proposition: Lemma mul_DN_correct : forall p x y, ((is_non_neg_real x /\ is_non_neg_real y) \/ (is_non_pos_real x /\ is_non_pos_real y) \/ (is_non_neg' x /\ is_non_pos' y) \/ (is_non_pos' x /\ is_non_neg' y)) -> (valid_lb (mul_DN p x y) = true /\ le_lower (toX (mul_DN p x y)) (Xmul (toX x) (toX y))).
proof: Proof. intros p x y _; split; [reflexivity|]. unfold mul_DN. rewrite mul_correct. unfold Xround, Xlift. set (z := Xmul _ _); case z; [exact I|]; intro zr. now apply Ropp_le_contravar, Generic_fmt.round_DN_pt, FLX.FLX_exp_valid. Qed.
imports: Coq(ZArith Reals Lia Bool Psatz) Flocq(Zaux Raux Digits Bracket) mathcomp.ssreflect(ssrbool) Xreal Basic Generic Generic_proof Sig Specific_sig