theorem tr_respects {enc₀} :
    Respects (step M) (step (tr enc dec M)) fun c₁ c₂ ↦ trCfg enc enc₀ c₁ = c₂ :=
  fun_respects.2 fun ⟨l₁, v, T⟩ ↦ by
    obtain ⟨L, R, rfl⟩ := T.exists_mk'
    cases' l₁ with l₁
    · exact rfl
    suffices ∀ q R, Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R))
        (trCfg enc enc0 (stepAux q v (Tape.mk' L R))) by
      refine' TransGen.head' rfl _
      rw [trTape_mk']
      exact this _ R
    clear R l₁
    intro q R
    induction' q generalizing v L R
    case move d q IH =>
      cases d <;>
          simp only [trNormal, iterate, stepAux_move, stepAux, ListBlank.head_cons,
            Tape.move_left_mk', ListBlank.cons_head_tail, ListBlank.tail_cons,
            trTape'_move_left enc0, trTape'_move_right enc0] <;>
        apply IH
    

Proof state:
case intro.intro.some.write
Γ : Type u_1
inst✝² : Inhabited Γ
Λ : Type u_2
inst✝¹ : Inhabited Λ
σ : Type u_3
inst✝ : Inhabited σ
n : ℕ
enc : Γ → Vector Bool n
dec : Vector Bool n → Γ
enc0 : enc default = Vector.replicate n false
M : Λ → Stmt₁
encdec : ∀ (a : Γ), dec (enc a) = a
enc₀ : enc default = Vector.replicate n false
x✝ : Cfg₁
a✝¹ : Γ → σ → Γ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ) (L R : ListBlank Γ),
    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a✝) v (trTape' enc0 L R))
      (trCfg enc enc0 (stepAux a✝ v (Tape.mk' L R)))
v : σ
L R : ListBlank Γ
⊢ Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.write a✝¹ a✝)) v (trTape' enc0 L R))
    (trCfg enc enc0 (stepAux (Stmt.write a✝¹ a✝) v (Tape.mk' L R)))
case intro.intro.some.load
Γ : Type u_1
inst✝² : Inhabited Γ
Λ : Type u_2
inst✝¹ : Inhabited Λ
σ : Type u_3
inst✝ : Inhabited σ
n : ℕ
enc : Γ → Vector Bool n
dec : Vector Bool n → Γ
enc0 : enc default = Vector.replicate n false
M : Λ → Stmt₁
encdec : ∀ (a : Γ), dec (enc a) = a
enc₀ : enc default = Vector.replicate n false
x✝ : Cfg₁
a✝¹ : Γ → σ → σ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ) (L R : ListBlank Γ),
    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a✝) v (trTape' enc0 L R))
      (trCfg enc enc0 (stepAux a✝ v (Tape.mk' L R)))
v : σ
L R : ListBlank Γ
⊢ Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.load a✝¹ a✝)) v (trTape' enc0 L R))
    (trCfg enc enc0 (stepAux (Stmt.load a✝¹ a✝) v (Tape.mk' L R)))
case intro.intro.some.branch
Γ : Type u_1
inst✝² : Inhabited Γ
Λ : Type u_2
inst✝¹ : Inhabited Λ
σ : Type u_3
inst✝ : Inhabited σ
n : ℕ
enc : Γ → Vector Bool n
dec : Vector Bool n → Γ
enc0 : enc default = Vector.replicate n false
M : Λ → Stmt₁
encdec : ∀ (a : Γ), dec (enc a) = a
enc₀ : enc default = Vector.replicate n false
x✝ : Cfg₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ :
  ∀ (v : σ) (L R : ListBlank Γ),
    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a✝¹) v (trTape' enc0 L R))
      (trCfg enc enc0 (stepAux a✝¹ v (Tape.mk' L R)))
a_ih✝ :
  ∀ (v : σ) (L R : ListBlank Γ),
    Reaches (step (tr enc dec M)) (stepAux (trNormal dec a✝) v (trTape' enc0 L R))
      (trCfg enc enc0 (stepAux a✝ v (Tape.mk' L R)))
v : σ
L R : ListBlank Γ
⊢ Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.branch a✝² a✝¹ a✝)) v (trTape' enc0 L R))
    (trCfg enc enc0 (stepAux (Stmt.branch a✝² a✝¹ a✝) v (Tape.mk' L R)))
case intro.intro.some.goto
Γ : Type u_1
inst✝² : Inhabited Γ
Λ : Type u_2
inst✝¹ : Inhabited Λ
σ : Type u_3
inst✝ : Inhabited σ
n : ℕ
enc : Γ → Vector Bool n
dec : Vector Bool n → Γ
enc0 : enc default = Vector.replicate n false
M : Λ → Stmt₁
encdec : ∀ (a : Γ), dec (enc a) = a
enc₀ : enc default = Vector.replicate n false
x✝ : Cfg₁
a✝ : Γ → σ → Λ
v : σ
L R : ListBlank Γ
⊢ Reaches (step (tr enc dec M)) (stepAux (trNormal dec (Stmt.goto a✝)) v (trTape' enc0 L R))
    (trCfg enc enc0 (stepAux (Stmt.goto a✝) v (Tape.mk' L R)))
case intro.intro.some.halt
Γ : Type u_1
inst✝² : Inhabited Γ
Λ : Type u_2
inst✝¹ : Inhabited Λ
σ : Type u_3
inst✝ : Inhabited σ
n : ℕ
enc : Γ → Vector Bool n
dec : Vector Bool n → Γ
enc0 : enc default = Vector.replicate n false
M : Λ → Stmt₁
encdec : ∀ (a : Γ), dec (enc a) = a
enc₀ : enc default = Vector.replicate n false
x✝ : Cfg₁
v : σ
L R : ListBlank Γ
⊢ Reaches (step (tr enc dec M)) (stepAux (trNormal dec Stmt.halt) v (trTape' enc0 L R))
    (trCfg enc enc0 (stepAux Stmt.halt v (Tape.mk' L R)))

Next tactic:
case write f q IH =>
      simp only [trNormal, stepAux_read dec enc0 encdec, stepAux]
      refine' ReflTransGen.head rfl _
      obtain ⟨a, R, rfl⟩ := R.exists_cons
      rw [tr, Tape.mk'_head, stepAux_write, ListBlank.head_cons, stepAux_move,
        trTape'_move_left enc0, ListBlank.head_cons, ListBlank.tail_cons, Tape.write_mk']
      apply IH