mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case coe
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
hτ : IsStoppingTime f τ
i : ι
ω : Ω
a : ι
⊢ ↑i < ↑a ∨ ↑i = ↑a ↔ ↑a = ↑i ∨ ↑i < ↑a

Tactic:
| coe a =>

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
hτ : IsStoppingTime f τ
i : ι
⊢ {ω | τ ω < ↑i} = {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
hτ : IsStoppingTime f τ
i : ι
ω : Ω
⊢ ω ∈ {ω | τ ω < ↑i} ↔ ω ∈ {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ {ω | ↑i ≤ τ ω} = {ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
⊢ ω ∈ {ω | ↑i ≤ τ ω} ↔ ω ∈ {ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case coe
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
a : ι
⊢ i < a ∨ a = i ↔ a = i ∨ i < a

Tactic:
or_comm

Goal state after:
case coe
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
a : ι
⊢ a = i ∨ i < a ↔ a = i ∨ i < a
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case coe
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
a : ι
⊢ ↑i < ↑a ∨ ↑i = ↑a ↔ ↑a = ↑i ∨ ↑i < ↑a

Tactic:
| coe a =>

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ {ω | τ ω < ↑i} = {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
ω : Ω
⊢ ω ∈ {ω | τ ω < ↑i} ↔ ω ∈ {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
i : WithTop ι
⊢ {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
i : WithTop ι
ω : Ω
⊢ ω ∈ {ω | min (τ ω) (π ω) ≤ i} ↔ ω ∈ {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i}
Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
⊢ ∫ (ω : Ω), stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω ∂μ ≤
    ∫ (x : Ω), stoppedValue f (fun x ↦ ↑n) x ∂μ

Tactic:
refine hsub.expected_stoppedValue_mono
        (hsub.stronglyAdapted.adapted.isStoppingTime_hittingBtwn measurableSet_Ici)
        (isStoppingTime_const _ _) (fun ω ↦ ?_) (fun _ => mod_cast le_rfl)

Goal state after:
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
ω : Ω
⊢ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω) ≤ ↑n
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
f : Filtration ι m
inst✝ : PseudoMetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x ↦ Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ ↑i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x ↦ ↑(↑x).1
j : ι
⊢ (fun x ↦ τ (↑x).2) ⁻¹' Set.Iic ↑j = (fun x ↦ (↑x).2) ⁻¹' {ω | τ ω ≤ ↑(min i j)}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
f : Filtration ι m
inst✝ : PseudoMetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x ↦ Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ ↑i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x ↦ ↑(↑x).1
j : ι
ω : ↑s
⊢ ω ∈ (fun x ↦ τ (↑x).2) ⁻¹' Set.Iic ↑j ↔ ω ∈ (fun x ↦ (↑x).2) ⁻¹' {ω | τ ω ≤ ↑(min i j)}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case refine_2
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
f : Filtration ι m
inst✝ : PseudoMetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x ↦ Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ ↑i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x ↦ ↑(↑x).1
sc : Set (↑(Set.Iic i) × Ω) := sᶜ
⊢ (fun x ↦ min (↑↑(↑x).1) (τ (↑x).2)) = fun x ↦ ↑↑(↑x).1

Tactic:
ext1 ω

Goal state after:
case refine_2
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
f : Filtration ι m
inst✝ : PseudoMetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x ↦ Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ ↑i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x ↦ ↑(↑x).1
sc : Set (↑(Set.Iic i) × Ω) := sᶜ
ω : ↑sc
⊢ min (↑↑(↑ω).1) (τ (↑ω).2) = ↑↑(↑ω).1
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
inst✝ : TopologicalSpace β
f : Filtration ι m
h : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
n : ι
hτ_le : ∀ (ω : Ω), τ ω ≤ ↑n
hτ_le' : ∀ (ω : Ω), (τ ω).untopA ≤ n
⊢ stoppedValue u τ = (fun p ↦ u (↑p.1) p.2) ∘ fun ω ↦ (⟨(τ ω).untopA, ⋯⟩, ω)

