Theorem: MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff
File: Mathlib/Probability/Process/Stopping.lean

tactic: constructor <;> intro h
state_before: 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
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
state_after: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})

case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})

tactic: have : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω} := by
  rw [Set.inter_assoc, Set.inter_self]
state_before: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
state_after: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
this : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})

tactic: rw [this]
state_before: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
this : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
state_after: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
this : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω})

tactic: exact measurableSet_inter_le _ hπ _ h
state_before: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
this : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω})
state_after: no goals

tactic: rw [Set.inter_assoc, Set.inter_self]
state_before: 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω}
state_after: no goals

tactic: rw [measurableSet_min_iff hτ hπ] at h
state_before: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
state_after: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ∧ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})

tactic: exact h.1
state_before: case 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
h : MeasurableSet (s ∩ {ω | τ ω ≤ π ω}) ∧ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω})
state_after: no goals