Module: Mathlib.Data.WSeq.Basic

Goal state before:
pp: α : Type u
l : List α
s✝ : WSeq α
s1 s2 : Computation (List α)
l' : List α
s : WSeq α
h :
  s1 =
      Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l' ++ l, s.think) ∧
    s2 =
      Computation.map (fun x ↦ l.reverse ++ x)
        (Computation.corec
          (fun x ↦
            match x with
            | (l, s) =>
              match Seq.destruct s with
              | none => Sum.inl l.reverse
              | some (none, s') => Sum.inr (l, s')
              | some (some a, s') => Sum.inr (a :: l, s'))
          (l', s.think))
⊢ BisimO
    (fun c1 c2 ↦
      ∃ l' s,
        c1 =
            Computation.corec
              (fun x ↦
                match x with
                | (l, s) =>
                  match Seq.destruct s with
                  | none => Sum.inl l.reverse
                  | some (none, s') => Sum.inr (l, s')
                  | some (some a, s') => Sum.inr (a :: l, s'))
              (l' ++ l, s) ∧
          c2 =
            Computation.map (fun x ↦ l.reverse ++ x)
              (Computation.corec
                (fun x ↦
                  match x with
                  | (l, s) =>
                    match Seq.destruct s with
                    | none => Sum.inl l.reverse
                    | some (none, s') => Sum.inr (l, s')
                    | some (some a, s') => Sum.inr (a :: l, s'))
                (l', s)))
    (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l' ++ l, s.think)).destruct
    (Computation.map (fun x ↦ l.reverse ++ x)
        (Computation.corec
          (fun x ↦
            match x with
            | (l, s) =>
              match Seq.destruct s with
              | none => Sum.inl l.reverse
              | some (none, s') => Sum.inr (l, s')
              | some (some a, s') => Sum.inr (a :: l, s'))
          (l', s.think))).destruct

Tactic:
refine ⟨l', s, ?_, ?_⟩

Goal state after:
case refine_1
α : Type u
l : List α
s✝ : WSeq α
s1 s2 : Computation (List α)
l' : List α
s : WSeq α
h :
  s1 =
      Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l' ++ l, s.think) ∧
    s2 =
      Computation.map (fun x ↦ l.reverse ++ x)
        (Computation.corec
          (fun x ↦
            match x with
            | (l, s) =>
              match Seq.destruct s with
              | none => Sum.inl l.reverse
              | some (none, s') => Sum.inr (l, s')
              | some (some a, s') => Sum.inr (a :: l, s'))
          (l', s.think))
⊢ (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l' ++ l, s.think)).tail =
    Computation.corec
      (fun x ↦
        match x with
        | (l, s) =>
          match Seq.destruct s with
          | none => Sum.inl l.reverse
          | some (none, s') => Sum.inr (l, s')
          | some (some a, s') => Sum.inr (a :: l, s'))
      (l' ++ l, s)

case refine_2
α : Type u
l : List α
s✝ : WSeq α
s1 s2 : Computation (List α)
l' : List α
s : WSeq α
h :
  s1 =
      Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l' ++ l, s.think) ∧
    s2 =
      Computation.map (fun x ↦ l.reverse ++ x)
        (Computation.corec
          (fun x ↦
            match x with
            | (l, s) =>
              match Seq.destruct s with
              | none => Sum.inl l.reverse
              | some (none, s') => Sum.inr (l, s')
              | some (some a, s') => Sum.inr (a :: l, s'))
          (l', s.think))
⊢ (Computation.map (fun x ↦ l.reverse ++ x)
        (Computation.corec
          (fun x ↦
            match x with
            | (l, s) =>
              match Seq.destruct s with
              | none => Sum.inl l.reverse
              | some (none, s') => Sum.inr (l, s')
              | some (some a, s') => Sum.inr (a :: l, s'))
          (l', s.think))).tail =
    Computation.map (fun x ↦ l.reverse ++ x)
      (Computation.corec
        (fun x ↦
          match x with
          | (l, s) =>
            match Seq.destruct s with
            | none => Sum.inl l.reverse
            | some (none, s') => Sum.inr (l, s')
            | some (some a, s') => Sum.inr (a :: l, s'))
        (l', s))