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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
ω : Ω
⊢ u (min (↑(σ ω).untopA) (τ ω)).untopA ω =
    if σ ω = ⊤ then u (min (↑(Classical.arbitrary ι)) (τ ω)).untopA ω else u (min (σ ω) (τ ω)).untopA ω

Tactic:
cases σ ω <;> cases τ ω

Goal state after:
case top.top
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
ω : Ω
⊢ u (min ↑⊤.untopA ⊤).untopA ω = if ⊤ = ⊤ then u (min ↑(Classical.arbitrary ι) ⊤).untopA ω else u (min ⊤ ⊤).untopA ω

case top.coe
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
ω : Ω
a✝ : ι
⊢ u (min ↑⊤.untopA ↑a✝).untopA ω =
    if ⊤ = ⊤ then u (min ↑(Classical.arbitrary ι) ↑a✝).untopA ω else u (min ⊤ ↑a✝).untopA ω

case coe.top
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
ω : Ω
a✝ : ι
⊢ u (min ↑(↑a✝).untopA ⊤).untopA ω =
    if ↑a✝ = ⊤ then u (min ↑(Classical.arbitrary ι) ⊤).untopA ω else u (min ↑a✝ ⊤).untopA ω

case coe.coe
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
ω : Ω
a✝¹ a✝ : ι
⊢ u (min ↑(↑a✝¹).untopA ↑a✝).untopA ω =
    if ↑a✝¹ = ⊤ then u (min ↑(Classical.arbitrary ι) ↑a✝).untopA ω else u (min ↑a✝¹ ↑a✝).untopA ω
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
r : ℝ
R : ℝ≥0
hr : 0 ≤ r
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
i : ℕ
⊢ ∀ᵐ (ω : Ω) ∂μ, stoppedAbove f r i ω ≤ r + ↑R

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
⊢ stoppedValue (stoppedProcess u τ) σ = fun ω ↦
    if σ ω ≠ ⊤ then stoppedValue u (fun ω ↦ min (σ ω) (τ ω)) ω
    else stoppedValue u (fun ω ↦ min (↑(Classical.arbitrary ι)) (τ ω)) ω

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
ω : Ω
hω : σ ω ≠ ⊤
⊢ stoppedValue (stoppedProcess u τ) σ ω = stoppedValue u (fun ω ↦ min (σ ω) (τ ω)) ω

Tactic:
by

Goal state after:
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✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
μ : Measure Ω
hσ✝ : ∀ᵐ (ω : Ω) ∂μ, σ ω ≠ ⊤
ω : Ω
hσ : σ ω ≠ ⊤
⊢ stoppedValue (stoppedProcess u τ) σ ω = stoppedValue u (fun ω ↦ min (σ ω) (τ ω)) ω

Tactic:
by

Goal state after:
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✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
μ : Measure Ω
hσ : ∀ᵐ (ω : Ω) ∂μ, σ ω ≠ ⊤
⊢ stoppedValue (stoppedProcess u τ) σ =ᵐ[μ] stoppedValue u fun ω ↦ min (σ ω) (τ ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝ : LinearOrder ι
i : ι
ω : Ω
h : ↑i ≤ τ ω
⊢ stoppedProcess u τ i ω = u i ω

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝ : LinearOrder ι
i : ι
ω : Ω
h : τ ω ≤ ↑i
⊢ stoppedProcess u τ i ω = u (τ ω).untopA ω

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hr : 0 ≤ r
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
i : ℕ
⊢ eLpNorm (stoppedAbove f r i) 1 μ ≤ 2 * μ Set.univ * ENNReal.ofReal (r + ↑R)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝¹ : LinearOrder ι
inst✝ : Zero β
s : Set Ω
i : ι
⊢ stoppedProcess (fun i ↦ s.indicator (u i)) τ i = s.indicator (stoppedProcess u τ i)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝² : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝¹ : LinearOrder ι
inst✝ : Zero β
s : Set Ω
⊢ stoppedProcess (fun i ↦ s.indicator (u i)) τ = fun i ↦ s.indicator (stoppedProcess u τ i)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: Ω : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hr : 0 ≤ r
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
i : ℕ
⊢ eLpNorm (stoppedAbove f r i) 1 μ ≤ ↑(2 * μ Set.univ * ENNReal.ofReal (r + ↑R)).toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω) atTop (𝓝 c)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
ht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω) atTop (𝓝 c)
ω : Ω
hω : ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω) atTop (𝓝 c)
hωb : (upperBounds (Set.range fun n ↦ f n ω)).Nonempty
i : ℕ
hi : hωb.some < ↑i
⊢ ∀ (n : ℕ), f n ω < ↑i

Tactic:
by

Goal state after:
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
⊢ ENNReal.ofReal
        (∫ (ω : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω},
          stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω ∂μ) +
      ENNReal.ofReal
        (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε},
          stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω ∂μ) =
    ENNReal.ofReal (∫ (ω : Ω), stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω ∂μ)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
⊢ stoppedProcess (stoppedProcess u τ) σ = stoppedProcess u (σ ⊓ τ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
ht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω) atTop (𝓝 c)
ω : Ω
hω : ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω) atTop (𝓝 c)
hωb : (upperBounds (Set.range fun n ↦ f n ω)).Nonempty
i : ℕ
hi : hωb.some < ↑i
hib : ∀ (n : ℕ), f n ω < ↑i
⊢ ∀ (n : ℕ), stoppedAbove f (↑i) n ω = f n ω

Tactic:
by

Goal state after:
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
⊢ ENNReal.ofReal (∫ (ω : Ω), stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
⊢ stoppedProcess (stoppedProcess u τ) σ = stoppedProcess u fun ω ↦ min (σ ω) (τ ω)

Tactic:
by

Goal state after:
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
⊢ ↑ε * μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} ≤
    ENNReal.ofReal (∫ (ω : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω}, f n ω ∂μ)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
h : σ ≤ τ
⊢ stoppedProcess (stoppedProcess u τ) σ = stoppedProcess u σ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : Nonempty ι
u : ι → Ω → β
τ σ : Ω → WithTop ι
inst✝ : LinearOrder ι
h : τ ≤ σ
⊢ stoppedProcess (stoppedProcess u τ) σ = stoppedProcess u τ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) → ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
β : Type u_3
inst✝² : LinearOrder ι
mβ : MeasurableSpace β
inst✝¹ : NormedAddCommGroup β
inst✝ : BorelSpace β
f : ι → Ω → β
i j : ι
hf : ∀ (i : ι), StronglyMeasurable (f i)
hfi : iIndepFun f μ
hij : i < j
this : Indep (⨆ k ∈ {j}, MeasurableSpace.comap (f k) mβ) (⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) μ
⊢ Indep (MeasurableSpace.comap (f j) mβ) (↑(Filtration.natural f hf) i) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
β : Type u_3
inst✝² : LinearOrder ι
mβ : MeasurableSpace β
inst✝¹ : NormedAddCommGroup β
inst✝ : BorelSpace β
f : ι → Ω → β
i j : ι
hf : ∀ (i : ι), StronglyMeasurable (f i)
hfi : iIndepFun f μ
hij : i < j
⊢ Disjoint {j} {k | k ≤ i}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
β : Type u_3
inst✝² : LinearOrder ι
mβ : MeasurableSpace β
inst✝¹ : NormedAddCommGroup β
inst✝ : BorelSpace β
f : ι → Ω → β
i j : ι
hf : ∀ (i : ι), StronglyMeasurable (f i)
hfi : iIndepFun f μ
hij : i < j
⊢ Indep (MeasurableSpace.comap (f j) mβ) (↑(Filtration.natural f hf) i) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
β : Type u_3
inst✝⁵ : LinearOrder ι
mβ : MeasurableSpace β
inst✝⁴ : NormedAddCommGroup β
inst✝³ : BorelSpace β
f : ι → Ω → β
i j : ι
inst✝² : SecondCountableTopology β
inst✝¹ : CompleteSpace β
inst✝ : NormedSpace ℝ β
hf : ∀ (i : ι), StronglyMeasurable (f i)
hfi : iIndepFun f μ
hij : i < j
⊢ μ[f j | ↑(Filtration.natural f hf) i] =ᵐ[μ] fun x ↦ ∫ (x : Ω), f j x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hf0 : f 0 = 0
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
inst✝ : LinearOrder ι
i j : ι
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s μ
hij : i < j
⊢ μ[(s j).indicator fun x ↦ 1 | ↑(filtrationOfSet hsm) i] =ᵐ[μ] fun x ↦ μ.real (s j)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
g : ℕ → Ω → ℝ := fun n ω ↦ f n ω - f 0 ω
hg : Submartingale g ℱ μ
⊢ g 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalExpectation

Goal state before:
pp: Ω : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m : MeasurableSpace Ω
μ : Measure Ω
f : Ω → E
hle₁ : m₁ ≤ m
hle₂ : m₂ ≤ m
inst✝ : SigmaFinite (μ.trim hle₂)
hf : StronglyMeasurable f
hindp : Indep m₁ m₂ μ
hfint : MemLp f 1 μ
s : Set Ω
hms : MeasurableSet s
hs : μ s < ∞
⊢ Continuous fun f ↦ ∫ (x : Ω), ↑↑↑f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalExpectation

Goal state before:
pp: Ω : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m : MeasurableSpace Ω
μ : Measure Ω
f : Ω → E
hle₁ : m₁ ≤ m
hle₂ : m₂ ≤ m
inst✝ : SigmaFinite (μ.trim hle₂)
hf : StronglyMeasurable f
hindp : Indep m₁ m₂ μ
hfint : MemLp f 1 μ
s : Set Ω
hms : MeasurableSet s
hs : μ s < ∞
h_integral : Continuous fun f ↦ ∫ (x : Ω), ↑↑↑f x ∂μ
⊢ Continuous fun f ↦ ∫ (x : Ω) in s, ↑↑↑f x ∂μ

Tactic:
by

Goal state after:
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:
by

Goal state after:
Module: Mathlib.Probability.Martingale.BorelCantelli

Goal state before:
pp: 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 : Submartingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
g : ℕ → Ω → ℝ := fun n ω ↦ f n ω - f 0 ω
hg : Submartingale g ℱ μ
hg0 : g 0 = 0
⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |g (i + 1) ω - g i ω| ≤ ↑R

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ∞
this✝ : IsProbabilityMeasure μ
this :
  {ω |
      Tendsto (fun n ↦ ∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator 1 | ↑(filtrationOfSet hsm) k] ω) atTop atTop} =ᵐ[μ]
    Set.univ
⊢ μ {ω | Tendsto (fun n ↦ ∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator 1 | ↑(filtrationOfSet hsm) k] ω) atTop atTop} =
    1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.ConditionalExpectation

Goal state before:
pp: Ω : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m : MeasurableSpace Ω
μ : Measure Ω
f : Ω → E
hle₁ : m₁ ≤ m
hle₂ : m₂ ≤ m
inst✝ : SigmaFinite (μ.trim hle₂)
hf : StronglyMeasurable f
hindp : Indep m₁ m₂ μ
⊢ μ[f | m₂] =ᵐ[μ] fun x ↦ ∫ (x : Ω), f x ∂μ

Tactic:
by

Goal state after:
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
sc : Set (↑(Set.Iic i) × Ω) := sᶜ
h_min_eq_left : (fun x ↦ min (↑↑(↑x).1) (τ (↑x).2)) = fun x ↦ ↑↑(↑x).1
⊢ Measurable (sᶜ.domRestrict fun p ↦ min (↑↑p.1) (τ p.2))

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
p : ℕ
μ : Measure Ω
hp : p ≠ 0
⊢ moment 0 p μ = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
p : ℕ
⊢ moment X p 0 = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
p : ℕ
μ : Measure Ω
hp : p ≠ 0
⊢ centralMoment 0 p μ = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
⊢ moment X 1 μ = ∫ (x : Ω), X x ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
p : ℕ
⊢ centralMoment X p 0 = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_int : Integrable X μ
⊢ centralMoment X 1 μ = (1 - μ.real Set.univ) * ∫ (x : Ω), X x ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsZeroOrProbabilityMeasure μ
h : IsProbabilityMeasure μ
h_int : ¬Integrable X μ
h_sub : Integrable (fun x ↦ X x - integral μ X) μ
⊢ X = (fun x ↦ X x - integral μ X) + fun x ↦ integral μ X

Tactic:
by

Goal state after:
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
⊢ IsStronglyProgressive f fun i ω ↦ (min (↑i) (τ ω)).untopA

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsZeroOrProbabilityMeasure μ
h : IsProbabilityMeasure μ
h_int : ¬Integrable X μ
⊢ ¬Integrable (fun x ↦ X x - integral μ X) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BorelCantelli

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
s : ℕ → Set Ω
hsm : ∀ (n : ℕ), MeasurableSet (s n)
hs : iIndepSet s μ
hs' : ∑' (n : ℕ), μ (s n) = ∞
this✝ : IsProbabilityMeasure μ
this :
  ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = (fun x ↦ μ.real (s (n + 1))) ω
ω : Ω
hω : ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = μ.real (s (n + 1))
⊢ Tendsto (fun n ↦ ∑ k ∈ Finset.range n, μ (s (k + 1))) atTop (𝓝 ∞)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsZeroOrProbabilityMeasure μ
⊢ centralMoment X 1 μ = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
p : ℕ
μ : Measure Ω
X Y : Ω → ℝ
hXY : X =ᵐ[μ] Y
x : Ω
hx : X x = Y x
⊢ ((X - fun x ↦ ∫ (a : Ω), Y a ∂μ) ^ p) x = ((Y - fun x ↦ ∫ (a : Ω), Y a ∂μ) ^ p) x

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
p : ℕ
μ : Measure Ω
X Y : Ω → ℝ
hXY : X =ᵐ[μ] Y
⊢ centralMoment X p μ = centralMoment Y p μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
⊢ mgf 0 μ t = μ.real Set.univ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
⊢ cgf 0 μ t = log (μ.real Set.univ)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
⊢ mgf X 0 = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
⊢ cgf X 0 = 0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t c : ℝ
⊢ mgf (fun x ↦ c) μ t = μ.real Set.univ * rexp (t * c)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t c : ℝ
inst✝ : IsProbabilityMeasure μ
⊢ mgf (fun x ↦ c) μ t = rexp (t * c)

Tactic:
by

Goal state after:
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✝⁸ : Nonempty ι
u : ι → Ω → β
τ : Ω → WithTop ι
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
f : Filtration ι m
inst✝ : PseudoMetrizableSpace ι
h : IsStronglyProgressive f u
hτ : IsStoppingTime f τ
⊢ IsStronglyProgressive f (MeasureTheory.stoppedProcess u τ)

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
c : ℝ
⊢ cgf (fun x ↦ c) μ t = log (μ.real Set.univ) + t * c

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
c : ℝ
⊢ cgf (fun x ↦ c) μ t = t * c

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
⊢ mgf X μ 0 = μ.real Set.univ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
⊢ mgf X μ 0 = 1

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
⊢ cgf X μ 0 = log (μ.real Set.univ)

Tactic:
by

Goal state after: