proposition: Lemma root_bigmul x (ps : seq {poly R}) : ~~root (\big[*%R/1]_(p <- ps) p) x = all (fun p => ~~ root p x) ps.
proof: Proof. elim: ps => [|p ps ihp]; first by rewrite big_nil root1. by rewrite big_cons /= rootM negb_or ihp. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq choice) mathcomp(fintype bigop ssralg poly)