Theorem: MeasureTheory.stoppedProcess_eq'
File: Mathlib/Probability/Process/Stopping.lean

tactic: rw [stoppedProcess_eq, this, Finset.sum_range_succ_comm, ← add_assoc]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
this : {a | n ≤ τ a}.indicator (u n) = {a | n + 1 ≤ τ a}.indicator (u n) + {a | τ a = n}.indicator (u n)
⊢ stoppedProcess u τ n = {a | n + 1 ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.range (n + 1), {a | τ a = i}.indicator (u i)
state_after: no goals

tactic: ext x
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
⊢ {a | n ≤ τ a}.indicator (u n) = {a | n + 1 ≤ τ a}.indicator (u n) + {a | τ a = n}.indicator (u n)
state_after: case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
⊢ {a | n ≤ τ a}.indicator (u n) x = ({a | n + 1 ≤ τ a}.indicator (u n) + {a | τ a = n}.indicator (u n)) x

tactic: rw [add_comm, Pi.add_apply, ← Set.indicator_union_of_not_mem_inter]
state_before: case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
⊢ {a | n ≤ τ a}.indicator (u n) x = ({a | n + 1 ≤ τ a}.indicator (u n) + {a | τ a = n}.indicator (u n)) x
state_after: case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
⊢ {a | n ≤ τ a}.indicator (u n) x = ({a | τ a = n} ∪ {a | n + 1 ≤ τ a}).indicator (u n) x

case h.h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
⊢ x ∉ {a | τ a = n} ∩ {a | n + 1 ≤ τ a}

tactic: simp_rw [@eq_comm _ _ n, @le_iff_eq_or_lt _ _ n, Nat.succ_le_iff, Set.setOf_or]
state_before: case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
⊢ {a | n ≤ τ a}.indicator (u n) x = ({a | τ a = n} ∪ {a | n + 1 ≤ τ a}).indicator (u n) x
state_after: no goals

tactic: rintro ⟨h₁, h₂⟩
state_before: case h.h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
⊢ x ∉ {a | τ a = n} ∩ {a | n + 1 ≤ τ a}
state_after: case h.h.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
h₁ : x ∈ {a | τ a = n}
h₂ : x ∈ {a | n + 1 ≤ τ a}
⊢ False

tactic: rw [Set.mem_setOf] at h₁ h₂
state_before: case h.h.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
h₁ : x ∈ {a | τ a = n}
h₂ : x ∈ {a | n + 1 ≤ τ a}
⊢ False
state_after: case h.h.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
h₁ : τ x = n
h₂ : n + 1 ≤ τ x
⊢ False

tactic: exact (Nat.succ_le_iff.1 h₂).ne h₁.symm
state_before: case h.h.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
f : Filtration ℕ m
u : ℕ → Ω → β
τ π : Ω → ℕ
inst✝ : AddCommMonoid β
n : ℕ
x : Ω
h₁ : τ x = n
h₂ : n + 1 ≤ τ x
⊢ False
state_after: no goals