contribution on relational arithmetics started

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma b/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma
new file mode 100644 (file)
index 0000000..4f95ad4
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/add_defs.ma
@@ -0,0 +1,22 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/add_defs".
+
+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/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma b/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
new file mode 100644 (file)
index 0000000..567bb1f
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/add_props.ma
@@ -0,0 +1,118 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/add_props".
+
+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_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.
+   rewrite > H0. clear H0. clear p
+ | lapply add_gen_S_2 to H1 using H0. clear H1.
+   decompose H0.
+   rewrite > H2. clear H2. clear 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 using H0. clear H.
+   rewrite > H0. clear H0. clear p
+ | lapply add_gen_S_2 to H1 using H0. clear H1.
+   decompose H0.
+   rewrite > H2. clear H2. clear 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 using H0. clear H.
+ decompose H0.
+ rewrite > H1. clear H1. clear 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 using H0. clear H.
+ decompose H0.
+ rewrite > H1. clear H1. clear r.
+ auto.
+qed.
+
+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.
+   rewrite > H0. clear H0. clear p
+ | lapply add_gen_S_2 to H1 using 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.
+   decompose H0.
+   rewrite > H2. clear H2. clear r2.
+   lapply H to H4, H3 using H0. clear H. clear H4. clear H3.
+   decompose H0.
+ ]; 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.
+   rewrite > H0. clear H0. clear p1
+ | lapply add_gen_S_2 to H1 using 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.
+   decompose H0.
+   rewrite > H2. clear H2. clear r2.
+   lapply H to H4, H3 using H0. clear H. clear H4. clear H3. 
+   decompose H0.
+ ]; 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.
+   rewrite > H0 in H1. clear H0. clear p
+ | lapply add_gen_S_2 to H1 using H0. clear H1.
+   decompose H0.
+   rewrite > H3. clear H3. clear r1.
+   lapply add_gen_S_2 to H2 using H0. clear H2.
+   decompose H0.
+   rewrite > H2. clear H2. clear r2.
+ ]; auto.

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/makefile b/matita/contribs/RELATIONAL-ARITHMETICS/makefile
new file mode 100644 (file)
index 0000000..dc30e25
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/makefile
@@ -0,0 +1,33 @@
+H=@
+
+RT_BASEDIR=/home/fguidi/svn/trunk/helm/software/matita/
+OPTIONS=-bench
+MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS)
+CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS) 
+MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS)
+CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS) 
+
+devel:=$(shell basename `pwd`)
+
+all: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel)
+clean: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel)
+cleanall: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all
+
+all.opt opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel)
+clean.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel)
+cleanall.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all
+
+%.mo: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@
+%.mo.opt: preall
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@
+       
+preall:
+       $(H)MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel)
+

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma b/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma
new file mode 100644 (file)
index 0000000..a188208
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/nat_defs.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_defs".
+
+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.
+