proposition: Lemma ltr_distl_creal (e : F) (i : nat) (x : creal) (j : nat) (a b : F) : 0 < e -> (cauchymod x e <= i)%N -> (cauchymod x e <= j)%N -> `| x i - a | <= b - e -> `| x j - a | < b.
proof: Proof. move=> e_gt0 hi hj hb. rewrite (le_lt_trans (ler_dist_add (x i) _ _)) ?ltr_le_add //. by rewrite -[b](addrNK e) addrC ler_lt_add ?cauchymodP. Qed.
imports: 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 bigenough)