mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ 0 ≤ (‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹

Tactic:
by

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 : ∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ ≤ ‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ
⊢ 0 ≤ (‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal

Tactic:
by

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
⊢ μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} = (μ.prod μ) ({x | ‖x‖ ≤ a} ×ˢ {y | b < ‖y‖})

Tactic:
by

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 : ∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ ≤ ‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ
⊢ 0 ≤ p.toReal⁻¹

Tactic:
by

Goal state after:
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 : E
⊢ (covarianceOperator μ) x = ∫ (y : E), ⟪x, y⟫ • y ∂μ

Tactic:
by

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
⊢ IsGaussian (Measure.map (fun x ↦ x + c) μ)

Tactic:
by

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
⊢ IsGaussian (Measure.map (fun x ↦ c + x) μ)

Tactic:
by

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
⊢ IsGaussian (Measure.map (fun x ↦ x - c) μ)

Tactic:
by

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
⊢ (μ.prod μ) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖} =
    (Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ)) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖}

Tactic:
by

Goal state after:
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
⊢ (↑(covarianceOperator μ)).IsPositive

Tactic:
by

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
⊢ IsGaussian (Measure.map (fun x ↦ -x) μ)

Tactic:
by

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
this : IsGaussian (Measure.map (fun x ↦ c + x) (Measure.map (fun x ↦ -x) μ))
⊢ Measurable fun x ↦ c + x

Tactic:
by

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
this : IsGaussian (Measure.map (fun x ↦ c + x) (Measure.map (fun x ↦ -x) μ))
⊢ Measurable fun x ↦ -x

Tactic:
by

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
⊢ Measurable ⇑(ContinuousLinearMap.rotation (-(π / 4)))

Tactic:
by

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
this : IsGaussian (Measure.map (fun x ↦ c + x) (Measure.map (fun x ↦ -x) μ))
⊢ IsGaussian (Measure.map (fun x ↦ c + -x) μ)

Tactic:
by

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
⊢ IsGaussian (Measure.map (fun x ↦ c - x) μ)

Tactic:
by

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 : ∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ ≤ ‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ
⊢ ‖L‖ₑ ^ p.toReal ≠ ∞

Tactic:
by

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 : ∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ ≤ ‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ
h : eLpNorm id p μ ≠ ∞
⊢ p ≠ 0

Tactic:
by

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
⊢ Measurable fun p ↦ ‖p.1‖

Tactic:
by

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 : ∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ ≤ ‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ
⊢ (∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹ ≤
    ENNReal.ofReal ((‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹)

Tactic:
by

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
⊢ Measurable fun p ↦ a

Tactic:
by

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
⊢ Measurable fun p ↦ b

Tactic:
by

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
⊢ Measurable fun p ↦ ‖p.2‖

Tactic:
by

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
⊢ ∫⁻ (x : E), ‖L x‖ₑ ^ p.toReal ∂μ ≤ ∫⁻ (x : E), ‖L‖ₑ ^ p.toReal * ‖x‖ₑ ^ p.toReal ∂μ

Tactic:
by

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
⊢ 0 ≤ √2

Tactic:
by

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
⊢ 0 ≤ 2

Tactic:
by

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
⊢ ∫⁻ (x : E), ‖L‖ₑ ^ p.toReal * ‖x‖ₑ ^ p.toReal ∂μ = ‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ

Tactic:
by

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
⊢ √2 ≠ 0

Tactic:
by

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
⊢ ‖2⁻¹ * √2‖ = (√2)⁻¹

Tactic:
by

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

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
hμ : ∀ (L : StrongDual ℝ E), ∫ (x : E), L x ∂μ = 0
L : StrongDual ℝ E
⊢ charFunDual μ L = cexp (-↑Var[⇑L; μ] / 2)

Tactic:
by

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
⊢ ‖(toLpₗ μ p) L‖ ≤ ‖L‖ * (eLpNorm id p μ).toReal

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
inst✝² : IsGaussian μ
inst✝¹ : SecondCountableTopology E
inst✝ : CompleteSpace E
hμ : ∀ (L : StrongDual ℝ E), ∫ (x : E), L x ∂μ = 0
θ : ℝ
L : StrongDual ℝ (E × E)
⊢ charFunDual (Measure.map (⇑(ContinuousLinearMap.rotation θ)) (μ.prod μ)) L = charFunDual (μ.prod μ) L

Tactic:
charFunDual_map,

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
inst✝² : IsGaussian μ
inst✝¹ : SecondCountableTopology E
inst✝ : CompleteSpace E
hμ : ∀ (L : StrongDual ℝ E), ∫ (x : E), L x ∂μ = 0
θ : ℝ
L : StrongDual ℝ (E × E)
⊢ charFunDual (μ.prod μ) (L ∘SL ContinuousLinearMap.rotation θ) = charFunDual (μ.prod μ) L
Module: Mathlib.Probability.Distributions.Gaussian.Basic

