mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: case mp
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
ω : Ω
hω₁ : BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hω₂ : BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
this : (∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
c : ℝ
hc : c ∈ upperBounds (Set.range fun n ↦ (-f) n ω)
⊢ BddBelow (Set.range fun n ↦ f n ω)

Tactic:
refine ⟨-c, fun ω hω => ?_⟩

Goal state after:
case mp
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
ω✝ : Ω
hω₁ : BddAbove (Set.range fun n ↦ f n ω✝) ↔ ∃ c, Tendsto (fun n ↦ f n ω✝) atTop (𝓝 c)
hω₂ : BddAbove (Set.range fun n ↦ (-f) n ω✝) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω✝) atTop (𝓝 c)
this : (∃ c, Tendsto (fun n ↦ f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω✝) atTop (𝓝 c)
c : ℝ
hc : c ∈ upperBounds (Set.range fun n ↦ (-f) n ω✝)
ω : ℝ
hω : ω ∈ Set.range fun n ↦ f n ω✝
⊢ -c ≤ ω
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: case mpr
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
ω : Ω
hω₁ : BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hω₂ : BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
this : (∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
c : ℝ
hc : c ∈ lowerBounds (Set.range fun n ↦ f n ω)
⊢ BddAbove (Set.range fun n ↦ (-f) n ω)

Tactic:
refine ⟨-c, fun ω hω => ?_⟩

Goal state after:
case mpr
Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ (-f) n ω) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω) atTop (𝓝 c)
ω✝ : Ω
hω₁ : BddAbove (Set.range fun n ↦ f n ω✝) ↔ ∃ c, Tendsto (fun n ↦ f n ω✝) atTop (𝓝 c)
hω₂ : BddAbove (Set.range fun n ↦ (-f) n ω✝) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω✝) atTop (𝓝 c)
this : (∃ c, Tendsto (fun n ↦ f n ω✝) atTop (𝓝 c)) ↔ ∃ c, Tendsto (fun n ↦ (-f) n ω✝) atTop (𝓝 c)
c : ℝ
hc : c ∈ lowerBounds (Set.range fun n ↦ f n ω✝)
ω : ℝ
hω : ω ∈ Set.range fun n ↦ (-f) n ω✝
⊢ ω ≤ -c
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
β : Type u_2
u : ℕ → Ω → β
τ : Ω → ℕ∞
inst✝ : AddCommMonoid β
N : ℕ
ω : Ω
t : ℕ
ht : ↑t = τ ω
hbdd : t ≤ N
⊢ t ∈ Set.Iio (N + 1)

Tactic:
by simpa

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

Goal state before:
pp: case neg
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
a b t : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ω : Ω
hl : a ≤ X ω
hr : X ω ≤ b
ht : ¬0 ≤ t
⊢ (a * t ≤ t * X ω ∨ b * t ≤ t * X ω) ∧ (t * X ω ≤ a * t ∨ t * X ω ≤ b * t)

Tactic:
· exact ⟨Or.inr (by nlinarith), Or.inl (by nlinarith)⟩

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
this : |t| ≠ 0
⊢ t ≠ 0

Tactic:
by simpa

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

Goal state before:
pp: case h₂
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
this : ε = z + (ε - z)
⊢ |(ε - z).re * X ω| ≤ t / 2 * |X ω|

Tactic:
rw [abs_mul]

Goal state after:
case h₂
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
hε : ε ∈ Metric.ball z (t / 2)
this : ε = z + (ε - z)
⊢ |(ε - z).re| * |X ω| ≤ t / 2 * |X ω|
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case hbc
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := min (z.re - l) (u - z.re) / 2
h_pos : 0 < min (z.re - l) (u - z.re)
ht : 0 < t
ω : Ω
ε : ℂ
this : ε = z + (ε - z)
hε : ‖ε - z‖ < t / 2
⊢ ‖ε - z‖ ≤ t / 2

Tactic:
exact hε.le

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

Goal state before:
pp: case e_a.e_a.e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ
⊢ (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v =
    ∫ (a : Ω), 2 * X a * deriv (cgf X μ) v * rexp (v * X a) ∂μ

Tactic:
← integral_const_mul,

Goal state after:
case e_a.e_a.e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ
⊢ (∫ (a : Ω), 2 * (X a * rexp (v * X a)) ∂μ) * deriv (cgf X μ) v =
    ∫ (a : Ω), 2 * X a * deriv (cgf X μ) v * rexp (v * X a) ∂μ
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: μ₁ μ₂ : ℝ
v₁ v₂ : ℝ≥0
h : gaussianReal μ₁ v₁ = gaussianReal μ₂ v₂
this : ↑v₁ = ↑v₂
⊢ v₁ = v₂

Tactic:
by simpa

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

Goal state before:
pp: E : Type u_1
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace ℝ E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
μ : Measure E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : SecondCountableTopology F
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
h : MemLp id 2 μ
L : E →L[ℝ] F
u v : F
⊢ ((covarianceBilin (Measure.map (⇑L) μ)) u) v =
    ((covarianceBilin μ) ((ContinuousLinearMap.adjoint L) u)) ((ContinuousLinearMap.adjoint L) v)

Tactic:
rw [covarianceBilin_apply, covarianceBilin_apply h]
  · simp_rw [id, L.integral_id_map (h.integrable (by simp))]
    rw [integral_map]
    · simp_rw [← map_sub, ← L.adjoint_inner_left]
    all_goals fun_prop
  · exact memLp_map_measure_iff (by fun_prop) (by fun_prop) |>.2 (L.comp_memLp' h)

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
hp : ¬p = 0
hp_top : p = ∞
h_Lp : MemLp id ∞ μ
this : (essSup (fun x ↦ ‖L x‖ₑ) μ).toReal ≤ ‖L‖ₑ.toReal * (essSup enorm μ).toReal
⊢ (essSup (fun x ↦ ‖L x‖ₑ) μ).toReal ≤ ‖L‖ * (essSup (fun x ↦ ‖x‖ₑ) μ).toReal

Tactic:
toReal_enorm

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
mE : MeasurableSpace E
μ : Measure E
p : ℝ≥0∞
𝕜 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : OpensMeasurableSpace E
L : StrongDual 𝕜 E
hp : ¬p = 0
hp_top : p = ∞
h_Lp : MemLp id ∞ μ
this : (essSup (fun x ↦ ‖L x‖ₑ) μ).toReal ≤ ‖L‖ * (essSup enorm μ).toReal
⊢ (essSup (fun x ↦ ‖L x‖ₑ) μ).toReal ≤ ‖L‖ * (essSup (fun x ↦ ‖x‖ₑ) μ).toReal
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case hX
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) i, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hY
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hZ
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
· exact Measurable.aestronglyMeasurable (by fun_prop)

Goal state after:
case hY
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

case hZ
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

case h
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: case hY
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEStronglyMeasurable (fun u ↦ ⟪(basisFun ι ℝ) j, u⟫) (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

pp: case hZ
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

pp: case h
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)

Tactic:
· exact Measurable.aestronglyMeasurable (by fun_prop)

Goal state after:
case hZ
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ AEMeasurable (fun ω ↦ toLp 2 fun x ↦ X x ω) μ

case h
ι : Type u_2
Ω : Type u_3
inst✝¹ : Fintype ι
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
hX : ∀ (i : ι), MemLp (X i) 2 μ
i j : ι
this : ∀ (i : ι), AEMeasurable (X i) μ
⊢ MemLp id 2 (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)
Module: Mathlib.Probability.Moments.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 : ‖(toLpₗ μ p) L‖ ≤ (‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹
⊢ ‖L‖ₑ.toReal * (∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹ = ‖L‖ * (eLpNorm id p μ).toReal

Tactic:
toReal_enorm

Goal state after:
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 : ‖(toLpₗ μ p) L‖ ≤ (‖L‖ₑ ^ p.toReal * ∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹
⊢ ‖L‖ * (∫⁻ (x : E), ‖x‖ₑ ^ p.toReal ∂μ).toReal ^ p.toReal⁻¹ = ‖L‖ * (eLpNorm id p μ).toReal
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
x✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)
m : E
f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
hf : f.toBilinForm.IsPosSemidef
h : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)
L : E →L[ℝ] ℝ
t : ℝ
⊢ cexp (↑(t • L m) * I - ↑((f (t • L)) (t • L)) / 2) = cexp (↑t * ↑(L m) * I - ↑↑((f L) L).toNNReal * ↑t ^ 2 / 2)

