X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FRELATIONAL%2FNLE%2Fprops.ma;h=a6d0ca7a88dbac1fb0b3232c984ffe5a5c0c67c6;hb=b0b69bb764af48d7f022477398f0beff5fb3f3f3;hp=066e056e0435f74c01f0dc43970bbce5a2575994;hpb=648f1521fa36f3c19cca3b3a29bbf3e8146eca4e;p=helm.git

diff --git a/matita/contribs/RELATIONAL/NLE/props.ma b/matita/contribs/RELATIONAL/NLE/props.ma
index 066e056e0..a6d0ca7a8 100644
--- a/matita/contribs/RELATIONAL/NLE/props.ma
+++ b/matita/contribs/RELATIONAL/NLE/props.ma
@@ -14,23 +14,65 @@
 
 set "baseuri" "cic:/matita/RELATIONAL/NLE/props".
 
-include "NLE/fwd.ma".
+include "NLE/inv.ma".
+
+theorem nle_zero_1: \forall q. zero <= q.
+ intros. auto.
+qed.
+
+theorem nle_succ_succ: \forall p,q. p <= q \to succ p <= succ q.
+ intros. elim H. clear H. auto.
+qed.
+
+theorem nle_ind: \forall P:(Nat \to Nat \to Prop).
+                 (\forall n:Nat.P zero n) \to 
+                 (\forall n,n1:Nat.
+                  n <= n1 \to P n n1 \to P (succ n) (succ n1)
+                 ) \to
+                \forall n,n1:Nat. n <= n1 \to P n n1.
+ intros 4. elim n; clear n;
+ [ auto
+ | lapply linear nle_inv_succ_1 to H3. decompose; subst.
+   auto
+ ].
+qed.
 
 theorem nle_refl: \forall x. x <= x.
- intros 1; elim x; clear x; intros; auto new.
+ intros 1; elim x; clear x; intros; auto.
 qed.
 
 theorem nle_trans_succ: \forall x,y. x <= y \to x <= succ y.
- intros. elim H; clear H x y; intros; auto new.
+ intros. elim H using nle_ind; clear H x y; auto.
+qed.
+
+theorem nle_false: \forall x,y. x <= y \to y < x \to False.
+ intros 3; elim H using nle_ind; clear H x y; auto.
+qed.
+
+theorem nle_trans: \forall x,y. x <= y \to
+                   \forall z. y <= z \to x <= z.
+ intros 3. elim H using nle_ind; clear H x y;
+ [ auto
+ | lapply linear nle_inv_succ_1 to H3. decompose. subst. 
+   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 new
- | lapply linear nle_gen_succ_2 to H1. decompose;
-   [ subst. auto new
-   | lapply linear H to H3 as H0. decompose;
-     subst; auto new
+ intros. elim H using nle_ind; clear H x y;
+ [ elim n; clear n; auto
+ | decompose; subst; auto
+ ].
+qed.
+
+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. decompose.
+     subst. auto depth=4
+   | lapply linear nle_lt_or_eq to H1; decompose;
+     subst; auto
    ]
  ].
 qed.