-- Mathlib/Probability/Process/Stopping.lean
theorem measurableSpace_le [IsCountablyGenerated (atTop : Filter ι)] [IsDirected ι (· ≤ ·)]
    (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m

Proof state:
case h.mp
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : ∀ (b : ι), ∀ᶠ (a : ℕ) in atTop, b ≤ seq a
ω : Ω
hx : ω ∈ s
⊢ ∃ i, τ ω ≤ seq i

Tactic:
exact (h_seq_tendsto (τ ω)).exists

Result:
no goals

Proof state:
case inr.intro
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
⊢ MeasurableSet s

Tactic:
rw [(_ : s = ⋃ n, s ∩ {ω | τ ω ≤ seq n})]

Result:
case inr.intro
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
⊢ MeasurableSet (⋃ n, s ∩ {ω | τ ω ≤ seq n})

Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
⊢ s = ⋃ n, s ∩ {ω | τ ω ≤ seq n}

Proof state:
case h
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
ω : Ω
⊢ ω ∈ s ↔ ω ∈ ⋃ n, s ∩ {ω | τ ω ≤ seq n}

Tactic:
constructor <;> rw [Set.mem_iUnion]

Result:
case h.mp
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
ω : Ω
⊢ ω ∈ s → ∃ i, ω ∈ s ∩ {ω | τ ω ≤ seq i}

case h.mpr
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
ω : Ω
⊢ (∃ i, ω ∈ s ∩ {ω | τ ω ≤ seq i}) → ω ∈ s

Proof state:
case inr
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
⊢ MeasurableSet s

Tactic:
obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := (atTop : Filter ι).exists_seq_tendsto

Result:
case inr.intro
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
h✝ : Nonempty ι
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
seq : ℕ → ι
h_seq_tendsto : Tendsto seq atTop atTop
⊢ MeasurableSet s

Proof state:
case inl
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
hs : MeasurableSet s
h✝ : IsEmpty ι
⊢ MeasurableSet s

Tactic:
haveI : IsEmpty Ω := ⟨fun ω => IsEmpty.false (τ ω)⟩

Result:
case inl
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝² : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝¹ : atTop.IsCountablyGenerated
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
hτ : IsStoppingTime f τ
s : Set Ω
hs : MeasurableSet s
h✝ : IsEmpty ι
this : IsEmpty Ω
⊢ MeasurableSet s