theorem evaln_complete {c n x} : x ∈ eval c n ↔ ∃ k, x ∈ evaln k c n :=
  ⟨fun h => by
    rsuffices ⟨k, h⟩ : ∃ k, x ∈ evaln (k + 1) c n
    · exact ⟨k + 1, h⟩
    induction c generalizing n x <;> simp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h ⊢
    iterate 4 exact ⟨⟨_, le_rfl⟩, h.symm⟩
    case pair cf cg hf hg =>
      rcases h with ⟨x, hx, y, hy, rfl⟩
      rcases hf hx with ⟨k₁, hk₁⟩; rcases hg hy with ⟨k₂, hk₂⟩
      refine' ⟨max k₁ k₂, _⟩
      refine'
        ⟨le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk₁, _,
          evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk₁, _,
          evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk₂, rfl⟩
    case comp cf cg hf hg =>
      rcases h with ⟨y, hy, hx⟩
      rcases hg hy with ⟨k₁, hk₁⟩; rcases hf hx with ⟨k₂, hk₂⟩
      refine' ⟨max k₁ k₂, _⟩
      exact
        ⟨le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk₁, _,
          evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk₁,
          evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk₂⟩
    case prec cf cg hf hg =>
      revert h
      generalize n.unpair.1 = n₁; generalize n.unpair.2 = n₂
      induction' n₂ with m IH generalizing x n <;> simp
      · intro h
        rcases hf h with ⟨k, hk⟩
        exact ⟨_, le_max_left _ _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk⟩
      · intro y hy hx
        rcases IH hy with ⟨k₁, nk₁, hk₁⟩
        rcases hg hx with ⟨k₂, hk₂⟩
        refine'
          ⟨(max k₁ k₂).succ,
            Nat.le_succ_of_le <| le_max_of_le_left <|
              le_trans (le_max_left _ (Nat.pair n₁ m)) nk₁, y,
            evaln_mono (Nat.succ_le_succ <| le_max_left _ _) _,
            evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_right _ _) hk₂⟩
        simp only [evaln._eq_8, bind, unpaired, unpair_pair, Option.mem_def, Option.bind_eq_some,
          Option.guard_eq_some', exists_and_left, exists_const]
        exact ⟨le_trans (le_max_right _ _) nk₁, hk₁⟩
    

Proof state:
cf : Code
hf : ∀ {n x : ℕ}, x ∈ eval cf n → ∃ k, x ∈ evaln (k + 1) cf n
n x : ℕ
h :
  ∃ a,
    (0 ∈ eval cf (Nat.pair (unpair n).1 (a + (unpair n).2)) ∧
        ∀ {m : ℕ}, m < a → ∃ a ∈ eval cf (Nat.pair (unpair n).1 (m + (unpair n).2)), ¬a = 0) ∧
      a + (unpair n).2 = x
⊢ ∃ k,
    n ≤ k ∧
      ∃ a,
        evaln (k + 1) cf n = some a ∧
          (if a = 0 then some (unpair n).2 else evaln k (rfind' cf) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some x

Next tactic:
case rfind' cf hf =>
      rcases h with ⟨y, ⟨hy₁, hy₂⟩, rfl⟩
      suffices ∃ k, y + n.unpair.2 ∈ evaln (k + 1) (rfind' cf) (Nat.pair n.unpair.1 n.unpair.2) by
        simpa [evaln, Bind.bind]
      revert hy₁ hy₂
      generalize n.unpair.2 = m
      intro hy₁ hy₂
      induction' y with y IH generalizing m <;> simp [evaln, Bind.bind]
      · simp at hy₁
        rcases hf hy₁ with ⟨k, hk⟩
        exact ⟨_, Nat.le_of_lt_succ <| evaln_bound hk, _, hk, by simp; rfl⟩
      · rcases hy₂ (Nat.succ_pos _) with ⟨a, ha, a0⟩
        rcases hf ha with ⟨k₁, hk₁⟩
        rcases IH m.succ (by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy₁)
            fun {i} hi => by
            simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using
              hy₂ (Nat.succ_lt_succ hi) with
          ⟨k₂, hk₂⟩
        use (max k₁ k₂).succ
        rw [zero_add] at hk₁
        use Nat.le_succ_of_le <| le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk₁
        use a
        use evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_left _ _) hk₁
        simpa [Nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using
          evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk₂