mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.ZeroOne

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
h : ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)
t : Set Ω
h_indep : IndepSet t t κ μα
a : α
h_0_1_top : (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞
h' : IsFiniteMeasure (κ a)
⊢ (κ a) t = 0 ∨ (κ a) t = 1

Tactic:
simpa using

Goal state after:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
h : ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)
t : Set Ω
h_indep : IndepSet t t κ μα
a : α
h_0_1_top : (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞
h' : IsFiniteMeasure (κ a)
⊢ (κ a) t = 0 ∨ (κ a) t = 1
Module: Mathlib.Probability.Independence.ZeroOne

Goal state before:
pp: Ω : Type u_2
m m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hm : Indep m m μ
t : Set Ω
ht : MeasurableSet t
⊢ μ t = 0 ∨ μ t = 1

Tactic:
simpa using

Goal state after:
Ω : Type u_2
m m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hm : Indep m m μ
t : Set Ω
ht : MeasurableSet t
⊢ μ t = 0 ∨ μ t = 1
Module: Mathlib.Probability.Kernel.Invariance

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
κ : Kernel α α
inst✝ : IsMarkovKernel κ
π : Measure α
h_rev : κ.IsReversible π
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α), (κ x) s ∂π = ∫⁻ (x : α) in s, (κ x) Set.univ ∂π

Tactic:
simpa using

Goal state after:
α : Type u_1
mα : MeasurableSpace α
κ : Kernel α α
inst✝ : IsMarkovKernel κ
π : Measure α
h_rev : κ.IsReversible π
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α), (κ x) s ∂π = π s
Module: Mathlib.Probability.Kernel.Irreducible

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
c : ℝ≥0∞
φ : Measure α
κ : Kernel α α
hκ : IsIrreducible φ κ
s : Set α
hs : MeasurableSet s
hsp : (c • φ) s > 0
⊢ ∀ (a : α), ∃ n, ((κ ^ n) a) s > 0

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
c : ℝ≥0∞
φ : Measure α
κ : Kernel α α
hκ : IsIrreducible φ κ
s : Set α
hs : MeasurableSet s
hsp : (c • φ) s > 0
⊢ ∀ (a : α), ∃ n, 0 < ((κ ^ n) a) s
Module: Mathlib.Probability.Kernel.Irreducible

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
φ₁ φ₂ : Measure α
hφ : φ₁ ≤ φ₂
κ : Kernel α α
hκ : IsIrreducible φ₂ κ
s : Set α
hs : MeasurableSet s
hsp : φ₁ s > 0
⊢ ∀ (a : α), ∃ n, ((κ ^ n) a) s > 0

Tactic:
simpa using

Goal state after:
α : Type u_1
mα : MeasurableSpace α
φ₁ φ₂ : Measure α
hφ : φ₁ ≤ φ₂
κ : Kernel α α
hκ : IsIrreducible φ₂ κ
s : Set α
hs : MeasurableSet s
hsp : φ₁ s > 0
⊢ ∀ (a : α), ∃ n, 0 < ((κ ^ n) a) s
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X : SFinKer
f₁ : X.carrier → PUnit.{?u.96 + 1} × X.carrier := fun x ↦ (PUnit.unit, x)
hf₁ : Measurable f₁
hf₂ : Measurable Prod.snd
⊢ { carrier := { carrier := PUnit.{u + 1}, str := PUnit.instMeasurableSpace }.carrier × X.carrier,
      str := Prod.instMeasurableSpace } ≅
    X

Tactic:
refine ⟨⟨Kernel.id.map Prod.snd, inferInstance⟩,
      ⟨Kernel.id.map f₁, inferInstance⟩, ?_, ?_⟩

Goal state after:
case refine_1
X : SFinKer
f₁ : X.carrier → PUnit.{u + 1} × X.carrier := fun x ↦ (PUnit.unit, x)
hf₁ : Measurable f₁
hf₂ : Measurable Prod.snd
⊢ { hom := Kernel.id.map Prod.snd, property := ⋯ } ≫ { hom := Kernel.id.map f₁, property := ⋯ } =
    𝟙
      { carrier := { carrier := PUnit.{u + 1}, str := PUnit.instMeasurableSpace }.carrier × X.carrier,
        str := Prod.instMeasurableSpace }

case refine_2
X : SFinKer
f₁ : X.carrier → PUnit.{u + 1} × X.carrier := fun x ↦ (PUnit.unit, x)
hf₁ : Measurable f₁
hf₂ : Measurable Prod.snd
⊢ { hom := Kernel.id.map f₁, property := ⋯ } ≫ { hom := Kernel.id.map Prod.snd, property := ⋯ } = 𝟙 X
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_1
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
h𝓑𝓧 : 𝓑 ≤ 𝓧
x✝ : π.IsProper
h : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x
⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x

Tactic:
simpa using

Goal state after:
case refine_1
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
h𝓑𝓧 : 𝓑 ≤ 𝓧
x✝ : π.IsProper
h : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x
⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
h𝓑𝓧 : 𝓑 ≤ 𝓧
h : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x
⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x

Tactic:
simpa using

Goal state after:
case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
h𝓑𝓧 : 𝓑 ≤ 𝓧
h : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x
⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
A B : Set X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
μ : Measure X
hA : MeasurableSet A
hB : MeasurableSet B
⊢ ∫⁻ (a : X) in B, (π a) A ∂μ = ∫⁻ (a : X), B.indicator (fun x ↦ (π a) A) a ∂μ

Tactic:
← lintegral_indicator (h𝓑𝓧 _ hB)

Goal state after:
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
A B : Set X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
μ : Measure X
hA : MeasurableSet A
hB : MeasurableSet B
⊢ ∫⁻ (a : X), B.indicator (fun a ↦ (π a) A) a ∂μ = ∫⁻ (a : X), B.indicator (fun x ↦ (π a) A) a ∂μ
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
i n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
inst✝ : SigmaFinite (μ.trim ⋯)
hin : i ≤ n
t : Set Ω
⊢ MeasurableSet (t ∩ {x | τ x = ↑i}) ↔ MeasurableSet (t ∩ {x | τ x = ↑i})

Tactic:
IsStoppingTime.measurableSet_inter_eq_iff

Goal state after:
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
i n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
inst✝ : SigmaFinite (μ.trim ⋯)
hin : i ≤ n
t : Set Ω
⊢ MeasurableSet (t ∩ {ω | τ ω = ↑i}) ↔ MeasurableSet (t ∩ {x | τ x = ↑i})
Module: Mathlib.Probability.Martingale.OptionalSampling

Goal state before:
pp: case pos
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
t : Set Ω
⊢ MeasurableSet (t ∩ {x | τ x = ↑i}) ↔ MeasurableSet (t ∩ {x | τ x = ↑i})

Tactic:
IsStoppingTime.measurableSet_inter_eq_iff

Goal state after:
case pos
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
t : Set Ω
⊢ MeasurableSet (t ∩ {ω | τ ω = ↑i}) ↔ MeasurableSet (t ∩ {x | τ x = ↑i})
Module: Mathlib.Probability.Martingale.OptionalSampling

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
ι : Type u_3
inst✝⁶ : LinearOrder ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝² : SigmaFiniteFiltration μ ℱ
τ : Ω → WithTop ι
f : ι → Ω → E
n : ι
inst✝¹ : Nonempty ι
h✝ : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
i : WithTop ι
hi : i ∈ Set.range τ
h : i = ⊤
⊢ False

Tactic:
simp only [h, Set.mem_range] at hi
    obtain ⟨ω, hω⟩ := hi
    specialize hτ_le ω
    simp [hω] at hτ_le

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

Goal state before:
pp: case e_f
Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
e : γ × β' × β ≃ᵐ (γ × β') × β := MeasurableEquiv.prodAssoc.symm
h :
  Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ) ∘ₖ (Kernel.id ×ₖ condDistrib g k μ) =
    Kernel.id ∘ₖ (Kernel.id ×ₖ condDistrib g k μ) ×ₖ (condDistrib f k μ ∘ₖ Kernel.id)
⊢ Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ =
    Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ) ∘ₖ (Kernel.id ×ₖ condDistrib g k μ)

Tactic:
simpa using

Goal state after:
case e_f
Ω : Type u_1
β : Type u_3
β' : Type u_4
mΩ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
γ : Type u_5
mγ : MeasurableSpace γ
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝³ : StandardBorelSpace β
inst✝² : Nonempty β
inst✝¹ : StandardBorelSpace β'
inst✝ : Nonempty β'
hf : Measurable f
hg : Measurable g
k : Ω → γ
hk : Measurable k
e : γ × β' × β ≃ᵐ (γ × β') × β := MeasurableEquiv.prodAssoc.symm
h :
  Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ) ∘ₖ (Kernel.id ×ₖ condDistrib g k μ) =
    Kernel.id ∘ₖ (Kernel.id ×ₖ condDistrib g k μ) ×ₖ (condDistrib f k μ ∘ₖ Kernel.id)
⊢ Kernel.id ×ₖ condDistrib g k μ ×ₖ condDistrib f k μ =
    Kernel.id ×ₖ Kernel.prodMkRight β' (condDistrib f k μ) ∘ₖ (Kernel.id ×ₖ condDistrib g k μ)
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
h : HasSubgaussianMGF X c κ ν
h_int : Integrable (fun ω ↦ rexp (1 * X ω)) (⇑κ ∘ₘ ν)
⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)

Tactic:
simpa using

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
h : HasSubgaussianMGF X c κ ν
h_int : Integrable (fun ω ↦ rexp (1 * X ω)) (⇑κ ∘ₘ ν)
⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)
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
h : HasSubgaussianMGF X c κ ν
h_int✝ : ∀ᵐ (ω' : Ω') ∂ν, Integrable (fun y ↦ rexp (1 * X y)) (κ ω')
ω : Ω'
h_int : Integrable (fun y ↦ rexp (1 * X y)) (κ ω)
⊢ AEStronglyMeasurable X (κ ω)

Tactic:
simpa using

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
h : HasSubgaussianMGF X c κ ν
h_int✝ : ∀ᵐ (ω' : Ω') ∂ν, Integrable (fun y ↦ rexp (1 * X y)) (κ ω')
ω : Ω'
h_int : Integrable (fun y ↦ rexp (1 * X y)) (κ ω)
⊢ AEStronglyMeasurable X (κ ω)
Module: Mathlib.Probability.Moments.Tilted

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

Tactic:
simpa using

Goal state after:
case pos
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
ht : t ∈ interior (integrableExpSet X μ)
p : ℝ≥0
hX : AEMeasurable X μ
hp : p = 0
⊢ AEStronglyMeasurable X (μ.tilted fun x ↦ t * X x)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case pos
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
h : HasSubgaussianMGF X c κ ν
t : ℝ
p : ℝ≥0
hp0 : p = 0
⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)

Tactic:
simpa using

Goal state after:
case pos
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
h : HasSubgaussianMGF X c κ ν
t : ℝ
p : ℝ≥0
hp0 : p = 0
⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)
Module: Mathlib.Probability.ProbabilityMassFunction.Binomial

Goal state before:
pp: k b : ℕ
hb : k ≤ b
x : ℝ≥0
h : x ≤ 1
⊢ k % (b + 1) = k

Tactic:
simpa using

Goal state after:
k b : ℕ
hb : k ≤ b
x : ℝ≥0
h : x ≤ 1
⊢ k ≤ b
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case pos
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
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
t : ℝ
h0 : κ ω' = 0
⊢ log (mgf X (κ ω') t) ≤ log (rexp (↑c * t ^ 2 / 2))

Tactic:
simpa using

Goal state after:
case pos
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
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
h_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')
t : ℝ
h0 : κ ω' = 0
⊢ 0 ≤ ↑c * t ^ 2 / 2
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
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
⊢ IsFiniteMeasure (κ ω')

Tactic:
simpa using

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
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
⊢ IsFiniteMeasure (κ ω')
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
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : IsFiniteMeasure (κ ω')
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
⊢ (κ ω').real Set.univ ≤ 1

Tactic:
simpa using

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
h✝ : HasSubgaussianMGF X c κ ν
ω' : Ω'
h : IsFiniteMeasure (κ ω')
h_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
⊢ (κ ω').real Set.univ ≤ 1
Module: Mathlib.Probability.Process.Kolmogorov

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝¹ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace E
p q : ℝ
M : ℝ≥0
P : Measure Ω
X : T → Ω → E
hX : IsAEKolmogorovProcess X P p q M
s t : T
h : edist s t = 0
this : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0
ω : Ω
hω : edist (X s ω) (X t ω) ^ p = 0 ω
⊢ edist (X s ω) (X t ω) = 0

Tactic:
simpa using

Goal state after:
T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝¹ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace E
p q : ℝ
M : ℝ≥0
P : Measure Ω
X : T → Ω → E
hX : IsAEKolmogorovProcess X P p q M
s t : T
h : edist s t = 0
this : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0
ω : Ω
hω : edist (X s ω) (X t ω) ^ p = 0 ω
⊢ edist (X s ω) (X t ω) = 0
Module: Mathlib.Probability.Process.Kolmogorov

Goal state before:
pp: T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝¹ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace E
p q : ℝ
P : Measure Ω
X : T → Ω → E
hX : IsAEKolmogorovProcess X P p q 0
s t : T
this : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0
ω : Ω
hω : edist (X s ω) (X t ω) ^ p = 0 ω
⊢ edist (X s ω) (X t ω) = 0

Tactic:
simpa using

Goal state after:
T : Type u_1
Ω : Type u_2
E : Type u_3
inst✝¹ : PseudoEMetricSpace T
mΩ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace E
p q : ℝ
P : Measure Ω
X : T → Ω → E
hX : IsAEKolmogorovProcess X P p q 0
s t : T
this : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0
ω : Ω
hω : edist (X s ω) (X t ω) ^ p = 0 ω
⊢ edist (X s ω) (X t ω) = 0
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
h : HasSubgaussianMGF X c κ ν
t : ℝ
⊢ Integrable (fun ω ↦ rexp (t * (-X) ω)) (⇑κ ∘ₘ ν)

