mathlib-initiative/mathlib-tactics huggingface dataset:
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 positivity

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 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
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 < ∞
hc_one_sub_lt_top : 1 - c < ∞
hc_ratio_pos : 0 < (c / (1 - c)).toReal
⊢ 0 < (μ {x | ‖x‖ ≤ a} / (1 - μ {x | ‖x‖ ≤ a})).toReal⁻¹

Tactic:
by positivity

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
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsGaussian μ
C C' : ℝ
hint : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)
hC'_pos : 0 < C'
hC'_lt : C' < C
h_int : ∀ᵐ (y : E) ∂μ, Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
⊢ Integrable (fun x ↦ rexp (C' * ‖x‖ ^ 2)) μ

Tactic:
obtain ⟨y, hy⟩ : ∃ y, Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ := h_int.exists

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsGaussian μ
C C' : ℝ
hint : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)
hC'_pos : 0 < C'
hC'_lt : C' < C
h_int : ∀ᵐ (y : E) ∂μ, Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
y : E
hy : Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
⊢ Integrable (fun x ↦ rexp (C' * ‖x‖ ^ 2)) μ
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 < ∞
hc_one_sub_lt_top : 1 - c < ∞
hc_ratio_pos : 0 < (c / (1 - c)).toReal
⊢ 0 < (μ {x | ‖x‖ ≤ a} / (1 - μ {x | ‖x‖ ≤ a})).toReal⁻¹ ^ 2 ^ n

Tactic:
by positivity

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
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsGaussian μ
C C' : ℝ
hint : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)
hC'_pos : 0 < C'
hC'_lt : C' < C
h_int : ∀ᵐ (y : E) ∂μ, Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
y : E
hy : Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
ε : ℝ := (C - C') / C'
⊢ 0 < C'

Tactic:
by 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
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 < ∞
hc_one_sub_lt_top : 1 - c < ∞
hc_ratio_pos : 0 < (c / (1 - c)).toReal
⊢ 0 ≤ (μ {x | ‖x‖ ≤ a} / (1 - μ {x | ‖x‖ ≤ a})).toReal⁻¹

Tactic:
by 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
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 < ∞
hc_one_sub_lt_top : 1 - c < ∞
hc_ratio_pos : 0 < (c / (1 - c)).toReal
⊢ 0 ≤ 2 ^ n

Tactic:
by positivity

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
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsGaussian μ
C C' : ℝ
hint : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)
hC'_pos : 0 < C'
hC'_lt : C' < C
h_int : ∀ᵐ (y : E) ∂μ, Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
y : E
hy : Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
ε : ℝ := (C - C') / C'
hε : 0 < ε
this : ∀ (x : E), rexp (C' * ‖x‖ ^ 2) ≤ rexp (C / ε * ‖y‖ ^ 2) * rexp (C * ‖x - y‖ ^ 2)
x✝ : E
⊢ 0 x✝ ≤ (fun x ↦ rexp (C' * ‖x‖ ^ 2)) x✝

Tactic:
by positivity

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

Goal state before:
pp: c : ℝ≥0∞
hc_gt : 2⁻¹ < c
hc_lt : c < 1
⊢ 0 < 8 * (1 + √2) ^ 2

Tactic:
by positivity

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
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsGaussian μ
C C' : ℝ
hint : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)
hC'_pos : 0 < C'
hC'_lt : C' < C
h_int : ∀ᵐ (y : E) ∂μ, Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
y : E
hy : Integrable (fun x ↦ rexp (C * ‖x - y‖ ^ 2)) μ
ε : ℝ := (C - C') / C'
hε : 0 < ε
x : E
⊢ 0 < ε

Tactic:
by positivity

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
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
μ : Measure E
inst✝ : IsGaussian μ
C : ℝ
hC_pos : 0 < C
hC : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) (μ ∗ Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ)
⊢ 0 < C / 2

Tactic:
by 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
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
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c * a⁻¹ ^ 2
x : E
hx : x ∈ closedBall 0 (t (n + 1)) \ closedBall 0 (t n)
⊢ 0 ≤ a⁻¹ ^ 2

Tactic:
by positivity

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
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
μ : Measure E
inst✝ : IsGaussian μ
p : ℝ≥0
C : ℝ
hC_pos : 0 < C
hC : Integrable (fun x ↦ rexp (C * ‖x‖ ^ 2)) μ
x✝ : E
⊢ 0 x✝ ≤ (fun x ↦ rexp (-C * ‖x‖ ^ 2)) x✝

