RELATIONAL-ARITHMETICS updated

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
index 86aa7c1925b1ee6b1e4bd51480ad114fa5816800..e04d8f616d2a26b30e8d7bda6cfa29d4e87407f7 100644 (file)
--- 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_gen.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_gen.ma
new file mode 100644 (file)
index 0000000..de3eb44
--- /dev/null
+++ b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_gen.ma
@@ -0,0 +1,30 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/nat_gen".
+
+include "library/logic/equality.ma".
+include "nat_defs.ma".
+
+theorem eq_gen_O_S: \forall (P:Prop). \forall m2. O = S m2 \to P.
+ intros. discriminate H.
+qed.
+
+theorem eq_gen_S_O: \forall (P:Prop). \forall m1. S m1 = O \to P.
+ intros. discriminate H.
+qed.
+
+theorem eq_gen_S_S: \forall m1,m2. S m1 = S m2 \to m1 = m2.
+ intros. injection H. assumption.
+qed.

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma
deleted file mode 100644 (file)
index 3e5dd52..0000000
--- a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma
+++ /dev/null
@@ -1,30 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/nat_props".
-
-include "library/logic/equality.ma".
-include "nat_defs.ma".
-
-theorem eq_gen_O_S: \forall (P:Prop). \forall m2. O = S m2 \to P.
- intros. discriminate H.
-qed.
-
-theorem eq_gen_S_O: \forall (P:Prop). \forall m1. S m1 = O \to P.
- intros. discriminate H.
-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