X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FRELATIONAL-ARITHMETICS%2Fadd_props.ma;h=ab59c3a516a7846272400b16194e4128511775d4;hb=818e4b4f4812ddb42a4647a205e4d14c0793b6aa;hp=86aa7c1925b1ee6b1e4bd51480ad114fa5816800;hpb=aebd5af5c7d3e422813159d81fde05b6a9778765;p=helm.git

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
index 86aa7c192..ab59c3a51 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_fwd.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.
@@ -41,45 +23,36 @@ qed.
 theorem add_S_1: \forall p,q,r. add p q r \to add (S p) q (S r).
  intros 2. elim q; clear q;
  [ lapply add_gen_O_2 to H as H0. clear H.
-   rewrite > H0. clear H0. clear p
+   rewrite > H0. clear H0 p
  | lapply add_gen_S_2 to H1 as H0. clear H1.
-   decompose H0.
-   rewrite > H2. clear H2. clear r
+   decompose.
+   rewrite > H2. clear H2 r
  ]; auto.
 qed.
 
 theorem add_sym: \forall p,q,r. add p q r \to add q p r.
  intros 2. elim q; clear q;
  [ lapply add_gen_O_2 to H as H0. clear H.
-   rewrite > H0. clear H0. clear p
+   rewrite > H0. clear H0 p
  | lapply add_gen_S_2 to H1 as H0. clear H1.
-   decompose H0.
-   rewrite > H2. clear H2. clear r
+   decompose.
+   rewrite > H2. clear H2 r
  ]; auto.
 qed.
 
 theorem add_shift_S_sx: \forall p,q,r. add p (S q) r \to add (S p) q r.
  intros.
  lapply add_gen_S_2 to H as H0. clear H.
- decompose H0.
- rewrite > H1. clear H1. clear r.
+ decompose.
+ rewrite > H1. clear H1 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.
  lapply add_gen_S_1 to H as H0. clear H.
- decompose H0.
- rewrite > H1. clear H1. clear r.
+ decompose.
+ rewrite > H1. clear H1 r.
  auto.
 qed.
 
@@ -88,15 +61,15 @@ theorem add_trans_1: \forall p,q1,r1. add p q1 r1 \to
                      \exists q. add q1 q2 q \land add p q r2.
  intros 2; elim q1; clear q1; intros;
  [ lapply add_gen_O_2 to H as H0. clear H.
-   rewrite > H0. clear H0. clear p
+   rewrite > H0. clear H0 p
  | lapply add_gen_S_2 to H1 as H0. clear H1.
-   decompose H0.
-   rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
+   decompose.
+   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
    lapply add_gen_S_1 to H2 as H0. clear H2.
-   decompose H0.
-   rewrite > H2. clear H2. clear r2.
-   lapply H to H4, H3 as H0. clear H. clear H4. clear H3.
-   decompose H0.
+   decompose.
+   rewrite > H2. clear H2 r2.
+   lapply H to H4, H3 as H0. clear H H4 H3.
+   decompose.
  ]; apply ex_intro; [| auto || auto ]. (**)
 qed.
 
@@ -105,15 +78,15 @@ theorem add_trans_2: \forall p1,q,r1. add p1 q r1 \to
                      \exists p. add p1 p2 p \land add p q r2.
  intros 2; elim q; clear q; intros;
  [ lapply add_gen_O_2 to H as H0. clear H.
-   rewrite > H0. clear H0. clear p1
+   rewrite > H0. clear H0 p1
  | lapply add_gen_S_2 to H1 as H0. clear H1.
-   decompose H0.
-   rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
+   decompose.
+   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
    lapply add_gen_S_2 to H2 as H0. clear H2.
-   decompose H0.
-   rewrite > H2. clear H2. clear r2.
-   lapply H to H4, H3 as H0. clear H. clear H4. clear H3. 
-   decompose H0.
+   decompose.
+   rewrite > H2. clear H2 r2.
+   lapply H to H4, H3 as H0. clear H H4 H3. 
+   decompose.
  ]; apply ex_intro; [| auto || auto ]. (**)
 qed.
 
@@ -121,12 +94,12 @@ theorem add_conf: \forall p,q,r1. add p q r1 \to
                   \forall r2. add p q r2 \to r1 = r2.
  intros 2. elim q; clear q; intros;
  [ lapply add_gen_O_2 to H as H0. clear H.
-   rewrite > H0 in H1. clear H0. clear p
+   rewrite > H0 in H1. clear H0 p
  | lapply add_gen_S_2 to H1 as H0. clear H1.
-   decompose H0.
-   rewrite > H3. clear H3. clear r1.
+   decompose.
+   rewrite > H3. clear H3 r1.
    lapply add_gen_S_2 to H2 as H0. clear H2.
-   decompose H0.
-   rewrite > H2. clear H2. clear r2.
+   decompose.
+   rewrite > H2. clear H2 r2.
  ]; auto.
 qed.