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : MeasurableSpace E
inst✝⁷ : BorelSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace F
inst✝³ : BorelSpace F
μ : Measure E
inst✝² : IsGaussian μ
inst✝¹ : SecondCountableTopologyEither E F
ν : Measure F
inst✝ : IsGaussian ν
L : StrongDual ℝ (E × F)
L₁ : E →L[ℝ] ℝ := L ∘SL ContinuousLinearMap.inl ℝ E F
hL₁ : L₁ = L ∘SL ContinuousLinearMap.inl ℝ E F
L₂ : F →L[ℝ] ℝ := L ∘SL ContinuousLinearMap.inr ℝ E F
hL₂ : L₂ = L ∘SL ContinuousLinearMap.inr ℝ E F
⊢ 2 ≠ ∞

Tactic:
by

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
⊢ (Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ)) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖} =
    (μ.prod μ) {p | ‖p.1 - p.2‖ / √2 ≤ a ∧ b < ‖p.1 + p.2‖ / √2}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
inst✝² : IsGaussian μ
inst✝¹ : SecondCountableTopology E
inst✝ : CompleteSpace E
hμ : ∀ (L : StrongDual ℝ E), ∫ (x : E), L x ∂μ = 0
θ : ℝ
L : StrongDual ℝ (E × E)
⊢ (L ∘SL ContinuousLinearMap.rotation θ) ∘SL ContinuousLinearMap.inl ℝ E E =
    Real.cos θ • L ∘SL ContinuousLinearMap.inl ℝ E E - Real.sin θ • L ∘SL ContinuousLinearMap.inr ℝ E E

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
mE : MeasurableSpace E
μ✝ : Measure E
p✝ : ℝ≥0∞
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : OpensMeasurableSpace E
μ : Measure E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
⊢ Continuous[ContinuousLinearMap.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (toLpₗ μ p).toFun

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
inst✝² : IsGaussian μ
inst✝¹ : SecondCountableTopology E
inst✝ : CompleteSpace E
hμ : ∀ (L : StrongDual ℝ E), ∫ (x : E), L x ∂μ = 0
θ : ℝ
L : StrongDual ℝ (E × E)
h1 :
  (L ∘SL ContinuousLinearMap.rotation θ) ∘SL ContinuousLinearMap.inl ℝ E E =
    Real.cos θ • L ∘SL ContinuousLinearMap.inl ℝ E E - Real.sin θ • L ∘SL ContinuousLinearMap.inr ℝ E E
⊢ (L ∘SL ContinuousLinearMap.rotation θ) ∘SL ContinuousLinearMap.inr ℝ E E =
    Real.sin θ • L ∘SL ContinuousLinearMap.inl ℝ E E + Real.cos θ • L ∘SL ContinuousLinearMap.inr ℝ E E

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
p : ℝ≥0∞
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Fact (1 ≤ p)
h_Lp : MemLp id p μ
L : StrongDual 𝕜 E
⊢ (toLp μ p) L = MemLp.toLp ⇑L ⋯

Tactic:
by

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

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : MeasurableSpace E
inst✝⁷ : BorelSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace F
inst✝³ : BorelSpace F
μ : Measure E
inst✝² : IsGaussian μ
inst✝¹ : SecondCountableTopologyEither E F
ν : Measure F
inst✝ : IsGaussian ν
⊢ IsGaussian (μ.prod ν)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
p : ℝ≥0∞
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Fact (1 ≤ p)
h_Lp : ¬MemLp id p μ
L : StrongDual 𝕜 E
⊢ (toLp μ p) L = 0

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
this : b - a < ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2
⊢ (b - a) / √2 < (‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2) / √2

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
this : b - a < ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2
⊢ 0 ≤ 2

Tactic:
by

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
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
h : MemLp id 2 μ
L₁ L₂ : StrongDual ℝ E
⊢ ((uncenteredCovarianceBilinDual μ) L₁) L₂ = ∫ (x : E), L₁ x * L₂ x ∂μ

Tactic:
by

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
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
h : ¬MemLp id 2 μ
L₁ L₂ : StrongDual ℝ E
⊢ ((uncenteredCovarianceBilinDual μ) L₁) L₂ = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
mE : MeasurableSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
⊢ uncenteredCovarianceBilinDual 0 = 0

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
this : b - a < ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2
⊢ (‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2) / √2 = (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2

Tactic:
by

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
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
L₁ L₂ : StrongDual ℝ E
h : MemLp id 2 μ
⊢ ‖((uncenteredCovarianceBilinDual μ) L₁) L₂‖ = ‖∫ (x : E), L₁ x * L₂ x ∂μ‖

Tactic:
by

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
inst✝¹ : NormedSpace ℝ E
inst✝ : OpensMeasurableSpace E
L₁ L₂ : StrongDual ℝ E
h : MemLp id 2 μ
⊢ ∫ (a : E), ‖L₁ a * L₂ a‖ ∂μ ≤ ∫ (x : E), ‖L₁ x‖ * ‖L₂ x‖ ∂μ

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
this : b - a < ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2
⊢ (b - a) / √2 < (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
⊢ b - a < ‖p.1 + p.2‖ / √2 - a

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
⊢ ‖p.1 + p.2‖ / √2 - a ≤ ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2

Tactic:
by

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
p : E × E
hp1 : ‖p.1 - p.2‖ / √2 ≤ a
hp2 : b < ‖p.1 + p.2‖ / √2
⊢ (b - a) / √2 < (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2

Tactic:
by

Goal state after: