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

Goal state before:
pp: case refine_2
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
i : ι
⊢ MeasurableSet {ω | τ ω ≤ ↑i}

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
i : ι
⊢ MeasurableSet {ω | τ ω ≤ ↑i}
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: case refine_1
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b - a ≤ (f ω i - a)⁺
⊢ b ≤ f ω i

pp: case refine_2
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b ≤ f ω i
⊢ b - a ≤ (f ω i - a)⁺

Tactic:
·

Goal state after:
case refine_1
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b - a ≤ (f ω i - a)⁺
⊢ b ≤ f 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✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
i : ι
⊢ sᶜ ∩ {ω | τ ω ≤ ↑i} = (sᶜ ∪ {ω | τ ω ≤ ↑i}ᶜ) ∩ {ω | τ ω ≤ ↑i}

Tactic:
·

Goal state after:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
s : Set Ω
hs : MeasurableSet s ∧ ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i})
i : ι
⊢ sᶜ ∩ {ω | τ ω ≤ ↑i} = (sᶜ ∪ {ω | τ ω ≤ ↑i}ᶜ) ∩ {ω | τ ω ≤ ↑i}
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: case refine_2
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b ≤ f ω i
⊢ b - a ≤ (f ω i - a)⁺

Tactic:
·

Goal state after:
case refine_2
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
h : b ≤ f ω i
⊢ b - a ≤ (f ω i - a)⁺
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
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)
hR' : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑‖f n ω - a‖₊ ∂μ ≤ ↑R + ↑‖a‖₊ * μ Set.univ
n : ℕ
ω : Ω
⊢ ↑‖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
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)
hR' : ∀ (n : ℕ), ∫⁻ (ω : Ω), ↑‖f n ω - a‖₊ ∂μ ≤ ↑R + ↑‖a‖₊ * μ Set.univ
n : ℕ
ω : Ω
⊢ ↑‖f n ω - a‖₊ ≠ ∞
Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
⊢ 0 ≤ ∑ i ∈ Finset.range (N + 1), ∫ (a : Ω), {ω | τ ω ≤ ↑i ∧ ↑i < π ω}.indicator (f (i + 1) - f i) a ∂μ

pp: case hf
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
⊢ ∀ i ∈ Finset.range (N + 1), Integrable ({ω | τ ω ≤ ↑i ∧ ↑i < π ω}.indicator (f (i + 1) - f i)) μ

Tactic:
·

Goal state after:
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
⊢ 0 ≤ ∑ i ∈ Finset.range (N + 1), ∫ (a : Ω), {ω | τ ω ≤ ↑i ∧ ↑i < π ω}.indicator (f (i + 1) - f i) a ∂μ
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
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)
⊢ ENNReal.ofReal (b - a) ≠ 0

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
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)
⊢ ENNReal.ofReal (b - a) ≠ 0
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: case h0
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) * ENNReal.ofReal (b - a) ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ
⊢ ENNReal.ofReal (b - a) ≠ 0 ∨ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ ≠ 0

pp: case ht
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) * ENNReal.ofReal (b - a) ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ
⊢ ENNReal.ofReal (b - a) ≠ ∞ ∨ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ ≠ ∞

Tactic:
·

Goal state after:
case h0
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) * ENNReal.ofReal (b - a) ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ
⊢ ENNReal.ofReal (b - a) ≠ 0 ∨ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ ≠ 0
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
hle : τ ≤ π
s : Set Ω
hs : MeasurableSet s
i : ι
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ ↑i} ∩ {ω | π ω ≤ ↑i})

pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
hle : τ ≤ π
s : Set Ω
hs : MeasurableSet s
i : ι
⊢ s ∩ {ω | π ω ≤ ↑i} = s ∩ {ω | τ ω ≤ ↑i} ∩ {ω | π ω ≤ ↑i}

Tactic:
·

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
hle : τ ≤ π
s : Set Ω
hs : MeasurableSet s
i : ι
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ ↑i} ∩ {ω | π ω ≤ ↑i})
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: case ht
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) * ENNReal.ofReal (b - a) ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ
⊢ ENNReal.ofReal (b - a) ≠ ∞ ∨ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ ≠ ∞

Tactic:
·

Goal state after:
case ht
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) * ENNReal.ofReal (b - a) ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ
⊢ ENNReal.ofReal (b - a) ≠ ∞ ∨ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ ≠ ∞
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
hle : τ ≤ π
s : Set Ω
hs : MeasurableSet s
i : ι
⊢ s ∩ {ω | π ω ≤ ↑i} = s ∩ {ω | τ ω ≤ ↑i} ∩ {ω | π ω ≤ ↑i}

Tactic:
·

Goal state after:
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
τ π : Ω → WithTop ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
hle : τ ≤ π
s : Set Ω
hs : MeasurableSet s
i : ι
⊢ s ∩ {ω | π ω ≤ ↑i} = s ∩ {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
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ ∫ (a : Ω) in {ω | τ ω ≤ ↑i ∧ ↑i < π ω}, f i a ∂μ ≤ ∫ (a : Ω) in {ω | τ ω ≤ ↑i ∧ ↑i < π ω}, f (i + 1) a ∂μ

pp: case hf
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f (i + 1)) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})

pp: case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f i) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})

Tactic:
·

Goal state after:
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ ∫ (a : Ω) in {ω | τ ω ≤ ↑i ∧ ↑i < π ω}, f i a ∂μ ≤ ∫ (a : Ω) in {ω | τ ω ≤ ↑i ∧ ↑i < π ω}, f (i + 1) a ∂μ
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case mp
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})
⊢ MeasurableSet s

pp: case mpr
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
⊢ MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})

Tactic:
·

Goal state after:
case mp
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})
⊢ MeasurableSet s
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
i : ι
s : Set Ω
h : MeasurableSet s
⊢ MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})

Tactic:
·

Goal state after:
case mpr
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
⊢ MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case pos
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
j : ι
hij : i ≤ j
⊢ MeasurableSet (s ∩ {ω | ↑i ≤ ↑j})

pp: case neg
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
j : ι
hij : ¬i ≤ j
⊢ MeasurableSet (s ∩ {ω | ↑i ≤ ↑j})

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
j : ι
hij : i ≤ j
⊢ MeasurableSet (s ∩ {ω | ↑i ≤ ↑j})
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case neg
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
j : ι
hij : ¬i ≤ j
⊢ MeasurableSet (s ∩ {ω | ↑i ≤ ↑j})

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : Preorder ι
f : Filtration ι m
i : ι
s : Set Ω
h : MeasurableSet s
j : ι
hij : ¬i ≤ j
⊢ MeasurableSet (s ∩ {ω | ↑i ≤ ↑j})
Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: case hf
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f (i + 1)) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})

pp: case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f i) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})

Tactic:
·

Goal state after:
case hf
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f (i + 1)) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})
Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f i) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})

Tactic:
·

Goal state after:
case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
this : ∀ (i : ℕ), MeasurableSet {ω | τ ω ≤ ↑i ∧ ↑i < π ω}
i : ℕ
x✝ : i ∈ Finset.range (N + 1)
⊢ Integrable (f i) (μ.restrict {ω | τ ω ≤ ↑i ∧ ↑i < π ω})
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
this : MeasurableSpace Ω := ⨆ n, ↑ℱ n
⊢ μ {a | ¬∃ c, Tendsto (fun n ↦ f n a) atTop (𝓝 c)} = 0

pp: case hs
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
this : MeasurableSpace Ω := ⨆ n, ↑ℱ n
⊢ MeasurableSet {a | ¬∃ c, Tendsto (fun n ↦ f n a) atTop (𝓝 c)}

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
this : MeasurableSpace Ω := ⨆ n, ↑ℱ n
⊢ μ {a | ¬∃ c, Tendsto (fun n ↦ f n a) atTop (𝓝 c)} = 0
Module: Mathlib.Probability.Martingale.Convergence

Goal state before:
pp: case hs
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
this : MeasurableSpace Ω := ⨆ n, ↑ℱ n
⊢ MeasurableSet {a | ¬∃ c, Tendsto (fun n ↦ f n a) atTop (𝓝 c)}

Tactic:
·

Goal state after:
case hs
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
this : MeasurableSpace Ω := ⨆ n, ↑ℱ n
⊢ MeasurableSet {a | ¬∃ c, Tendsto (fun n ↦ f n a) atTop (𝓝 c)}
Module: Mathlib.Probability.Process.Stopping

Goal state before:
pp: case mp
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})
⊢ MeasurableSet (s ∩ {ω | τ ω = ↑i})

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})
⊢ MeasurableSet (s ∩ {ω | τ ω = ↑i})

Tactic:
·

Goal state after:
case mp
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})
⊢ MeasurableSet (s ∩ {ω | τ ω = ↑i})
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})
⊢ MeasurableSet (s ∩ {ω | τ ω = ↑i})

Tactic:
·

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})
⊢ MeasurableSet (s ∩ {ω | τ ω = ↑i})
Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: case hf
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
⊢ Integrable (stoppedValue f π) μ

pp: case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
⊢ Integrable (stoppedValue f τ) μ

Tactic:
·

Goal state after:
case hf
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
⊢ Integrable (stoppedValue f π) μ
Module: Mathlib.Probability.Martingale.OptionalStopping

Goal state before:
pp: case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
⊢ Integrable (stoppedValue f τ) μ

Tactic:
·

Goal state after:
case hg
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
τ π : Ω → ℕ∞
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
inst✝ : SigmaFiniteFiltration μ 𝒢
f : ℕ → Ω → E
hf : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
hπ : IsStoppingTime 𝒢 π
hle : τ ≤ π
N : ℕ
hbdd : ∀ (ω : Ω), π ω ≤ ↑N
⊢ Integrable (stoppedValue f τ) μ