Tactic:
map_smul,

Goal state after:
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
x✝ : ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)
m : E
f : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
hf : f.toBilinForm.IsPosSemidef
h : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)
L : E →L[ℝ] ℝ
t : ℝ
⊢ cexp (↑(t • L m) * I - ↑(t • (t • f L) L) / 2) = cexp (↑t * ↑(L m) * I - ↑↑((f L) L).toNNReal * ↑t ^ 2 / 2)
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
⊢ IsGaussian μ ↔ ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = cexp (↑⟪t, m⟫ * I - ↑((f t) t) / 2)

Tactic:
isGaussian_iff_gaussian_charFunDual

Goal state after:
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 f, f.toBilinForm.IsPosSemidef ∧ ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp (↑(L m) * I - ↑((f L) L) / 2)) ↔
    ∃ m f, f.toBilinForm.IsPosSemidef ∧ ∀ (t : E), charFun μ t = cexp (↑⟪t, m⟫ * I - ↑((f t) t) / 2)
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: c : ℝ≥0∞
hc_ge : 2⁻¹ ≤ c
hc_le : c ≤ 1
⊢ 0 ≤ logRatio c

Tactic:
by
  cases hc_ge.eq_or_lt'
  · simp [logRatio, *]
  cases hc_le.eq_or_lt'
  · simp [logRatio, *]
  exact (logRatio_pos ‹_› ‹_›).le

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

Goal state before:
pp: case pos
a : ℝ
c : ℝ≥0∞
hc_gt : 2⁻¹ < c
hc_lt : c < 1
n : ℕ
ha : a = 0
⊢ 1 - c ≤ c

Tactic:
refine tsub_le_iff_left.mpr ?_

Goal state after:
case pos
a : ℝ
c : ℝ≥0∞
hc_gt : 2⁻¹ < c
hc_lt : c < 1
n : ℕ
ha : a = 0
⊢ 1 ≤ c + c
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:
(IsGaussian.hasGaussianLaw_id.eval s).map_eq_gaussianReal,

Goal state after:
I : Finset ℝ≥0
s : ↥I
⊢ gaussianReal (∫ (x : ↥I → ℝ), x s ∂projectiveFamily I) Var[fun x ↦ x s; projectiveFamily I].toNNReal =
    gaussianReal 0 ↑s
Module: Mathlib.Probability.Distributions.Fernique

Goal state before:
pp: case neg
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), ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    c *
      (ENNReal.ofReal (rexp (logRatio c')) +
        ∑' (n : ℕ), ENNReal.ofReal (rexp (-2⁻¹ * Real.log (c' / (1 - c')).toReal * 2 ^ n)))

Tactic:
rw [← setLIntegral_univ, h_iUnion]

Goal state after:
case neg
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 a ∪ ⋃ n, closedBall 0 (t (n + 1)) \ closedBall 0 (t n),
      ENNReal.ofReal (rexp (C * ‖x‖ ^ 2)) ∂μ ≤
    c *
      (ENNReal.ofReal (rexp (logRatio c')) +
        ∑' (n : ℕ), ENNReal.ofReal (rexp (-2⁻¹ * Real.log (c' / (1 - c')).toReal * 2 ^ n)))
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X : (t : T) → Ω → 𝓧 t
P : Measure Ω
hX : ∀ (t : T), AEMeasurable (X t) P
⊢ IsProjectiveMeasureFamily fun I ↦ Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P

Tactic:
by
  intro I J hJI
  rw [AEMeasurable.map_map_of_aemeasurable (Finset.measurable_restrict₂ _).aemeasurable]
  · simp [Finset.restrict_def, Finset.restrict₂_def, Function.comp_def]
  · exact aemeasurable_pi_lambda _ fun _ ↦ hX _

Goal state after:
Module: Mathlib.Probability.Process.FiniteDimensionalLaws

Goal state before:
pp: T : Type u_1
Ω : Type u_2
𝓧 : T → Type u_3
mΩ : MeasurableSpace Ω
mα : (t : T) → MeasurableSpace (𝓧 t)
X Y : (t : T) → Ω → 𝓧 t
P : Measure Ω
h : ∀ (t : T), X t =ᵐ[P] Y t
I : Finset T
h' : ∀ᵐ (ω : Ω) ∂P, ∀ (i : ↥I), X (↑i) ω = Y (↑i) ω
⊢ (fun ω ↦ I.restrict fun x ↦ X x ω) =ᵐ[P] fun ω ↦ I.restrict fun x ↦ Y x ω

Tactic:
filter_upwards [h'] with ω h using funext h

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
s✝ : Finset ι
E : Type u_3
inst✝⁴ : MeasurableSpace E
inst✝³ : NormedAddCommGroup E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
X : ι → Ω → E
inst✝ : NormedSpace ℝ E
this : IsProbabilityMeasure P
i : ι
s : Finset ι
hi : i ∉ s
hs :
  (∀ i ∈ s, AEMeasurable (X i) P) →
    iIndepFun (s.restrict X) P → charFunDual (Measure.map (∑ i ∈ s, X i) P) = ∏ i ∈ s, charFunDual (Measure.map (X i) P)
mX : ∀ i_1 ∈ insert i s, AEMeasurable (X i_1) P
hX : iIndepFun ((insert i s).restrict X) P
⊢ ⟨i, ⋯⟩ ∉ {x | ↑x ∈ s}

Tactic:
by simpa

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
X : ι → Ω → E
inst✝¹ : Fintype ι
inst✝ : NormedSpace ℝ E
mX : ∀ (i : ι), AEMeasurable (X i) P
hX : iIndepFun X P
⊢ ∀ i ∈ univ, AEMeasurable (X i) P

Tactic:
by simpa

Goal state after:
Module: Mathlib.Probability.Independence.CharacteristicFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
E : Type u_3
inst✝⁵ : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
X : ι → Ω → E
inst✝¹ : Fintype ι
inst✝ : InnerProductSpace ℝ E
mX : ∀ (i : ι), AEMeasurable (X i) P
hX : iIndepFun X P
⊢ ∀ i ∈ univ, AEMeasurable (X i) P

Tactic:
by simpa

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

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
S : T → Type u_4
X : (t : T) → S t → Ω → E
inst✝ : NormedSpace ℝ E
hX : IsGaussianProcess (fun p ω ↦ X p.fst p.snd ω) P
mX : ∀ (t : T) (s : S t), AEMeasurable (X t s) P
h :
  ∀ (t₁ t₂ : T), t₁ ≠ t₂ → ∀ (s₁ : S t₁) (s₂ : S t₂) (L₁ L₂ : StrongDual ℝ E), cov[⇑L₁ ∘ X t₁ s₁, ⇑L₂ ∘ X t₂ s₂; P] = 0
this : IsProbabilityMeasure P
I : Finset T
J : (i : ↥I) → Finset (S ↑i)
i j : ↥I
hij : i ≠ j
L₁ : StrongDual ℝ (↥(J i) → E)
L₂ : StrongDual ℝ (↥(J j) → E)
h1 : (⇑L₁ ∘ fun ω k ↦ X (↑i) (↑k) ω) = ∑ k, ⇑(L₁ ∘SL ContinuousLinearMap.single ℝ (fun j ↦ E) k) ∘ X ↑i ↑k
h2 : (⇑L₂ ∘ fun ω k ↦ X (↑j) (↑k) ω) = ∑ k, ⇑(L₂ ∘SL ContinuousLinearMap.single ℝ (fun j ↦ E) k) ∘ X ↑j ↑k
x✝³ : ↥(J i)
x✝² : x✝³ ∈ univ
x✝¹ : ↥(J j)
x✝ : x✝¹ ∈ univ
⊢ ↑i ≠ ↑j

Tactic:
by simpa

Goal state after: