X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FRELATIONAL-ARITHMETICS%2Fadd_props.ma;h=86aa7c1925b1ee6b1e4bd51480ad114fa5816800;hb=ad82cd48e0083ff34cbde75b2aa46891e2c2893c;hp=567bb1fed7b1c4f56a7c24e305813fa68f6f8af7;hpb=110578d269f4ced34c4317eed5171a7f45884c15;p=helm.git

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma b/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
index 567bb1fed..86aa7c192 100644
--- a/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
@@ -14,22 +14,35 @@
 
 set "baseuri" "cic:/matita/RELATIONAL-ARITHMETICS/add_props".
 
+include "nat_props.ma".
 include "add_defs.ma".
 
-axiom add_gen_O_2: \forall p,r. add p O r \to p = r.
-
-axiom add_gen_S_2: \forall p,q,r. add p (S q) r \to 
-                   \exists s. r = (S s) \land add p q s.
+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.
 
 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 using H0. clear H.
+ [ lapply add_gen_O_2 to H as H0. clear H.
    rewrite > H0. clear H0. clear p
- | lapply add_gen_S_2 to H1 using H0. clear H1.
+ | lapply add_gen_S_2 to H1 as H0. clear H1.
    decompose H0.
    rewrite > H2. clear H2. clear r
  ]; auto.
@@ -37,9 +50,9 @@ 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 using H0. clear H.
+ [ lapply add_gen_O_2 to H as H0. clear H.
    rewrite > H0. clear H0. clear p
- | lapply add_gen_S_2 to H1 using H0. clear H1.
+ | lapply add_gen_S_2 to H1 as H0. clear H1.
    decompose H0.
    rewrite > H2. clear H2. clear r
  ]; auto.
@@ -47,7 +60,7 @@ 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 using H0. clear H.
+ lapply add_gen_S_2 to H as H0. clear H.
  decompose H0.
  rewrite > H1. clear H1. clear r.
  auto.
@@ -64,7 +77,7 @@ 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 using H0. clear H.
+ lapply add_gen_S_1 to H as H0. clear H.
  decompose H0.
  rewrite > H1. clear H1. clear r.
  auto.
@@ -74,45 +87,46 @@ theorem add_trans_1: \forall p,q1,r1. add p q1 r1 \to
                      \forall q2,r2. add r1 q2 r2 \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 using H0. clear H.
+ [ lapply add_gen_O_2 to H as H0. clear H.
    rewrite > H0. clear H0. clear p
- | lapply add_gen_S_2 to H1 using H0. clear H1.
+ | lapply add_gen_S_2 to H1 as H0. clear H1.
    decompose H0.
    rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
-   lapply add_gen_S_1 to H2 using H0. clear H2.
+   lapply add_gen_S_1 to H2 as H0. clear H2.
    decompose H0.
    rewrite > H2. clear H2. clear r2.
-   lapply H to H4, H3 using H0. clear H. clear H4. clear H3.
+   lapply H to H4, H3 as H0. clear H. clear H4. clear H3.
    decompose H0.
- ]; apply ex_intro; [| auto || auto ].
+ ]; apply ex_intro; [| auto || auto ]. (**)
 qed.
 
 theorem add_trans_2: \forall p1,q,r1. add p1 q r1 \to 
                      \forall p2,r2. add p2 r1 r2 \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 using H0. clear H.
+ [ lapply add_gen_O_2 to H as H0. clear H.
    rewrite > H0. clear H0. clear p1
- | lapply add_gen_S_2 to H1 using H0. clear H1.
+ | lapply add_gen_S_2 to H1 as H0. clear H1.
    decompose H0.
    rewrite > H3. rewrite > H3 in H2. clear H3. clear r1.
-   lapply add_gen_S_2 to H2 using H0. clear H2.
+   lapply add_gen_S_2 to H2 as H0. clear H2.
    decompose H0.
    rewrite > H2. clear H2. clear r2.
-   lapply H to H4, H3 using H0. clear H. clear H4. clear H3. 
+   lapply H to H4, H3 as H0. clear H. clear H4. clear H3. 
    decompose H0.
- ]; apply ex_intro; [| auto || auto ].
+ ]; apply ex_intro; [| auto || auto ]. (**)
 qed.
 
 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 using H0. clear H.
+ [ lapply add_gen_O_2 to H as H0. clear H.
    rewrite > H0 in H1. clear H0. clear p
- | lapply add_gen_S_2 to H1 using H0. clear H1.
+ | lapply add_gen_S_2 to H1 as H0. clear H1.
    decompose H0.
    rewrite > H3. clear H3. clear r1.
-   lapply add_gen_S_2 to H2 using H0. clear H2.
+   lapply add_gen_S_2 to H2 as H0. clear H2.
    decompose H0.
    rewrite > H2. clear H2. clear r2.
  ]; auto.
+qed.