X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FRELATIONAL%2FNPlus%2Fprops.ma;h=5e7cc1b7f592c8bca10f3516ad9d1398cf4fb5c3;hb=45d71beffd253ffd767a9afbfcec5c4f44afd8a8;hp=56146a3bf602d702fe969329930a23f70db6588d;hpb=c86f1a6c4d00f66a81c576589f215cb667595710;p=helm.git

diff --git a/matita/contribs/RELATIONAL/NPlus/props.ma b/matita/contribs/RELATIONAL/NPlus/props.ma
index 56146a3bf..5e7cc1b7f 100644
--- a/matita/contribs/RELATIONAL/NPlus/props.ma
+++ b/matita/contribs/RELATIONAL/NPlus/props.ma
@@ -16,92 +16,83 @@ set "baseuri" "cic:/matita/RELATIONAL/NPlus/props".
 
 include "NPlus/fwd.ma".
 
-theorem nplus_zero_1: \forall q. NPlus zero q q.
+theorem nplus_zero_1: \forall q. zero + q == q.
  intros. elim q; clear q; auto.
 qed.
 
-theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to NPlus (succ p) q (succ r).
+theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to 
+                      (succ p) + q == (succ r).
  intros 2. elim q; clear q;
  [ lapply linear nplus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
+   subst
  | lapply linear nplus_gen_succ_2 to H1 as H0.
    decompose.
-   rewrite > H2. clear H2 r
+   subst
  ]; auto.
 qed.
 
-theorem nplus_sym: \forall p,q,r. NPlus p q r \to NPlus q p r.
+theorem nplus_sym: \forall p,q,r. (p + q == r) \to q + p == r.
  intros 2. elim q; clear q;
  [ lapply linear nplus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
+   subst
  | lapply linear nplus_gen_succ_2 to H1 as H0.
    decompose.
-   rewrite > H2. clear H2 r
+   subst
  ]; auto.
 qed.
 
 theorem nplus_shift_succ_sx: \forall p,q,r. 
-                             NPlus p (succ q) r \to NPlus (succ p) q r.
+                             (p + (succ q) == r) \to (succ p) + q == r.
  intros.
  lapply linear nplus_gen_succ_2 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
+ decompose. subst. auto.
 qed.
 
 theorem nplus_shift_succ_dx: \forall p,q,r. 
-                             NPlus (succ p) q r \to NPlus p (succ q) r.
+                             ((succ p) + q == r) \to p + (succ q) == r.
  intros.
  lapply linear nplus_gen_succ_1 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
+ decompose. subst. auto.
 qed.
 
-theorem nplus_trans_1: \forall p,q1,r1. NPlus p q1 r1 \to 
-                       \forall q2,r2. NPlus r1 q2 r2 \to
-                       \exists q. NPlus q1 q2 q \land NPlus p q r2.
+theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to 
+                       \forall q2,r2. (r1 + q2 == r2) \to
+                       \exists q. (q1 + q2 == q) \land p + q == r2.
  intros 2; elim q1; clear q1; intros;
  [ lapply linear nplus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
+   subst.
  | lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+   decompose. subst.
    lapply linear nplus_gen_succ_1 to H2 as H0.
-   decompose.
-   rewrite > H2. clear H2 r2.
+   decompose. subst.
    lapply linear H to H4, H3 as H0.
    decompose.
  ]; apply ex_intro; [| auto || auto ]. (**)
 qed.
 
-theorem nplus_trans_2: \forall p1,q,r1. NPlus p1 q r1 \to 
-                       \forall p2,r2. NPlus p2 r1 r2 \to
-                       \exists p. NPlus p1 p2 p \land NPlus p q r2.
+theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to 
+                       \forall p2,r2. (p2 + r1 == r2) \to
+                       \exists p. (p1 + p2 == p) \land p + q == r2.
  intros 2; elim q; clear q; intros;
  [ lapply linear nplus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p1
+   subst
  | lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+   decompose. subst.
    lapply linear nplus_gen_succ_2 to H2 as H0.
-   decompose.
-   rewrite > H2. clear H2 r2.
+   decompose. subst.
    lapply linear H to H4, H3 as H0.
    decompose.
  ]; apply ex_intro; [| auto || auto ]. (**)
 qed.
 
-theorem nplus_conf: \forall p,q,r1. NPlus p q r1 \to 
-                    \forall r2. NPlus p q r2 \to r1 = r2.
+theorem nplus_conf: \forall p,q,r1. (p + q == r1) \to 
+                    \forall r2. (p + q == r2) \to r1 = r2.
  intros 2. elim q; clear q; intros;
  [ lapply linear nplus_gen_zero_2 to H as H0.
-   rewrite > H0 in H1. clear H0 p
+   subst
  | lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H3. clear H3 r1.
+   decompose. subst.
    lapply linear nplus_gen_succ_2 to H2 as H0.
-   decompose.
-   rewrite > H2. clear H2 r2.
+   decompose. subst.
  ]; auto.
 qed.