theorem cont_eval_fix {f k v} (fok : Code.Ok f) :
    Turing.eval step (stepNormal f (Cont.fix f k) v) =
      f.fix.eval v >>= fun v => Turing.eval step (Cfg.ret k v) := by
  refine' Part.ext fun x => _
  simp only [Part.bind_eq_bind, Part.mem_bind_iff]
  constructor
  · suffices ∀ c, x ∈ eval step c → ∀ v c', c = Cfg.then c' (Cont.fix f k) →
      Reaches step (stepNormal f Cont.halt v) c' →
        ∃ v₁ ∈ f.eval v, ∃ v₂ ∈ if List.headI v₁ = 0 then pure v₁.tail else f.fix.eval v₁.tail,
          x ∈ eval step (Cfg.ret k v₂) by
      intro h
      obtain ⟨v₁, hv₁, v₂, hv₂, h₃⟩ :=
        this _ h _ _ (stepNormal_then _ Cont.halt _ _) ReflTransGen.refl
      refine' ⟨v₂, PFun.mem_fix_iff.2 _, h₃⟩
      simp only [Part.eq_some_iff.2 hv₁, Part.map_some]
      split_ifs at hv₂ ⊢
      · rw [Part.mem_some_iff.1 hv₂]
        exact Or.inl (Part.mem_some _)
      · exact Or.inr ⟨_, Part.mem_some _, hv₂⟩
    refine' fun c he => evalInduction he fun y h IH => _
    rintro v (⟨v'⟩ | ⟨k', v'⟩) rfl hr <;> rw [Cfg.then] at h IH <;> simp only [] at h IH
    · have := mem_eval.2 ⟨hr, rfl⟩
      rw [fok, Part.bind_eq_bind, Part.mem_bind_iff] at this
      obtain ⟨v'', h₁, h₂⟩ := this
      rw [reaches_eval] at h₂
      swap
      exact ReflTransGen.single rfl
      cases Part.mem_unique h₂ (mem_eval.2 ⟨ReflTransGen.refl, rfl⟩)
      refine' ⟨v', h₁, _⟩
      rw [stepRet] at h
      revert h
      by_cases he : v'.headI = 0 <;> simp only [exists_prop, if_pos, if_false, he] <;> intro h
      · refine' ⟨_, Part.mem_some _, _⟩
        rw [reaches_eval]
        

Proof state:
case pos
f : Code
k : Cont
v✝ : List ℕ
fok : Code.Ok f
x c : Cfg
he✝ : x ∈ eval step c
v v' : List ℕ
IH :
  ∀ (a' : Cfg),
    step (stepRet (Cont.fix f k) v') = some a' →
      ∀ (v : List ℕ) (c' : Cfg),
        a' = Cfg.then c' (Cont.fix f k) →
          Reaches step (stepNormal f Cont.halt v) c' →
            ∃ v₁ ∈ Code.eval f v,
              ∃ v₂ ∈ if List.headI v₁ = 0 then pure (List.tail v₁) else Code.eval (Code.fix f) (List.tail v₁),
                x ∈ eval step (Cfg.ret k v₂)
hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')
h₁ : v' ∈ Code.eval f v
h₂✝ : Cfg.halt v' ∈ eval step (Cfg.ret Cont.halt v')
h₂ : Cfg.halt v' ∈ eval step (stepRet Cont.halt v')
he : List.headI v' = 0
h : x ∈ eval step (stepRet k (List.tail v'))
⊢ x ∈ eval step ?m.221625
case pos
f : Code
k : Cont
v✝ : List ℕ
fok : Code.Ok f
x c : Cfg
he✝ : x ∈ eval step c
v v' : List ℕ
IH :
  ∀ (a' : Cfg),
    step (stepRet (Cont.fix f k) v') = some a' →
      ∀ (v : List ℕ) (c' : Cfg),
        a' = Cfg.then c' (Cont.fix f k) →
          Reaches step (stepNormal f Cont.halt v) c' →
            ∃ v₁ ∈ Code.eval f v,
              ∃ v₂ ∈ if List.headI v₁ = 0 then pure (List.tail v₁) else Code.eval (Code.fix f) (List.tail v₁),
                x ∈ eval step (Cfg.ret k v₂)
hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')
h₁ : v' ∈ Code.eval f v
h₂✝ : Cfg.halt v' ∈ eval step (Cfg.ret Cont.halt v')
h₂ : Cfg.halt v' ∈ eval step (stepRet Cont.halt v')
he : List.headI v' = 0
h : x ∈ eval step (stepRet k (List.tail v'))
⊢ Reaches step (Cfg.ret k (List.tail v')) ?m.221625
f : Code
k : Cont
v✝ : List ℕ
fok : Code.Ok f
x c : Cfg
he✝ : x ∈ eval step c
v v' : List ℕ
IH :
  ∀ (a' : Cfg),
    step (stepRet (Cont.fix f k) v') = some a' →
      ∀ (v : List ℕ) (c' : Cfg),
        a' = Cfg.then c' (Cont.fix f k) →
          Reaches step (stepNormal f Cont.halt v) c' →
            ∃ v₁ ∈ Code.eval f v,
              ∃ v₂ ∈ if List.headI v₁ = 0 then pure (List.tail v₁) else Code.eval (Code.fix f) (List.tail v₁),
                x ∈ eval step (Cfg.ret k v₂)
hr : Reaches step (stepNormal f Cont.halt v) (Cfg.halt v')
h₁ : v' ∈ Code.eval f v
h₂✝ : Cfg.halt v' ∈ eval step (Cfg.ret Cont.halt v')
h₂ : Cfg.halt v' ∈ eval step (stepRet Cont.halt v')
he : List.headI v' = 0
h : x ∈ eval step (stepRet k (List.tail v'))
⊢ Cfg

Next tactic:
exact h