Definition double_round_sqrt_beta_ge_4_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 - 1)%Z).

Theorem double_round_sqrt_beta_ge_4_FTZ : (4 <= beta)%Z -> forall choice1 choice2, (2 * (emin' + prec') <= emin + prec <= 1)%Z -> (2 * prec + 1 <= prec')%Z -> forall x, FTZ_format beta emin prec x -> double_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (sqrt x).