proposition: Lemma up_pos r : 0 <= r -> (0 < up r)%Z.
proof: Proof. move=> Hr. apply lt_IZR => /=. move/Rgt_lt : (proj1 (archimed r)) => Hr'. exact: (leR_ltR_trans Hr). Qed.
imports: mathcomp(all_ssreflect) mathcomp(all_algebra vector reals normedtype) mathcomp(mathcomp_extra boolp) mathcomp(Rstruct) ssrR realType_ext Reals Lra mathcomp(lra)