Definition cond_incr (b : bool) m := if b then (m + 1)%Z else m.

Theorem inbetween_float_round_sign :
forall rnd choice,
( forall x m l, inbetween_int m (Rabs x) l ->
rnd x = cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l) ) ->
forall x m l,
let e := canonic_exp beta fexp x in
inbetween_float beta m e (Rabs x) l ->
round beta fexp rnd x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l)) e).