theorem trNormal_respects (c k v s) :
    ∃ b₂,
      TrCfg (stepNormal c k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          ⟨some (trNormal c (trCont k)), s, K'.elim (trList v) [] [] (trContStack k)⟩ b₂ := by
  induction c generalizing k v s
  case zero' => refine' ⟨_, ⟨s, rfl⟩, TransGen.single _⟩; simp
  case succ => refine' ⟨_, ⟨none, rfl⟩, head_main_ok.trans succ_ok⟩
  case tail =>
    let o : Option Γ' := List.casesOn v none fun _ _ => some Γ'.cons
    refine' ⟨_, ⟨o, rfl⟩, _⟩; convert clear_ok _ using 2; simp; rfl; swap
    refine' splitAtPred_eq _ _ (trNat v.headI) _ _ (trNat_natEnd _) _
    cases v <;> simp
  

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

Next tactic:
case
    cons f fs IHf _ =>
    obtain ⟨c, h₁, h₂⟩ := IHf (Cont.cons₁ fs v k) v none
    refine' ⟨c, h₁, TransGen.head rfl <| (move_ok (by decide) (splitAtPred_false _)).trans _⟩
    simp only [TM2.step, Option.mem_def, elim_stack, elim_update_stack, elim_update_main, ne_eq,
      Function.update_noteq, elim_main, elim_rev, elim_update_rev]
    refine' (copy_ok _ none [] (trList v).reverse _ _).trans _
    convert h₂ using 2
    simp [List.reverseAux_eq, trContStack]