Tactic:
simpa using

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
h : HasSubgaussianMGF X c κ ν
t : ℝ
⊢ Integrable (fun ω ↦ rexp (-(t * X ω))) (⇑κ ∘ₘ ν)
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
h : HasSubgaussianMGF X c κ ν
ω' : Ω'
hm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
t : ℝ
⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
simpa using

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
h : HasSubgaussianMGF X c κ ν
ω' : Ω'
hm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)
t : ℝ
⊢ ∫ (x : Ω), rexp (-(t * X x)) ∂κ ω' ≤ rexp (↑c * t ^ 2 / 2)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case refine_3
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
hX : Measurable X
h : HasSubgaussianMGF X c κ ν
⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf id ((κ.map X) ω') t ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
simpa using

Goal state after:
case refine_3
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
hX : Measurable X
h : HasSubgaussianMGF X c κ ν
⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (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
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : LinearOrder ι
𝓕 : Filtration ι mΩ
p : (ι → Ω → E) → Prop
inst✝³ : OrderBot ι
inst✝² : Zero E
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
hp : IsStable 𝓕 p
X : ι → Ω → E
hX : (fun Y ↦ Locally p 𝓕 Y P) X
τ : Ω → WithTop ι
hτ : IsStoppingTime 𝓕 τ
n : ℕ
⊢ p
    (stoppedProcess (stoppedProcess (fun i ↦ ({ω | ⊥ < Locally.localSeq hX n ω} ∩ {ω | ⊥ < τ ω}).indicator (X i)) τ)
      (Locally.localSeq hX n))

Tactic:
Set.inter_comm,

Goal state after:
ι : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : LinearOrder ι
𝓕 : Filtration ι mΩ
p : (ι → Ω → E) → Prop
inst✝³ : OrderBot ι
inst✝² : Zero E
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
hp : IsStable 𝓕 p
X : ι → Ω → E
hX : (fun Y ↦ Locally p 𝓕 Y P) X
τ : Ω → WithTop ι
hτ : IsStoppingTime 𝓕 τ
n : ℕ
⊢ p
    (stoppedProcess (stoppedProcess (fun i ↦ ({ω | ⊥ < τ ω} ∩ {ω | ⊥ < Locally.localSeq hX n ω}).indicator (X i)) τ)
      (Locally.localSeq hX n))
Module: Mathlib.RepresentationTheory.Subrepresentation

Goal state before:
pp: case single
A : Type u_1
G : Type u_2
W : Type u_3
M : Type u_4
inst✝⁵ : CommSemiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid W
inst✝² : Module A W
ρ : Representation A G W
inst✝¹ : AddCommMonoid M
inst✝ : Module A[G] M
σ : Subrepresentation (Representation.ofModule M)
m : M
hm : m ∈ σ.toSubmodule.carrier
g : G
a : A
⊢ MonoidAlgebra.single g a • m ∈ σ.toSubmodule.carrier

Tactic:
← mul_one a,

Goal state after:
case single
A : Type u_1
G : Type u_2
W : Type u_3
M : Type u_4
inst✝⁵ : CommSemiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid W
inst✝² : Module A W
ρ : Representation A G W
inst✝¹ : AddCommMonoid M
inst✝ : Module A[G] M
σ : Subrepresentation (Representation.ofModule M)
m : M
hm : m ∈ σ.toSubmodule.carrier
g : G
a : A
⊢ MonoidAlgebra.single g (a * 1) • m ∈ σ.toSubmodule.carrier
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case hf
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := ⋯
q : ℝ≥0 := ⋯
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ MemLp (fun a ↦ rexp (t * X a)) (ENNReal.ofReal ↑p) (κ ω')

Tactic:
simpa using

Goal state after:
case hf
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX
q : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cY
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ MemLp (fun a ↦ rexp (t * X a)) (↑p) (κ ω')
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case hg
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := ⋯
q : ℝ≥0 := ⋯
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ MemLp (fun a ↦ rexp (t * Y a)) (ENNReal.ofReal ↑q) (κ ω')

Tactic:
simpa using

Goal state after:
case hg
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX
q : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cY
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ MemLp (fun a ↦ rexp (t * Y a)) (↑q) (κ ω')
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
hA : 0 ≤ A
⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂Measure.map f μ

Tactic:
simpa using

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
hA : 0 ≤ A
⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂Measure.map f μ