mathlib-initiative/mathlib-tactics huggingface dataset:
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‖ - ‖p.1 - p.2‖) / 2 ≤ min ‖p.1‖ ‖p.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
⊢ 2 ≠ 0

Tactic:
by

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

Goal state before:
pp: case refine_3
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
x : E
⊢ (fun x ↦ ‖L₁ x‖ * ‖L₂ x‖) x ≤ (fun x ↦ ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖) x

Tactic:
simp only

Goal state after:
case refine_3
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
x : E
⊢ ‖L₁ x‖ * ‖L₂ x‖ ≤ ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖
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 - p.2‖ / √2 ≤ a ∧ b < ‖p.1 + p.2‖ / √2} ≤
    (μ.prod μ) {p | (b - a) / √2 < ‖p.1‖ ∧ (b - a) / √2 < ‖p.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
⊢ ∫ (x : E), ‖L₁ x‖ * ‖L₂ x‖ ∂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
⊢ (μ.prod μ) ({x | (b - a) / √2 < ‖x‖} ×ˢ {y | (b - a) / √2 < ‖y‖}) ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 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
⊢ μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 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
⊢ ∫ (x : E), ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖ ∂μ = ‖L₁‖ * ‖L₂‖ * ∫ (x : E), ‖x‖ ^ 2 ∂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
⊢ ‖((uncenteredCovarianceBilinDual μ) L₁) L₂‖ ≤ ‖L₁‖ * ‖L₂‖ * ∫ (x : E), ‖x‖ ^ 2 ∂μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
p : ℝ≥0∞
h_Lp : MemLp (fun x ↦ x - ∫ (y : E), y ∂μ) p μ
⊢ id = fun x ↦ x - ∫ (x : E), x ∂μ + ∫ (x : E), 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_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
a : ℝ
n : ℕ
⊢ μ {x | ‖x‖ ≤ a} * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2

Tactic:
by

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

Goal state before:
pp: a : ℝ
ha_pos : 0 < a
⊢ 1 - √2 ≤ 0

Tactic:
by

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

Goal state before:
pp: a : ℝ
ha_pos : 0 < a
⊢ a / (1 - √2) < normThreshold a 0

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ √2 ≠ 1

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
p : ℝ≥0∞
c : E := ∫ (x : E), x ∂μ
h_Lp : MemLp (fun x ↦ x - c) p μ
this : id = fun x ↦ x - c + c
hx : ¬c = 0
hp0 : p ≠ 0
hptop : p ≠ ∞
⊢ AEStronglyMeasurable (fun x ↦ c) μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
p : ℝ≥0∞
c : E := ⋯
h_Lp : MemLp (fun x ↦ x - c) p μ
this : id = fun x ↦ x - c + c
hx : ¬c = 0
hp0 : p ≠ 0
hptop : p ≠ ∞
⊢ Integrable (fun a ↦ a) μ

Tactic:
contrapose hx

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
p : ℝ≥0∞
c : E := ∫ (x : E), x ∂μ
h_Lp : MemLp (fun x ↦ x - c) p μ
this : id = fun x ↦ x - c + c
hp0 : p ≠ 0
hptop : p ≠ ∞
hx : ¬Integrable (fun a ↦ a) μ
⊢ c = 0
Module: Mathlib.Probability.Moments.CovarianceBilinDual

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
p : ℝ≥0∞
c : E := ∫ (x : E), x ∂μ
h_Lp : MemLp (fun x ↦ x - c) p μ
this : id = fun x ↦ x - c + c
hx : ¬c = 0
hp0 : p ≠ 0
hptop : p ≠ ∞
⊢ Integrable (fun x ↦ ‖x‖) μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : CompleteSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsGaussian μ
L : StrongDual ℝ E
⊢ charFunDual μ L = cexp (↑(L (∫ (x : E), id x ∂μ)) * I - ↑(((covarianceBilinDual μ) L) L) / 2)

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ normThreshold a n = a * (1 + √2) * (√2 ^ (n + 1) - 1)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
p : ℝ≥0∞
c : E := ∫ (x : E), x ∂μ
h_Lp : MemLp (fun x ↦ x - c) p μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
hp0 : p ≠ 0
hptop : p ≠ ∞
I : Integrable (fun x ↦ ‖x‖) μ
this : Integrable (fun x ↦ 2 ^ p.toReal * ‖x - c‖ ^ p.toReal) μ
⊢ AEStronglyMeasurable (fun x ↦ ‖c‖ ^ p.toReal) μ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
⊢ 0 ≤ ‖c‖ ^ ↑p

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ 0 ≤ √2 ^ (0 + 2) - 1

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ √2 ^ (0 + 2) - 1 ≤ √2 ^ (n + 2) - 1

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ 0 ≤ 1

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ (√2 ^ (n + 2) - 1) ^ 2 ≤ (√2 ^ (n + 2)) ^ 2

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ 0 ≤ 2

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖y‖ ≤ ‖c‖ / 2
⊢ ‖c‖ = ‖y - (y - c)‖

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ (√2 ^ (n + 2)) ^ 2 = 2 ^ (n + 2)

Tactic:
by

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

Goal state before:
pp: a : ℝ
n : ℕ
⊢ normThreshold a (n + 1) ^ 2 ≤ a ^ 2 * (1 + √2) ^ 2 * 2 ^ (n + 2)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝¹ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this✝ : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖y‖ ≤ ‖c‖ / 2
this : ‖c‖ ≤ ‖y‖ + ‖y - c‖
⊢ ‖c‖ ^ ↑p ≤ (2 * ‖y - c‖) ^ ↑p

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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
n : ℕ
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
⊢ 0 < 2⁻¹

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝¹ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this✝ : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖y‖ ≤ ‖c‖ / 2
this : ‖c‖ ≤ ‖y‖ + ‖y - c‖
⊢ 0 ≤ 2

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝¹ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this✝ : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖y‖ ≤ ‖c‖ / 2
this : ‖c‖ ≤ ‖y‖ + ‖y - c‖
⊢ 0 ≤ ‖y - c‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝¹ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this✝ : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖y‖ ≤ ‖c‖ / 2
this : ‖c‖ ≤ ‖y‖ + ‖y - c‖
⊢ (2 * ‖y - c‖) ^ ↑p = 0 + 2 ^ ↑p * ‖y - c‖ ^ ↑p

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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
⊢ Measurable norm

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝¹ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this✝ : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖y‖ ≤ ‖c‖ / 2
this : ‖c‖ ≤ ‖y‖ + ‖y - c‖
⊢ 0 + 2 ^ ↑p * ‖y - c‖ ^ ↑p ≤ 2 * ‖c‖ ^ (↑p - 1) * ‖y‖ + 2 ^ ↑p * ‖y - c‖ ^ ↑p

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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
⊢ Measurable fun x ↦ 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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ μ {x | normThreshold a (n + 1) < ‖x‖} = c⁻¹ * (c * μ {x | normThreshold a (n + 1) < ‖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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * (c * μ {x | normThreshold a (n + 1) < ‖x‖}) ≤ c⁻¹ * μ {x | normThreshold a n < ‖x‖} ^ 2

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖c‖ / 2 ≤ ‖y‖
⊢ ‖c‖ ^ ↑p = ‖c‖ ^ (↑p - 1 + 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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * μ {x | normThreshold a n < ‖x‖} ^ 2 ≤ c⁻¹ * (c * ((1 - c) / c) ^ 2 ^ n) ^ 2

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖c‖ / 2 ≤ ‖y‖
⊢ 0 < ‖c‖

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
h2 :
  (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
⊢ 2 ≠ ∞

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖c‖ / 2 ≤ ‖y‖
⊢ ‖c‖ ^ (↑p - 1 + 1) = ‖c‖ ^ (↑p - 1) * ‖c‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : CompleteSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
⊢ IsGaussian μ ↔
    ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)

Tactic:
refine ⟨fun h ↦ ⟨μ[id], covarianceBilinDual μ, isPosSemidef_covarianceBilinDual,
    h.charFunDual_eq'⟩,
    fun ⟨m, f, hf, h⟩ ↦ isGaussian_of_map_eq_gaussianReal fun L ↦ ⟨L m, (f L L).toNNReal, ?_⟩⟩
  apply Measure.ext_of_charFun
  ext t
  simp_rw [charFun_map_eq_charFunDual_smul, h, charFun_gaussianReal,
    smul_apply, map_smul, smul_apply, smul_eq_mul]
  norm_cast
  congrm exp (_ - ofReal ?_)
  rw [Real.coe_toNNReal]
  · ring
  exact hf.nonneg L

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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_gt : 2⁻¹ < c
hc_pos : 0 < c
hc_lt_top : c < ∞
n : ℕ
hn : μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n
h_mul_le : c * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2
⊢ c⁻¹ * (c * ((1 - c) / c) ^ 2 ^ n) ^ 2 = c * ((1 - c) / c) ^ 2 ^ (n + 1)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopology E
inst✝⁴ : CompleteSpace E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
⊢ IsGaussian μ ↔
    ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖c‖ / 2 ≤ ‖y‖
⊢ ‖c‖ ^ (↑p - 1) * ‖c‖ ≤ ‖c‖ ^ (↑p - 1) * (2 * ‖y‖)

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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
n : ℕ
⊢ μ {x | normThreshold a n < ‖x‖} ≤ μ {x | ‖x‖ ≤ a} * ((1 - μ {x | ‖x‖ ≤ a}) / μ {x | ‖x‖ ≤ a}) ^ 2 ^ n

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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
ha_lt : μ {x | ‖x‖ ≤ a} < 1
n : ℕ
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
⊢ 0 < 2⁻¹

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
inst✝ : NormedSpace ℝ E
c : E := ∫ (x : E), x ∂μ
this✝ : id = fun x ↦ x - c + c
hx : ¬c = 0
I : Integrable (fun x ↦ ‖x‖) μ
y : E
p : ℝ≥0
h_Lp : MemLp (fun x ↦ x - c) (↑p) μ
hp0 : ↑p ≠ 0
this : Integrable (fun x ↦ 2 ^ (↑p).toReal * ‖x - c‖ ^ (↑p).toReal) μ
h✝ : ‖c‖ / 2 ≤ ‖y‖
⊢ ‖c‖ ^ (↑p - 1) * (2 * ‖y‖) = 2 * ‖c‖ ^ (↑p - 1) * ‖y‖ + 0

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
⊢ Measure.map (⇑(ContinuousLinearMap.rotation θ)) (μ.prod μ) = μ.prod 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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_gt : 2⁻¹ < μ {x | ‖x‖ ≤ a}
ha_lt : μ {x | ‖x‖ ≤ a} < 1
n : ℕ
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
hc_pos : 0 < c
hc_lt : c < 1
hc_lt_top : c < ∞
⊢ 1 < 2

Tactic:
by

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

Goal state before:
pp: 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 : IsGaussian μ
inst✝ : SecondCountableTopology E
L : StrongDual ℝ E
⊢ Integrable (⇑L) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)

Tactic:
by

Goal state after: