Section Double_round_plus.
Definition round_round_plus_hyp fexp1 fexp2 :=
(forall ex ey, (fexp1 (ex + 1) - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z)
/\ (forall ex ey, (fexp1 (ex - 1) + 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z)
/\ (forall ex ey, (fexp1 ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z)
/\ (forall ex ey, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z).

(* fexp1 (mag x) - 1 <= mag y : * addition is exact in the largest precision (fexp2). *)
Lemma round_round_plus_aux0 :
forall (fexp1 fexp2 : Z -> Z), Valid_exp fexp1 ->
round_round_plus_hyp fexp1 fexp2 ->
forall x y,
(0 < x)%R -> (0 < y)%R -> (y <= x)%R ->
(fexp1 (mag x) - 1 <= mag y)%Z ->
generic_format beta fexp1 x -> generic_format beta fexp1 y ->
generic_format beta fexp2 (x + y).