-- Mathlib/Computability/PartrecCode.lean
theorem evaln_sound : ∀ {k c n x}, x ∈ evaln k c n → x ∈ eval c n
  | 0, _, n, x, h => by simp [evaln] at h
  | k + 1, c, n, x, h => by
    induction' c with cf cg hf hg cf cg hf hg cf cg hf hg cf hf generalizing x n <;>
        simp [eval, evaln, Option.bind_eq_some, Seq.seq] at h ⊢ <;>
      obtain ⟨_, h⟩ := h
    iterate 4 simpa [pure, PFun.pure, eq_comm] using h
    · -- pair cf cg
      rcases h with ⟨y, ef, z, eg, rfl⟩
      exact ⟨_, hf _ _ ef, _, hg _ _ eg, rfl⟩
    · --comp hf hg
      rcases h with ⟨y, eg, ef⟩
      exact ⟨_, hg _ _ eg, hf _ _ ef⟩
    · -- prec cf cg
      revert h
      induction' n.unpair.2 with m IH generalizing x <;> simp [Option.bind_eq_some]
      · apply hf
      · refine fun y h₁ h₂ => ⟨y, IH _ ?_, ?_⟩
        · have := evaln_mono k.le_succ h₁
          simp [evaln, Option.bind_eq_some] at this
          exact this.2
        · exact hg _ _ h₂
    · -- rfind' cf
      rcases h with ⟨m, h₁, h₂⟩
      by_cases m0 : m = 0 <;> simp [m0] at h₂
      · exact
          ⟨0, ⟨by simpa [m0] using hf _ _ h₁, fun {m} => (Nat.not_lt_zero _).elim⟩, by simp [h₂]⟩
      · have := evaln_sound h₂
        simp [eval] at this
        rcases this with ⟨y, ⟨hy₁, hy₂⟩, rfl⟩
        refine
          ⟨y + 1, ⟨by simpa [add_comm, add_left_comm] using hy₁, fun {i} im => ?_⟩, by
            simp [add_comm, add_left_comm]⟩
        rcases i with - | i
        · exact ⟨m, by simpa using hf _ _ h₁, m0⟩
        · rcases hy₂ (Nat.lt_of_succ_lt_succ im) with ⟨z, hz, z0⟩
          exact ⟨z, by simpa [add_comm, add_left_comm] using hz, z0⟩

Proof state:
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n : ℕ
left✝ : n ≤ k
m : ℕ
h₁ : evaln (k + 1) cf n = some m
m0 : ¬m = 0
y : ℕ
hy₁ : 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + ((unpair n).2 + 1)))
hy₂ : ∀ {m : ℕ}, m < y → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + ((unpair n).2 + 1))), ¬a = 0
h₂ : evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1)) = some (y + ((unpair n).2 + 1))
i : ℕ
im : i + 1 < y + 1
z : ℕ
hz : z ∈ cf.eval (Nat.pair (unpair n).1 (i + ((unpair n).2 + 1)))
z0 : ¬z = 0
⊢ z ∈ cf.eval (Nat.pair (unpair n).1 (i + 1 + (unpair n).2))

Tactic:
simpa [add_comm, add_left_comm] using hz

Result:
no goals

Proof state:
case prec.intro.succ.refine_1
k : ℕ
cf cg : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
hg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ cg.eval n
n : ℕ
left✝ : n ≤ k
m : ℕ
IH :
  ∀ (x : ℕ),
    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)))
          m =
        some x →
      x ∈ Nat.rec (cf.eval (unpair n).1) (fun y IH => IH.bind fun i => cg.eval (Nat.pair (unpair n).1 (Nat.pair y i))) m
x y : ℕ
h₁ : evaln k (cf.prec cg) (Nat.pair (unpair n).1 m) = some y
h₂ : evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair m y)) = 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)))
      m =
    some y

Tactic:
have := evaln_mono k.le_succ h₁

Result:
case prec.intro.succ.refine_1
k : ℕ
cf cg : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
hg : ∀ (n x : ℕ), x ∈ evaln (k + 1) cg n → x ∈ cg.eval n
n : ℕ
left✝ : n ≤ k
m : ℕ
IH :
  ∀ (x : ℕ),
    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)))
          m =
        some x →
      x ∈ Nat.rec (cf.eval (unpair n).1) (fun y IH => IH.bind fun i => cg.eval (Nat.pair (unpair n).1 (Nat.pair y i))) m
x y : ℕ
h₁ : evaln k (cf.prec cg) (Nat.pair (unpair n).1 m) = some y
h₂ : evaln (k + 1) cg (Nat.pair (unpair n).1 (Nat.pair m y)) = some x
this : y ∈ evaln k.succ (cf.prec cg) (Nat.pair (unpair n).1 m)
⊢ 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)))
      m =
    some y

Proof state:
case neg
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n x : ℕ
left✝ : n ≤ k
m : ℕ
h₁ : evaln (k + 1) cf n = some m
m0 : ¬m = 0
h₂ : evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1)) = some x
this : x ∈ cf.rfind'.eval (Nat.pair (unpair n).1 ((unpair n).2 + 1))
⊢ ∃ a,
    (0 ∈ cf.eval (Nat.pair (unpair n).1 (a + (unpair n).2)) ∧
        ∀ {m : ℕ}, m < a → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + (unpair n).2)), ¬a = 0) ∧
      a + (unpair n).2 = x

Tactic:
simp [eval] at this

Result:
case neg
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n x : ℕ
left✝ : n ≤ k
m : ℕ
h₁ : evaln (k + 1) cf n = some m
m0 : ¬m = 0
h₂ : evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1)) = some x
this :
  ∃ a,
    (0 ∈ cf.eval (Nat.pair (unpair n).1 (a + ((unpair n).2 + 1))) ∧
        ∀ {m : ℕ}, m < a → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + ((unpair n).2 + 1))), ¬a = 0) ∧
      a + ((unpair n).2 + 1) = x
⊢ ∃ a,
    (0 ∈ cf.eval (Nat.pair (unpair n).1 (a + (unpair n).2)) ∧
        ∀ {m : ℕ}, m < a → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + (unpair n).2)), ¬a = 0) ∧
      a + (unpair n).2 = x

Proof state:
case neg.intro.intro.intro.succ
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n : ℕ
left✝ : n ≤ k
m : ℕ
h₁ : evaln (k + 1) cf n = some m
m0 : ¬m = 0
y : ℕ
hy₁ : 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + ((unpair n).2 + 1)))
hy₂ : ∀ {m : ℕ}, m < y → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + ((unpair n).2 + 1))), ¬a = 0
h₂ : evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1)) = some (y + ((unpair n).2 + 1))
i : ℕ
im : i + 1 < y + 1
⊢ ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (i + 1 + (unpair n).2)), ¬a = 0

Tactic:
rcases hy₂ (Nat.lt_of_succ_lt_succ im) with ⟨z, hz, z0⟩

Result:
case neg.intro.intro.intro.succ.intro.intro
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n : ℕ
left✝ : n ≤ k
m : ℕ
h₁ : evaln (k + 1) cf n = some m
m0 : ¬m = 0
y : ℕ
hy₁ : 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + ((unpair n).2 + 1)))
hy₂ : ∀ {m : ℕ}, m < y → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + ((unpair n).2 + 1))), ¬a = 0
h₂ : evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1)) = some (y + ((unpair n).2 + 1))
i : ℕ
im : i + 1 < y + 1
z : ℕ
hz : z ∈ cf.eval (Nat.pair (unpair n).1 (i + ((unpair n).2 + 1)))
z0 : ¬z = 0
⊢ ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (i + 1 + (unpair n).2)), ¬a = 0