proposition: Lemma pos_up_mod_3 s : (s < 4)%positive -> (Z.pred (Zpos (pos_up s)) mod 3 = (1 + Z.pred (Z.pos s)) mod 3)%Z.
proof: Proof. case s; simpl; auto; intros p; case p; simpl; auto; intros p1; case p1; unfold Pos.lt; intros; discriminate. Qed.
imports: ZArith List