removed the impredicativity of falsum

[helm.git] / helm / software / matita / contribs / RELATIONAL / NLE / fwd.ma

diff --git a/helm/software/matita/contribs/RELATIONAL/NLE/fwd.ma b/helm/software/matita/contribs/RELATIONAL/NLE/fwd.ma
index 7d8a9e3b3528b127769d3d9e130401baf3453554..5ddeb41aff1b1efe28b47c639310f49cd8373f50 100644 (file)
--- a/helm/software/matita/contribs/RELATIONAL/NLE/fwd.ma
+++ b/helm/software/matita/contribs/RELATIONAL/NLE/fwd.ma
@@ -16,46 +16,44 @@ set "baseuri" "cic:/matita/RELATIONAL/NLE/fwd".
 
 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.