Definition round_sign_DN s l :=
match l with
| loc_Exact => false
| _ => s
end.

Theorem inbetween_float_DN_sign :
forall x m l,
let e := canonic_exp beta fexp x in
inbetween_float beta m e (Rabs x) l ->
round beta fexp Zfloor x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_DN (Rlt_bool x 0) l) m)) e).