(* Build a polynomial from an infinite sequence of coefficients and a bound. *)
Fact poly_key : unit. Proof. by []. Qed.
Definition poly := locked_with poly_key poly_expanded_def.
Canonical poly_unlockable := [unlockable fun poly].
Local Notation "\poly_ ( i < n ) E" := (poly n (fun i : nat => E)).

Lemma coefK p : \poly_(i < size p) p`_i = p.
Definition add_poly_def p q := \poly_(i < maxn (size p) (size q)) (p`_i + q`_i).