mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Distributions.TwoValued

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
m₀ : MeasurableSpace Ω
hm : m ≤ m₀
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXmeas✝ : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
this :
  ∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {m₀ : MeasurableSpace Ω},
    m ≤ m₀ →
      ∀ {μ : Measure Ω} [IsFiniteMeasure μ],
        AEMeasurable X μ →
          (∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1) → Measurable X → Var[X; μ | m] =ᵐ[μ] μ[X | m] * μ[1 - X | m]
hXmeas : ¬Measurable X
Y : Ω → ℝ
hYmeas : Measurable Y
hXY : X =ᵐ[μ] Y
ω : Ω
hXω : X ω = 0 ∨ X ω = 1
⊢ X ω = Y ω → Y ω = 0 ∨ Y ω = 1

Tactic:
hXYω

Goal state after:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
m₀ : MeasurableSpace Ω
hm : m ≤ m₀
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXmeas✝ : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
this :
  ∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {m₀ : MeasurableSpace Ω},
    m ≤ m₀ →
      ∀ {μ : Measure Ω} [IsFiniteMeasure μ],
        AEMeasurable X μ →
          (∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1) → Measurable X → Var[X; μ | m] =ᵐ[μ] μ[X | m] * μ[1 - X | m]
hXmeas : ¬Measurable X
Y : Ω → ℝ
hYmeas : Measurable Y
hXY : X =ᵐ[μ] Y
ω : Ω
hXω : X ω = 0 ∨ X ω = 1
hXYω : X ω = Y ω
⊢ Y ω = 0 ∨ Y ω = 1
Module: Mathlib.Probability.Distributions.TwoValued

Goal state before:
pp: Ω✝ : Type u_1
m✝ : MeasurableSpace Ω✝
X✝ : Ω✝ → ℝ
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
m₀ : MeasurableSpace Ω
hm : m ≤ m₀
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hXmeas✝ : AEMeasurable X μ
hX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1
hXmeas : Measurable X
⊢ μ[X ^ 2 | m] - μ[X | m] ^ 2 =ᵐ[μ] μ[X | m] - μ[X | m] ^ 2

Tactic:
by
      refine .sub ?_ ae_eq_rfl
      exact condExp_congr_ae <| by filter_upwards [hX]; rintro ω (hω | hω) <;> simp [hω]

Goal state after:
Module: Mathlib.Probability.Kernel.Invariance

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
κ : Kernel α α
μ : Measure α
hκ : κ.Invariant μ
⊢ const α (⇑κ ∘ₘ μ) = const α μ

Tactic:
hκ.def

Goal state after:
α : Type u_1
mα : MeasurableSpace α
κ : Kernel α α
μ : Measure α
hκ : κ.Invariant μ
⊢ const α μ = const α μ
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ ⨆ n, ∫⁻ (a : X), B.indicator 1 a * f' n a ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), ⨆ n, f' n x ∂π x₀

pp: case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ Monotone fun i x ↦ B.indicator 1 x * f' i x

Tactic:
lintegral_iSup hf'_meas hf'_mono,

Goal state after:
case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ ⨆ n, ∫⁻ (a : X), B.indicator 1 a * f' n a ∂π x₀ = B.indicator 1 x₀ * ⨆ n, ∫⁻ (a : X), f' n a ∂π x₀

case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ Monotone fun i x ↦ B.indicator 1 x * f' i x
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : (s : S) → E s →ᵇ ℝ), ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
ω : Ω
⊢ f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
    A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
      A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω

Tactic:
rw [Set.indicator_apply]

Goal state after:
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : (s : S) → E s →ᵇ ℝ), ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
ω : Ω
⊢ f (if ω ∈ A then 1 ω else 0) * ∏ s, (g s) (X s ω) =
    A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
      A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : (s : S) → E s →ᵇ ℝ), ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
h1 :
  ∀ (ω : Ω),
    f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
ω : Ω
⊢ f (A.indicator 1 ω) = A.indicator (fun x ↦ f 1) ω + Aᶜ.indicator (fun x ↦ f 0) ω

