mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝³ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : AddCommMonoid E
N : ι
ω : Ω
hbdd : τ ω ≤ ↑N
h_contra : τ ω = ⊤
⊢ False

Tactic:
simp [h_contra] at hbdd

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

Goal state before:
pp: case pos
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : X ⟂ᵢ[μ] Y
h_int_X : Integrable (fun ω ↦ rexp (t * X ω)) μ
h_int_Y : Integrable (fun ω ↦ rexp (t * Y ω)) μ
hμ : μ = 0
⊢ cgf (X + Y) μ t = cgf X μ t + cgf Y μ t

Tactic:
simp [hμ]

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

Goal state before:
pp: case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t a b : ℝ
ha : Integrable (fun ω ↦ rexp (a * X ω)) μ
hb : Integrable (fun ω ↦ rexp (b * X ω)) μ
hat : a ≤ t
htb : t ≤ b
hab : a = b
ha_eq_t : a = t
⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) μ

Tactic:
← ha_eq_t

Goal state after:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t a b : ℝ
ha : Integrable (fun ω ↦ rexp (a * X ω)) μ
hb : Integrable (fun ω ↦ rexp (b * X ω)) μ
hat : a ≤ t
htb : t ≤ b
hab : a = b
ha_eq_t : a = t
⊢ AEStronglyMeasurable (fun ω ↦ rexp (a * X ω)) μ
Module: Mathlib.Probability.Moments.Basic

Goal state before:
pp: case empty
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
h_meas : ∀ (i : ι), AEMeasurable (X i) μ
this : IsProbabilityMeasure μ
⊢ mgf (∑ i ∈ ∅, X i) μ t = ∏ i ∈ ∅, mgf (X i) μ t

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
inst✝ : IsFiniteMeasure μ
hu : Integrable (fun ω ↦ rexp (u * X ω)) μ
h_nonneg : 0 ≤ t
htu : t ≤ u
⊢ Integrable (fun ω ↦ rexp (0 * X ω)) μ

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
inst✝ : IsFiniteMeasure μ
hu : Integrable (fun ω ↦ rexp (u * X ω)) μ
h_nonpos : t ≤ 0
htu : u ≤ t
⊢ Integrable (fun ω ↦ rexp (0 * X ω)) μ

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case e_a.e_s
Ω : Type u_1
ι : Type u_3
inst✝³ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : AddCommMonoid E
n : ι
h_mem : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑(Finset.Iio n)
i : ι
⊢ i ∈ {i ∈ Finset.Iio n | i < n} ↔ i ∈ Finset.Iio n

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : Nonempty ι
μ : Measure Ω
τ : Ω → WithTop ι
E : Type u_4
p : ℝ≥0∞
u : ι → Ω → E
inst✝² : PartialOrder ι
ℱ : Filtration ι m
inst✝¹ : NormedAddCommGroup E
inst✝ : LocallyFiniteOrderBot ι
hτ : IsStoppingTime ℱ τ
hu : ∀ (n : ι), MemLp (u n) p μ
N : ι
ω : Ω
hbdd : τ ω ≤ ↑N
h_contra : τ ω = ⊤
⊢ False

Tactic:
simp [h_contra] at hbdd

Goal state after:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : Nonempty ι
μ : Measure Ω
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝² : PartialOrder ι
ℱ : Filtration ι m
inst✝¹ : NormedAddCommGroup E
inst✝ : LocallyFiniteOrderBot ι
hτ : IsStoppingTime ℱ τ
hu : ∀ (n : ι), Integrable (u n) μ
N : ι
ω : Ω
hbdd : τ ω ≤ ↑N
h_contra : τ ω = ⊤
⊢ False

