-- 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
c : Code
n x : ℕ
h : x ∈ evaln (k + 1) c n
⊢ x ∈ c.eval n

Tactic:
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

Result:
case zero.intro
k n x : ℕ
left✝ : n ≤ k
h : 0 = x
⊢ x ∈ pure 0 n

case succ.intro
k n x : ℕ
left✝ : n ≤ k
h : n + 1 = x
⊢ x = n + 1

case left.intro
k n x : ℕ
left✝ : n ≤ k
h : (unpair n).1 = x
⊢ x = (unpair n).1

case right.intro
k n x : ℕ
left✝ : n ≤ k
h : (unpair n).2 = x
⊢ x = (unpair n).2

case pair.intro
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 x : ℕ
left✝ : n ≤ k
h : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x
⊢ ∃ a ∈ cf.eval n, ∃ a_1 ∈ cg.eval n, Nat.pair a a_1 = x

case comp.intro
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 x : ℕ
left✝ : n ≤ k
h : ∃ a, evaln (k + 1) cg n = some a ∧ evaln (k + 1) cf a = some x
⊢ ∃ a ∈ cg.eval n, x ∈ cf.eval a

case prec.intro
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 x : ℕ
left✝ : n ≤ k
h :
  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
⊢ 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)))
      (unpair n).2

case rfind'.intro
k : ℕ
cf : Code
hf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n
n x : ℕ
left✝ : n ≤ k
h :
  ∃ a,
    evaln (k + 1) cf n = some a ∧
      (if a = 0 then some (unpair n).2 else evaln k cf.rfind' (Nat.pair (unpair n).1 ((unpair n).2 + 1))) = some 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:
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))
⊢ y + 1 + (unpair n).2 = y + ((unpair n).2 + 1)

Tactic:
simp [add_comm, add_left_comm]

Result:
no goals

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))
⊢ 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + 1 + (unpair n).2))

Tactic:
simpa [add_comm, add_left_comm] using hy₁

Result:
no goals

Proof state:
case pos
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 : (unpair n).2 = 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

Tactic:
exact
  ⟨0, ⟨by simpa [m0] using hf _ _ h₁, fun {m} => (Nat.not_lt_zero _).elim⟩, by simp [h₂]⟩

Result:
no goals

Proof state:
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

Tactic:
exact ⟨z, by simpa [add_comm, add_left_comm] using hz, z0⟩

Result:
no goals

Proof state:
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 : (unpair n).2 = x
⊢ 0 + (unpair n).2 = x

Tactic:
simp [h₂]

Result:
no goals

Proof state:
case comp.intro.intro.intro
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 x : ℕ
left✝ : n ≤ k
y : ℕ
eg : evaln (k + 1) cg n = some y
ef : evaln (k + 1) cf y = some x
⊢ ∃ a ∈ cg.eval n, x ∈ cf.eval a

Tactic:
exact ⟨_, hg _ _ eg, hf _ _ ef⟩

Result:
no goals

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))
im : 0 < y + 1
⊢ m ∈ cf.eval (Nat.pair (unpair n).1 (0 + (unpair n).2))

Tactic:
simpa using hf _ _ h₁

Result:
no goals

Proof state:
case pair.intro
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 x : ℕ
left✝ : n ≤ k
h : ∃ a, evaln (k + 1) cf n = some a ∧ ∃ a_1, evaln (k + 1) cg n = some a_1 ∧ Nat.pair a a_1 = x
⊢ ∃ a ∈ cf.eval n, ∃ a_1 ∈ cg.eval n, Nat.pair a a_1 = x

Tactic:
rcases h with ⟨y, ef, z, eg, rfl⟩

Result:
case pair.intro.intro.intro.intro.intro
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
y : ℕ
ef : evaln (k + 1) cf n = some y
z : ℕ
eg : evaln (k + 1) cg n = some z
⊢ ∃ a ∈ cf.eval n, ∃ a_1 ∈ cg.eval n, Nat.pair a a_1 = Nat.pair y z