mem_rec_on — Mathlib

Lean4
theorem mem_rec_on {C : Seq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s')) : C s :=
  by
  obtain ⟨k, e⟩ := M; unfold Stream'.get at e
  induction k generalizing s with
  |
    zero =>
    have TH : s = cons a (tail s) := by
      apply destruct_eq_cons
      unfold destruct get? Functor.map
      rw [← e]
      rfl
    rw [TH]
    apply h1 _ _ (Or.inl rfl)
  | succ k IH =>
    cases s with
    | nil => injection e
    | cons b s' =>
      have h_eq : (cons b s').val (Nat.succ k) = s'.val k := by cases s' using Subtype.recOn; rfl
      rw [h_eq] at e
      apply h1 _ _ (Or.inr (IH e))
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