Tactic:
simp [h_contra] at hbdd

Goal state after:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
β : Type u_2
u : ℕ → Ω → β
τ π : Ω → ℕ∞
inst✝ : AddCommGroup β
hle : τ ≤ π
N : ℕ
hπ_top : ∀ (ω : Ω), π ω ≠ ⊤
hτ_top : ∀ (ω : Ω), τ ω ≠ ⊤
ω : Ω
i t : ℕ
ht : ↑t = τ ω
b : ℕ
hb : ↑b = π ω
hbdd : b ≤ N
⊢ untopA ↑t ≤ i ∧ i < untopA ↑b ↔ i ∈ {x ∈ Finset.range (N + 1) | ↑t ≤ ↑x ∧ ↑x < ↑b}

Tactic:
simp

Goal state after:
Ω : Type u_1
β : Type u_2
u : ℕ → Ω → β
τ π : Ω → ℕ∞
inst✝ : AddCommGroup β
hle : τ ≤ π
N : ℕ
hπ_top : ∀ (ω : Ω), π ω ≠ ⊤
hτ_top : ∀ (ω : Ω), τ ω ≠ ⊤
ω : Ω
i t : ℕ
ht : ↑t = τ ω
b : ℕ
hb : ↑b = π ω
hbdd : b ≤ N
⊢ t ≤ i → i < b → i ≤ N
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
β : Type u_2
u : ℕ → Ω → β
τ : Ω → ℕ∞
inst✝ : AddCommMonoid β
N : ℕ
ω : Ω
hbdd : τ ω ≤ ↑N
h_contra : τ ω = ⊤
⊢ False

Tactic:
simp [h_contra] at hbdd

Goal state after:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case top
Ω : Type u_1
β : Type u_2
u : ℕ → Ω → β
τ : Ω → ℕ∞
inst✝ : AddCommMonoid β
n : ℕ
x ω : Ω
⊢ ↑n + 1 ≤ ⊤ ↔ ↑n < ⊤

Tactic:
simp

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

Goal state before:
pp: case e_a.e_a
x t p : ℝ
hp : 0 ≤ p
ht : 0 < t
hp_zero : ¬p = 0
h_x_le : ∀ (c : ℝ), 0 < c → x ≤ c⁻¹ * rexp (c * x)
h_neg_x_le : ∀ (c : ℝ), 0 < c → -x ≤ c⁻¹ * rexp (-c * x)
h_abs_le : ∀ (c : ℝ), 0 < c → |x| ≤ c⁻¹ * max (rexp (c * x)) (rexp (-c * x))
⊢ (t / p)⁻¹ = p / t

Tactic:
simp

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

Goal state before:
pp: case e'_9
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
h : t ∈ interior (integrableExpSet X μ)
x✝¹ x✝ : Ω
⊢ X x✝ = X x✝ ^ (0 + 1)

Tactic:
simp

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

Goal state before:
pp: case e'_4
x t p : ℝ
hp : 0 ≤ p
ht : t ≠ 0
ht_neg : t < 0
⊢ t = - -t

Tactic:
simp

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

Goal state before:
pp: case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
⊢ 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

Tactic:
simp [hμ]

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

Goal state before:
pp: 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
⊢ |t| = |t| - x + x

Tactic:
simp

Goal state after:
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)
⊢ ε = z + (ε - z)

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ |c⁻¹| * gaussianPDFReal (c⁻¹ * μ) (NNReal.mk (c⁻¹ ^ 2) ⋯ * v) x =
    |c⁻¹| * gaussianPDFReal (c⁻¹ * μ) (NNReal.mk (c ^ 2)⁻¹ ⋯ * v) x

Tactic:
simp

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

Goal state before:
pp: 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|
n : ℕ
x✝ : Ω
⊢ |X x✝| ^ n = |X x✝| ^ ↑n

Tactic:
simp

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

Goal state before:
pp: 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
x✝ : Ω
⊢ v * X x✝ = v * X x✝ + 0 * |X x✝|

Tactic:
simp

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

Goal state before:
pp: 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
n : ℕ
ω : Ω
⊢ |X ω| ^ n = |X ω| ^ ↑n

Tactic:
simp

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

Goal state before:
pp: 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
n : ℕ
ω : Ω
⊢ X ω ^ n = X ω ^ ↑n

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ
n : ℕ
ω : Ω
⊢ |X ω| ^ n = |X ω| ^ ↑n

Tactic:
simp

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

Goal state before:
pp: case e'_9
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
x✝¹ x✝ : Ω
⊢ ↑(X x✝) = ↑(X x✝) ^ (0 + 1)

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ
ht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ
n : ℕ
ω : Ω
⊢ X ω ^ n = X ω ^ ↑n

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
inst✝ : IsProbabilityMeasure μ
hXY : mgf X μ = mgf Y μ'
⊢ Integrable (fun ω ↦ rexp (0 * X ω)) μ

Tactic:
simp

Goal state after:
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 =
    deriv (fun t ↦ ∫ (x : Ω), (fun ω ↦ X ω ^ 1 * rexp (t * X ω)) x ∂μ) v

Tactic:
simp

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

Goal state before:
pp: case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
⊢ iteratedDeriv 2 (cgf X μ) v = (∫ (x : Ω), (fun ω ↦ (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v

Tactic:
simp [hμ]

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

Goal state before:
pp: case e'_3.e'_2
μ : ℝ
v : ℝ≥0
c : ℝ
⊢ ↑((NNReal.mk (c ^ 2) ⋯)⁻¹ * v) = ↑(NNReal.mk (c⁻¹ ^ 2) ⋯ * v)

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
y x✝ : ℝ
⊢ y + -x✝ = ((fun x ↦ y + x) ∘ fun x ↦ -x) x✝

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
hv : v ∈ interior (integrableExpSet X μ)
n : ℕ
ω : Ω
⊢ |X ω| ^ n = |X ω| ^ ↑n

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
hv : v ∈ interior (integrableExpSet X μ)
n : ℕ
ω : Ω
⊢ X ω ^ n = X ω ^ ↑n

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
h : 0 ∈ interior (integrableExpSet X μ)
p : ℝ
hp : 0 ≤ p
⊢ (fun ω ↦ |X ω| ^ p) = fun ω ↦ |X ω| ^ p * rexp (0 * X ω)

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
h : 0 ∈ interior (integrableExpSet X μ)
n : ℕ
x✝ : Ω
⊢ |X x✝| ^ n = |X x✝| ^ n * rexp (0 * X x✝)

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
h : 0 ∈ interior (integrableExpSet X μ)
p : ℝ
hp : 0 ≤ p
⊢ (fun ω ↦ X ω ^ p) = fun ω ↦ X ω ^ p * rexp (0 * X ω)

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
h : 0 ∈ interior (integrableExpSet X μ)
n : ℕ
x✝ : Ω
⊢ X x✝ ^ n = X x✝ ^ n * rexp (0 * X x✝)

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
h : 0 ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
hp_zero : ¬p = 0
⊢ ↑p ≠ ∞

Tactic:
simp

Goal state after:
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
x✝ : Ω
⊢ X x✝ = X x✝ ^ 1

Tactic:
simp

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

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
x✝ : Ω
⊢ ↑|X x✝| ^ n = ↑(|X x✝| ^ ↑n)

Tactic:
simp

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

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
x✝ : Ω
⊢ ↑(X x✝) ^ n = ↑(X x✝ ^ ↑n)

Tactic:
simp

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

Goal state before:
pp: case e_v
m : ℝ
v : ℝ≥0
L : StrongDual ℝ ℝ
⊢ (L 1 ^ 2).toNNReal * v = (max (L 1 ^ 2) 0 * ↑v).toNNReal

Tactic:
rw [Real.toNNReal_mul (by positivity), Real.toNNReal_coe]

Goal state after:
case e_v
m : ℝ
v : ℝ≥0
L : StrongDual ℝ ℝ
⊢ (L 1 ^ 2).toNNReal * v = (max (L 1 ^ 2) 0).toNNReal * v
Module: Mathlib.Probability.Distributions.Gaussian.Basic

Goal state before:
pp: μ : Measure ℝ
h : IsGaussian μ
⊢ μ = Measure.map (⇑(ContinuousLinearMap.id ℝ ℝ)) μ

