Theorem: MeasureTheory.progMeasurable_min_stopping_time
File: Mathlib/Probability/Process/Stopping.lean

Proof steps:
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
⊢ ProgMeasurable f fun i ω => min i (τ ω)
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: intro i
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: let m_prod : MeasurableSpace (Set.Iic i × Ω) := Subtype.instMeasurableSpace.prod (f i)
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: let m_set : ∀ t : Set (Set.Iic i × Ω), MeasurableSpace t := fun _ =>
    @Subtype.instMeasurableSpace (Set.Iic i × Ω) _ m_prod
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: let s := {p : Set.Iic i × Ω | τ p.2 ≤ i}
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: have hs : MeasurableSet[m_prod] s := @measurable_snd (Set.Iic i) Ω _ (f i) _ (hτ i)
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x => ↑(↑x).1
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: have h_meas_fst : ∀ t : Set (Set.Iic i × Ω),
      Measurable[m_set t] fun x : t => ((x : Set.Iic i × Ω).fst : ι) :=
    fun t => (@measurable_subtype_coe (Set.Iic i × Ω) m_prod _).fst.subtype_val
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x => ↑(↑x).1
⊢ StronglyMeasurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: case hf
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x => ↑(↑x).1
⊢ Measurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
tactic: apply Measurable.stronglyMeasurable
state_before: case hf
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x => ↑(↑x).1
⊢ Measurable fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2
state_after: case hf.refine_1
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x => ↑(↑x).1
⊢ Measurable (s.restrict fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2)

case hf.refine_2
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁷ : LinearOrder ι
inst✝⁶ : MeasurableSpace ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : SecondCountableTopology ι
inst✝² : BorelSpace ι
inst✝¹ : TopologicalSpace β
u : ι → Ω → β
τ : Ω → ι
f : Filtration ι m
inst✝ : MetrizableSpace ι
hτ : IsStoppingTime f τ
i : ι
m_prod : MeasurableSpace (↑(Set.Iic i) × Ω) := Subtype.instMeasurableSpace.prod (↑f i)
m_set : (t : Set (↑(Set.Iic i) × Ω)) → MeasurableSpace ↑t := fun x => Subtype.instMeasurableSpace
s : Set (↑(Set.Iic i) × Ω) := {p | τ p.2 ≤ i}
hs : MeasurableSet s
h_meas_fst : ∀ (t : Set (↑(Set.Iic i) × Ω)), Measurable fun x => ↑(↑x).1
⊢ Measurable (sᶜ.restrict fun p => (fun i ω => min i (τ ω)) (↑p.1) p.2)
tactic: refine measurable_of_restrict_of_restrict_compl hs ?_ ?_