proposition: Lemma oplus_bounded_phi0 alpha beta gamma : nf alpha -> nf beta -> nf gamma -> lt alpha (phi0 gamma) -> lt beta (phi0 gamma) -> lt (alpha o+ beta) (phi0 gamma).
proof: Proof. intros H H0 H1 H2 H3. apply lt_a_phi0_b_phi0; auto. apply lt_a_phi0_b_oplus; auto. eapply lt_a_phi0_b_intro with 0; auto. eapply nf_intro;auto. 1,2 :now apply lt_a_phi0_b_phi0R. Qed.
imports: Coq(Arith ArithRing Lia) hydras(Prelude.More_Arith Prelude.Merge_Sort Epsilon0.T1)