Tactic:
simp

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

Goal state before:
pp: μ : Measure ℝ
h : IsGaussian μ
⊢ gaussianReal (∫ (x : ℝ), (ContinuousLinearMap.id ℝ ℝ) x ∂μ) Var[⇑(ContinuousLinearMap.id ℝ ℝ); μ].toNNReal =
    gaussianReal (∫ (x : ℝ), id x ∂μ) Var[id; μ].toNNReal

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : AddCommMonoid E
inst✝¹ : Module ℝ E
mE : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
h : ∀ (L : E →L[ℝ] ℝ), IsGaussian (Measure.map (⇑L) μ)
L : StrongDual ℝ E
⊢ gaussianReal (∫ (x : E), id (L x) ∂μ) Var[id ∘ ⇑L; μ].toNNReal = gaussianReal (∫ (x : E), L x ∂μ) Var[⇑L; μ].toNNReal

Tactic:
simp

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

Goal state before:
pp: case hX
μ : ℝ
v : ℝ≥0
t : ℝ
⊢ Measure.map (fun x ↦ x) (gaussianReal μ v) = gaussianReal μ v

Tactic:
simp

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
p : Measure Ω
μ : ℝ
v : ℝ≥0
X : Ω → ℝ
hX : Measure.map X p = gaussianReal μ v
t : ℝ
⊢ Real.log (mgf X p t) = μ * t + ↑v * t ^ 2 / 2

Tactic:
mgf_gaussianReal hX t,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
p : Measure Ω
μ : ℝ
v : ℝ≥0
X : Ω → ℝ
hX : Measure.map X p = gaussianReal μ v
t : ℝ
⊢ Real.log (rexp (μ * t + ↑v * t ^ 2 / 2)) = μ * t + ↑v * t ^ 2 / 2
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ : ℝ
v : ℝ≥0
t : ℝ
⊢ Measure.map (fun x ↦ x) (gaussianReal μ v) = gaussianReal μ v

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ (covarianceBilinDual 0).bilinearComp (toDualMap ℝ E).toContinuousLinearMap (toDualMap ℝ E).toContinuousLinearMap = 0

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
⊢ 0 ∈ interior (integrableExpSet (fun x ↦ x) (gaussianReal μ v))

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure E
inst✝ : IsGaussian μ
x : E
L : StrongDual ℝ E
⊢ Measure.map (⇑L) (Measure.dirac x) = gaussianReal (∫ (x : E), L x ∂Measure.dirac x) Var[⇑L; Measure.dirac x].toNNReal

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
⊢ deriv (fun y ↦ μ) 0 * 0 + μ * deriv (fun y ↦ y) 0 + deriv (fun t ↦ ↑v * t ^ 2 / 2) 0 = μ

Tactic:
simp

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

Goal state before:
pp: case h0
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : Subsingleton E
inst✝ : IsProbabilityMeasure μ
s : Set E
⊢ μ ∅ = (Measure.dirac 0) ∅

Tactic:
simp

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

Goal state before:
pp: case h1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : Subsingleton E
inst✝ : IsProbabilityMeasure μ
s : Set E
⊢ μ Set.univ = (Measure.dirac 0) Set.univ

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
⊢ ∫ (ω : ℝ), ω ^ 2 ∂gaussianReal 0 v = ∫ (x : ℝ), ((fun x ↦ x) ^ 2) x ∂gaussianReal 0 v

Tactic:
simp

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

Goal state before:
pp: case h
μ : ℝ
v : ℝ≥0
⊢ 0 ∈ interior (integrableExpSet (fun x ↦ x) (gaussianReal 0 v))

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : IsGaussian μ
L : StrongDual ℝ E
⊢ 1 ≠ ∞

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
this : (deriv fun t ↦ rexp (↑v * t ^ 2 / 2)) = fun t ↦ ↑v * t * rexp (↑v * t ^ 2 / 2)
⊢ (deriv (fun y ↦ ↑v) 0 * 0 + ↑v * deriv (fun y ↦ y) 0) * rexp (↑v * 0 ^ 2 / 2) +
      ↑v * 0 * deriv (fun t ↦ rexp (↑v * t ^ 2 / 2)) 0 =
    ↑v

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v p : ℝ≥0
⊢ 0 ∈ interior (integrableExpSet id (gaussianReal μ v))

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
μ : Measure E
L : E ≃L[ℝ] F
h : IsGaussian (Measure.map (⇑L) μ)
⊢ μ = Measure.map (⇑L.symm ∘ ⇑L) μ

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
μ : Measure E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
h : MemLp id 2 μ
L : E →L[ℝ] F
u v : F
⊢ 1 ≤ 2

Tactic:
simp

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

Goal state before:
pp: μ : ℝ
v : ℝ≥0
L : ℝ →ₗ[ℝ] ℝ
⊢ ⇑L = fun x ↦ L 1 * x

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
mE : MeasurableSpace E
μ✝ : Measure E
p✝ : ℝ≥0∞
𝕜 : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
μ : Measure E
p : ℝ≥0∞
h_Lp : MemLp id p μ
u v : StrongDual 𝕜 E
⊢ MemLp.toLp ⇑(u + v) ⋯ = MemLp.toLp ⇑u ⋯ + MemLp.toLp ⇑v ⋯

Tactic:
by push_cast; rw [MemLp.toLp_add]

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

Goal state before:
pp: case pos.e_f
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : CompleteSpace E
inst✝ : IsProbabilityMeasure μ
c : E
h : MemLp id 2 μ
x y : E
h_Lp : MemLp id 2 (Measure.map (fun x ↦ c + x) μ)
z : E
⊢ ⟪x, c + z - (∫ (a : E), c ∂μ + ∫ (a : E), a ∂μ)⟫ * ⟪y, c + z - (∫ (a : E), c ∂μ + ∫ (a : E), a ∂μ)⟫ =
    ⟪x, z - ∫ (x : E), x ∂μ⟫ * ⟪y, z - ∫ (x : E), x ∂μ⟫

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : CompleteSpace E
inst✝ : IsProbabilityMeasure μ
c : E
h : MemLp id 2 μ
x y : E
h_Lp : MemLp id 2 (Measure.map (fun x ↦ c + x) μ)
z : E
⊢ 1 ≤ 2

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : CompleteSpace E
inst✝ : IsProbabilityMeasure μ
c x✝² x✝¹ : E
h : MemLp (id ∘ fun x ↦ c + x) 2 μ
x✝ : E
⊢ id x✝ = ((fun x ↦ -c) + id ∘ fun x ↦ c + x) x✝

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
p : ℝ≥0∞
𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : OpensMeasurableSpace E
L : StrongDual 𝕜 E
hp : ¬p = 0
hp_top : p = ∞
h_Lp : MemLp id ∞ μ
⊢ ‖L‖ₑ ≠ ∞

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure E
inst✝ : IsGaussian μ
c : E
L : StrongDual ℝ E
x✝ : E
⊢ (⇑L ∘ fun x ↦ x + c) x✝ = L x✝ + L c

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
p : ℝ≥0∞
𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : OpensMeasurableSpace E
L : StrongDual 𝕜 E
h_Lp : MemLp id p μ
hp : ¬p = 0
hp_top : ¬p = ∞
h0 : 0 < p.toReal
this : ‖(toLpₗ μ p) L‖ ≤ (‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹
⊢ ‖L‖ * ((∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹).toReal =
    ‖L‖ * ((∫⁻ (x : E), ‖id x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)).toReal

Tactic:
simp

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

Goal state before:
pp: case hX
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
x y : EuclideanSpace ℝ ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ ∀ (i : ι), MemLp (fun ω ↦ x.ofLp i * X i ω) 2 μ

pp: case hY
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
x y : EuclideanSpace ℝ ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ ∀ (i : ι), MemLp (fun ω ↦ y.ofLp i * X i ω) 2 μ

Tactic:
all_goals exact fun i ↦ (hX i).const_mul _

Goal state after: