induction n <;> simp at *
case a
a : String
n : ℕ
⊢ Ø ⋁ (¬ₛ _ a↗ =ₛ O) [a ≔ fromNat n]
case a.zero
a : String
⊢ Ø ⋁
    ¬ₛ
      (match String.decEq a a with
          | isTrue h => O
          | isFalse h => _ a)↗ =ₛ
        O

case a.succ
a : String
n✝ : ℕ
n_ih✝ :
  Ø ⋁
    ¬ₛ
      (match String.decEq a a with
          | isTrue h => fromNat n✝
          | isFalse h => _ a)↗ =ₛ
        O
⊢ Ø ⋁
    ¬ₛ
      (match String.decEq a a with
          | isTrue h => fromNat n✝↗
          | isFalse h => _ a)↗ =ₛ
        O