proposition: Lemma abs_neq_lt : forall x : Z, x <> 0%Z -> 0 < Z.abs_nat x.
proof: Proof. simple induction x. intro. elim H. reflexivity. intros. change (Z.abs_nat 0 < Z.abs_nat (Zpos p)) in |- *. apply ltzlt. unfold Z.le in |- *. simpl in |- *. discriminate. unfold Z.le in |- *. simpl in |- *. discriminate. unfold Z.lt in |- *. simpl in |- *. reflexivity. intros. change (Z.abs_nat 0 < Z.abs_nat (Zpos p)) in |- *. apply ltzlt. unfold Z.le in |- *. simpl in |- *. discriminate. unfold Z.le in |- *. simpl in |- *. discriminate. unfold Z.lt in |- *. simpl in |- *. reflexivity. Qed.
imports: ZArith EqNat lemmas