Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case e_f
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ cexp
      (↑((L ∘SL single ℝ E i) (∫ (x : E i), x ∂Measure.map (X i) P)) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2) =
    cexp
      (↑(L (Pi.single i (∫ (x : E i), x ∂Measure.map (X i) P))) * I -
        ↑(((covarianceBilinDual (Measure.map (X i) P)) (L ∘SL single ℝ E i)) (L ∘SL single ℝ E i)) / 2)

pp: case e_f.h
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ Integrable id (Measure.map (X i) P)

Tactic:
covarianceBilinDual_self_eq_variance

Goal state after:
case e_f
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ cexp
      (↑((L ∘SL single ℝ E i) (∫ (x : E i), x ∂Measure.map (X i) P)) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2) =
    cexp
      (↑(L (Pi.single i (∫ (x : E i), x ∂Measure.map (X i) P))) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2)

case e_f.h
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ MemLp id 2 (Measure.map (X i) P)

case e_f.h
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ Integrable id (Measure.map (X i) P)