-- Mathlib/Data/Seq/Parallel.lean
theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : Terminates c] :
    Terminates (parallel S)

Proof state:
case succ.inr.inr
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
⊢ (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think.Terminates

Tactic:
refine @Computation.think_terminates _ _ ?_

Result:
case succ.inr.inr
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
⊢ (corec parallel.aux1
      (match S.destruct with
      | none => (l', Seq.nil)
      | some (none, S') => (l', S')
      | some (some c, S') => (c :: l', S'))).Terminates

Proof state:
case zero.inr.inr
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? 0
H : S.destruct = some (some c, S.tail)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match some (some c, S.tail) with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
⊢ (corec parallel.aux1
        (match some (some c, S.tail) with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think.Terminates

Tactic:
refine @Computation.think_terminates _ _ ?_

Result:
case zero.inr.inr
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? 0
H : S.destruct = some (some c, S.tail)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match some (some c, S.tail) with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
⊢ (corec parallel.aux1
      (match some (some c, S.tail) with
      | none => (l', Seq.nil)
      | some (none, S') => (l', S')
      | some (some c, S') => (c :: l', S'))).Terminates

Proof state:
case succ.inr.inl
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a✝ : some (some c) = S.get? (n + 1)
a : α
h : parallel.aux2 l = Sum.inl a
⊢ (corec parallel.aux1 (l, S)).Terminates

Tactic:
have C : corec parallel.aux1 (l, S) = pure a := by
  apply destruct_eq_pure
  rw [corec_eq, parallel.aux1]
  rw [h]
  simp only [rmap]

Result:
case succ.inr.inl
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a✝ : some (some c) = S.get? (n + 1)
a : α
h : parallel.aux2 l = Sum.inl a
C : corec parallel.aux1 (l, S) = pure a
⊢ (corec parallel.aux1 (l, S)).Terminates

Proof state:
case succ.inr.inr.none
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
TT : ∀ (l' : List (Computation α)), (corec parallel.aux1 (l', S.tail)).Terminates
e : S.get? 0 = none
⊢ (corec parallel.aux1
      (match S.destruct with
      | none => (l', Seq.nil)
      | some (none, S') => (l', S')
      | some (some c, S') => (c :: l', S'))).Terminates

Tactic:
have D : Seq.destruct S = none := by
  dsimp [Seq.destruct]
  rw [e]
  rfl

Result:
case succ.inr.inr.none
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
  ∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
    c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
  corec parallel.aux1 (l, S) =
    (corec parallel.aux1
        (match S.destruct with
        | none => (l', Seq.nil)
        | some (none, S') => (l', S')
        | some (some c, S') => (c :: l', S'))).think
TT : ∀ (l' : List (Computation α)), (corec parallel.aux1 (l', S.tail)).Terminates
e : S.get? 0 = none
D : S.destruct = none
⊢ (corec parallel.aux1
      (match S.destruct with
      | none => (l', Seq.nil)
      | some (none, S') => (l', S')
      | some (some c, S') => (c :: l', S'))).Terminates

Proof state:
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? 0
H : S.destruct = some (some c, S.tail)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
⊢ (match
      match parallel.aux2 l with
      | Sum.inl a => Sum.inl a
      | Sum.inr b =>
        Sum.inr
          (match S.destruct with
          | none => (b, Seq.nil)
          | some (none, S') => (b, S')
          | some (some c, S') => (c :: b, S')) with
    | Sum.inl a => Sum.inl a
    | Sum.inr b => Sum.inr (corec parallel.aux1 b)) =
    Sum.inr ?m.10546

Tactic:
rw [h, H]

Result:
no goals