Tactic:
rw [Set.indicator_apply]

Goal state after:
Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
inst✝⁵ : (s : S) → TopologicalSpace (E s)
inst✝⁴ : (s : S) → MeasurableSpace (E s)
inst✝³ : ∀ (s : S), BorelSpace (E s)
inst✝² : ∀ (s : S), HasOuterApproxClosed (E s)
X : (s : S) → Ω → E s
inst✝¹ : Fintype S
inst✝ : IsProbabilityMeasure P
A : Set Ω
mA : NullMeasurableSet A P
mX : ∀ (s : S), AEMeasurable (X s) P
h : ∀ (f : (s : S) → E s →ᵇ ℝ), ∫ (ω : Ω) in A, ∏ s, (f s) (X s ω) ∂P = P.real A * ∫ (ω : Ω), ∏ s, (f s) (X s ω) ∂P
f : ℝ →ᵇ ℝ
g : (t : S) → E t →ᵇ ℝ
h1 :
  ∀ (ω : Ω),
    f (A.indicator 1 ω) * ∏ s, (g s) (X s ω) =
      A.indicator (fun ω ↦ f 1 * ∏ s, (g s) (X s ω)) ω + f 0 * ∏ s, (g s) (X s ω) -
        A.indicator (fun ω ↦ f 0 * ∏ s, (g s) (X s ω)) ω
ω : Ω
⊢ f (if ω ∈ A then 1 ω else 0) = A.indicator (fun x ↦ f 1) ω + Aᶜ.indicator (fun x ↦ f 0) ω
Module: Mathlib.Probability.Martingale.OptionalSampling

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → WithTop ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
inst✝ : SigmaFinite (μ.trim ⋯)
i : ι
hin : ¬i ≤ n
this : {x | τ x = ↑i} = ∅
⊢ μ[f n | hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = ↑i}] f i

Tactic:
simp [this]; norm_cast

Goal state after:
Module: Mathlib.Probability.Kernel.Representation

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
κ : Kernel X ↑I
inst✝ : IsMarkovKernel κ
f : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}
measurable_f : Measurable (uncurry f)
a : X
x : ↑I
Iic_to_Icc : Iic x = Icc 0 x
κ_in_I : (κ a).real (Icc 0 x) ∈ I
ξ : ↑I
hξ : f a ξ ≤ x
hx : ¬x = 1
g : ↑I → ℝ := fun y ↦ (κ a).real (Icc 0 y)
⊢ (𝓝[>] x).NeBot

Tactic:
refine nhdsGT_neBot_of_exists_gt ?_

Goal state after:
X : Type u_1
mX : MeasurableSpace X
κ : Kernel X ↑I
inst✝ : IsMarkovKernel κ
f : X → ↑I → ↑I := fun s t ↦ sSup {x | (κ s).real (Icc 0 x) < ↑t}
measurable_f : Measurable (uncurry f)
a : X
x : ↑I
Iic_to_Icc : Iic x = Icc 0 x
κ_in_I : (κ a).real (Icc 0 x) ∈ I
ξ : ↑I
hξ : f a ξ ≤ x
hx : ¬x = 1
g : ↑I → ℝ := fun y ↦ (κ a).real (Icc 0 y)
⊢ ∃ b, x < b
Module: Mathlib.Probability.Moments.Tilted

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)

