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✝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}ᶜ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m i : ι
hi : i ≤ m
⊢ hittingBtwn u s n m ω < i ↔ ∃ j ∈ Set.Ico n i, u j ω ∈ s

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 : ℕ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
⊢ 0 ≤ ∫ (x : Ω), (∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) x ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
h' : hittingAfter u s n ω < ↑i
h : hittingAfter u s n ω = ⊤
⊢ False

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
⊢ MeasurableSet {ω | ↑i ≤ τ ω}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
h' : hittingAfter u s n ω < ↑i
h_top : hittingAfter u s n ω ≠ ⊤
⊢ ∃ j, n ≤ j ∧ u j ω ∈ 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✝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.Upcrossing

Goal state before:
pp: Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
h₁ : 0 ≤ ∫ (x : Ω), (∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) x ∂μ
⊢ ∫ (x : Ω), (∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) x ∂μ =
    ∫ (x : Ω), (∑ k ∈ Finset.range n, (f (k + 1) - f k)) x ∂μ -
      ∫ (x : Ω), (∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) 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✝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
⊢ MeasurableSet {ω | τ ω = ↑i}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
h_top : hittingAfter u s n ω ≠ ⊤
h_exists : ∃ j, n ≤ j ∧ u j ω ∈ s
h_le : ↑n ≤ hittingAfter u s n ω
h' : sInf {i | n ≤ i ∧ u i ω ∈ s} < i
⊢ n ∈ lowerBounds {i | n ≤ i ∧ u i ω ∈ 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✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = ↑i}
i : ι
⊢ {ω | τ ω ≤ ↑i} = ⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = ↑k}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n i : ι
ω : Ω
⊢ hittingAfter u s n ω < ↑i ↔ ∃ j ∈ Set.Ico n i, u j ω ∈ 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✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → WithTop ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = ↑i}
⊢ IsStoppingTime f τ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : m₁ ≤ m₂
j : ι
hj₁ : j ∈ Set.Icc n m₁
hj₂ : u j ω ∈ s
⊢ sInf (Set.Icc n m₂ ∩ {i | u i ω ∈ s}) ≤ sInf (Set.Icc n m₁ ∩ {i | u i ω ∈ s})

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 : ℕ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
⊢ ∫ (x : Ω), (∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) x ∂μ ≤
    ∫ (x : Ω), f n x ∂μ - ∫ (x : Ω), f 0 x ∂μ

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
inst✝² : Preorder ι
inst✝¹ : OrderBot ι
inst✝ : InfSet ι
a b : ℝ
f : ι → Ω → ℝ
ω : Ω
⊢ upcrossingsBefore a b f ⊥ ω = ⊥

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
⊢ upcrossingsBefore a b f 0 ω = 0

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
⊢ upcrossingsBefore a b f 0 = 0

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
ω : Ω
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ upperCrossingTime a b f N n ω = N

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
N : ℕ
f : ℕ → Ω → ℝ
ω : Ω
hab : a < b
⊢ upcrossingsBefore a b f N ω ≤ N

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
h : 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:
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 : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
t : ι
ht : 𝓝[>] t = ⊥
⊢ {ω | τ ω ≤ ↑t} = {ω | τ ω < ↑t} ∪ {ω | τ ω = ↑t}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
n m : ι
E F : Set β
hEF : E ⊆ F
ω : Ω
hF : ∃ j ∈ Set.Icc n m, u j ω ∈ F
hE : ∃ j ∈ Set.Icc n m, u j ω ∈ E
⊢ n ∈ lowerBounds (Set.Icc n m ∩ {i | u i ω ∈ F})

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
n m : ι
E F : Set β
hEF : E ⊆ F
ω : Ω
hE : ¬∃ j ∈ Set.Icc n m, u j ω ∈ E
t : ι
ht : t ∈ Set.Icc n m ∧ u t ω ∈ F
⊢ n ∈ lowerBounds (Set.Icc n m ∩ {i | u i ω ∈ F})

