End Zle_bool.
Section Zlt_bool.
Inductive Zlt_bool_prop (x y : Z) : bool -> Prop :=
| Zlt_bool_true_ : (x < y)%Z -> Zlt_bool_prop x y true
| Zlt_bool_false_ : (y <= x)%Z -> Zlt_bool_prop x y false.

Theorem Zlt_bool_false :
forall x y : Z,
(y <= x)%Z -> Zlt_bool x y = false.