proposition: Lemma symfP (p : {mpoly R[n]}) : p \is symmetric -> p = symf p \mPo S.
proof: Proof. move=> sym_p; rewrite /symf; set k := tag _. case/and3P: (symfnP k sym_p)=> /= _ _ /eqP {1}->. by rewrite {}/k; case: symfnS=> /= k -> //; rewrite !simpm. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq path) mathcomp(choice fintype tuple finfun bigop finset binomial) mathcomp(order fingroup perm ssralg zmodp poly ssrint) mathcomp(matrix vector) mathcomp(bigenough) ssrcomplements freeg