theorem tr_ret_respects (k v s) : ∃ b₂,
    TrCfg (stepRet k v) b₂ ∧
      Reaches₁ (TM2.step tr)
        ⟨some (Λ'.ret (trCont k)), s, K'.elim (trList v) [] [] (trContStack k)⟩ b₂ := by
  induction k generalizing v s
  case halt => exact ⟨_, rfl, TransGen.single rfl⟩
  case
    cons₁ fs as k _ =>
    obtain ⟨s', h₁, h₂⟩ := trNormal_respects fs (Cont.cons₂ v k) as none
    refine' ⟨s', h₁, TransGen.head rfl _⟩; simp
    refine' (move₂_ok (by decide) _ (splitAtPred_false _)).trans _; · rfl
    simp only [TM2.step, Option.mem_def, Option.elim, id_eq, elim_update_main, elim_main, elim_aux,
      List.append_nil, elim_update_aux]
    refine' (move₂_ok (by decide) _ _).trans _; pick_goal 4; · rfl
    pick_goal 4;
    · exact
        splitAtPred_eq _ _ _ (some Γ'.consₗ) _
          (fun x h => Bool.decide_false (trList_ne_consₗ _ _ h)) ⟨rfl, rfl⟩
    refine' (move₂_ok (by decide) _ (splitAtPred_false _)).trans _; · rfl
    simp only [TM2.step, Option.mem_def, Option.elim, elim_update_stack, elim_main,
      List.append_nil, elim_update_main,  id_eq, elim_update_aux, ne_eq, Function.update_noteq,
      elim_aux, elim_stack]
    exact h₂
  case cons₂ ns k IH =>
    obtain ⟨c, h₁, h₂⟩ := IH (ns.headI :: v) none
    exact ⟨c, h₁, TransGen.head rfl <| head_stack_ok.trans h₂⟩
  case comp f k _ =>
    obtain ⟨s', h₁, h₂⟩ := trNormal_respects f k v s
    exact ⟨_, h₁, TransGen.head rfl h₂⟩
  

Proof state:
case fix
a✝¹ : Code
a✝ : Cont
a_ih✝ :
  ∀ (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepRet a✝ v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (Λ'.ret (trCont a✝)), var := s, stk := elim (trList v) [] [] (trContStack a✝) } b₂
v : List ℕ
s : Option Γ'
⊢ ∃ b₂,
    TrCfg (stepRet (Cont.fix a✝¹ a✝) v) b₂ ∧
      Reaches₁ (TM2.step tr)
        { l := some (Λ'.ret (trCont (Cont.fix a✝¹ a✝))), var := s,
          stk := elim (trList v) [] [] (trContStack (Cont.fix a✝¹ a✝)) }
        b₂

Next tactic:
case fix f k IH =>
    rw [stepRet]
    have :
      if v.headI = 0 then natEnd (trList v).head?.iget = true ∧ (trList v).tail = trList v.tail
      else
        natEnd (trList v).head?.iget = false ∧
          (trList v).tail = (trNat v.headI).tail ++ Γ'.cons :: trList v.tail := by
      cases' v with n
      · exact ⟨rfl, rfl⟩
      cases' n with n
      · simp
      rw [trList, List.headI, trNat, Nat.cast_succ, Num.add_one, Num.succ, List.tail]
      cases (n : Num).succ' <;> exact ⟨rfl, rfl⟩
    by_cases h : v.headI = 0 <;> simp only [h, ite_true, ite_false] at this ⊢
    · obtain ⟨c, h₁, h₂⟩ := IH v.tail (trList v).head?
      refine' ⟨c, h₁, TransGen.head rfl _⟩
      simp only [Option.mem_def, TM2.stepAux, trContStack, contStack, elim_main, this, cond_true,
        elim_update_main]
      exact h₂
    · obtain ⟨s', h₁, h₂⟩ := trNormal_respects f (Cont.fix f k) v.tail (some Γ'.cons)
      refine' ⟨_, h₁, TransGen.head rfl <| TransGen.trans _ h₂⟩
      simp only [Option.mem_def, TM2.stepAux, elim_main, this.1, cond_false, elim_update_main,
        trCont]
      convert clear_ok (splitAtPred_eq _ _ (trNat v.headI).tail (some Γ'.cons) _ _ _) using 2
      · simp
        convert rfl
      · exact fun x h => trNat_natEnd _ _ (List.tail_subset _ h)
      · exact ⟨rfl, this.2⟩