Section Zcompare.
Inductive Zcompare_prop (x y : Z) : comparison -> Prop :=
| Zcompare_Lt_ : (x < y)%Z -> Zcompare_prop x y Lt
| Zcompare_Eq_ : x = y -> Zcompare_prop x y Eq
| Zcompare_Gt_ : (y < x)%Z -> Zcompare_prop x y Gt.

Theorem Zcompare_Lt :
forall x y,
(x < y)%Z -> Z.compare x y = Lt.