proposition: Lemma leqifD_distz m1 m2 m3 : `|m1 - m3| <= `|m1 - m2| + `|m2 - m3| ?= iff (m1 <= m2 <= m3)%R || (m3 <= m2 <= m1)%R.
proof: Proof. apply/leqifP; rewrite -ltz_nat -eqz_nat PoszD !abszE; apply/leifP. wlog le_m31 : m1 m3 / (m3 <= m1)%R. move=> IH; case/orP: (le_total m1 m3) => /IH //. by rewrite (addrC `|_|)%R orbC !(distrC m1) !(distrC m3). rewrite ger0_norm ?subr_ge0 // orb_idl => [|/andP[le_m12 le_m23]]; last first. by have /eqP->: m2 == m3; rewrite ?lexx // eq_le le_m23 (le_trans le_m31). rewrite -{1}(subrK m2 m1) -(addrA _ m2) -subr_ge0 andbC -[X in X && _]subr_ge0. by apply: leifD; apply/real_leif_norm/num_real. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat choice seq) mathcomp(fintype finfun bigop order ssralg countalg ssrnum) mathcomp(poly)