊢ gaussianPDFReal μ v (x + y) = gaussianPDFReal μ v (x + y)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |X a| ^ p * rexp (x * |X a|) ≤ (p / (|t| - x)) ^ p * rexp ((|t| - x + x) * |X a|)

Tactic:
]

Goal state after:
case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |X a| ^ p * rexp (x * |X a|) ≤ (p / (|t| - x)) ^ p * rexp ((|t| - x + x) * |X a|)
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (√↑v)⁻¹ * (√(2 * π))⁻¹ * rexp (-(c⁻¹ * x - μ) ^ 2 / (2 * ↑v)) =
    |c| * (√(2 * π * (c ^ 2 * ↑v)))⁻¹ * rexp (-(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)))

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (√↑v)⁻¹ * (√(2 * π))⁻¹ * rexp (-(c⁻¹ * x - μ) ^ 2 / (2 * ↑v)) =
    |c| * (√(2 * π * (c ^ 2 * ↑v)))⁻¹ * rexp (-(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)))
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |X a| ^ p * rexp (x * |X a|) ≤ (p / (|t| - x)) ^ p * rexp ((|t| - x) * |X a|) * rexp (x * |X a|)

Tactic:
]

Goal state after:
case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |X a| ^ p * rexp (x * |X a|) ≤ (p / (|t| - x)) ^ p * rexp ((|t| - x) * |X a|) * rexp (x * |X a|)
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: case refine_1
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ |c| / √↑v = |c| / √↑v

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case e'_4.e'_5
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |t| - x = |t| - x

pp: case e'_4.e'_5
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ 0 ≤ |t| - x

Tactic:
]

Goal state after:
case e'_4.e'_5
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ 0 ≤ |t| - x
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case e'_4.e'_6
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |t| - x = |t| - x

pp: case e'_4.e'_6
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ 0 ≤ |t| - x

Tactic:
]

Goal state after:
case e'_4.e'_6
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ 0 ≤ |t| - x
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |t| ≠ x

Tactic:
]

Goal state after:
case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
hX : AEMeasurable X μ
a : Ω
this : |t| = |t| - x + x
⊢ |t| ≠ x
Module: Mathlib.Probability.Moments.MGFAnalytic

Goal state before:
pp: case hc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ DifferentiableAt ℝ (iteratedDeriv 1 (mgf X μ)) v

Tactic:
]

Goal state after:
case hc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ DifferentiableAt ℝ (iteratedDeriv 1 (mgf X μ)) v
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
this : ε = z + (ε - z)
⊢ z.re * X ω + (ε - z).re * X ω ≤ z.re * X ω + t / 2 * |X ω|

Tactic:
]

Goal state after:
case a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
this : ε = z + (ε - z)
⊢ z.re * X ω + (ε - z).re * X ω ≤ z.re * X ω + t / 2 * |X ω|
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
x : ℝ
⊢ gaussianPDFReal μ v x = gaussianPDFReal μ v x

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
x : ℝ
⊢ 0 < gaussianPDFReal μ v x

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
x : ℝ
⊢ 0 < gaussianPDFReal μ v x
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
h : v ≠ 0
⊢ NullSingletonClass (ℙ.withDensity (gaussianPDF μ v))

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
h : v ≠ 0
⊢ NullSingletonClass (ℙ.withDensity (gaussianPDF μ v))
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case h₂
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
this : ε = z + (ε - z)
⊢ |(ε - z).re| * |X ω| ≤ t / 2 * |X ω|

Tactic:
]

Goal state after:
case h₂
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
this : ε = z + (ε - z)
⊢ |(ε - z).re| * |X ω| ≤ t / 2 * |X ω|
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ ∫⁻ (a : ℝ) in s, gaussianPDF μ v a = ∫⁻ (x : ℝ) in s, gaussianPDF μ v x

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ ∫⁻ (x : ℝ) in s, gaussianPDF μ v x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal (gaussianPDFReal μ v x)

pp: case hfi
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ Integrable (gaussianPDFReal μ v) (ℙ.restrict s)

pp: case f_nn
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ 0 ≤ᵐ[ℙ.restrict s] gaussianPDFReal μ v

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ ∫⁻ (x : ℝ) in s, gaussianPDF μ v x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal (gaussianPDFReal μ v x)

case hfi
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ Integrable (gaussianPDFReal μ v) (ℙ.restrict s)

case f_nn
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ 0 ≤ᵐ[ℙ.restrict s] gaussianPDFReal μ v
Module: Mathlib.Probability.Moments.MGFAnalytic

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ (deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v * mgf X μ v -
        (∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * ∫ (x : Ω), (fun ω ↦ X ω * rexp (v * X ω)) x ∂μ) /
      mgf X μ v ^ 2 =
    (deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v * mgf X μ v -
        (∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) /
      mgf X μ v ^ 2

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ℙ.withDensity (gaussianPDF μ v) ≪ ℙ

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ℙ.withDensity (gaussianPDF μ v) ≪ ℙ
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ℙ ≪ ℙ.withDensity (gaussianPDF μ v)

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ℙ ≪ ℙ.withDensity (gaussianPDF μ v)
Module: Mathlib.Probability.Moments.MGFAnalytic

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v * mgf X μ v / mgf X μ v ^ 2 -
      ((∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) / mgf X μ v ^ 2 =
    deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v / mgf X μ v - deriv (cgf X μ) v ^ 2

Tactic:
]

Goal state after:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v * mgf X μ v / mgf X μ v ^ 2 -
      ((∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) / mgf X μ v ^ 2 =
    deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v / mgf X μ v - deriv (cgf X μ) v ^ 2
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case refine_1
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
p : ℝ
hp : 0 ≤ p
⊢ v + t - (v - t) ≠ 0

Tactic:
]

Goal state after:
case refine_1
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
p : ℝ
hp : 0 ≤ p
⊢ v + t - (v - t) ≠ 0
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: case pos
μ : ℝ
v : ℝ≥0
hv : v = 0
⊢ Measure.dirac μ = Measure.dirac μ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Moments.IntegrableExpMul

Goal state before:
pp: case refine_2
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
p : ℝ
hp : 0 ≤ p
hX : AEMeasurable X μ
⊢ Integrable (fun a ↦ ‖X a ^ p * rexp (v * X a)‖) μ

pp: case refine_2
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
p : ℝ
hp : 0 ≤ p
hX : AEMeasurable X μ
⊢ AEStronglyMeasurable (fun ω ↦ X ω ^ p * rexp (v * X ω)) μ

Tactic:
]

Goal state after:
case refine_2
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
p : ℝ
hp : 0 ≤ p
hX : AEMeasurable X μ
⊢ Integrable (fun a ↦ ‖X a ^ p * rexp (v * X a)‖) μ

case refine_2
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
p : ℝ
hp : 0 ≤ p
hX : AEMeasurable X μ
⊢ AEStronglyMeasurable (fun ω ↦ X ω ^ p * rexp (v * X ω)) μ
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
⊢ ∂ℙ.withDensity (gaussianPDF μ v)/∂ℙ =ᵐ[ℙ] gaussianPDF μ v

Tactic:
]

Goal state after:
case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
⊢ ∂ℙ.withDensity (gaussianPDF μ v)/∂ℙ =ᵐ[ℙ] gaussianPDF μ v
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ → ℝ
hf : MeasurableEmbedding f
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt f (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ (Measure.comap f (ℙ.withDensity fun x ↦ ENNReal.ofReal (gaussianPDFReal μ v x))) s =
    ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPDFReal μ v (f x))

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ → ℝ
hf : MeasurableEmbedding f
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt f (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ (Measure.comap f (ℙ.withDensity fun x ↦ ENNReal.ofReal (gaussianPDFReal μ v x))) s =
    ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPDFReal μ v (f x))
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
⊢ NormedSpace.exp (ε • ↑(X ω)) * ↑(X ω) = NormedSpace.exp (ε • ↑(X ω)) * ↑(X ω)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ ≃ᵐ ℝ
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ (Measure.map (⇑f.symm) (ℙ.withDensity fun x ↦ ENNReal.ofReal (gaussianPDFReal μ v x))) s =
    ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPDFReal μ v (f x))