Tactic:
by 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
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
t : ℕ → ℝ := normThreshold a
c : ℝ≥0∞ := μ {x | ‖x‖ ≤ a}
C : ℝ := logRatio c * a⁻¹ ^ 2
⊢ 0 ≤ rexp (2⁻¹ * Real.log (c.toReal / (1 - c).toReal) * 2 ^ n)

Tactic:
by 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
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 positivity

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✝ : SubNegZeroMonoid E
hX : HasIndepIncrements X P
r s t : T
hrs : r ≤ s
hst : s ≤ t
h : ∀ᵐ (ω : Ω) ∂P, X r ω = 0
⊢ X s ⟂ᵢ[P] X t - X s

Tactic:
refine (hX.indepFun_sub_sub hrs hst).congr ?_ .rfl
  filter_upwards [h] with ω hω using by simp [hω]

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

Tactic:
by 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✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ∀ (a : ℝ), 0 < a → 2⁻¹ < μ {x | ‖x‖ ≤ a} → 1 ≤ μ {x | ‖x‖ ≤ a}
h_ne : ∀ (b : ℝ), μ {x | ‖x‖ ≤ b} ≠ 1
a : ℝ
h_of_pos : ∀ (a' : ℝ), 0 < a' → μ {x | ‖x‖ ≤ a'} ≤ 2⁻¹
ha : a ≤ 0
x : E
hx : x ∈ {x | ‖x‖ ≤ a}
⊢ 0 ≤ 1

Tactic:
by 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✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ∀ (a : ℝ), 0 < a → 2⁻¹ < μ {x | ‖x‖ ≤ a} → 1 ≤ μ {x | ‖x‖ ≤ a}
h_ne : ∀ (b : ℝ), μ {x | ‖x‖ ≤ b} ≠ 1
a : ℝ
h_of_pos : ∀ (a' : ℝ), 0 < a' → μ {x | ‖x‖ ≤ a'} ≤ 2⁻¹
ha : a ≤ 0
⊢ 0 < 1

Tactic:
by 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✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ¬∃ a, 0 < a ∧ 2⁻¹ < μ {x | ‖x‖ ≤ a} ∧ μ {x | ‖x‖ ≤ a} < 1
b : ℝ
hb : μ {x | ‖x‖ ≤ b} = 1
⊢ ∀ᵐ (x : E) ∂μ, ‖x‖ ≤ b

Tactic:
rwa [ae_iff_prob_eq_one]

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✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ¬∃ a, 0 < a ∧ 2⁻¹ < μ {x | ‖x‖ ≤ a} ∧ μ {x | ‖x‖ ≤ a} < 1
b : ℝ
hb : μ {x | ‖x‖ ≤ b} = 1
⊢ Measurable fun x ↦ ‖x‖ ≤ b
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✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ¬∃ a, 0 < a ∧ 2⁻¹ < μ {x | ‖x‖ ≤ a} ∧ μ {x | ‖x‖ ≤ a} < 1
b : ℝ
hb : μ {x | ‖x‖ ≤ b} = 1
hb' : ∀ᵐ (x : E) ∂μ, ‖x‖ ≤ b
⊢ 0 < 1

Tactic:
by 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✝ : IsProbabilityMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
h_meas_Ioo : ¬∃ a, 0 < a ∧ 2⁻¹ < μ {x | ‖x‖ ≤ a} ∧ μ {x | ‖x‖ ≤ a} < 1
b : ℝ
hb : μ {x | ‖x‖ ≤ b} = 1
hb' : ∀ᵐ (x : E) ∂μ, ‖x‖ ≤ b
x✝ : E
⊢ 0 x✝ ≤ (fun x ↦ rexp (1 * ‖x‖ ^ 2)) x✝

Tactic:
by 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✝ : IsFiniteMeasure μ
h_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ
hμ_zero : μ = 0
⊢ 0 < 1

Tactic:
by positivity

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

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
hL : ∀ (i : ι), (L i).toBilinForm.IsPosSemidef
x : StrongDual ℝ ((i : ι) → E i)
⊢ 0 ≤ ((diagonalStrongDualPi L).toBilinForm x) x

Tactic:
rw [toBilinForm_diagonalStrongDualPi_apply]

