set "baseuri" "cic:/matita/RELATIONAL/NLE/props".
-include "NLE/fwd.ma".
-
-theorem nle_refl: \forall x. x <= x.
- intros 1; elim x; clear x; intros; auto.
-qed.
+include "NLE/order.ma".
theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
- intros. elim H; clear H x y; intros; auto.
+ intros. elim H; clear H x y; auto.
qed.
-theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.
- intros 1. elim y; clear y; intros;
- [ lapply linear nle_gen_zero_2 to H. auto
- | lapply linear nle_gen_succ_2 to H1. decompose;
- [ rewrite > H1. clear H1. auto
- | lapply linear H to H3 as H0. decompose;
- [ rewrite > H1. clear H1 x. auto
- | rewrite < H. clear H n. auto
- ]
- ]
+theorem nle_gt_or_le: \forall x, y. y > x \lor y <= x.
+ intros 2; elim y; clear y;
+ [ auto
+ | decompose;
+ [ lapply linear nle_inv_succ_1 to H1
+ | lapply linear nle_lt_or_eq to H1
+ ]; decompose; subst; auto depth = 4
].
qed.