mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ Integrable (fun x ↦ x) μ

Tactic:
·

Goal state after:
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
⊢ Integrable (fun x ↦ x) μ
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case neg
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
⊢ covarianceBilin (Measure.map (fun x ↦ c + x) μ) = covarianceBilin μ

Tactic:
·

Goal state after:
case neg
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
⊢ covarianceBilin (Measure.map (fun x ↦ c + x) μ) = covarianceBilin μ
Module: Mathlib.Probability.Moments.CovarianceBilinDual

Goal state before:
pp: case pos
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 = ∞
⊢ ‖(toLpₗ μ p) L‖ ≤ ‖L‖ * (eLpNorm id p μ).toReal

pp: case neg
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 = ∞
⊢ ‖(toLpₗ μ p) L‖ ≤ ‖L‖ * (eLpNorm id p μ).toReal

Tactic:
·

Goal state after:
case pos
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 = ∞
⊢ ‖(toLpₗ μ p) L‖ ≤ ‖L‖ * (eLpNorm id p μ).toReal
Module: Mathlib.Probability.Moments.CovarianceBilinDual

Goal state before:
pp: case hb
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
⊢ essSup (fun x ↦ ‖L‖ₑ * ‖x‖ₑ) μ ≠ ∞

pp: case h
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
⊢ essSup (fun x ↦ ‖L x‖ₑ) μ ≤ essSup (fun x ↦ ‖L‖ₑ * ‖x‖ₑ) μ

Tactic:
·

Goal state after:
case hb
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
⊢ essSup (fun x ↦ ‖L‖ₑ * ‖x‖ₑ) μ ≠ ∞
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: 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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ cov[(fun u ↦ ⟪(basisFun ι ℝ) i, u⟫) ∘ fun ω ↦ toLp 2 fun x ↦ X x ω,
      (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) ∘ fun ω ↦ toLp 2 fun x ↦ X x ω; μ] =
    cov[X i, X j; μ]

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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) i, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) 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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hZ
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ cov[(fun u ↦ ⟪(basisFun ι ℝ) i, u⟫) ∘ fun ω ↦ toLp 2 fun x ↦ X x ω,
      (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) ∘ fun ω ↦ toLp 2 fun x ↦ X x ω; μ] =
    cov[X i, X j; μ]
Module: Mathlib.Probability.Moments.CovarianceBilinDual

Goal state before:
pp: case h
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
⊢ essSup (fun x ↦ ‖L x‖ₑ) μ ≤ essSup (fun x ↦ ‖L‖ₑ * ‖x‖ₑ) μ

Tactic:
·

Goal state after:
case h
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
⊢ essSup (fun x ↦ ‖L x‖ₑ) μ ≤ essSup (fun x ↦ ‖L‖ₑ * ‖x‖ₑ) μ
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) i, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) 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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hZ
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) i, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hZ
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case hZ
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
case hZ
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case h
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
case h
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
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: 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
⊢ cov[fun ω ↦ ⟪∑ i, ⟪(basisFun ι ℝ) i, x⟫ • (basisFun ι ℝ) i, toLp 2 fun x ↦ X x ω⟫, fun ω ↦
      ⟪∑ i, ⟪(basisFun ι ℝ) i, y⟫ • (basisFun ι ℝ) i, toLp 2 fun x ↦ X x ω⟫; μ] =
    ∑ i, ∑ j, x.ofLp i * y.ofLp j * cov[X i, X j; μ]

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
⊢ AEStronglyMeasurable (fun u ↦ ⟪x, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) 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
⊢ AEStronglyMeasurable (fun u ↦ ⟪y, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hZ
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
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
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
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
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
⊢ cov[fun ω ↦ ⟪∑ i, ⟪(basisFun ι ℝ) i, x⟫ • (basisFun ι ℝ) i, toLp 2 fun x ↦ X x ω⟫, fun ω ↦
      ⟪∑ i, ⟪(basisFun ι ℝ) i, y⟫ • (basisFun ι ℝ) i, toLp 2 fun x ↦ X x ω⟫; μ] =
    ∑ i, ∑ j, x.ofLp i * y.ofLp j * cov[X i, X j; μ]
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: 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, ∑ j, cov[fun ω ↦ x.ofLp i * X i ω, fun ω ↦ y.ofLp j * X j ω; μ] =
    ∑ x_1, ∑ x_2, x.ofLp x_1 * y.ofLp x_2 * cov[X x_1, X x_2; μ]

