Section Double_round_sqrt.
Definition round_round_sqrt_hyp fexp1 fexp2 :=
(forall ex, (2 * fexp1 ex <= fexp1 (2 * ex))%Z)
/\ (forall ex, (2 * fexp1 ex <= fexp1 (2 * ex - 1))%Z)
/\ (forall ex, (fexp1 (2 * ex) < 2 * ex)%Z ->
(fexp2 ex + ex <= 2 * fexp1 ex - 2)%Z).

Lemma round_round_sqrt_aux :
forall fexp1 fexp2 : Z -> Z,
Valid_exp fexp1 -> Valid_exp fexp2 ->
round_round_sqrt_hyp fexp1 fexp2 ->
forall x,
0 < x ->
(fexp2 (mag (sqrt x)) <= fexp1 (mag (sqrt x)) - 1)%Z ->
generic_format beta fexp1 x ->
/ 2 * ulp beta fexp2 (sqrt x) < Rabs (sqrt x - midp fexp1 (sqrt x)).