mathlib-initiative/mathlib-tactics huggingface dataset:
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
i j : ι
⊢ {ω | τ ω ≤ ↑i} ∩ {ω | τ ω ≤ ↑j} = {ω | τ ω ≤ ↑(min i j)}

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
τ : Ω → ℕ∞
inst✝ : SigmaFiniteFiltration μ 𝒢
h : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
σ π : Ω → ℕ∞
hσ : IsStoppingTime 𝒢 σ
hπ : IsStoppingTime 𝒢 π
hσ_le_π : σ ≤ π
n : ℕ
hπ_le_n : ∀ (x : Ω), π x ≤ ↑n
ω : Ω
⊢ ↑n ≠ ⊤

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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
i : ι
⊢ MeasurableSet {ω | τ ω ≤ ↑i}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n ↦ μ[g | ↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n ↦ μ[g | ↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n ↦ μ[g | ↑ℱ n]) ℱ μ) μ
this : g =ᵐ[μ] limitProcess (fun n x ↦ μ[g | ↑ℱ n] x) ℱ μ
⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n ↦ μ[g | ↑ℱ n] x) atTop (𝓝 (g x))

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
τ : Ω → ℕ∞
inst✝ : SigmaFiniteFiltration μ 𝒢
h : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
⊢ Submartingale (stoppedProcess f τ) 𝒢 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✝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 | ↑i ≤ τ ω} = {ω | τ ω = ↑i} ∪ {ω | ↑i < 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
ε : ℝ≥0
n : ℕ
⊢ ∀ (ω : Ω),
    (↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω) → ↑ε ≤ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω

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✝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:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n ↦ μ[g | ↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n ↦ μ[g | ↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n ↦ μ[g | ↑ℱ n]) ℱ μ) μ
⊢ ∀ (n : ℕ) (s : Set Ω),
    MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x ↦ μ[g | ↑ℱ n] x) ℱ μ x ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
⊢ upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
    lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N n = lowerCrossingTime a b f N n

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
⊢ upcrossingsBefore 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N ω = upcrossingsBefore a b f N ω

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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ MeasurableSet {ω | τ ω = ↑i}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω ↦ (f N ω - a)⁺) 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✝ : 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:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n ↦ μ[g | ↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n ↦ μ[g | ↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n ↦ μ[g | ↑ℱ n]) ℱ μ) μ
this :
  ∀ (n : ℕ) (s : Set Ω),
    MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x ↦ μ[g | ↑ℱ n] x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ IsPiSystem {s | ∃ n, MeasurableSet s}

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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ MeasurableSet {ω | ↑i ≤ τ ω}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
N : ℕ
hab : b - a ≤ 0
⊢ 0 ≤ ∫ (x : Ω), ↑(upcrossingsBefore a b f N 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✝ : 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:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
N : ℕ
⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω ↦ (f N ω - a)⁺) 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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
h_countable : (Set.range τ).Countable
i : ι
⊢ MeasurableSet {ω | τ ω < ↑i}

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✝³ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : SecondCountableTopology ι
hτ : IsStoppingTime f τ
⊢ Measurable τ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
⊢ ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1

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
ε : ℝ≥0
n : ℕ
this :
  ∀ (ω : Ω),
    (↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω) → ↑ε ≤ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω
h :
  ↑ε * μ.real {a | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k a} ≤
    ∫ (x : Ω) in {a | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k a},
      stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) x ∂μ
⊢ ↑ε ≠ ∞

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✝ : 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:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1
⊢ ∀ k ∈ Finset.Ico (upcrossingsBefore a b f N ω + 1) (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 0

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
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},
        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
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
⊢ ⋯.measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace

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✝ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
⊢ MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s

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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
i : ι
⊢ ⋯.measurableSpace = hτ.measurableSpace ⊓ ↑f i

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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
i : ι
⊢ MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
⊢ upcrossingsBefore a b f N ω = ∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ℱ : Filtration ℕ m0
hf : StronglyAdapted ℱ f
hab : a < b
⊢ upcrossingsBefore a b f N = fun ω ↦ ∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ℱ : Filtration ℕ m0
hf : StronglyAdapted ℱ f
hab : a < b
⊢ Measurable (upcrossingsBefore a b f N)

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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
i : WithTop ι
⊢ s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ i} =
    s ∩ {ω | τ ω ≤ i} ∩ {ω | min (τ ω) (π ω) ≤ i} ∩ {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : StronglyAdapted ℱ f
hab : a < b
⊢ ∀ᵐ (ω : Ω) ∂μ, ‖↑(upcrossingsBefore a b f N ω)‖ ≤ ↑N

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
⊢ upcrossings a b f ω < ∞ ↔ ∃ k, upcrossings a b f ω ≤ ↑k

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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
h_eq :
  ∀ (i : WithTop ι),
    s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ i} =
      s ∩ {ω | τ ω ≤ i} ∩ {ω | min (τ ω) (π ω) ≤ i} ∩ {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i}
i : ι
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ ↑i} ∩ {ω | min (τ ω) (π ω) ≤ ↑i} ∩ {ω | min (τ ω) ↑i ≤ min (min (τ ω) (π ω)) ↑i})

Tactic:
refine ((hs.2 i).inter ((hτ.min hπ) i)).inter ?_

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
h_eq :
  ∀ (i : WithTop ι),
    s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ i} =
      s ∩ {ω | τ ω ≤ i} ∩ {ω | min (τ ω) (π ω) ≤ i} ∩ {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i}
i : ι
⊢ MeasurableSet {ω | min (τ ω) ↑i ≤ min (min (τ ω) (π ω)) ↑i}
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
⊢ upcrossings a b f ω < ∞ ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k

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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : MeasurableSet s
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})

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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
⊢ ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ π 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
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ENNReal.ofReal (∫ (ω : Ω), f n ω ∂μ) =
    ENNReal.ofReal (∫ (ω : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω}, f n ω ∂μ) +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
hab : a < b
this : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)
N : ℕ
⊢ ∫⁻ (ω : Ω), ↑(upcrossingsBefore a b f N ω) ∂μ = ∫⁻ (ω : Ω), ↑↑(upcrossingsBefore a b f N ω) ∂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✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
i : ι
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ ↑i}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})

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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
⊢ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ ↑j} = {ω | min (τ ω) ↑j ≤ min (π ω) ↑j} ∩ {ω | τ ω ≤ ↑j}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n ↦ μ[g | ↑ℱ n] x) atTop (𝓝 (g x))

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
this :
  ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ↑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
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : SecondCountableTopology ι
inst✝ : OrderTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
⊢ MeasurableSet {ω | τ ω ≤ π ω}

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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : SecondCountableTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
⊢ {ω | τ ω = π ω} = {ω | τ ω ≤ π ω} ∩ {ω | π ω ≤ τ ω}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n ↦ μ[g | ↑ℱ n] x) atTop (𝓝 (μ[g | ⨆ n, ↑ℱ n] x))

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
f : ℕ → Ω → ℝ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
a b : ℝ
hf : Submartingale f ℱ μ
⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂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✝³ : LinearOrder ι
f : Filtration ι m
τ π : Ω → WithTop ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : SecondCountableTopology ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
⊢ MeasurableSet {ω | τ ω = π 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
inst✝ : Bot ι
⊢ (stoppedValue u fun x ↦ ⊥) = u ⊥

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
⊢ Tendsto (fun n ↦ eLpNorm (μ[g | ↑ℱ n] - μ[g | ⨆ n, ↑ℱ n]) 1 μ) atTop (𝓝 0)

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
inst✝¹ : LinearOrder ι
inst✝ : OrderBot ι
⊢ (stoppedProcess u fun x ↦ ⊥) = fun x ↦ u ⊥

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
inst✝ : LinearOrder ι
⊢ (stoppedProcess u fun x ↦ ⊤) = u

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
ω : Ω
hω : ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε
⊢ hittingBtwn f {y | ↑ε ≤ y} 0 n ω = n

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 : 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
ω : Ω
hbddω : ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
h_zero : ¬WithTop.untopA (min (↑i) (leastGE f r ω)) = 0
k : ℕ
hk : WithTop.untopA (min (↑i) (leastGE f r ω)) = k + 1
⊢ ↑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
⊢ ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +
      ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε}, f n ω ∂μ) ≤
    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)

Tactic:
by

Goal state after: