Module: Mathlib.Computability.PartrecCode

Goal state before:
pp: k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
  ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
    k ≤ k₂ →
      (x ∈ o₁ → x ∈ o₂) →
        (x ∈ do
            guard (n ≤ k)
            o₁) →
          x ∈ do
            guard (n ≤ k₂)
            o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h :
  (fun n ↦ do
        guard (n ≤ k)
        unpaired
            (fun a n ↦
              Nat.casesOn n (evaln (k + 1) cf a) fun y ↦ do
                let i ← evaln k (cf.prec cg) (Nat.pair a y)
                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
            n)
      n =
    some x
⊢ Nat.rec (evaln (k + 1) cf (unpair n).1)
        (fun n_1 n_ih ↦
          (evaln k (cf.prec cg) (Nat.pair (unpair n).1 n_1)).bind fun i ↦
            evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
        (unpair n).2 =
      some x →
    Nat.rec (evaln (k₂ + 1) cf (unpair n).1)
        (fun n_1 n_ih ↦
          (evaln k₂ (cf.prec cg) (Nat.pair (unpair n).1 n_1)).bind fun i ↦
            evaln (k₂ + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
        (unpair n).2 =
      some x

Tactic:
induction n.unpair.2 <;> simp [Option.bind_eq_some_iff]

Goal state after:
case zero
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
  ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
    k ≤ k₂ →
      (x ∈ o₁ → x ∈ o₂) →
        (x ∈ do
            guard (n ≤ k)
            o₁) →
          x ∈ do
            guard (n ≤ k₂)
            o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h :
  (fun n ↦ do
        guard (n ≤ k)
        unpaired
            (fun a n ↦
              Nat.casesOn n (evaln (k + 1) cf a) fun y ↦ do
                let i ← evaln k (cf.prec cg) (Nat.pair a y)
                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
            n)
      n =
    some x
⊢ evaln (k + 1) cf (unpair n).1 = some x → evaln (k₂ + 1) cf (unpair n).1 = some x

case succ
k k₂ : ℕ
hl : k + 1 ≤ k₂ + 1
hl' : k ≤ k₂
this :
  ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
    k ≤ k₂ →
      (x ∈ o₁ → x ∈ o₂) →
        (x ∈ do
            guard (n ≤ k)
            o₁) →
          x ∈ do
            guard (n ≤ k₂)
            o₂
cf cg : Code
hf : ∀ (n x : ℕ), evaln (k + 1) cf n = some x → evaln (k₂ + 1) cf n = some x
hg : ∀ (n x : ℕ), evaln (k + 1) cg n = some x → evaln (k₂ + 1) cg n = some x
n x : ℕ
h :
  (fun n ↦ do
        guard (n ≤ k)
        unpaired
            (fun a n ↦
              Nat.casesOn n (evaln (k + 1) cf a) fun y ↦ do
                let i ← evaln k (cf.prec cg) (Nat.pair a y)
                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))
            n)
      n =
    some x
n✝ : ℕ
a✝ :
  Nat.rec (evaln (k + 1) cf (unpair n).1)
        (fun n_1 n_ih ↦
          (evaln k (cf.prec cg) (Nat.pair (unpair n).1 n_1)).bind fun i ↦
            evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
        n✝ =
      some x →
    Nat.rec (evaln (k₂ + 1) cf (unpair n).1)
        (fun n_1 n_ih ↦
          (evaln k₂ (cf.prec cg) (Nat.pair (unpair n).1 n_1)).bind fun i ↦
            evaln (k₂ + 1) cg (Nat.pair (unpair n).1 (Nat.pair n_1 i)))
        n✝ =
      some x
⊢ ∀ (x_1 : ℕ),
    evaln k (cf.prec cg) (Nat.pair (unpair n).1 n✝) = some x_1 →
      evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair n✝ x_1)) = some x →
        ∃ a,
          evaln k₂ (cf.prec cg) (Nat.pair (unpair n).1 n✝) = some a ∧
            evaln (k₂ + 1) cg (Nat.pair (unpair n).1 (Nat.pair n✝ a)) = some x