axioms removed

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma
index 4f95ad4e85e4c6daaa8d979c389d82fe34a1e099..77d6921a97a943a722bb1420c16f2ddcde33e3c9 100644 (file)
--- a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma
+++ b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma
@@ -19,4 +19,3 @@ include "nat_defs.ma".
 inductive add (p:nat): nat \to nat \to Prop \def
    | add_O_2: add p O p
    | add_S_2: \forall q, r. add p q r \to add p (S q) (S r).
-

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
index 567bb1fed7b1c4f56a7c24e305813fa68f6f8af7..86aa7c1925b1ee6b1e4bd51480ad114fa5816800 100644 (file)
--- a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
+++ b/helm/software/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.

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma
index a188208f2f76c69c35c6d2fde0b3a4c02e2c5918..a78886780fa09cb9f2b4f251110f166f4fa1dd98 100644 (file)
--- a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma
+++ b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma
@@ -19,7 +19,3 @@ include "library/logic/equality.ma".
 inductive nat: Set \def
    | O: nat
    | S: nat \to nat.
-
-theorem eq_gen_O_S: \forall (P:Prop). \forall m2. O = S m2 \to P.
- intros. inversion H.
-

diff --git a/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.ma
new file mode 100644 (file)
index 0000000..3e5dd52
--- /dev/null
+++ b/helm/software/matita/contribs/RELATIONAL-ARITHMETICS/nat_props.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_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