Tactic:
by
  have hX : AEMeasurable X μ := aemeasurable_of_mem_interior_integrableExpSet ht
  by_cases hp : p = 0
  · simpa [hp] using hX.aestronglyMeasurable.mono_ac (tilted_absolutelyContinuous _ _)
  refine ⟨hX.aestronglyMeasurable.mono_ac (tilted_absolutelyContinuous _ _), ?_⟩
  rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top]
  rotate_left
  · simp [hp]
  · simp
  simp_rw [ENNReal.coe_toReal, ← ofReal_norm, norm_eq_abs,
    ENNReal.ofReal_rpow_of_nonneg (x := |X _|) (p := p) (abs_nonneg (X _)) p.2]
  refine Integrable.lintegral_lt_top ?_
  simp_rw [integrable_tilted_iff (interior_subset (s := integrableExpSet X μ) ht),
    smul_eq_mul, mul_comm]
  exact integrable_rpow_abs_mul_exp_of_mem_interior_integrableExpSet ht p.2

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Y : Ω → ℝ
h : HasSubgaussianMGF X c κ ν
h'✝ : X =ᵐ[⇑κ ∘ₘ ν] Y
h'' : ∀ᵐ (ω' : Ω') ∂ν, ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
ω' : Ω'
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h' : ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
t : ℝ
⊢ mgf Y (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
rw [mgf_congr (Filter.EventuallyEq.symm h')]

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Y : Ω → ℝ
h : HasSubgaussianMGF X c κ ν
h'✝ : X =ᵐ[⇑κ ∘ₘ ν] Y
h'' : ∀ᵐ (ω' : Ω') ∂ν, ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
ω' : Ω'
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h' : ∀ᵐ (ω : Ω) ∂κ ω', X ω = Y ω
t : ℝ
⊢ mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
⊢ (1 / 2) ^ n ≤ P (⋂ k, {ω | σ n k ω < min (τ n ω) ↑(T n)})

Tactic:
rw [measure_iInter_of_ae_antitone, le_iInf_iff]

Goal state after:
ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
⊢ ∀ (i : ℕ), (1 / 2) ^ n ≤ P {ω | σ n i ω < min (τ n ω) ↑(T n)}

case hs
ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
⊢ ∀ᵐ (ω : Ω) ∂P, Antitone fun x ↦ ω ∈ {ω | σ n x ω < min (τ n ω) ↑(T n)}

case hsm
ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
⊢ ∀ (i : ℕ), NullMeasurableSet {ω | σ n i ω < min (τ n ω) ↑(T n)} P

case hfin
ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : ConditionallyCompleteLinearOrderBot ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
𝓕 : Filtration ι mΩ
inst✝¹ : SecondCountableTopology ι
inst✝ : IsFiniteMeasure P
τ : ℕ → Ω → WithTop ι
σ : ℕ → ℕ → Ω → WithTop ι
hτ : IsLocalizingSequence 𝓕 τ P
hσ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (σ n) P
T : ℕ → ι
hT : Tendsto T atTop atTop
n : ℕ
hn : ∀ (k : ℕ), (1 / 2) ^ n < P {ω | σ n k ω < min (τ n ω) ↑(T n)}
⊢ ∃ i, P {ω | σ n i ω < min (τ n ω) ↑(T n)} ≠ ∞
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁸ : ConditionallyCompleteLinearOrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
𝓕 : Filtration ι mΩ
X : ι → Ω → E
p : (ι → Ω → E) → Prop
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : IsFiniteMeasure P
inst✝³ : DenselyOrdered ι
inst✝² : NoMaxOrder ι
inst✝¹ : Zero E
inst✝ : 𝓕.IsRightContinuous
hp : IsStable 𝓕 p
hL : Locally (fun Y ↦ Locally p 𝓕 Y P) 𝓕 X P
τ : ℕ → ℕ → Ω → WithTop ι
hτ₁ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (τ n) P
hτ₂ :
  ∀ (n n_1 : ℕ),
    p
      (stoppedProcess
        (fun i ↦
          {ω | ⊥ < τ n n_1 ω}.indicator
            (stoppedProcess (fun i ↦ {ω | ⊥ < hL.localSeq n ω}.indicator (X i)) (hL.localSeq n) i))
        (τ n n_1))
⊢ Locally p 𝓕 X P

Tactic:
obtain ⟨nk, hnk, hpre⟩ :=
    hL.isLocalizingSequence_localSeq.isPrelocalizingSequence_inf_extraction hτ₁

Goal state after:
ι : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁸ : ConditionallyCompleteLinearOrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
𝓕 : Filtration ι mΩ
X : ι → Ω → E
p : (ι → Ω → E) → Prop
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : IsFiniteMeasure P
inst✝³ : DenselyOrdered ι
inst✝² : NoMaxOrder ι
inst✝¹ : Zero E
inst✝ : 𝓕.IsRightContinuous
hp : IsStable 𝓕 p
hL : Locally (fun Y ↦ Locally p 𝓕 Y P) 𝓕 X P
τ : ℕ → ℕ → Ω → WithTop ι
hτ₁ : ∀ (n : ℕ), IsLocalizingSequence 𝓕 (τ n) P
hτ₂ :
  ∀ (n n_1 : ℕ),
    p
      (stoppedProcess
        (fun i ↦
          {ω | ⊥ < τ n n_1 ω}.indicator
            (stoppedProcess (fun i ↦ {ω | ⊥ < hL.localSeq n ω}.indicator (X i)) (hL.localSeq n) i))
        (τ n n_1))
nk : ℕ → ℕ
hnk : StrictMono nk
hpre : IsPreLocalizingSequence 𝓕 (fun i ω ↦ min (hL.localSeq i ω) (τ i (nk i) ω)) P
⊢ Locally p 𝓕 X P
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∑ k ∈ range j, ∫ (x : ℝ) in ↑k..↑(k + 1), x ^ 2 ∂ρ =
    ∑ k ∈ range K, (∑ j ∈ Ioo k K, (↑j ^ 2)⁻¹) * ∫ (x : ℝ) in ↑k..↑(k + 1), x ^ 2 ∂ρ

Tactic:
simp_rw [mul_sum, sum_mul, sum_sigma']

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
⊢ ∑ x ∈ (range K).sigma range, (↑x.fst ^ 2)⁻¹ * ∫ (x : ℝ) in ↑x.snd..↑(x.snd + 1), x ^ 2 ∂ρ =
    ∑ x ∈ (range K).sigma fun a ↦ Ioo a K, (↑x.snd ^ 2)⁻¹ * ∫ (x : ℝ) in ↑x.fst..↑(x.fst + 1), x ^ 2 ∂ρ
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
inst✝¹ : SFinite ν
η : Kernel (Ω' × Ω) Ω''
inst✝ : IsZeroOrMarkovKernel η
hX : HasSubgaussianMGF X c κ ν
hY : HasSubgaussianMGF Y cY η (ν ⊗ₘ κ)
hκ : IsSFiniteKernel κ
q : ℚ
ω' : Ω'
hX_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
hX_int : Integrable (fun y ↦ rexp (↑q * X y)) (κ ω')
hY_mgf : ∀ᵐ (b : Ω) ∂κ ω', ∀ (t : ℝ), mgf Y (η (ω', b)) t ≤ rexp (↑cY * t ^ 2 / 2)
h_int_mul : Integrable (fun ω ↦ rexp (↑q * (X ω.1 + Y ω.2))) ((κ ⊗ₖ η) ω')
⊢ (∫ (a : Ω), rexp (↑q * X a) ∂κ ω') * rexp (↑cY * ↑q ^ 2 / 2) ≤ rexp ((↑c + ↑cY) * ↑q ^ 2 / 2)

