proposition: Lemma cst_pi (x : algcreal F) : ((\pi_{alg F} x)%:CR == to_alg_creal x)%CR.
proof: Proof. apply: eq_crealP; exists_big_modulus m {alg F}. move=> e i e_gt0 hi /=; 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 ?divrn_gt0 ?inf_alg_gt0. rewrite /= {2}[inf_alg _](splitf 2) ler_add2r ltW // distrC. by rewrite cauchymodP ?divrn_gt0 ?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)