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
    

Proof state:
case intro.intro.refine'_2.refine'_1
fs : Code
as : List ℕ
k : Cont
a_ih✝ :
  ∀ (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepRet k v) b₂ ∧
        Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
          b₂
v : List ℕ
s : Option Γ'
s' : Cfg'
h₁ : TrCfg (stepNormal fs (Cont.cons₂ v k) as) s'
h₂ :
  Reaches₁ (TM2.step tr)
    { l := some (trNormal fs (trCont (Cont.cons₂ v k))), var := none,
      stk := elim (trList as) [] [] (trContStack (Cont.cons₂ v k)) }
    s'
⊢ List Γ'
case intro.intro.refine'_2.refine'_2
fs : Code
as : List ℕ
k : Cont
a_ih✝ :
  ∀ (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepRet k v) b₂ ∧
        Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
          b₂
v : List ℕ
s : Option Γ'
s' : Cfg'
h₁ : TrCfg (stepNormal fs (Cont.cons₂ v k) as) s'
h₂ :
  Reaches₁ (TM2.step tr)
    { l := some (trNormal fs (trCont (Cont.cons₂ v k))), var := none,
      stk := elim (trList as) [] [] (trContStack (Cont.cons₂ v k)) }
    s'
⊢ Option Γ'
case intro.intro.refine'_2.refine'_3
fs : Code
as : List ℕ
k : Cont
a_ih✝ :
  ∀ (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepRet k v) b₂ ∧
        Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
          b₂
v : List ℕ
s : Option Γ'
s' : Cfg'
h₁ : TrCfg (stepNormal fs (Cont.cons₂ v k) as) s'
h₂ :
  Reaches₁ (TM2.step tr)
    { l := some (trNormal fs (trCont (Cont.cons₂ v k))), var := none,
      stk := elim (trList as) [] [] (trContStack (Cont.cons₂ v k)) }
    s'
⊢ List Γ'
case intro.intro.refine'_2.refine'_5
fs : Code
as : List ℕ
k : Cont
a_ih✝ :
  ∀ (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepRet k v) b₂ ∧
        Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
          b₂
v : List ℕ
s : Option Γ'
s' : Cfg'
h₁ : TrCfg (stepNormal fs (Cont.cons₂ v k) as) s'
h₂ :
  Reaches₁ (TM2.step tr)
    { l := some (trNormal fs (trCont (Cont.cons₂ v k))), var := none,
      stk := elim (trList as) [] [] (trContStack (Cont.cons₂ v k)) }
    s'
⊢ splitAtPred (fun s => decide (s = Γ'.consₗ)) (elim [] [] (trList v) (trContStack (Cont.cons₁ fs as k)) stack) =
    (?intro.intro.refine'_2.refine'_1, ?intro.intro.refine'_2.refine'_2, ?intro.intro.refine'_2.refine'_3)
case intro.intro.refine'_2.refine'_6
fs : Code
as : List ℕ
k : Cont
a_ih✝ :
  ∀ (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepRet k v) b₂ ∧
        Reaches₁ (TM2.step tr) { l := some (Λ'.ret (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) }
          b₂
v : List ℕ
s : Option Γ'
s' : Cfg'
h₁ : TrCfg (stepNormal fs (Cont.cons₂ v k) as) s'
h₂ :
  Reaches₁ (TM2.step tr)
    { l := some (trNormal fs (trCont (Cont.cons₂ v k))), var := none,
      stk := elim (trList as) [] [] (trContStack (Cont.cons₂ v k)) }
    s'
⊢ TransGen (fun a b => b ∈ TM2.step tr a)
    { l := some (move₂ (fun x => false) aux stack (trNormal fs (Cont'.cons₂ (trCont k)))), var := none,
      stk :=
        update
          (update (elim [] [] (trList v) (trContStack (Cont.cons₁ fs as k))) stack
            (Option.elim ?intro.intro.refine'_2.refine'_2 id List.cons ?intro.intro.refine'_2.refine'_3))
          main (?intro.intro.refine'_2.refine'_1 ++ elim [] [] (trList v) (trContStack (Cont.cons₁ fs as k)) main) }
    s'

Next tactic:
pick_goal 4