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.

(* Specifications: *)
Theorem Zlt_bool_spec :
forall x y, Zlt_bool_prop x y (Zlt_bool x y).
Theorem Zlt_bool_true :
forall x y : Z,
(x < y)%Z -> Zlt_bool x y = true.
Theorem Zlt_bool_false :
forall x y : Z,
(y <= x)%Z -> Zlt_bool x y = false.
Theorem negb_Zle_bool :
forall x y : Z,
negb (Zle_bool x y) = Zlt_bool y x.
Theorem negb_Zlt_bool :
forall x y : Z,
negb (Zlt_bool x y) = Zle_bool y x.