proposition: Lemma eqyP x : x = +oo <-> (forall A, (0 < A)%R -> A%:E <= x).
proof: Proof. split=> [-> // A A0|Ax]; first by rewrite leey. apply/eqP; rewrite eq_le leey /= leNgt; apply/negP. case: x Ax => [x Ax _|//|/(_ _ ltr01)//]. suff: ~ x%:E < (Order.max 0 x + 1)%:E. by apply; rewrite lte_fin ltr_pwDr// le_maxr lexx orbT. by apply/negP; rewrite -leNgt; apply/Ax/ltr_pwDr; rewrite // le_maxr lexx. Qed.
imports: HB(structures) mathcomp(all_ssreflect all_algebra finmap) mathcomp(mathcomp_extra) signed