proposition: Lemma Rle_up_pos r : 0 <= r -> r <= IZR (Z.abs (up r)).
proof: Proof. move=> Hr. rewrite Z.abs_eq; last first. apply up_pos in Hr. by apply Z.lt_le_incl. case: (base_Int_part r). rewrite /Int_part minus_IZR => _ ?; Lra.lra. Qed.
imports: mathcomp(all_ssreflect) mathcomp(all_algebra vector reals normedtype) mathcomp(mathcomp_extra boolp) mathcomp(Rstruct) ssrR realType_ext Reals Lra mathcomp(lra)