theorem move_ok {p k₁ k₂ q s L₁ o L₂} {S : K' → List Γ'} (h₁ : k₁ ≠ k₂)
    (e : splitAtPred p (S k₁) = (L₁, o, L₂)) :
    Reaches₁ (TM2.step tr) ⟨some (Λ'.move p k₁ k₂ q), s, S⟩
      ⟨some q, o, update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂))⟩ := by
  induction' L₁ with a L₁ IH generalizing S s
  · rw [(_ : [].reverseAux _ = _), Function.update_eq_self]
    swap
    · rw [Function.update_noteq h₁.symm]
      rfl
    refine' TransGen.head' rfl _
    simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq]
    revert e; cases' S k₁ with a Sk <;> intro e
    · cases e
      rfl
    simp only [splitAtPred, Option.elim, List.head?, List.tail_cons, Option.iget_some] at e ⊢
    revert e; cases p a <;> intro e <;>
      simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e ⊢
    · simp only [e]
      rfl
  · refine' TransGen.head rfl _
    simp only [TM2.step, Option.mem_def, TM2.stepAux, Option.elim, ne_eq, List.reverseAux_cons]
    cases' e₁ : S k₁ with a' Sk <;> rw [e₁, splitAtPred] at e
    · cases e
    cases e₂ : p a' <;> simp only [e₂, cond] at e
    

Proof state:
case cons.cons.false
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (List.reverseAux L₁ (S k₂)) }
s : Option Γ'
S : K' → List Γ'
a' : Γ'
Sk : List Γ'
e₁ : S k₁ = a' :: Sk
e₂ : p a' = false
e : (a' :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2) = (a :: L₁, o, L₂)
⊢ TransGen
    (fun a b =>
      (match a with
        | { l := none, var := var, stk := stk } => none
        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =
        some b)
    (bif
        match List.head? (a' :: Sk), true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := List.head? (a' :: Sk), stk := update S k₁ (List.tail (a' :: Sk)) }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := List.head? (a' :: Sk),
        stk :=
          update (update S k₁ (List.tail (a' :: Sk))) k₂
            (Option.iget (List.head? (a' :: Sk)) :: update S k₁ (List.tail (a' :: Sk)) k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (List.reverseAux L₁ (a :: S k₂)) }
case cons.cons.true
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (List.reverseAux L₁ (S k₂)) }
s : Option Γ'
S : K' → List Γ'
a' : Γ'
Sk : List Γ'
e₁ : S k₁ = a' :: Sk
e₂ : p a' = true
e : ([], some a', Sk) = (a :: L₁, o, L₂)
⊢ TransGen
    (fun a b =>
      (match a with
        | { l := none, var := var, stk := stk } => none
        | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =
        some b)
    (bif
        match List.head? (a' :: Sk), true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := List.head? (a' :: Sk), stk := update S k₁ (List.tail (a' :: Sk)) }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := List.head? (a' :: Sk),
        stk :=
          update (update S k₁ (List.tail (a' :: Sk))) k₂
            (Option.iget (List.head? (a' :: Sk)) :: update S k₁ (List.tail (a' :: Sk)) k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (List.reverseAux L₁ (a :: S k₂)) }

Next tactic:
swap