From 0b740a1706924024ca4d0ba25ca2dc28cf6f45d6 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Wed, 21 Jun 2006 13:17:36 +0000
Subject: [PATCH] RELATIONAL-ARITHMETICS updated

---
 .../RELATIONAL-ARITHMETICS/add_props.ma       | 47 ++++++++-----------
 .../{nat_props.ma => nat_gen.ma}              |  4 +-
 2 files changed, 22 insertions(+), 29 deletions(-)
 rename helm/software/matita/contribs/RELATIONAL-ARITHMETICS/{nat_props.ma => nat_gen.ma} (95%)

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
index 86aa7c192..e04d8f616 100644
--- a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
+++ b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
@@ -14,26 +14,8 @@
 
 set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/add_props".
 
-include "nat_props.ma".
-include "add_defs.ma".
+include "add_gen.ma".
 
-theorem add_gen_O_2: \forall p,r. add p O r \to p = r.
- intros. inversion H; clear H; intros;
- [ reflexivity
- | lapply eq_gen_O_S to H2 as H0. apply H0
- ].
-qed.
-
-theorem add_gen_S_2: \forall p,q,r. add p (S q) r \to 
-                     \exists s. r = (S s) \land add p q s.
- intros. inversion H; clear H; intros;
- [ lapply eq_gen_S_O to H as H0. apply H0
- | lapply eq_gen_S_S to H2 as H0. clear H2.
-   rewrite > H0. clear H0.
-   apply ex_intro; [| auto ] (**)
- ].
-qed.
-  
 theorem add_O_1: \forall q. add O q q.
  intros. elim q; clear q; auto.
 qed.
@@ -66,14 +48,6 @@ theorem add_shift_S_sx: \forall p,q,r. add p (S q) r \to add (S p) q r.
  auto.
 qed.
 
-theorem add_gen_O_1: \forall q,r. add O q r \to q = r.
- intros. auto.
-qed.
-
-theorem add_gen_S_1: \forall p,q,r. add (S p) q r \to 
-                     \exists s. r = (S s) \land add p q s.
- intros. auto.
-qed.
 
 theorem add_shift_S_dx: \forall p,q,r. add (S p) q r \to add p (S q) r.
  intros.
@@ -130,3 +104,22 @@ theorem add_conf: \forall p,q,r1. add p q r1 \to
    rewrite > H2. clear H2. clear r2.
  ]; auto.
 qed.
+
+
+
+theorem add_gen_eq_2_3: \forall p,q. add p q q \to p = O.
+ intros 2. elim q; clear q; intros;
+ [ lapply add_gen_O_2 to H as H0. clear H.
+   rewrite > H0. clear H0. clear p
+ | lapply add_gen_S_2 to H1 as H0. clear H1.
+   decompose H0.
+   lapply eq_gen_S_S to H2 as H0. clear H2.
+   rewrite < H0 in H3. clear H0. clear a
+ ]; auto.
+qed.
+
+theorem add_gen_eq_1_3: \forall p,q. add p q p \to q = O.
+ intros. 
+ lapply add_sym to H. clear H.
+ auto.
+qed.
diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_gen.ma
similarity index 95%
rename from helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma
rename to helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_gen.ma
index 3e5dd5220..de3eb4487 100644
--- a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma
+++ b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_gen.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/nat_props".
+set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/nat_gen".
 
 include "library/logic/equality.ma".
 include "nat_defs.ma".
@@ -27,4 +27,4 @@ qed.
 
 theorem eq_gen_S_S: \forall m1,m2. S m1 = S m2 \to m1 = m2.
  intros. injection H. assumption.
-qed.
\ No newline at end of file
+qed.
-- 
2.39.2