l3lab/ntp-mathlib huggingface dataset:
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
      and_congr_left_iff]
    intro h
    simp only [h, or_self_iff, and_true_iff]
    by_cases hj : j ≤ π ω
    · simp only [hj, h.trans hj, or_self_iff]
    · 

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
j : ι
ω : Ω
h : τ ω ≤ j
hj : ¬j ≤ π ω
⊢ τ ω ≤ π ω ↔ τ ω ≤ π ω ∨ j ≤ π ω

Next tactic:
simp only [hj, or_false_iff]
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
      and_congr_left_iff]
    intro h
    simp only [h, or_self_iff, and_true_iff]
    by_cases hj : j ≤ π ω
    · simp only [hj, h.trans hj, or_self_iff]
    · simp only [hj, or_false_iff]
  

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

Next tactic:
rw [this]
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
      and_congr_left_iff]
    intro h
    simp only [h, or_self_iff, and_true_iff]
    by_cases hj : j ≤ π ω
    · simp only [hj, h.trans hj, or_self_iff]
    · simp only [hj, or_false_iff]
  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
j : ι
this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j}
⊢ MeasurableSet ({ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j})

Next tactic:
refine' MeasurableSet.inter _ (hτ.measurableSet_le j)
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
      and_congr_left_iff]
    intro h
    simp only [h, or_self_iff, and_true_iff]
    by_cases hj : j ≤ π ω
    · simp only [hj, h.trans hj, or_self_iff]
    · simp only [hj, or_false_iff]
  rw [this]
  refine' MeasurableSet.inter _ (hτ.measurableSet_le j)
  

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

Next tactic:
apply @measurableSet_le _ _ _ _ _ (Filtration.seq f j) _ _ _ _ _ ?_ ?_
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
      and_congr_left_iff]
    intro h
    simp only [h, or_self_iff, and_true_iff]
    by_cases hj : j ≤ π ω
    · simp only [hj, h.trans hj, or_self_iff]
    · simp only [hj, or_false_iff]
  rw [this]
  refine' MeasurableSet.inter _ (hτ.measurableSet_le j)
  apply @measurableSet_le _ _ _ _ _ (Filtration.seq f j) _ _ _ _ _ ?_ ?_
  · 

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
j : ι
this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j}
⊢ Measurable fun a => min (τ a) j

Next tactic:
exact (hτ.min_const j).measurable_of_le fun _ => min_le_right _ _
theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
      and_congr_left_iff]
    intro h
    simp only [h, or_self_iff, and_true_iff]
    by_cases hj : j ≤ π ω
    · simp only [hj, h.trans hj, or_self_iff]
    · simp only [hj, or_false_iff]
  rw [this]
  refine' MeasurableSet.inter _ (hτ.measurableSet_le j)
  apply @measurableSet_le _ _ _ _ _ (Filtration.seq f j) _ _ _ _ _ ?_ ?_
  · exact (hτ.min_const j).measurable_of_le fun _ => min_le_right _ _
  · 

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
j : ι
this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j}
⊢ Measurable fun a => min (π a) j

Next tactic:
exact (hπ.min_const j).measurable_of_le fun _ => min_le_right _ _
theorem measurableSet_stopping_time_le [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hπ.measurableSpace] {ω | τ ω ≤ π ω} := by
  

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
⊢ MeasurableSet {ω | τ ω ≤ π ω}

Next tactic:
suffices MeasurableSet[(hτ.min hπ).measurableSpace] {ω : Ω | τ ω ≤ π ω} by
    rw [measurableSet_min_iff hτ hπ] at this; exact this.2
theorem measurableSet_stopping_time_le [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hπ.measurableSpace] {ω | τ ω ≤ π ω} := by
  suffices MeasurableSet[(hτ.min hπ).measurableSpace] {ω : Ω | τ ω ≤ π ω} by
    

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
this : MeasurableSet {ω | τ ω ≤ π ω}
⊢ MeasurableSet {ω | τ ω ≤ π ω}

Next tactic:
rw [measurableSet_min_iff hτ hπ] at this
theorem measurableSet_stopping_time_le [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hπ.measurableSpace] {ω | τ ω ≤ π ω} := by
  suffices MeasurableSet[(hτ.min hπ).measurableSpace] {ω : Ω | τ ω ≤ π ω} by
    rw [measurableSet_min_iff hτ hπ] at 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
this : MeasurableSet {ω | τ ω ≤ π ω} ∧ MeasurableSet {ω | τ ω ≤ π ω}
⊢ MeasurableSet {ω | τ ω ≤ π ω}

Next tactic:
exact this.2
theorem measurableSet_stopping_time_le [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hπ.measurableSpace] {ω | τ ω ≤ π ω} := by
  suffices MeasurableSet[(hτ.min hπ).measurableSpace] {ω : Ω | τ ω ≤ π ω} by
    rw [measurableSet_min_iff hτ hπ] at this; exact this.2
  

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
⊢ MeasurableSet {ω | τ ω ≤ π ω}

Next tactic:
rw [← Set.univ_inter {ω : Ω | τ ω ≤ π ω}, ← hτ.measurableSet_inter_le_iff hπ, Set.univ_inter]
theorem measurableSet_stopping_time_le [TopologicalSpace ι] [SecondCountableTopology ι]
    [OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
    (hπ : IsStoppingTime f π) : MeasurableSet[hπ.measurableSpace] {ω | τ ω ≤ π ω} := by
  suffices MeasurableSet[(hτ.min hπ).measurableSpace] {ω : Ω | τ ω ≤ π ω} by
    rw [measurableSet_min_iff hτ hπ] at this; exact this.2
  rw [← Set.univ_inter {ω : Ω | τ ω ≤ π ω}, ← hτ.measurableSet_inter_le_iff hπ, Set.univ_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
⊢ MeasurableSet {ω | τ ω ≤ π ω}

Next tactic:
exact measurableSet_le_stopping_time hτ hπ
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
⊢ MeasurableSet {ω | τ ω = π ω}

Next tactic:
rw [hτ.measurableSet]
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
⊢ ∀ (i : ι), MeasurableSet ({ω | τ ω = π ω} ∩ {ω | τ ω ≤ i})

Next tactic:
intro j
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
⊢ MeasurableSet ({ω | τ ω = π ω} ∩ {ω | τ ω ≤ j})

Next tactic:
have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
    · rw [h.1]
    · rw [← h.1]; exact h.2
    · cases' h with h' hσ_le
      cases' h' with h_eq hτ_le
      rwa [min_eq_left hτ_le, min_eq_left hσ_le] at h_eq
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    

Proof state:
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
⊢ {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}

Next tactic:
ext1 ω
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    

Proof state:
case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
⊢ ω ∈ {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} ↔ ω ∈ {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j}

Next tactic:
simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    

Proof state:
case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
⊢ τ ω = π ω ∧ τ ω ≤ j ↔ (min (τ ω) j = min (π ω) j ∧ τ ω ≤ j) ∧ π ω ≤ j

Next tactic:
refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
    · 

Proof state:
case h.refine'_1
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
h : τ ω = π ω ∧ τ ω ≤ j
⊢ min (τ ω) j = min (π ω) j

Next tactic:
rw [h.1]
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
    · rw [h.1]
    · 

Proof state:
case h.refine'_2
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
h : τ ω = π ω ∧ τ ω ≤ j
⊢ π ω ≤ j

Next tactic:
rw [← h.1]
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
    · rw [h.1]
    · rw [← h.1]; 

Proof state:
case h.refine'_2
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
h : τ ω = π ω ∧ τ ω ≤ j
⊢ τ ω ≤ j

Next tactic:
exact h.2
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
    · rw [h.1]
    · rw [← h.1]; exact h.2
    · 

Proof state:
case h.refine'_3
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
h : (min (τ ω) j = min (π ω) j ∧ τ ω ≤ j) ∧ π ω ≤ j
⊢ τ ω = π ω

Next tactic:
cases' h with h' hσ_le
theorem measurableSet_eq_stopping_time [AddGroup ι] [TopologicalSpace ι] [MeasurableSpace ι]
    [BorelSpace ι] [OrderTopology ι] [MeasurableSingletonClass ι] [SecondCountableTopology ι]
    [MeasurableSub₂ ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
    MeasurableSet[hτ.measurableSpace] {ω | τ ω = π ω} := by
  rw [hτ.measurableSet]
  intro j
  have : {ω | τ ω = π ω} ∩ {ω | τ ω ≤ j} =
      {ω | min (τ ω) j = min (π ω) j} ∩ {ω | τ ω ≤ j} ∩ {ω | π ω ≤ j} := by
    ext1 ω
    simp only [Set.mem_inter_iff, Set.mem_setOf_eq]
    refine' ⟨fun h => ⟨⟨_, h.2⟩, _⟩, fun h => ⟨_, h.1.2⟩⟩
    · rw [h.1]
    · rw [← h.1]; exact h.2
    · cases' h with h' hσ_le
      

Proof state:
case h.refine'_3.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁸ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁷ : AddGroup ι
inst✝⁶ : TopologicalSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : OrderTopology ι
inst✝² : MeasurableSingletonClass ι
inst✝¹ : SecondCountableTopology ι
inst✝ : MeasurableSub₂ ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
ω : Ω
h' : min (τ ω) j = min (π ω) j ∧ τ ω ≤ j
hσ_le : π ω ≤ j
⊢ τ ω = π ω

Next tactic:
cases' h' with h_eq hτ_le