include "logic/connectives.ma".
-include "NPlus/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.
- unfold NLE.
- intros. decompose.
- lapply linear nplus_gen_succ_2 to H1 as H.
- decompose. subst.
- apply ex_intro; auto. (**)
+theorem nle_inv_succ_1: \forall x,y. x < y \to
+ \exists z. y = succ z \land x <= z.
+ intros. elim H.
+ lapply linear nplus_gen_succ_2 to H1.
+ decompose. subst. auto depth = 4.
qed.
-
-theorem nle_gen_succ_succ: \forall x,y. x < succ y \to x <= y.
+theorem nle_inv_succ_succ: \forall x,y. x < succ y \to x <= y.
intros.
- lapply linear nle_gen_succ_1 to H as H0. decompose H0.
- lapply linear eq_gen_succ_succ to H1 as H. subst.
+ lapply linear nle_inv_succ_1 to H. decompose.
+ lapply linear eq_gen_succ_succ to H1. subst.
auto.
qed.
-theorem nle_gen_succ_zero: \forall (P:Prop). \forall x. x < zero \to P.
+theorem nle_inv_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 nle_inv_succ_1 to H. decompose.
+ lapply linear eq_gen_zero_succ to H1. decompose.
qed.
-theorem nle_gen_zero_2: \forall x. x <= zero \to x = zero.
+theorem nle_inv_zero_2: \forall x. x <= zero \to x = zero.
intros 1. elim x; clear x; intros;
- [ auto new timeout=30
- | apply (nle_gen_succ_zero ? ? H1)
+ [ auto
+ | lapply linear nle_inv_succ_zero to H1. decompose.
].
qed.
-theorem nle_gen_succ_2: \forall y,x. x <= succ y \to
+theorem nle_inv_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 timeout=30
- | lapply linear nle_gen_succ_succ to H1.
- right. apply ex_intro; [|auto new timeout=30] (**)
+ [ auto
+ | lapply linear nle_inv_succ_succ to H1.
+ auto depth = 4.
].
qed.