proposition: Lemma sqrt_DN_correct : forall p x, valid_lb x = true -> (valid_lb (sqrt_DN p x) = true /\ le_lower (toX (sqrt_DN p x)) (Xsqrt (toX x))).
proof: Proof. intros p x _; split; [reflexivity|]. unfold sqrt_DN. rewrite sqrt_correct. unfold Xround, Xlift. case toX; [easy|intro rx]. unfold Xsqrt', Xsqrt_nan'. case is_negative_spec; [easy|intros _]. 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