include "logic/connectives.ma".
-include "Nat/fwd.ma".
+include "NPlus/fwd.ma".
include "NLE/defs.ma".
theorem nle_gen_succ_1: \forall x,y. x < y \to
\exists z. y = succ z \land x <= z.
- intros. inversion H; clear H; intros;
- [ apply (eq_gen_succ_zero ? ? H)
- | lapply linear eq_gen_succ_succ to H2 as H0.
- subst.
- apply ex_intro; [|auto new timeout=30] (**)
- ].
+ unfold NLE.
+ intros. decompose.
+ lapply linear nplus_gen_succ_2 to H1 as H.
+ decompose. subst.
+ apply ex_intro; auto. (**)
qed.
+
theorem nle_gen_succ_succ: \forall x,y. x < succ y \to x <= y.
- intros; inversion H; clear H; intros;
- [ apply (eq_gen_succ_zero ? ? H)
- | lapply linear eq_gen_succ_succ to H2 as H0.
- lapply linear eq_gen_succ_succ to H3 as H2.
- subst. auto new timeout=30
- ].
+ intros.
+ lapply linear nle_gen_succ_1 to H as H0. decompose H0.
+ lapply linear eq_gen_succ_succ to H1 as H. subst.
+ auto.
qed.
theorem nle_gen_succ_zero: \forall (P:Prop). \forall x. x < zero \to P.