Tactic:
add_mul,

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
inst✝¹ : SFinite ν
η : Kernel (Ω' × Ω) Ω''
inst✝ : IsZeroOrMarkovKernel η
hX : HasSubgaussianMGF X c κ ν
hY : HasSubgaussianMGF Y cY η (ν ⊗ₘ κ)
hκ : IsSFiniteKernel κ
q : ℚ
ω' : Ω'
hX_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
hX_int : Integrable (fun y ↦ rexp (↑q * X y)) (κ ω')
hY_mgf : ∀ᵐ (b : Ω) ∂κ ω', ∀ (t : ℝ), mgf Y (η (ω', b)) t ≤ rexp (↑cY * t ^ 2 / 2)
h_int_mul : Integrable (fun ω ↦ rexp (↑q * (X ω.1 + Y ω.2))) ((κ ⊗ₖ η) ω')
⊢ (∫ (a : Ω), rexp (↑q * X a) ∂κ ω') * rexp (↑cY * ↑q ^ 2 / 2) ≤ rexp ((↑c * ↑q ^ 2 + ↑cY * ↑q ^ 2) / 2)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : HasSubgaussianMGF X c μ
⊢ integrableExpSet X μ = Set.univ

Tactic:
ext t
  simpa using! hX.integrable_exp_mul t

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c μ
ε : ℝ
hε : 0 ≤ ε
⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))

Tactic:
by
  rw [HasSubgaussianMGF_iff_kernel] at h
  simpa using h.measure_ge_le hε

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: case mk
A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
g : ρ.IntertwiningMap σ
toLinearMap✝ : V →ₗ[A] W
isIntertwining'✝ : ∀ (g : G), toLinearMap✝ ∘ₗ ρ g = σ g ∘ₗ toLinearMap✝
h : { toLinearMap := toLinearMap✝, isIntertwining' := isIntertwining'✝ }.toLinearMap = g.toLinearMap
⊢ { toLinearMap := toLinearMap✝, isIntertwining' := isIntertwining'✝ } = g

Tactic:
cases g

Goal state after:
case mk.mk
A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid V
inst✝² : AddCommMonoid W
inst✝¹ : Module A V
inst✝ : Module A W
ρ : Representation A G V
σ : Representation A G W
toLinearMap✝¹ : V →ₗ[A] W
isIntertwining'✝¹ : ∀ (g : G), toLinearMap✝¹ ∘ₗ ρ g = σ g ∘ₗ toLinearMap✝¹
toLinearMap✝ : V →ₗ[A] W
isIntertwining'✝ : ∀ (g : G), toLinearMap✝ ∘ₗ ρ g = σ g ∘ₗ toLinearMap✝
h :
  { toLinearMap := toLinearMap✝¹, isIntertwining' := isIntertwining'✝¹ }.toLinearMap =
    { toLinearMap := toLinearMap✝, isIntertwining' := isIntertwining'✝ }.toLinearMap
⊢ { toLinearMap := toLinearMap✝¹, isIntertwining' := isIntertwining'✝¹ } =
    { toLinearMap := toLinearMap✝, isIntertwining' := isIntertwining'✝ }
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * ∑ j ∈ range (u i), Var[Y j; ℙ] =
    ∑ j ∈ range (u (N - 1)), (∑ i ∈ range N with j < u i, (↑(u i) ^ 2)⁻¹) * Var[Y j; ℙ]

Tactic:
simp_rw [mul_sum, sum_mul, sum_sigma']

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∑ x ∈ (range N).sigma fun a ↦ range (u a), (↑(u x.fst) ^ 2)⁻¹ * Var[Y x.snd; ℙ] =
    ∑ x ∈ (range (u (N - 1))).sigma fun a ↦ {i ∈ range N | a < u i}, (↑(u x.snd) ^ 2)⁻¹ * Var[Y x.fst; ℙ]
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case hc
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
t : ℝ
ht : t < 0
⊢ ∫ (x : Ω), (-X) x ∂μ = 0

Tactic:
simp only [Pi.neg_apply]

Goal state after:
case hc
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
inst✝ : IsProbabilityMeasure μ
a b : ℝ
hm : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hc : ∫ (x : Ω), X x ∂μ = 0
t : ℝ
ht : t < 0
⊢ ∫ (x : Ω), -X x ∂μ = 0