proposition: Lemma inf_out E : ~ has_inf E -> inf E = 0.
proof: Proof. move=> ninfE; rewrite -oppr0 -(@sup_out _ (-%R @` E)) => // supNE; apply: ninfE. exact/has_inf_supN. Qed.
imports: HB(structures) mathcomp(all_ssreflect all_algebra) mathcomp(mathcomp_extra boolp classical_sets set_interval) Setoid