theorem map_parallel (f : α → β) (S) : map f (parallel S) = parallel (S.map (map f)) := by
  refine'
    eq_of_bisim
      (fun c1 c2 =>
        ∃ l S,
          c1 = map f (corec parallel.aux1 (l, S)) ∧
            c2 = corec parallel.aux1 (l.map (map f), S.map (map f)))
      _ ⟨[], S, rfl, rfl⟩
  intro c1 c2 h
  exact
    match c1, c2, h with
    | _, _, ⟨l, S, rfl, rfl⟩ => by
      have : parallel.aux2 (l.map (map f))
          = lmap f (rmap (List.map (map f)) (parallel.aux2 l)) := by
        simp only [parallel.aux2, rmap, lmap]
        induction' l with c l IH <;> simp
        rw [IH]
        

Proof state:
case cons
α : Type u
β : Type v
f : α → β
S✝ : WSeq (Computation α)
c1 c2 : Computation β
h : ∃ l S, c1 = map f (corec parallel.aux1 (l, S)) ∧ c2 = corec parallel.aux1 (List.map (map f) l, WSeq.map (map f) S)
S : WSeq (Computation α)
c : Computation α
l : List (Computation α)
IH :
  List.foldr
      (fun c o =>
        match o with
        | Sum.inl a => Sum.inl a
        | Sum.inr ls =>
          match destruct c with
          | Sum.inl a => Sum.inl a
          | Sum.inr b => Sum.inr (b :: ls))
      (Sum.inr []) (List.map (map f) l) =
    match
      match
        List.foldr
          (fun c o =>
            match o with
            | Sum.inl a => Sum.inl a
            | Sum.inr ls =>
              match destruct c with
              | Sum.inl a => Sum.inl a
              | Sum.inr b => Sum.inr (b :: ls))
          (Sum.inr []) l with
      | Sum.inl a => Sum.inl a
      | Sum.inr b => Sum.inr (List.map (map f) b) with
    | Sum.inl a => Sum.inl (f a)
    | Sum.inr b => Sum.inr b
⊢ (match
      match
        match
          List.foldr
            (fun c o =>
              match o with
              | Sum.inl a => Sum.inl a
              | Sum.inr ls =>
                match destruct c with
                | Sum.inl a => Sum.inl a
                | Sum.inr b => Sum.inr (b :: ls))
            (Sum.inr []) l with
        | Sum.inl a => Sum.inl a
        | Sum.inr b => Sum.inr (List.map (map f) b) with
      | Sum.inl a => Sum.inl (f a)
      | Sum.inr b => Sum.inr b with
    | Sum.inl a => Sum.inl a
    | Sum.inr ls =>
      match
        match
          match destruct c with
          | Sum.inl a => Sum.inl a
          | Sum.inr b => Sum.inr (map f b) with
        | Sum.inl a => Sum.inl (f a)
        | Sum.inr b => Sum.inr b with
      | Sum.inl a => Sum.inl a
      | Sum.inr b => Sum.inr (b :: ls)) =
    match
      match
        match
          List.foldr
            (fun c o =>
              match o with
              | Sum.inl a => Sum.inl a
              | Sum.inr ls =>
                match destruct c with
                | Sum.inl a => Sum.inl a
                | Sum.inr b => Sum.inr (b :: ls))
            (Sum.inr []) l with
        | Sum.inl a => Sum.inl a
        | Sum.inr ls =>
          match destruct c with
          | Sum.inl a => Sum.inl a
          | Sum.inr b => Sum.inr (b :: ls) with
      | Sum.inl a => Sum.inl a
      | Sum.inr b => Sum.inr (List.map (map f) b) with
    | Sum.inl a => Sum.inl (f a)
    | Sum.inr b => Sum.inr b

Next tactic:
cases List.foldr (fun c o =>
            match o with
            | Sum.inl a => Sum.inl a
            | Sum.inr ls => rmap (fun c' => c' :: ls) (destruct c)) (Sum.inr List.nil) l