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:
·

Goal state after:
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, ∑ j, cov[fun ω ↦ x.ofLp i * X i ω, fun ω ↦ y.ofLp j * X j ω; μ] =
    ∑ x_1, ∑ x_2, x.ofLp x_1 * y.ofLp x_2 * cov[X x_1, X x_2; μ]
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case hZ
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
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
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
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
case hZ
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
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case h
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
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
·

Goal state after:
case h
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
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x y : E
⊢ ∫ (x_1 : E), ⟪y, ⟪x, x_1⟫ • x_1⟫ ∂μ =
    (((uncenteredCovarianceBilinDual μ).bilinearComp (toDualMap ℝ E).toContinuousLinearMap
          (toDualMap ℝ E).toContinuousLinearMap)
        x)
      y

pp: case hf
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x y : E
⊢ Integrable (fun y ↦ ⟪x, y⟫ • y) μ

Tactic:
·

Goal state after:
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x y : E
⊢ ∫ (x_1 : E), ⟪y, ⟪x, x_1⟫ • x_1⟫ ∂μ =
    (((uncenteredCovarianceBilinDual μ).bilinearComp (toDualMap ℝ E).toContinuousLinearMap
          (toDualMap ℝ E).toContinuousLinearMap)
        x)
      y
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case neg
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : ¬MemLp id 2 μ
⊢ (↑(covarianceOperator μ)).IsPositive

pp: case pos
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
⊢ (↑(covarianceOperator μ)).IsPositive

Tactic:
·

Goal state after:
case neg
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : ¬MemLp id 2 μ
⊢ (↑(covarianceOperator μ)).IsPositive
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case pos.refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x y : E
⊢ ⟪↑(covarianceOperator μ) x, y⟫ = ⟪x, ↑(covarianceOperator μ) y⟫

pp: case pos.refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x : E
⊢ 0 ≤ RCLike.re ⟪↑(covarianceOperator μ) x, x⟫

Tactic:
·

Goal state after:
case pos.refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x y : E
⊢ ⟪↑(covarianceOperator μ) x, y⟫ = ⟪x, ↑(covarianceOperator μ) y⟫
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case pos.refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x : E
⊢ 0 ≤ RCLike.re ⟪↑(covarianceOperator μ) x, x⟫

Tactic:
·

Goal state after:
case pos.refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x : E
⊢ 0 ≤ RCLike.re ⟪↑(covarianceOperator μ) x, x⟫
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case pos.refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : CompleteSpace E
hμ : MemLp id 2 μ
x : E
⊢ 0 ≤ RCLike.re ⟪↑(covarianceOperator μ) x, x⟫

Tactic:
· simp only [ContinuousLinearMap.coe_coe, covarianceOperator_inner hμ, ← pow_two,
      RCLike.re_to_real]
    positivity

Goal state after:
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖}

pp: E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ (μ.prod μ) (⇑(ContinuousLinearMap.rotation (-(π / 4))) ⁻¹' {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖}) =
    (μ.prod μ) {p | ‖p.1 - p.2‖ / √2 ≤ a ∧ b < ‖p.1 + p.2‖ / √2}

Tactic:
·

Goal state after:
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖}
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet fun p ↦ Real.le✝ ‖p.1‖ a

pp: case refine_2
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet fun p ↦ Real.lt✝ b ‖p.2‖

Tactic:
·

Goal state after:
case refine_1
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet fun p ↦ Real.le✝ ‖p.1‖ a
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet fun p ↦ Real.lt✝ b ‖p.2‖

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : SecondCountableTopology E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : SFinite μ
h : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a b : ℝ
⊢ MeasurableSet fun p ↦ Real.lt✝ b ‖p.2‖
Module: Mathlib.Probability.Moments.CovarianceBilinDual

Goal state before:
pp: case hz
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
x : E
⊢ 0 ≤ p.toReal

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
x : E
⊢ ‖L x‖ₑ ^ p.toReal ≤ (‖L‖ₑ * ‖x‖ₑ) ^ p.toReal

Tactic:
·

Goal state after:
case hz
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
x : E
⊢ 0 ≤ p.toReal