include "NLE/fwd.ma".
+theorem nle_zero: \forall q. zero <= q.
+ unfold NLE.
+ intros. apply ex_intro; auto. (**)
+qed.
+
+theorem nle_succ: \forall p,q. p <= q \to succ p <= succ q.
+ unfold NLE.
+ intros. decompose.
+ apply ex_intro; auto. (**)
+qed.
+
theorem nle_refl: \forall x. x <= x.
intros 1; elim x; clear x; intros; auto new timeout=100.
qed.
theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
- intros. elim H; clear H x y; intros; auto new timeout=100.
+ intros 1. elim x; clear x; intros;
+ [ auto new timeout=100.
+ | lapply linear nle_gen_succ_1 to H1 as H0. decompose H0. subst.
+ auto new timeout=100.
+ ].
qed.
theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.