mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ hittingBtwn f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω =
    hittingBtwn f (Set.Ici b) (lowerCrossingTime a b f M n ω) M ω

Tactic:
this

Goal state after:
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ hittingBtwn f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω =
    hittingBtwn f (Set.Ici b) (lowerCrossingTime a b f N n ω) M ω
Module: Mathlib.Probability.Process.HittingTime

Goal state before:
pp: case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
ω : Ω
u : ι → Ω → β
s : Set β
n m₁ m₂ : ι
hm : m₁ ≤ m₂
h : ¬∃ j ∈ Set.Icc n m₁, u j ω ∈ s
⊢ hittingBtwn u s n m₁ ω ≤ hittingBtwn u s n m₂ ω

Tactic:
[

Goal state after:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
ω : Ω
u : ι → Ω → β
s : Set β
n m₁ m₂ : ι
hm : m₁ ≤ m₂
h : ¬∃ j ∈ Set.Icc n m₁, u j ω ∈ s
⊢ hittingBtwn u s n m₁ ω ≤ hittingBtwn u s n m₂ ω
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: case neg
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ k ∈ Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω

Tactic:
[

Goal state after:
case neg
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ k ∈ Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
hab : a < b
this : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ) / ENNReal.ofReal (b - a)
⊢ ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑‖f n ω - a‖₊ ∂μ ≤ ↑R + ↑‖a‖₊ * μ Set.univ

Tactic:
[

Goal state after:
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
hab : a < b
this : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ) / ENNReal.ofReal (b - a)
⊢ ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑‖f n ω - a‖₊ ∂μ ≤ ↑R + ↑‖a‖₊ * μ Set.univ
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: case neg
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
  ∀ (k : ℕ),
    ∑ n ∈ Finset.range N,
        (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator
          (fun m ↦ f (m + 1) ω - f m ω) n =
      stoppedValue f (fun ω ↦ ↑(upperCrossingTime a b f N (k + 1) ω)) ω -
        stoppedValue f (fun ω ↦ ↑(lowerCrossingTime a b f N k ω)) ω
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
    ∑ y ∈ Finset.range N,
      ∑ x ∈ Finset.range N,
        (Set.Ico (lowerCrossingTime a b f N y ω) (upperCrossingTime a b f N (y + 1) ω)).indicator
          (fun j ↦ f (j + 1) ω - f j ω) x

Tactic:
[

Goal state after:
case neg
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
  ∀ (k : ℕ),
    ∑ n ∈ Finset.range N,
        (Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator
          (fun m ↦ f (m + 1) ω - f m ω) n =
      stoppedValue f (fun ω ↦ ↑(upperCrossingTime a b f N (k + 1) ω)) ω -
        stoppedValue f (fun ω ↦ ↑(lowerCrossingTime a b f N k ω)) ω
⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤
    ∑ y ∈ Finset.range N,
      ∑ x ∈ Finset.range N,
        (Set.Ico (lowerCrossingTime a b f N y ω) (upperCrossingTime a b f N (y + 1) ω)).indicator
          (fun j ↦ f (j + 1) ω - f j ω) x
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: case refine_1
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
this : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ) / ENNReal.ofReal (b - a)
hbdd : ∀ (n : ℕ), ∫⁻ (x : Ω), ‖f n x‖ₑ ∂μ ≤ ↑R
n : ℕ
ω : Ω
⊢ ↑‖f n ω - a‖₊ ≤ ↑‖f n ω‖₊ + ↑‖a‖₊

Tactic:
[

Goal state after:
case refine_1
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
this : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ) / ENNReal.ofReal (b - a)
hbdd : ∀ (n : ℕ), ∫⁻ (x : Ω), ‖f n x‖ₑ ∂μ ≤ ↑R
n : ℕ
ω : Ω
⊢ ↑‖f n ω - a‖₊ ≤ ↑‖f n ω‖₊ + ↑‖a‖₊
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: case refine_2
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
this : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ) / ENNReal.ofReal (b - a)
hbdd : ∀ (n : ℕ), ∫⁻ (x : Ω), ‖f n x‖ₑ ∂μ ≤ ↑R
n : ℕ
⊢ ∫⁻ (a_1 : Ω), ↑‖f n a_1‖₊ + ↑‖a‖₊ ∂μ ≤ ↑R + ↑‖a‖₊ * μ Set.univ

Tactic:
[

Goal state after:
case refine_2
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hab : a < b
this : ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ (⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ) / ENNReal.ofReal (b - a)
hbdd : ∀ (n : ℕ), ∫⁻ (x : Ω), ‖f n x‖ₑ ∂μ ≤ ↑R
n : ℕ
⊢ ∫⁻ (a_1 : Ω), ↑‖f n a_1‖₊ + ↑‖a‖₊ ∂μ ≤ ↑R + ↑‖a‖₊ * μ Set.univ
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
i : ι
⊢ MeasurableSet {ω | max (τ ω) (π ω) ≤ ↑i}

Tactic:
[

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
i : ι
⊢ MeasurableSet {ω | max (τ ω) (π 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
i : ι
⊢ MeasurableSet {ω | min (τ ω) (π ω) ≤ ↑i}

Tactic:
[

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
i : ι
⊢ MeasurableSet {ω | min (τ ω) (π ω) ≤ ↑i}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case coe
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : AddGroup ι
inst✝² : Preorder ι
inst✝¹ : AddRightMono ι
inst✝ : AddLeftMono ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
i : ι
hi : 0 ≤ i
j : ι
ω : Ω
a : ι
⊢ a + i ≤ j ↔ a ≤ j - i

Tactic:
[

Goal state after:
case coe
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝³ : AddGroup ι
inst✝² : Preorder ι
inst✝¹ : AddRightMono ι
inst✝ : AddLeftMono ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
i : ι
hi : 0 ≤ i
j : ι
ω : Ω
a : ι
⊢ a + i ≤ j ↔ a ≤ j - i
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: case pos
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
  upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
    lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ (f i ω - a)⁺ + a}) =
    sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ f i ω})

Tactic:
[

Goal state after:
case pos
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
  upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
    lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
ω : Ω
h₁ : ∃ j ∈ Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k ω) N, (f j ω - a)⁺ ∈ Set.Ici (b - a)
h₂ : ∃ j ∈ Set.Icc (lowerCrossingTime a b f N k ω) N, f j ω ∈ Set.Ici b
⊢ sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ (f i ω - a)⁺ + a}) =
    sInf (Set.Icc (lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k ω) N ∩ {i | b ≤ f i ω})
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case mpr
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
i : ι
this : ∀ (j : WithTop ι), {ω | τ ω = ↑i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = ↑i} ∩ {_ω | ↑i ≤ j}
h : MeasurableSet (s ∩ {ω | τ ω = ↑i})
j : ι
⊢ MeasurableSet (s ∩ ({ω | τ ω = ↑i} ∩ {ω | τ ω ≤ ↑j}))

Tactic:
this

Goal state after:
case mpr
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
i : ι
this : ∀ (j : WithTop ι), {ω | τ ω = ↑i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = ↑i} ∩ {_ω | ↑i ≤ j}
h : MeasurableSet (s ∩ {ω | τ ω = ↑i})
j : ι
⊢ MeasurableSet (s ∩ ({ω | τ ω = ↑i} ∩ {_ω | ↑i ≤ ↑j}))
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: Ω : Type u_1
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 : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
g' : Ω → ℝ := fun ω ↦ if h : ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c) then h.choose else 0
hle : ⨆ n, ↑ℱ n ≤ m0
ω : Ω
hω : ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
⊢ Tendsto (fun n ↦ f n ω) atTop (𝓝 (g' ω))

Tactic:
[

Goal state after:
Ω : Type u_1
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 : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
g' : Ω → ℝ := fun ω ↦ if h : ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c) then h.choose else 0
hle : ⨆ n, ↑ℱ n ≤ m0
ω : Ω
hω : ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
⊢ Tendsto (fun n ↦ f n ω) atTop (𝓝 (g' ω))
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
hπ_bdd : ∃ N, ∀ (x : Ω), π x ≤ ↑N
⊢ ∫ (x : Ω), stoppedValue (stoppedProcess f τ) σ x ∂μ ≤ ∫ (x : Ω), stoppedValue (stoppedProcess f τ) π x ∂μ

Tactic:
[

Goal state after:
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
hπ_bdd : ∃ N, ∀ (x : Ω), π x ≤ ↑N
⊢ ∫ (x : Ω), stoppedValue (stoppedProcess f τ) σ x ∂μ ≤ ∫ (x : Ω), stoppedValue (stoppedProcess f τ) π x ∂μ
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: case pos
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
  upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
    lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
this : upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N (k + 1) = upperCrossingTime a b f N (k + 1)
ω : Ω
h₁ : ∃ j ∈ Set.Icc (upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N (k + 1) ω) N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ∃ j ∈ Set.Icc (upperCrossingTime a b f N (k + 1) ω) N, f j ω ∈ Set.Iic a
⊢ sInf (Set.Icc (upperCrossingTime a b f N (k + 1) ω) N ∩ {i | (f i ω - a)⁺ ≤ 0}) =
    sInf (Set.Icc (upperCrossingTime a b f N (k + 1) ω) N ∩ {i | f i ω ≤ a})

Tactic:
[

Goal state after:
case pos
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
hf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a
k : ℕ
ih :
  upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ∧
    lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = lowerCrossingTime a b f N k
this : upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N (k + 1) = upperCrossingTime a b f N (k + 1)
ω : Ω
h₁ : ∃ j ∈ Set.Icc (upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N (k + 1) ω) N, (f j ω - a)⁺ ∈ Set.Iic 0
h₂ : ∃ j ∈ Set.Icc (upperCrossingTime a b f N (k + 1) ω) N, f j ω ∈ Set.Iic a
⊢ sInf (Set.Icc (upperCrossingTime a b f N (k + 1) ω) N ∩ {i | (f i ω - a)⁺ ≤ 0}) =
    sInf (Set.Icc (upperCrossingTime a b f N (k + 1) ω) N ∩ {i | f i ω ≤ a})
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 : Ω
⊢ ω ∈ {ω | τ ω ≤ ↑i} ∩ {ω | τ ω ≤ ↑j} ↔ ω ∈ {ω | τ ω ≤ ↑(min i j)}

Tactic:
simp [Set.mem_inter_iff, Set.mem_ofPred_eq]

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 j : ι
this : {ω | τ ω ≤ ↑i} ∩ {ω | τ ω ≤ ↑j} = {ω | τ ω ≤ ↑(min i j)}
⊢ MeasurableSet ({ω | τ ω ≤ ↑i} ∩ {ω | τ ω ≤ ↑j})

Tactic:
this

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
i j : ι
this : {ω | τ ω ≤ ↑i} ∩ {ω | τ ω ≤ ↑j} = {ω | τ ω ≤ ↑(min i j)}
⊢ MeasurableSet {ω | τ ω ≤ ↑(min i j)}
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
x : Ω
hx : ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k x
⊢ ↑ε ≤ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) x

Tactic:
[

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
ε : ℝ≥0
n : ℕ
x : Ω
hx : ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k x
⊢ ↑ε ≤ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) x
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
this : {ω | ↑i ≤ τ ω} = {ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω}
⊢ MeasurableSet {ω | ↑i ≤ τ ω}

Tactic:
this

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
this : {ω | ↑i ≤ τ ω} = {ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω}
⊢ MeasurableSet ({ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω})
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:
[

Goal state after:
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
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
this : {ω | τ ω < ↑i} = {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}
⊢ MeasurableSet {ω | τ ω < ↑i}

Tactic:
this

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
this : {ω | τ ω < ↑i} = {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}
⊢ MeasurableSet ({ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i})
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 =
    (b - a - 0) * ∫ (x : Ω), ↑(upcrossingsBefore 0 (b - a) (f - fun x x_1 ↦ a)⁺ N x) ∂μ

Tactic:
[

Goal state after:
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 =
    (b - a - 0) * ∫ (x : Ω), ↑(upcrossingsBefore 0 (b - a) (f - fun x x_1 ↦ a)⁺ N x) ∂μ
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
this : {ω | ↑i ≤ τ ω} = {ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω}
⊢ MeasurableSet {ω | ↑i ≤ τ ω}

Tactic:
this

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
this : {ω | ↑i ≤ τ ω} = {ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω}
⊢ MeasurableSet ({ω | τ ω = ↑i} ∪ {ω | ↑i < τ ω})
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
this : {ω | τ ω < ↑i} = {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}
⊢ MeasurableSet {ω | τ ω < ↑i}

Tactic:
this

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
this : {ω | τ ω < ↑i} = {ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i}
⊢ MeasurableSet ({ω | τ ω ≤ ↑i} \ {ω | τ ω = ↑i})
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case refine_2
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:
[

Goal state after:
case refine_2
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
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case refine_2
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
this : ∀ (i : WithTop ι), {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i}
⊢ ((MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})) ∧
      MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | π ω ≤ ↑i})) →
    MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | min (τ ω) (π ω) ≤ ↑i})

Tactic:
[

Goal state after:
case refine_2
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
this : ∀ (i : WithTop ι), {ω | min (τ ω) (π ω) ≤ i} = {ω | τ ω ≤ i} ∪ {ω | π ω ≤ i}
⊢ ((MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})) ∧
      MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | π ω ≤ ↑i})) →
    MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | min (τ ω) (π ω) ≤ ↑i})
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:
[

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
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
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
this :
  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
⊢ Measurable (upcrossingsBefore a b f N)

Tactic:
this

Goal state after:
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
this :
  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
⊢ Measurable fun ω ↦ ∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i
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}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ ↑i})

Tactic:
[

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}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | 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
this : upcrossings a b f ω < ∞ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
⊢ upcrossings a b f ω < ∞ ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k

Tactic:
[

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