proposition: Lemma next_noroot p a b : {in `]a, (next_root p a b)[, noroot p}.
proof: Proof. move=> z; case: next_rootP; first by rewrite itv_xx. by move=> y np0 py0 hy hp hz; rewrite (negPf (hp _ _)). move=> c p0 -> hp hz; rewrite (negPf (hp _ _)) //. by case: leP hz; first rewrite itv_xx. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrfun ssrbool eqtype ssrnat seq choice fintype) mathcomp(bigop order ssralg poly polydiv ssrnum zmodp) mathcomp(polyorder path interval ssrint)