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁶ : LinearOrder ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : SecondCountableTopology ι
inst✝¹ : BorelSpace ι
inst✝ : TopologicalSpace β
f : Filtration ι m
h : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
n : ι
hτ_le : ∀ (ω : Ω), τ ω ≤ ↑n
hτ_le' : ∀ (ω : Ω), (τ ω).untopA ≤ n
ω : Ω
⊢ stoppedValue u τ ω = ((fun p ↦ u (↑p.1) p.2) ∘ fun ω ↦ (⟨(τ ω).untopA, ⋯⟩, ω)) ω
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
t : Set β
ht : MeasurableSet t
i : ι
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
⊢ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑i} = (stoppedValue u fun ω ↦ min (τ ω) ↑i) ⁻¹' t ∩ {ω | τ ω ≤ ↑i}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
t : Set β
ht : MeasurableSet t
i : ι
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
ω : Ω
⊢ ω ∈ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑i} ↔ ω ∈ (stoppedValue u fun ω ↦ min (τ ω) ↑i) ⁻¹' t ∩ {ω | τ ω ≤ ↑i}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
t : Set β
ht : MeasurableSet t
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
⊢ stoppedValue u τ ⁻¹' t = (⋃ n, stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑(seq n)}) ∪ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω = ⊤}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹⁰ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁹ : LinearOrder ι
inst✝⁸ : MeasurableSpace ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : BorelSpace ι
inst✝³ : TopologicalSpace β
f : Filtration ι m
inst✝² : PseudoMetrizableSpace β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
hf_prog : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω ↦ min (τ ω) ↑i)
t : Set β
ht : MeasurableSet t
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
ω : Ω
⊢ ω ∈ stoppedValue u τ ⁻¹' t ↔
    ω ∈ (⋃ n, stoppedValue u τ ⁻¹' t ∩ {ω | τ ω ≤ ↑(seq n)}) ∪ stoppedValue u τ ⁻¹' t ∩ {ω | τ ω = ⊤}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝¹ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝ : AddCommMonoid E
s : Finset ι
hbdd : ∀ (ω : Ω), τ ω ∈ WithTop.some '' ↑s
y : Ω
⊢ {i ∈ s | y ∈ {ω | τ ω = ↑i}} = {(τ y).untopA}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
inst✝¹ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝ : AddCommMonoid E
s : Finset ι
hbdd : ∀ (ω : Ω), τ ω ∈ WithTop.some '' ↑s
y : Ω
ω : ι
⊢ ω ∈ {i ∈ s | y ∈ {ω | τ ω = ↑i}} ↔ ω ∈ {(τ y).untopA}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝² : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑s
ω : Ω
h : τ ω < ↑n
h_contra : τ ω = ⊤
⊢ False

Tactic:
by simp [h_contra] at h

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

Goal state before:
pp: case refine_2.inr
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω ↦ rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω ↦ rexp (-u * X ω)) μ
htu : |t| ≤ |u|
hu : u ≤ 0
⊢ Integrable (fun ω ↦ rexp (|u| * X ω)) μ

Tactic:
rwa [abs_of_nonpos hu]

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
inst✝³ : Nonempty ι
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : AddCommMonoid E
n : ι
ω : Ω
h : τ ω < ↑n
h_contra : τ ω = ⊤
⊢ False

Tactic:
by simp [h_contra] at h

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

Goal state before:
pp: case refine_1
Ω✝ : Type u_1
m✝ : MeasurableSpace Ω✝
X✝ : Ω✝ → ℝ
μ✝ : Measure Ω✝
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t₁ : ℝ
ht₁ : t₁ ∈ integrableExpSet X μ
t₂ : ℝ
ht₂ : t₂ ∈ integrableExpSet X μ
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
h_le : t₁ ≤ t₂
⊢ t₁ ≤ a * t₁ + b * t₂

Tactic:
calc t₁
    _ = a * t₁ + b * t₁ := by rw [← add_mul, hab, one_mul]
    _ ≤ a * t₁ + b * t₂ := by gcongr

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : Nonempty ι
μ : Measure Ω
τ : Ω → WithTop ι
E : Type u_4
p : ℝ≥0∞
u : ι → Ω → E
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝ : NormedAddCommGroup E
hτ : IsStoppingTime ℱ τ
hu : ∀ (n : ι), MemLp (u n) p μ
n : ι
s : Finset ι
hbdd : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑s
this : MemLp (fun ω ↦ ∑ i ∈ s with i < n, {a | τ a = ↑i}.indicator (u i) ω) p μ
⊢ ∑ i ∈ s with i < n, {ω | τ ω = ↑i}.indicator (u i) = fun ω ↦ ∑ i ∈ s with i < n, {a | τ a = ↑i}.indicator (u i) ω

