mathlib-initiative/mathlib-tactics huggingface dataset:
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 : InnerProductSpace ℝ E
inst✝ : IsFiniteMeasure μ
m : E
f : E →L[ℝ] E →L[ℝ] ℝ
hf : f.toBilinForm.IsPosSemidef
h : ∀ (t : E), charFun μ t = cexp (↑⟪t, m⟫ * I - ↑((f t) t) / 2)
⊢ m = ∫ (x : E), x ∂μ ∧ f = covarianceBilin 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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
⊢ Measurable fun x ↦ ENNReal.ofReal (rexp (C * a ^ 2))

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SecondCountableTopology E
inst✝⁵ : CompleteSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
inst✝² : InnerProductSpace ℝ E
ν : Measure E
inst✝¹ : IsGaussian μ
inst✝ : IsGaussian ν
hm : ∫ (x : E), id x ∂μ = ∫ (x : E), id x ∂ν
hv : covarianceBilin μ = covarianceBilin ν
⊢ μ = ν

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SecondCountableTopology E
inst✝⁵ : CompleteSpace E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
μ : Measure E
inst✝² : InnerProductSpace ℝ E
ν : Measure E
inst✝¹ : IsGaussian μ
inst✝ : IsGaussian ν
h : μ = ν
⊢ ∫ (x : E), id x ∂μ = ∫ (x : E), id x ∂ν ∧ covarianceBilin μ = covarianceBilin ν

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
x y : EuclideanSpace ℝ ι
⊢ ((covarianceBilin (multivariateGaussian μ S)) x) y = x.ofLp ⬝ᵥ S *ᵥ y.ofLp

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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
x : E
hx : x ∈ closedBall 0 a
⊢ 0 ≤ a⁻¹ ^ 2

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
i✝ j i : ι
⊢ (fun x ↦ x.ofLp i) = fun x ↦ ⟪(EuclideanSpace.basisFun ι ℝ) i, x⟫

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ covMatrix I = Matrix.of fun s t ↦ ℙ.real (Icc 0 ↑↑s ∩ Icc 0 ↑↑t)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ (covMatrix I).PosSemidef

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ Measurable ofLp

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ Measurable (toLp 2)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ Measure.map (toLp 2) (projectiveFamily I) = multivariateGaussian 0 (covMatrix I)

Tactic:
rw [projectiveFamily, Measure.map_map]
    · simp [← MeasurableEquiv.coe_toLp]
    all_goals fun_prop

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ Measure.map (toLp 2) (projectiveFamily I) = multivariateGaussian 0 (covMatrix I)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
I : Finset ℝ≥0
f : (↥I → ℝ) → E
⊢ ∫ (x : ↥I → ℝ), f x ∂projectiveFamily I = ∫ (x : EuclideanSpace ℝ ↥I), f x.ofLp ∂multivariateGaussian 0 (covMatrix I)

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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
⊢ ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * a ^ 2)) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
f g : (↥I → ℝ) → ℝ
⊢ cov[f, g; projectiveFamily I] = cov[f ∘ ofLp, g ∘ ofLp; multivariateGaussian 0 (covMatrix I)]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
f : (↥I → ℝ) → ℝ
⊢ Var[f; projectiveFamily I] = Var[f ∘ ofLp; multivariateGaussian 0 (covMatrix I)]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I✝ J I : Finset ℝ≥0
⊢ IsGaussian (projectiveFamily I)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
i j : ι
⊢ cov[fun x ↦ x.ofLp i, fun x ↦ x.ofLp j; multivariateGaussian μ S] = S i j

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : CompleteSpace E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : IsFiniteMeasure P
hX : AEMeasurable X P
h :
  ∀ (t : E),
    charFun (Measure.map X P) t = cexp (↑(∫ (x : Ω), (fun ω ↦ ⟪t, X ω⟫) x ∂P) * I - ↑Var[fun ω ↦ ⟪t, X ω⟫; P] / 2)
⊢ HasGaussianLaw X P

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
L : StrongDual ℝ E
hX : HasGaussianLaw X P
⊢ AEStronglyMeasurable (fun x ↦ ↑(L x)) (Measure.map X P)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
L : StrongDual ℝ E
hX : HasGaussianLaw X P
⊢ AEMeasurable (⇑L) (Measure.map X P)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
⊢ ∫ (x : ↥I → ℝ), x ∂projectiveFamily I = 0

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
i : ι
⊢ Measurable fun x ↦ x.ofLp i

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
i : ι
⊢ Var[fun x ↦ x.ofLp i; multivariateGaussian μ S] = S i i

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
i : ι
⊢ Var[fun x ↦ x.ofLp i; multivariateGaussian μ S] = S i i

Tactic:
by
  rw [← covariance_self, covariance_eval_multivariateGaussian hS]
  exact Measurable.aemeasurable <| by fun_prop

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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
⊢ ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * a ^ 2)) ∂μ =
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
L : StrongDual ℝ E
hX : HasGaussianLaw X P
⊢ charFunDual (Measure.map X P) L = cexp (↑(∫ (x : Ω), (⇑L ∘ X) x ∂P) * I - ↑Var[⇑L ∘ X; P] / 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
a : ℝ
inst✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
ha_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
⊢ ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s : ↥I
⊢ ∫ (x : ↥I → ℝ), x s ∂projectiveFamily I = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s t : ↥I
⊢ cov[fun x ↦ x s, fun x ↦ x t; projectiveFamily I] = ↑(min ↑s ↑t)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
i : ι
⊢ Measure.map (fun x ↦ x.ofLp i) (multivariateGaussian μ S) = gaussianReal (μ.ofLp i) (S i i).toNNReal

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
μ : EuclideanSpace ℝ ι
S : Matrix ι ι ℝ
hS : S.PosSemidef
x : EuclideanSpace ℝ ι
⊢ charFun (multivariateGaussian μ S) x = cexp (↑⟪x, μ⟫ * I - ↑(x.ofLp ⬝ᵥ S *ᵥ x.ofLp) / 2)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s : ↥I
⊢ Var[fun x ↦ x s; projectiveFamily I] = ↑↑s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s : ↥I
⊢ Measurable fun x ↦ x s

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
⊢ Measurable ⇑(EuclideanSpace.restrict₂ hJI)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s : ↥I
⊢ Measure.map (fun x ↦ x s) (projectiveFamily I) = gaussianReal 0 ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s t : ↥I
⊢ Measurable fun x ↦ x s - x t

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : NormedAddCommGroup E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
X : Ω → E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure P
hX : AEMeasurable X P
h :
  ∀ (L : StrongDual ℝ E), charFunDual (Measure.map X P) L = cexp (↑(∫ (x : Ω), (⇑L ∘ X) x ∂P) * I - ↑Var[⇑L ∘ X; P] / 2)
⊢ HasGaussianLaw X P

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
L : StrongDual ℝ E
hX : HasGaussianLaw X P
⊢ charFunDual (Measure.map X P) L = cexp (↑(∫ (ω : Ω), L (X ω) ∂P) * I - ↑Var[fun ω ↦ L (X ω); P] / 2)

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i✝ j i : ↥J
⊢ (fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis i, u⟫) ∘ ⇑(EuclideanSpace.restrict₂ hJI) = fun u ↦ u.ofLp ⟨↑i, ⋯⟩

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
p : ℝ≥0∞
hp : p ≠ ∞
⊢ MemLp X p P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s t : ↥I
hst : ↑s ≤ ↑t
⊢ ↑s - ↑t ≤ ↑t - ↑s

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
⊢ 2 ≠ ∞

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
X : Ω → E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
hX : HasGaussianLaw X P
⊢ 1 ≠ ∞

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
F : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
X : Ω → E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
hX : HasGaussianLaw X P
L : E →L[ℝ] F
⊢ Measurable ⇑L

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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ha : ∀ᵐ (a_1 : E) ∂μ, ‖a_1‖ ≤ a
ht_int_zero : ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤ ENNReal.ofReal (rexp (logRatio c'))
x : E
hx : ‖x‖ ≤ a
⊢ x ∈ closedBall 0 a

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
X : Ω → E
inst✝ : NormedSpace ℝ E
hX : HasGaussianLaw X P
⊢ HasGaussianLaw (-X) P

