proposition: Lemma approx_proof x e : {y | 0 < e -> `|x - y%:RA| < e}.
proof: Proof. elim/quotW:x => x; pose_big_enough i. exists (x i)=> e_gt0; rewrite (lt_trans _ (inf_lt_alg _)) //. rewrite !piE sub_pi norm_pi -(rwP lt_algP) /= norm_algcrealE /=. pose_big_enough j. apply: (@lt_crealP _ (inf_alg e / 2%:R) j j) => //. by rewrite pmulr_rgt0 ?gtr0E ?inf_alg_gt0. rewrite /= {2}[inf_alg e](splitf 2) /= ler_add2r. by rewrite ltW // cauchymodP ?pmulr_rgt0 ?gtr0E ?inf_alg_gt0. by close. 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)