proposition: Lemma mcomp_sym k (p : {mpoly R[n]}) (t : n.-tuple {mpoly R[k]}) : (forall i : 'I_n, t`_i \is symmetric) -> p \mPo t \is symmetric.
proof: Proof. move=> sym_t; pose_big_enough l. rewrite (comp_mpolywE _ (w := l)) //. 2: by close. apply/rpred_sum=> m _; apply/rpredZ/rpred_prod=> i _. by rewrite (tnth_nth 0); apply/rpredX/sym_t. 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