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_4
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : Fintype ι
E : Type u_6
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology E
X : ι → Ω → E
hX : HasGaussianLaw (fun ω x ↦ X x ω) P
⊢ HasGaussianLaw (∑ i, X i) P

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_4
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : Fintype ι
E : Type u_6
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology E
X : ι → Ω → E
hX : HasGaussianLaw (fun ω x ↦ X x ω) P
⊢ HasGaussianLaw (fun ω ↦ ∑ i, X i ω) P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I : Finset ℝ≥0
s t : ↥I
⊢ Measure.map (fun x ↦ x s - x t) (projectiveFamily I) = gaussianReal 0 (nndist ↑s ↑t)

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i j : ↥J
⊢ Measurable fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis i, u⟫

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: I J : Finset ℝ≥0
hJI : J ⊆ I
⊢ Finset.restrict₂ hJI ∘ ⇑(MeasurableEquiv.toLp 2 (↥I → ℝ)).symm = ofLp ∘ ⇑(EuclideanSpace.restrict₂ hJI)

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i j : ↥J
⊢ Measurable fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis j, u⟫

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily

Goal state before:
pp: ⊢ IsProjectiveMeasureFamily projectiveFamily

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i j : ↥J
⊢ Measurable ?m.409

Tactic:
by

Goal state after:
ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
i j : ↥J
⊢ Measurable ?m.409
Module: Mathlib.Probability.Distributions.Gaussian.Multivariate

Goal state before:
pp: ι : Type u_2
inst✝ : DecidableEq ι
I J : Finset ι
μ : EuclideanSpace ℝ ↥I
S : Matrix ↥I ↥I ℝ
hS : S.PosSemidef
hJI : J ⊆ I
⊢ Measure.map (⇑(EuclideanSpace.restrict₂ hJI)) (multivariateGaussian μ S) =
    multivariateGaussian ((EuclideanSpace.restrict₂ hJI) μ) (S.submatrix (fun i ↦ ⟨↑i, ⋯⟩) fun i ↦ ⟨↑i, ⋯⟩)

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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
⊢ Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
x✝ : Fin (0 + 1)
⊢ ∀ ⦃b : Fin (0 + 1)⦄, x✝ ≤ b → (fun x ↦ t 0) x✝ ≤ (fun x ↦ t 0) b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
hX : HasIndepIncrements X P
t : ℕ → T
ht : Monotone t
⊢ iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) 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_pos : 0 < a
c' : ℝ≥0∞
hc' : c' ≤ μ {x | ‖x‖ ≤ a}
hc'_gt : 2⁻¹ < c'
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c' * a⁻¹ ^ 2
hc'_le : c' ≤ 1
ht_int_zero :
  ∫⁻ (x : E) in closedBall 0 a, ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    μ {x | ‖x‖ ≤ a} * ENNReal.ofReal (rexp (logRatio c'))
ha : ¬μ {x | ‖x‖ ≤ a} = 1
ha_lt : μ {x | ‖x‖ ≤ a} < 1
hc'_lt : c' < 1
h_iUnion : Set.univ = closedBall 0 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
⊢ ∫⁻ (x : E) in closedBall 0 (t 0) ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    ∫⁻ (x : E) in closedBall 0 (t 0), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ +
      ∑' (i : ℕ), ∫⁻ (x : E) in closedBall 0 (t (i + 1)) \ closedBall 0 (t i), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
k : ℕ
⊢ min n k < n + 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
X : T → Ω → E
inst✝² : Preorder T
inst✝¹ : MeasurableSpace E
inst✝ : Sub E
h : ∀ (t : ℕ → T), Monotone t → EventuallyConst t atTop → iIndepFun (fun i ω ↦ X (t (i + 1)) ω - X (t i) ω) P
n : ℕ
t : Fin (n + 1) → T
ht : Monotone t
t' : ℕ → T := fun k ↦ t ⟨min n k, ⋯⟩
a b : ℕ
hab : a ≤ b
⊢ ⟨min n a, ⋯⟩ ≤ ⟨min n b, ⋯⟩

Tactic:
by

Goal state after: