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]
        exact h
        exact ReflTransGen.single rfl
      · obtain ⟨k₀, v₀, e₀⟩ := stepNormal.is_ret f Cont.halt v'.tail
        have e₁ := stepNormal_then f Cont.halt (Cont.fix f k) v'.tail
        rw [e₀, Cont.then, Cfg.then] at e₁
        simp only [] at e₁
        obtain ⟨v₁, hv₁, v₂, hv₂, h₃⟩ :=
          IH (stepRet (k₀.then (Cont.fix f k)) v₀) (by rw [stepRet, if_neg he, e₁]; rfl)
            v'.tail _ stepRet_then (by apply ReflTransGen.single; rw [e₀]; rfl)
        · refine' ⟨_, PFun.mem_fix_iff.2 _, h₃⟩
          simp only [Part.eq_some_iff.2 hv₁, Part.map_some, Part.mem_some_iff]
          split_ifs at hv₂ ⊢ <;> [exact Or.inl (congr_arg Sum.inl (Part.mem_some_iff.1 hv₂));
            

Proof state:
case neg
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 (stepNormal f (Cont.fix f k) (List.tail v'))
k₀ : Cont
v₀ : List ℕ
e₀ : stepNormal f Cont.halt (List.tail v') = Cfg.ret k₀ v₀
e₁ : stepNormal f (Cont.fix f k) (List.tail v') = Cfg.ret (Cont.then k₀ (Cont.fix f k)) v₀
v₁ : List ℕ
hv₁ : v₁ ∈ Code.eval f (List.tail v')
v₂ : List ℕ
h₃ : x ∈ eval step (Cfg.ret k v₂)
h✝ : ¬List.headI v₁ = 0
hv₂ : v₂ ∈ Code.eval (Code.fix f) (List.tail v₁)
⊢ False ∨
    ∃ a',
      Sum.inr a' = Sum.inr (List.tail v₁) ∧
        v₂ ∈
          PFun.fix
            (fun v =>
              Part.map (fun v => if List.headI v = 0 then Sum.inl (List.tail v) else Sum.inr (List.tail v))
                (Code.eval f v))
            a'

Next tactic:
exact Or.inr ⟨_, rfl, hv₂⟩