proposition: Lemma lt0_alg_total (x : alg) : x != zero_alg -> lt_alg zero_alg x || lt_alg x zero_alg.
proof: Proof. rewrite -[x]reprK !piE => /neq_algcrealP /= x_neq0. apply/orP; rewrite -!(rwP lt_algcrealP). case/neq_creal_ltVgt: x_neq0=> /= [lt_x0|lt_0x]; [right|by left]. pose_big_enough i. by apply: (@lt_crealP _ (diff lt_x0) i i); rewrite ?diff_gt0 ?diffP. 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)