proposition: Lemma ing_nring : forall n : nat, nring n [=] inject_Q (nring n).
proof: Proof. intros. apply not_ap_imp_eq; intro Hap. elim Hap; intro Hlt; elim Hlt; intros N HN; elim HN; clear Hap Hlt HN; intros e He HN. apply (less_irreflexive_unfolded F [0]). apply less_leEq_trans with e; auto. eapply leEq_wdr. apply (HN N); auto. apply x_minus_x; apply eq_symmetric_unfolded; apply ing_nring'. apply (less_irreflexive_unfolded F [0]). apply less_leEq_trans with e; auto. eapply leEq_wdr. apply (HN N); auto. apply x_minus_x; apply ing_nring'. Qed.
imports: CoRN.algebra.Cauchy_COF CoRN.reals.CReals