Goal state after:
ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
hL : ∀ (i : ι), (L i).toBilinForm.IsPosSemidef
x : StrongDual ℝ ((i : ι) → E i)
⊢ 0 ≤ ∑ i, ((L i).toBilinForm (x ∘SL single ℝ E i)) (x ∘SL single ℝ E i)
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case neg.refine_2
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : ¬t < 0
⊢ 1 ≤ ⨅ r, if ↑r < 0 then 0 else 1

Tactic:
refine le_ciInf fun x ↦ ?_

Goal state after:
case neg.refine_2
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : ¬t < 0
x : ↑(Ioi t)
⊢ 1 ≤ if ↑x < 0 then 0 else 1
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
r : ℚ
t : { q' // ↑r < ↑q' }
⊢ (fun t ↦ ⟨↑t, ⋯⟩) ((fun t ↦ ⟨↑t, ⋯⟩) t) = t

Tactic:
simp only [Subtype.coe_eta]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
r : ℚ
t : ↑(Ioi r)
⊢ (fun t ↦ ⟨↑t, ⋯⟩) ((fun t ↦ ⟨↑t, ⋯⟩) t) = t

Tactic:
simp only [Subtype.coe_eta]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
⊢ (hf.stieltjesFunction a).measure univ = 1

Tactic:
by
  rw [← ENNReal.ofReal_one, ← sub_zero (1 : ℝ)]
  exact StieltjesFunction.measure_univ _ (tendsto_stieltjesFunction_atBot hf a)
    (tendsto_stieltjesFunction_atTop hf a)

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
⊢ x ∈ ⋂ r, s ×ˢ Iic ↑(-r) ↔ x ∈ ⋂ r, s ×ˢ Iic ↑r

Tactic:
push _ ∈ _

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
⊢ (∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑(-i)) ↔ ∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑i
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
⊢ Measurable fun a r ↦ (preCDF ρ r a).toReal

Tactic:
rw [measurable_pi_iff]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
⊢ ∀ (a : ℚ), Measurable fun x ↦ (preCDF ρ a x).toReal
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ (∫⁻ (a : α) in s, preCDF ρ r a ∂ρ.fst).toReal = (ρ.IicSnd ↑r).real s

pp: case hfm
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ AEMeasurable (preCDF ρ r) (ρ.fst.restrict s)

pp: case hf
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst.restrict s, preCDF ρ r x < ∞

Tactic:
· rw [setLIntegral_preCDF_fst _ _ hs, measureReal_def]

Goal state after:
case hfm
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ AEMeasurable (preCDF ρ r) (ρ.fst.restrict s)

case hf
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst.restrict s, preCDF ρ r x < ∞
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
⊢ ∫⁻ (b : β), ((ξ ×ₖ prodMkLeft β η) (b, f ω)) s ∂κ ω =
    ∫⁻ (b : δ), ((η ∘ₖ deterministic f hf) ω) (Prod.mk b ⁻¹' s) ∂(ξ ∘ₖ (κ ×ₖ deterministic f hf)) ω

pp: case hg
β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
⊢ Measurable fun b ↦ ((ξ ×ₖ prodMkLeft β η) b) s

Tactic:
lintegral_comp,

Goal state after:
β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
⊢ ∫⁻ (b : β), ((ξ ×ₖ prodMkLeft β η) (b, f ω)) s ∂κ ω =
    ∫⁻ (b : β × ε), ∫⁻ (c : δ), ((η ∘ₖ deterministic f hf) ω) (Prod.mk c ⁻¹' s) ∂ξ b ∂(κ ×ₖ deterministic f hf) ω

case hg
β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
⊢ Measurable fun b ↦ ((η ∘ₖ deterministic f hf) ω) (Prod.mk b ⁻¹' s)

case hg
β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
⊢ Measurable fun b ↦ ((ξ ×ₖ prodMkLeft β η) b) s
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
⊢ ∫⁻ (b : β), ((ξ ×ₖ prodMkLeft β η) (b, f ω)) s ∂κ ω =
    ∫⁻ (b : β), ∫⁻ (c : δ), ((η ∘ₖ deterministic f hf) ω) (Prod.mk c ⁻¹' s) ∂ξ (b, f ω) ∂κ ω

Tactic:
congr with b
    rw [prod_apply' _ _ _ hs, prodMkLeft_apply, comp_deterministic_eq_comap, comap_apply]

Goal state after: