Module: Mathlib.Computability.PartrecCode

Goal state before:
pp: case succ.rfind'
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode cf.rfind') →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode cf.rfind'))))
          (k', c') n =
        evaln k' c' n
lf : encode cf < encode cf.rfind'
⊢ ((evaln k cf n).bind fun x ↦
      Nat.rec (some (unpair n).2)
        (fun n_1 n_ih ↦
          lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k', cf.rfind') (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
        x) =
    (evaln (k' + 1) cf n).bind fun a ↦
      if a = 0 then some (unpair n).2 else evaln k' cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1))

Tactic:
rcases evaln k cf n with - | x

Goal state after:
case succ.rfind'.none
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode cf.rfind') →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode cf.rfind'))))
          (k', c') n =
        evaln k' c' n
lf : encode cf < encode cf.rfind'
⊢ (Option.none.bind fun x ↦
      Nat.rec (some (unpair n).2)
        (fun n_1 n_ih ↦
          lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k', cf.rfind') (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
        x) =
    Option.none.bind fun a ↦
      if a = 0 then some (unpair n).2 else evaln k' cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1))

case succ.rfind'.some
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
cf : Code
hg :
  ∀ {k' : ℕ} {c' : Code} {n : ℕ},
    Nat.pair k' (encode c') < Nat.pair k (encode cf.rfind') →
      lup
          (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
            (List.range (Nat.pair k (encode cf.rfind'))))
          (k', c') n =
        evaln k' c' n
lf : encode cf < encode cf.rfind'
x : ℕ
⊢ ((some x).bind fun x ↦
      Nat.rec (some (unpair n).2)
        (fun n_1 n_ih ↦
          lup
            (List.map (fun n ↦ List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1))
              (List.range (Nat.pair (k' + 1) (encode cf.rfind'))))
            (k', cf.rfind') (Nat.pair (unpair n).1 ((unpair n).2 + 1)))
        x) =
    (some x).bind fun a ↦
      if a = 0 then some (unpair n).2 else evaln k' cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1))