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] (**)
- ].
+ 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
- ].
+ 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.
+theorem nle_gen_succ_zero: \forall x. x < zero \to False.
intros.
lapply linear nle_gen_succ_1 to H. decompose.
- apply (eq_gen_zero_succ ? ? H1).
+ lapply linear eq_gen_zero_succ to H1. decompose.
qed.
theorem nle_gen_zero_2: \forall x. x <= zero \to x = zero.
intros 1. elim x; clear x; intros;
- [ auto new
- | apply (nle_gen_succ_zero ? ? H1)
+ [ auto new timeout=30
+ | lapply linear nle_gen_succ_zero to H1. decompose.
].
qed.
theorem nle_gen_succ_2: \forall y,x. x <= succ y \to
x = zero \lor \exists z. x = succ z \land z <= y.
intros 2; elim x; clear x; intros;
- [ auto new
+ [ auto new timeout=30
| lapply linear nle_gen_succ_succ to H1.
- right. apply ex_intro; [|auto new] (**)
+ right. apply ex_intro; [|auto new timeout=30] (**)
].
qed.