proposition: Lemma horner_mul_mixed a p x : horner tt (mul_mixed tt a p) x = (a * horner tt p x)%R.
proof: Proof. rewrite !hornerE size_mul_mixed. rewrite big_endo; first last. by rewrite Rmult_0_r. by move=> *; rewrite Rmult_plus_distr_l. apply: eq_bigr => i _. by rewrite nth_mul_mixed Rmult_assoc. Qed.
imports: Coq(ZArith Reals) Coquelicot(Coquelicot) mathcomp.ssreflect(ssreflect ssrfun ssrbool eqtype ssrnat seq fintype bigop) Flocq(Core) Stdlib MathComp Interval Xreal Basic_rec