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

Theorem inbetween_int_DN_sign :
forall x m l,
inbetween_int m (Rabs x) l ->
Zfloor x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_DN (Rlt_bool x 0) l) m).