proposition: Lemma ap_plus : forall a, ap a -> forall b c, nf b -> nf c -> lt b a -> lt c a -> lt (b + c) a.
proof: Proof. destruct 1. intro b; elim b. - intro c; elim c;intros. + simpl; auto with T1. + simpl; auto. - intros c H0 n t H c0 H1 H2 H3 H4; generalize c0 H2 H4. destruct c1. + rewrite (plus_zero_r);auto. + intros H5 H6; case (lt_eq_lt_dec c c1_1). * destruct 1. { rewrite (plus_cons_cons_rw1 n t n0 c1_2 l); auto. } subst c1_1; rewrite (plus_cons_cons_rw2 H1 H5). apply lt_a_phi0_b_inv in H6. auto with T1. * intro H7; rewrite plus_cons_cons_rw3;auto. apply lt_a_phi0_b_inv in H3; auto with T1. Qed.
imports: Coq(Arith Max Bool Lia Compare_dec Relations Ensembles Wellfounded Bool RelationClasses Operators_Properties ArithRing Logic.Eqdep_dec) hydras(More_Arith Restriction DecPreOrder) hydras(OrdNotations)