proposition: Lemma inj_abs_neg : forall x : Z, (x < 0)%Z -> Z_of_nat (Z.abs_nat x) = (- x)%Z.
proof: Proof. intros. rewrite abs_opp. apply inj_abs_pos. apply Z.le_ge. apply Zplus_le_reg_l with x. rewrite <- Zplus_0_r_reverse. rewrite Zplus_opp_r. apply Zlt_le_weak. assumption. Qed.
imports: ZArith EqNat lemmas