End Double_round_div_FLT.
Section Double_round_div_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_div_hyp :
(emin' + prec' <= emin - 1)%Z ->
(2 * prec <= prec')%Z ->
double_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec').
Theorem double_round_div_FTZ :
forall choice1 choice2,
(exists n, (beta = 2 * n :> Z)%Z) ->
(emin' + prec' <= emin - 1)%Z ->
(2 * prec <= prec')%Z ->
forall x y,
y <> 0 ->
FTZ_format beta emin prec x -> FTZ_format beta emin prec y ->
double_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec')
choice1 choice2 (x / y).
Lemma FTZ_double_round_div_hyp :
(emin' + prec' <= emin - 1)%Z ->
(2 * prec <= prec')%Z ->
double_round_div_hyp (FTZ_exp emin prec) (FTZ_exp emin' prec').
Theorem double_round_div_FTZ :
forall choice1 choice2,
(exists n, (beta = 2 * n :> Z)%Z) ->
(emin' + prec' <= emin - 1)%Z ->
(2 * prec <= prec')%Z ->
forall x y,
y <> 0 ->
FTZ_format beta emin prec x -> FTZ_format beta emin prec y ->
double_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec')
choice1 choice2 (x / y).