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
n m : ι
⊢ Antitone fun x ↦ hittingBtwn u x n m

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
ω : Ω
M : ℕ
hNM : N ≤ M
k : ℕ
ih :
  upperCrossingTime a b f M k ω = upperCrossingTime a b f N k ω ∧
    lowerCrossingTime a b f M k ω = lowerCrossingTime a b f N k ω
h : lowerCrossingTime a b f N (k + 1) ω < N
h' : upperCrossingTime a b f N (k + 1) ω < N
⊢ upperCrossingTime a b f M k.succ ω = upperCrossingTime a b f N k.succ ω

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
n : ι
E F : Set β
hEF : E ⊆ F
ω : Ω
hF : ∃ j, n ≤ j ∧ u j ω ∈ F
hE : ∃ j, n ≤ j ∧ u j ω ∈ E
⊢ n ∈ lowerBounds {j | n ≤ j ∧ u j ω ∈ F}

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
ω : Ω
M : ℕ
hNM : N ≤ M
h : lowerCrossingTime a b f N n ω < N
⊢ upperCrossingTime a b f M n ω = upperCrossingTime a b f N n ω ∧
    lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
n : ι
⊢ Antitone fun x ↦ hittingAfter u x 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 : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
t : ι
ht : ¬𝓝[>] t = ⊥
this : (𝓝[>] t).NeBot
⊢ ∃ s, (∀ (n : ℕ), t < s n) ∧ Tendsto s atTop (𝓝 t)

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
ω : Ω
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N (n + 1) ω < N
⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
    lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
𝒢 : Filtration ℕ m0
inst✝⁵ : CompleteSpace E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : IsFiniteMeasure μ
R : ℝ
f : ℕ → Ω → E
ξ : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : StronglyAdapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
⊢ Submartingale (fun n ↦ ∑ k ∈ Finset.range n, ξ k • (f (k + 1) - f k)) 𝒢 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
ω : Ω
M : ℕ
hNM : N ≤ M
h : upperCrossingTime a b f N n ω < N
⊢ upperCrossingTime a b f M n ω = upperCrossingTime a b f N 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✝³ : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
t : ι
ht : ¬𝓝[>] t = ⊥
this : (𝓝[>] t).NeBot
s : ℕ → ι
hs_gt : ∀ (n : ℕ), t < s n
hs_tendsto : Tendsto s atTop (𝓝 t)
h_exists_lt : ∀ (u : ι), t < u → ∃ i, s i < u
u : WithTop ι
hu : ↑t < u
⊢ ∃ i, ↑(s i) < u

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
hab : a < b
hω : upcrossings a b f ω ≠ ∞
⊢ ¬((∃ᶠ (n : ℕ) in atTop, f n ω < a) ∧ ∃ᶠ (n : ℕ) in atTop, b < f n ω)

Tactic:
by

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

Goal state before:
pp: 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 : ι
⊢ Monotone fun x ↦ hittingBtwn u s n x ω

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
f : ℕ → Ω → ℝ
ω : Ω
hf₁ : liminf (fun n ↦ ↑‖f n ω‖₊) atTop < ∞
hf₂ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ∞
⊢ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)

Tactic:
by_cases h : IsBoundedUnder (· ≤ ·) atTop fun n => |f n ω|

Goal state after:
case pos
Ω : Type u_1
f : ℕ → Ω → ℝ
ω : Ω
hf₁ : liminf (fun n ↦ ↑‖f n ω‖₊) atTop < ∞
hf₂ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ∞
h : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦ |f n ω|
⊢ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)

case neg
Ω : Type u_1
f : ℕ → Ω → ℝ
ω : Ω
hf₁ : liminf (fun n ↦ ↑‖f n ω‖₊) atTop < ∞
hf₂ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ∞
h : ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦ |f n ω|
⊢ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)
Module: Mathlib.Probability.Martingale.Upcrossing

Goal state before:
pp: Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
hab : a < b
⊢ Monotone fun 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✝³ : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
t : ι
ht : ¬𝓝[>] t = ⊥
this : (𝓝[>] t).NeBot
s : ℕ → ι
hs_gt : ∀ (n : ℕ), t < s n
hs_tendsto : Tendsto s atTop (𝓝 t)
h_exists_lt : ∀ (u : ι), t < u → ∃ i, s i < u
h_exists_lt' : ∀ (u : WithTop ι), ↑t < u → ∃ i, ↑(s i) < u
⊢ {ω | τ ω ≤ ↑t} = ⋂ m, {ω | τ ω < ↑(s m)}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
m n n' : ι
hnn' : n ≤ n'
ω : Ω
h_n : ∃ j ∈ Set.Icc n m, u j ω ∈ s
h_n' : ∃ j ∈ Set.Icc n' m, u j ω ∈ s
⊢ n ∈ lowerBounds (Set.Icc n m ∩ {i | u i ω ∈ s})

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
m n n' : ι
hnn' : n ≤ n'
ω : Ω
h_n' : ¬∃ j ∈ Set.Icc n' m, u j ω ∈ s
t : ι
ht : t ∈ Set.Icc n m ∧ u t ω ∈ s
⊢ n ∈ lowerBounds (Set.Icc n m ∩ {i | u i ω ∈ s})

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
f : ℕ → Ω → ℝ
ω : Ω
hf₁ : liminf (fun n ↦ ↑‖f n ω‖₊) atTop < ∞
hf₂ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ∞
⊢ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
m : ι
⊢ Monotone fun x ↦ hittingBtwn u s x m

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 : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
t : ι
ht : ¬𝓝[>] t = ⊥
this : (𝓝[>] t).NeBot
s : ℕ → ι
hs_gt : ∀ (n : ℕ), t < s n
hs_tendsto : Tendsto s atTop (𝓝 t)
h_exists_lt : ∀ (u : ι), t < u → ∃ i, s i < u
h_exists_lt' : ∀ (u : WithTop ι), ↑t < u → ∃ i, ↑(s i) < u
h_eq_iInter : {ω | τ ω ≤ ↑t} = ⋂ m, {ω | τ ω < ↑(s m)}
⊢ ↑f t = ⨅ m_1, ↑f (s m_1)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n m : ι
hnm : n ≤ m
ω : Ω
h_n : ∃ j, n ≤ j ∧ u j ω ∈ s
h_m : ∃ j, m ≤ j ∧ u j ω ∈ s
⊢ n ∈ lowerBounds {j | n ≤ j ∧ u j ω ∈ 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
N₁ N₂ : ℕ
hN₁ : N ≤ N₁
hN₁' : f N₁ ω < a
hN₂ : N₁ ≤ N₂
hN₂' : b < f N₂ ω
⊢ upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω

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
ω : Ω
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ N ≤ upperCrossingTime a b f N n ω

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
ω : Ω
hab : a < b
hn : upcrossingsBefore a b f N ω < n
⊢ stoppedValue f (fun ω ↦ ↑(upperCrossingTime a b f N (n + 1) ω)) ω -
      stoppedValue f (fun ω ↦ ↑(lowerCrossingTime a b f N n ω)) ω =
    0

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
⊢ Monotone (hittingAfter u s)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : ConditionallyCompleteLinearOrder ι
inst✝¹ : WellFoundedLT ι
inst✝ : Countable ι
x✝ : MeasurableSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n n' : ι
hu : Adapted f u
hs : MeasurableSet s
i : ι
hi : i < n'
⊢ {ω | hittingBtwn u s n n' ω ≤ i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : ConditionallyCompleteLinearOrder ι
inst✝¹ : WellFoundedLT ι
inst✝ : Countable ι
x✝ : MeasurableSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n n' : ι
hu : Adapted f u
hs : MeasurableSet s
⊢ IsStoppingTime f fun ω ↦ ↑(hittingBtwn u s n n' ω)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : ConditionallyCompleteLinearOrder ι
inst✝¹ : WellFoundedLT ι
inst✝ : Countable ι
x✝ : MeasurableSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n : ι
hu : Adapted f u
hs : MeasurableSet s
i : ι
⊢ {ω | hittingAfter u s n ω ≤ ↑i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : ConditionallyCompleteLinearOrder ι
inst✝¹ : WellFoundedLT ι
inst✝ : Countable ι
x✝ : MeasurableSpace β
f : Filtration ι m
u : ι → Ω → β
s : Set β
n : ι
hu : Adapted f u
hs : MeasurableSet s
⊢ IsStoppingTime f (hittingAfter u s 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
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
⊢ ∀ (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

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 : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
t : ι
ht : ¬𝓝[>] t = ⊥
this : (𝓝[>] t).NeBot
s : ℕ → ι
hs_gt : ∀ (n : ℕ), t < s n
hs_tendsto : Tendsto s atTop (𝓝 t)
h_exists_lt : ∀ (u : ι), t < u → ∃ i, s i < u
h_exists_lt' : ∀ (u : WithTop ι), ↑t < u → ∃ i, ↑(s i) < u
h_eq_iInter : {ω | τ ω ≤ ↑t} = ⋂ m, {ω | τ ω < ↑(s m)}
h𝓕_eq_iInf : ↑f t = ⨅ m_1, ↑f (s m_1)
k : ℕ
⊢ ⋂ m, {ω | τ ω < ↑(s m)} = ⋂ m, ⋂ (_ : s m ≤ s k), {ω | τ ω < ↑(s m)}

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : ConditionallyCompleteLinearOrder ι
inst✝ : WellFoundedLT ι
u : ι → Ω → β
s : Set β
n m : ι
ω : Ω
h : ∃ j ∈ Set.Icc n m, u j ω ∈ s
⊢ stoppedValue u (fun ω ↦ ↑(hittingBtwn u s n m ω)) ω ∈ s

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : ConditionallyCompleteLinearOrder ι
inst✝⁵ : WellFoundedLT ι
inst✝⁴ : Countable ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : FirstCountableTopology ι
inst✝ : MeasurableSpace β
f : Filtration ι m
u : ι → Ω → β
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
N : ι
s : Set β
hs : MeasurableSet s
hf : Adapted f u
n : ι
x : Ω
hτbdd : τ x ≤ ↑N
h : τ x = ⊤
⊢ False

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
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
⊢ ∑ _k ∈ Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
    ∑ k ∈ Finset.range (upcrossingsBefore a b f N ω),
      (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)

Tactic:
by

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

Goal state before:
pp: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁶ : ConditionallyCompleteLinearOrder ι
inst✝⁵ : WellFoundedLT ι
inst✝⁴ : Countable ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : FirstCountableTopology ι
inst✝ : MeasurableSpace β
f : Filtration ι m
u : ι → Ω → β
τ : Ω → WithTop ι
hτ : IsStoppingTime f τ
N : ι
hτbdd : ∀ (x : Ω), τ x ≤ ↑N
s : Set β
hs : MeasurableSet s
hf : Adapted f u
n : ι
⊢ {x | hittingBtwn u s (τ x).untopA N x ≤ n} =
    (⋃ i, ⋃ (_ : i ≤ n), {x | τ x = ↑i} ∩ {x | hittingBtwn u s i N x ≤ n}) ∪
      ⋃ i, ⋃ (_ : i > n), {x | τ x = ↑i} ∩ {x | hittingBtwn u s i N x ≤ 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 : ConditionallyCompleteLinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : FirstCountableTopology ι
f : Filtration ι m
τ : Ω → WithTop ι
hf : f.IsRightContinuous
hτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}
hτ2 : ∀ (i : ι), 𝓝[>] i = ⊥ → MeasurableSet {ω | τ ω = ↑i}
⊢ IsStoppingTime f τ

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
⊢ IsStoppingTime f fun ω ↦ max (τ ω) (π ω)

Tactic:
by

Goal state after: