l3lab/ntp-mathlib huggingface dataset:
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      

Proof state:
case pos
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
⊢ ω ∈ s → (τ ω ≤ π ω ↔ τ ω ≤ π ω ∨ i ≤ π ω)

Next tactic:
intro
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      

Proof state:
case pos
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
⊢ τ ω ≤ π ω ↔ τ ω ≤ π ω ∨ i ≤ π ω

Next tactic:
constructor
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> 

Proof state:
case pos.mp
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
⊢ τ ω ≤ π ω → τ ω ≤ π ω ∨ i ≤ π ω

Next tactic:
intro h
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> 

Proof state:
case pos.mpr
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
⊢ τ ω ≤ π ω ∨ i ≤ π ω → τ ω ≤ π ω

Next tactic:
intro h
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · 

Proof state:
case pos.mp
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
h : τ ω ≤ π ω
⊢ τ ω ≤ π ω ∨ i ≤ π ω

Next tactic:
exact Or.inl h
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · 

Proof state:
case pos.mpr
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
h : τ ω ≤ π ω ∨ i ≤ π ω
⊢ τ ω ≤ π ω

Next tactic:
cases' h with h h
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · 

Proof state:
case pos.mpr.inl
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
h : τ ω ≤ π ω
⊢ τ ω ≤ π ω

Next tactic:
exact h
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · 

Proof state:
case pos.mpr.inr
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : τ ω ≤ i
a✝ : ω ∈ s
h : i ≤ π ω
⊢ τ ω ≤ π ω

Next tactic:
exact hτi.trans h
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    

Proof state:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : ¬τ ω ≤ i
⊢ (ω ∈ s ∧ τ ω ≤ π ω) ∧ (τ ω ≤ i ∨ π ω ≤ i) ↔ ((ω ∈ s ∧ τ ω ≤ i) ∧ (τ ω ≤ i ∨ π ω ≤ i)) ∧ (τ ω ≤ π ω ∨ i ≤ π ω)

Next tactic:
simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
    

Proof state:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : ¬τ ω ≤ i
⊢ ω ∈ s → τ ω ≤ π ω → i < π ω

Next tactic:
refine' fun _ hτ_le_π => lt_of_lt_of_le _ hτ_le_π
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
    refine' fun _ hτ_le_π => lt_of_lt_of_le _ hτ_le_π
    

Proof state:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : ¬τ ω ≤ i
x✝ : ω ∈ s
hτ_le_π : τ ω ≤ π ω
⊢ i < τ ω

Next tactic:
rw [← not_le]
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
    refine' fun _ hτ_le_π => lt_of_lt_of_le _ hτ_le_π
    rw [← not_le]
    

Proof state:
case neg
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
ω : Ω
hτi : ¬τ ω ≤ i
x✝ : ω ∈ s
hτ_le_π : τ ω ≤ π ω
⊢ ¬τ ω ≤ i

Next tactic:
exact hτi
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
    refine' fun _ hτ_le_π => lt_of_lt_of_le _ hτ_le_π
    rw [← not_le]
    exact hτi
  

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
this :
  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} ∩ {ω | min (τ ω) (π ω) ≤ i})

Next tactic:
rw [this]
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
    refine' fun _ hτ_le_π => lt_of_lt_of_le _ hτ_le_π
    rw [← not_le]
    exact hτi
  rw [this]
  

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
this :
  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 ≤ i} ∩ {ω | min (τ ω) (π ω) ≤ i} ∩ {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i})

Next tactic:
refine' ((hs i).inter ((hτ.min hπ) i)).inter _
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
    [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
    (s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
    MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
  simp_rw [IsStoppingTime.measurableSet] at hs ⊢
  intro i
  have : 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} := by
    ext1 ω
    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
      and_true_iff, true_or_iff, or_true_iff]
    by_cases hτi : τ ω ≤ i
    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
      intro
      constructor <;> intro h
      · exact Or.inl h
      · cases' h with h h
        · exact h
        · exact hτi.trans h
    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
      and_imp]
    refine' fun _ hτ_le_π => lt_of_lt_of_le _ hτ_le_π
    rw [← not_le]
    exact hτi
  rw [this]
  refine' ((hs i).inter ((hτ.min hπ) i)).inter _
  

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
this :
  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 {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i}

Next tactic:
apply @measurableSet_le _ _ _ _ _ (Filtration.seq f i) _ _ _ _ _ ?_ ?_