proposition: Lemma neq_to_alg_creal x y : (to_alg_creal x != to_alg_creal y)%CR <-> (x != y)%CR.
proof: Proof. split=> neq_xy. pose_big_enough i. apply: (@neq_crealP _ (inf_alg (lbound neq_xy)) i i) => //. by rewrite inf_alg_gt0. rewrite -ler_to_alg normr_to_alg rmorphB /= ltW //. by rewrite (lt_le_trans (inf_lt_alg _)) ?lbound_gt0 ?lboundP. by close. pose_big_enough i. apply: (@neq_crealP _ (lbound neq_xy)%:RA i i) => //. by rewrite ltr_to_alg lbound_gt0. by rewrite -rmorphB -normr_to_alg ler_to_alg lboundP. by close. Qed.
imports: HB(structures) mathcomp.ssreflect.ssreflect mathcomp(ssrfun ssrbool eqtype ssrnat seq choice fintype) mathcomp(bigop order ssralg ssrnum ssrint rat poly polydiv polyorder) mathcomp(perm matrix mxpoly polyXY binomial generic_quotient) mathcomp(cauchyreals separable zmodp bigenough)