Section PolynomialTheory.
Variable R : ringType.
Implicit Types (a b c x y z : R) (p q r d : {poly R}).
Canonical poly_subType := Eval hnf in [subType of {poly R}].
Canonical poly_eqType := Eval hnf in [eqType of {poly R}].
Canonical poly_choiceType := Eval hnf in [choiceType of {poly R}].
Definition lead_coef p := p`_(size p).-1.
Definition poly_nil := @Polynomial R [::] (oner_neq0 R).
Definition polyC c : {poly R} := insubd poly_nil [:: c].
Local Notation "c %:P" := (polyC c).
(* Builds a polynomial by extension. *)
Definition cons_poly c p : {poly R} :=
if p is Polynomial ((_ :: _) as s) ns then
@Polynomial R (c :: s) ns
else c%:P.
(* Build a polynomial directly from a list of coefficients. *)
Definition Poly := foldr cons_poly 0%:P.
(* Build a polynomial from an infinite sequence of coefficients and a bound. *)
Definition poly_expanded_def n E := Poly (mkseq E n).
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 lead_coef_poly n E :
n > 0 -> E n.-1 != 0 -> lead_coef (\poly_(i < n) E i) = E n.-1.