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : Nonempty ι
μ : Measure Ω
τ : Ω → WithTop ι
E : Type u_4
p : ℝ≥0∞
u : ι → Ω → E
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝ : NormedAddCommGroup E
hτ : IsStoppingTime ℱ τ
hu : ∀ (n : ι), MemLp (u n) p μ
n : ι
s : Finset ι
hbdd : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑s
this : MemLp (fun ω ↦ ∑ i ∈ s with i < n, {a | τ a = ↑i}.indicator (u i) ω) p μ
ω : Ω
⊢ (∑ i ∈ s with i < n, {ω | τ ω = ↑i}.indicator (u i)) ω = ∑ i ∈ s with i < n, {a | τ a = ↑i}.indicator (u i) ω
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : Nonempty ι
μ : Measure Ω
τ : Ω → WithTop ι
E : Type u_4
p : ℝ≥0∞
u : ι → Ω → E
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : NormedAddCommGroup E
inst✝ : LocallyFiniteOrderBot ι
hτ : IsStoppingTime ℱ τ
hu : ∀ (n : ι), MemLp (u n) p μ
n : ι
ω : Ω
h : τ ω < ↑n
h_contra : τ ω = ⊤
⊢ False

Tactic:
by simp [h_contra] at h

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : Nonempty ι
μ : Measure Ω
τ : Ω → WithTop ι
E : Type u_4
u : ι → Ω → E
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : NormedAddCommGroup E
inst✝ : LocallyFiniteOrderBot ι
hτ : IsStoppingTime ℱ τ
hu : ∀ (n : ι), Integrable (u n) μ
n : ι
ω : Ω
h : τ ω < ↑n
h_contra : τ ω = ⊤
⊢ False

Tactic:
by simp [h_contra] at h

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
𝒢 : Filtration ι m
τ η : Ω → WithTop ι
i : ι
s : Set Ω
inst✝ : DecidablePred fun x ↦ x ∈ s
hτ_st : IsStoppingTime 𝒢 τ
hη_st : IsStoppingTime 𝒢 η
hτ : ∀ (ω : Ω), ↑i ≤ τ ω
hη : ∀ (ω : Ω), ↑i ≤ η ω
hs : MeasurableSet s
n : ι
⊢ {ω | s.piecewise τ η ω ≤ ↑n} = s ∩ {ω | τ ω ≤ ↑n} ∪ sᶜ ∩ {ω | η ω ≤ ↑n}

Tactic:
ext1 ω

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
𝒢 : Filtration ι m
τ η : Ω → WithTop ι
i : ι
s : Set Ω
inst✝ : DecidablePred fun x ↦ x ∈ s
hτ_st : IsStoppingTime 𝒢 τ
hη_st : IsStoppingTime 𝒢 η
hτ : ∀ (ω : Ω), ↑i ≤ τ ω
hη : ∀ (ω : Ω), ↑i ≤ η ω
hs : MeasurableSet s
n : ι
ω : Ω
⊢ ω ∈ {ω | s.piecewise τ η ω ≤ ↑n} ↔ ω ∈ s ∩ {ω | τ ω ≤ ↑n} ∪ sᶜ ∩ {ω | η ω ≤ ↑n}
Module: Mathlib.Probability.Moments.ComplexMGF

Goal state before:
pp: case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
⊢ ∃ᶠ (z : ℂ) in 𝓝[≠] ↑t, complexMGF X μ z = complexMGF Y μ' z

Tactic:
have h_real : ∃ᶠ (x : ℝ) in 𝓝[≠] t, complexMGF X μ x = complexMGF Y μ' x := by
    refine .of_forall fun y ↦ ?_
    rw [complexMGF_ofReal, complexMGF_ofReal, hXY]

Goal state after:
case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')}
h_real : ∃ᶠ (x : ℝ) in 𝓝[≠] t, complexMGF X μ ↑x = complexMGF Y μ' ↑x
⊢ ∃ᶠ (z : ℂ) in 𝓝[≠] ↑t, complexMGF X μ z = complexMGF Y μ' z
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: case pos
μ : ℝ
v : ℝ≥0
z : ℂ
hv : v = 0
⊢ complexMGF id (gaussianReal μ v) z = cexp (z * ↑μ + ↑↑v * z ^ 2 / 2)

pp: case neg
μ : ℝ
v : ℝ≥0
z : ℂ
hv : ¬v = 0
⊢ complexMGF id (gaussianReal μ v) z = cexp (z * ↑μ + ↑↑v * z ^ 2 / 2)

Tactic:
· simp [complexMGF, hv]

Goal state after:
case neg
μ : ℝ
v : ℝ≥0
z : ℂ
hv : ¬v = 0
⊢ complexMGF id (gaussianReal μ v) z = cexp (z * ↑μ + ↑↑v * z ^ 2 / 2)
Module: Mathlib.Probability.Moments.MGFAnalytic

Goal state before:
pp: case 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
⊢ ((∫ (x : Ω), (fun ω ↦ X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v) ^ 2 =
    (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * ((∫ (x : Ω), (fun ω ↦ X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v) /
        mgf X μ v -
      ((∫ (x : Ω), (fun ω ↦ X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v) ^ 2

pp: case e_a.hb
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ mgf X μ v ≠ 0

Tactic:
· ring

Goal state after:
case e_a.hb
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ mgf X μ v ≠ 0
Module: Mathlib.Probability.Distributions.Gaussian.Basic

Goal state before:
pp: μ : Measure ℝ
h : IsGaussian μ
⊢ Measure.map (⇑(ContinuousLinearMap.id ℝ ℝ)) μ = gaussianReal (∫ (x : ℝ), id x ∂μ) Var[id; μ].toNNReal

Tactic:
rw [h.map_eq_gaussianReal]

Goal state after:
μ : Measure ℝ
h : IsGaussian μ
⊢ gaussianReal (∫ (x : ℝ), (ContinuousLinearMap.id ℝ ℝ) x ∂μ) Var[⇑(ContinuousLinearMap.id ℝ ℝ); μ].toNNReal =
    gaussianReal (∫ (x : ℝ), id x ∂μ) Var[id; μ].toNNReal
Module: Mathlib.Probability.Distributions.Gaussian.Real

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
p : Measure Ω
μ : ℝ
v : ℝ≥0
X : Ω → ℝ
hX : Measure.map X p = gaussianReal μ v
z : ℂ
⊢ complexMGF X p z = cexp (z * ↑μ + ↑↑v * z ^ 2 / 2)

Tactic:
have hX_meas : AEMeasurable X p := aemeasurable_of_map_neZero (by rw [hX]; infer_instance)
  rw [← complexMGF_id_map hX_meas, hX, complexMGF_id_gaussianReal]

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
inst✝ : IsZeroOrProbabilityMeasure μ
ht : 0 < t
hc : ∫ (x : Ω), X x ∂μ = 0
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
hu : UniqueDiffOn ℝ (Set.Icc 0 t)
⊢ ∃ u ∈ Set.Ioo 0 t, cgf X μ t - 0 = iteratedDeriv 2 (cgf X μ) u * t ^ 2 / 2

Tactic:
← sub_zero t

Goal state after:
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
inst✝ : IsZeroOrProbabilityMeasure μ
ht : 0 < t
hc : ∫ (x : Ω), X x ∂μ = 0
hs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)
hu : UniqueDiffOn ℝ (Set.Icc 0 t)
⊢ ∃ u ∈ Set.Ioo 0 t, cgf X μ t - 0 = iteratedDeriv 2 (cgf X μ) u * (t - 0) ^ 2 / 2
Module: Mathlib.Probability.Moments.CovarianceBilin

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
h : MemLp id 2 μ
x y : E
⊢ ((covarianceBilin μ) x) y = ∫ (z : E), ⟪x, z - ∫ (x : E), id x ∂μ⟫ * ⟪y, z - ∫ (x : E), id x ∂μ⟫ ∂μ

Tactic:
by
  simp [covarianceBilin, covarianceBilinDual_apply' h]

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

Goal state before:
pp: case pos
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
x y : ℝ
h : MemLp id 2 μ
⊢ ((covarianceBilin μ) x) y = x * y * Var[id; μ]

pp: case neg
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
x y : ℝ
h : ¬MemLp id 2 μ
⊢ ((covarianceBilin μ) x) y = x * y * Var[id; μ]

Tactic:
· simp only [covarianceBilin_apply_eq_cov h, RCLike.inner_apply, conj_trivial, mul_comm]
    rw [covariance_const_mul_left, covariance_const_mul_right, ← mul_assoc,
      covariance_self aemeasurable_id', Function.id_def]

Goal state after:
case neg
μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
x y : ℝ
h : ¬MemLp id 2 μ
⊢ ((covarianceBilin μ) x) y = x * y * Var[id; 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
⊢ ∫ (x : ℝ), x ∂gaussianReal μ₁ v₁ = μ₂ ∧ v₁ = v₂

Tactic:
← integral_id_gaussianReal (μ := μ₂) (v := v₂),

Goal state after:
μ₁ μ₂ : ℝ
v₁ v₂ : ℝ≥0
h : gaussianReal μ₁ v₁ = gaussianReal μ₂ v₂
⊢ ∫ (x : ℝ), x ∂gaussianReal μ₁ v₁ = ∫ (x : ℝ), x ∂gaussianReal μ₂ v₂ ∧ v₁ = v₂
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
⊢ p ≠ 0

Tactic:
by simp [hp]

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
h_Lp : MemLp id p μ
hp : ¬p = 0
hp_top : ¬p = ∞
h0 : 0 < p.toReal
⊢ p ≠ 0

Tactic:
by simp [hp]

Goal state after: