proposition: Lemma nth_coeff_c_mult_p : forall (p : RX) c n, nth_coeff n (_C_ c[*]p) [=] c[*]nth_coeff n p.
proof: Proof. do 2 intro. pattern p in |- *. apply cpoly_induc. intros. astepl (nth_coeff n ([0]:RX)). astepr (c[*][0]). astepl ([0]:R). algebra. intros. elim n. simpl in |- *. algebra. intros. astepl (nth_coeff (S n0) (c[*]c0[+X*]_C_ c[*]p0)). astepl (nth_coeff n0 (_C_ c[*]p0)). astepl (c[*]nth_coeff n0 p0). algebra. Qed.
imports: CoRN.algebra.CPolynomials Coq.Classes.Morphisms CoRN.algebra.CRings