-- Mathlib/Probability/Martingale/OptionalSampling.lean
theorem stoppedValue_ae_eq_condExp_of_le_const_of_countable_range (h : Martingale f ℱ μ)
    (hτ : IsStoppingTime ℱ τ) (hτ_le : ∀ x, τ x ≤ n) (h_countable_range : (Set.range τ).Countable)
    [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] :
    stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace]

Proof state:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
⊢ ∀ i ∈ Set.range τ, stoppedValue f τ =ᶠ[ae (μ.restrict {x | τ x = i})] μ[f n|hτ.measurableSpace]

Tactic:
exact fun i _ => stoppedValue_ae_eq_restrict_eq h _ hτ_le i

Result:
no goals

Proof state:
case h
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
x : Ω
⊢ x ∈ Set.univ ↔ x ∈ ⋃ i ∈ Set.range τ, {x | τ x = i}

Tactic:
simp only [Set.mem_univ, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',
  Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply']

Result:
no goals

Proof state:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
⊢ stoppedValue f τ =ᶠ[ae μ] μ[f n|hτ.measurableSpace]

Tactic:
nth_rw 1 [← @Measure.restrict_univ Ω _ μ]

Result:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
⊢ stoppedValue f τ =ᶠ[ae (μ.restrict Set.univ)] μ[f n|hτ.measurableSpace]

Proof state:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
⊢ stoppedValue f τ =ᶠ[ae μ] μ[f n|hτ.measurableSpace]

Tactic:
have : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i} := by
  ext1 x
  simp only [Set.mem_univ, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',
    Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply']

Result:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
⊢ stoppedValue f τ =ᶠ[ae μ] μ[f n|hτ.measurableSpace]

Proof state:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
⊢ Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}

Tactic:
ext1 x

Result:
case h
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
x : Ω
⊢ x ∈ Set.univ ↔ x ∈ ⋃ i ∈ Set.range τ, {x | τ x = i}

Proof state:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
⊢ stoppedValue f τ =ᶠ[ae (μ.restrict Set.univ)] μ[f n|hτ.measurableSpace]

Tactic:
rw [this, ae_eq_restrict_biUnion_iff _ h_countable_range]

Result:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ : Ω → ι
f : ι → Ω → E
n : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
⊢ ∀ i ∈ Set.range τ, stoppedValue f τ =ᶠ[ae (μ.restrict {x | τ x = i})] μ[f n|hτ.measurableSpace]