End Double_round_sqrt_beta_ge_4_FLT.
Section Double_round_sqrt_beta_ge_4_FTZ.
Variable emin prec : Z.
Variable emin' prec' : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.
Context { prec_gt_0_' : Prec_gt_0 prec' }.

(* Specifications: *)
Lemma FTZ_double_round_sqrt_beta_ge_4_hyp :
(2 * (emin' + prec') <= emin + prec <= 1)%Z ->
(2 * prec + 1 <= prec')%Z ->
double_round_sqrt_beta_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec').
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).
Lemma FTZ_double_round_sqrt_beta_ge_4_hyp :
(2 * (emin' + prec') <= emin + prec <= 1)%Z ->
(2 * prec + 1 <= prec')%Z ->
double_round_sqrt_beta_ge_4_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec').
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).