Tactic:
]

Goal state after:
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ ≃ᵐ ℝ
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ (Measure.map (⇑f.symm) (ℙ.withDensity fun x ↦ ENNReal.ofReal (gaussianPDFReal μ v x))) s =
    ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPDFReal μ v (f x))
Module: Mathlib.Probability.Moments.MGFAnalytic

Goal state before:
pp: case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v / mgf X μ v =
    deriv (fun u ↦ ∫ (ω : Ω), X ω * rexp (u * X ω) ∂μ) v / mgf X μ v

pp: case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ mgf X μ v ≠ 0

Tactic:
]

Goal state after:
case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ) =ᶠ[𝓝 v] fun u ↦ ∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ
⊢ mgf X μ v ≠ 0
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z ∈ {z | z.re ∈ interior (integrableExpSet X μ)}
h :
  HasFDerivAt (complexMGF X μ)
    (ContinuousLinearMap.toSpanSingleton ℂ (∫ (x : Ω), (fun ω ↦ ↑(X ω) * cexp (z * ↑(X ω))) x ∂μ)) z
⊢ DifferentiableWithinAt ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet X μ)} z

Tactic:
]

Goal state after:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z ∈ {z | z.re ∈ interior (integrableExpSet X μ)}
h :
  HasFDerivAt (complexMGF X μ)
    (ContinuousLinearMap.toSpanSingleton ℂ (∫ (x : Ω), (fun ω ↦ ↑(X ω) * cexp (z * ↑(X ω))) x ∂μ)) z
⊢ DifferentiableWithinAt ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet X μ)} z
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case succ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
n : ℕ
hn :
  ∀ {z : ℂ},
    z.re ∈ interior (integrableExpSet X μ) →
      HasDerivAt (iteratedDeriv n (complexMGF X μ)) (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ) z
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
⊢ HasDerivAt (deriv (iteratedDeriv n (complexMGF X μ)))
    (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1 + 1) * cexp (z * ↑(X ω))) x ∂μ) z

Tactic:
]

Goal state after:
case succ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
n : ℕ
hn :
  ∀ {z : ℂ},
    z.re ∈ interior (integrableExpSet X μ) →
      HasDerivAt (iteratedDeriv n (complexMGF X μ)) (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ) z
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
⊢ HasDerivAt (deriv (iteratedDeriv n (complexMGF X μ)))
    (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1 + 1) * cexp (z * ↑(X ω))) x ∂μ) z
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case succ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
n : ℕ
hn :
  ∀ {z : ℂ},
    z.re ∈ interior (integrableExpSet X μ) →
      HasDerivAt (iteratedDeriv n (complexMGF X μ)) (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ) z
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
this :
  deriv (iteratedDeriv n (complexMGF X μ)) =ᶠ[𝓝 z] fun z ↦
    ∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ
⊢ HasDerivAt (fun z ↦ ∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ)
    (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1 + 1) * cexp (z * ↑(X ω))) x ∂μ) z

Tactic:
]

Goal state after:
case succ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
n : ℕ
hn :
  ∀ {z : ℂ},
    z.re ∈ interior (integrableExpSet X μ) →
      HasDerivAt (iteratedDeriv n (complexMGF X μ)) (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ) z
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
this :
  deriv (iteratedDeriv n (complexMGF X μ)) =ᶠ[𝓝 z] fun z ↦
    ∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ
⊢ HasDerivAt (fun z ↦ ∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ)
    (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1 + 1) * cexp (z * ↑(X ω))) x ∂μ) z
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: case neg
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.addRight y).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ 1) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |1| * gaussianPDFReal μ v (e x)) = (gaussianReal (μ + y) v) s'

Tactic:
]

Goal state after:
case neg
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.addRight y).symm.toMeasurableEquiv
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x ↦ 1) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |1| * gaussianPDFReal μ v (e x)) = (gaussianReal (μ + y) v) s'