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));
            exact Or.inr ⟨_, rfl, hv₂⟩]
    · exact IH _ rfl _ _ stepRet_then (ReflTransGen.tail hr rfl)
  · rintro ⟨v', he, hr⟩
    rw [reaches_eval] at hr
    swap
    exact ReflTransGen.single rfl
    refine' PFun.fixInduction he fun v (he : v' ∈ f.fix.eval v) IH => _
    rw [fok, Part.bind_eq_bind, Part.mem_bind_iff]
    obtain he | ⟨v'', he₁', _⟩ := PFun.mem_fix_iff.1 he
    · obtain ⟨v', he₁, he₂⟩ := (Part.mem_map_iff _).1 he
      split_ifs at he₂ with h; cases he₂
      refine' ⟨_, he₁, _⟩
      rw [reaches_eval]
      swap
      exact ReflTransGen.single rfl
      

Proof state:
case pos.refl
f : Code
k : Cont
v✝ : List ℕ
fok : Code.Ok f
x : Cfg
v : List ℕ
IH :
  ∀ (a'' : List ℕ),
    Sum.inr a'' ∈
        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) →
      x ∈ eval step (stepNormal f (Cont.fix f k) a'')
v' : List ℕ
he₁ : v' ∈ Code.eval f v
h : List.headI v' = 0
he✝¹ : List.tail v' ∈ Code.eval (Code.fix f) v✝
hr✝ : x ∈ eval step (Cfg.ret k (List.tail v'))
hr : x ∈ eval step (stepRet k (List.tail v'))
he✝ : List.tail v' ∈ Code.eval (Code.fix f) v
he :
  Sum.inl (List.tail 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)
⊢ x ∈ eval step (stepRet (Cont.fix f k) v')

Next tactic:
rwa [stepRet, if_pos h]