theorem tr_respects_aux₂ {k : K} {q : Stmt₂₁} {v : σ} {S : ∀ k, List (Γ k)}
    {L : ListBlank (∀ k, Option (Γ k))}
    (hL : ∀ k, L.map (proj k) = ListBlank.mk ((S k).map some).reverse) (o : StAct₂ k) :
    let v' := stVar v (S k) o
    let Sk' := stWrite v (S k) o
    let S' := update S k Sk'
    ∃ L' : ListBlank (∀ k, Option (Γ k)),
      (∀ k, L'.map (proj k) = ListBlank.mk ((S' k).map some).reverse) ∧
        TM1.stepAux (trStAct q o) v
            ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom L))) =
          TM1.stepAux q v' ((Tape.move Dir.right)^[(S' k).length] (Tape.mk' ∅ (addBottom L'))) := by
  dsimp only; simp; cases o <;> simp only [stWrite, stVar, trStAct, TM1.stepAux]
  case push f =>
    have := Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k (some (f v)))
    refine'
      ⟨_, fun k' ↦ _, by
        -- Porting note: `rw [...]` to `erw [...]; rfl`.
        -- https://github.com/leanprover-community/mathlib4/issues/5164
        erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this,
          addBottom_modifyNth fun a ↦ update a k (some (f v)), Nat.add_one, iterate_succ']
        rfl⟩
    refine' ListBlank.ext fun i ↦ _
    rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]
    by_cases h' : k' = k
    · subst k'
      split_ifs with h
        <;> simp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map]
      -- Porting note: `le_refl` is required.
      · rw [List.getI_eq_get, List.get_append_right'] <;>
          simp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos',
            List.length_append, lt_add_iff_pos_right, List.length, le_refl]
      rw [← proj_map_nth, hL, ListBlank.nth_mk]
      cases' lt_or_gt_of_ne h with h h
      · rw [List.getI_append]
        simpa only [List.length_map, List.length_reverse] using h
      · rw [gt_iff_lt] at h
        rw [List.getI_eq_default, List.getI_eq_default] <;>
          simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse,
            List.length_append, List.length_map]
    · split_ifs <;> rw [Function.update_noteq h', ← proj_map_nth, hL]
      rw [Function.update_noteq h']
  case peek f =>
    rw [Function.update_eq_self]
    use L, hL; rw [Tape.move_left_right]; congr
    cases e : S k; · rfl
    rw [List.length_cons, iterate_succ', Function.comp, Tape.move_right_left,
      Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd, stk_nth_val _ (hL k), e,
      List.reverse_cons, ← List.length_reverse, List.get?_concat_length]
    rfl
  case pop f =>
    cases' e : S k with hd tl
    · simp only [Tape.mk'_head, ListBlank.head_cons, Tape.move_left_mk', List.length,
        Tape.write_mk', List.head?, iterate_zero_apply, List.tail_nil]
      

Proof state:
case nil
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
k : K
q : TM1.Stmt Γ' Λ' σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k)))
f : σ → Option (Γ k) → σ
e : S k = []
⊢ ∃ L',
    (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k [] k_1)))) ∧
      (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L))
        else
          TM1.stepAux q (f v ((ListBlank.head ∅).2 k))
            (Tape.mk' (ListBlank.tail ∅)
              (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) =
        TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L'))

Next tactic:
rw [← e, Function.update_eq_self]