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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_gen".
+
+include "nat_gen.ma".
+include "add_defs.ma".
+
+(* primitive generation lemmas proved by elimination and inversion *)
+
+theorem add_gen_O_1: \forall q,r. add O q r \to q = r.
+  intros. elim H; clear H; clear q; clear r; intros;
+ [ reflexivity
+ | clear H1. 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. elim H; clear H; clear q; clear r; intros;
+ [
+ | clear H1. 
+  decompose H2.
+  rewrite > H1. clear H1. clear n2
+ ]; apply ex_intro; [| auto || auto ]. (**)
+qed.
+
+theorem add_gen_O_2: \forall p,r. add p O r \to p = r.
+ intros. inversion H; clear H; intros;
+ [ auto
+ | clear H. clear H1.
+   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
+ | clear H1. clear H3. clear r.
+   lapply eq_gen_S_S to H2 as H0. clear H2.
+   rewrite > H0. clear H0. clear q.
+   apply ex_intro; [| auto ] (**)
+ ].
+qed.
+
+theorem add_gen_O_3: \forall p,q. add p q O \to p = O \land q = O.
+ intros. inversion H; clear H; intros;
+ [ rewrite < H1. clear H1. clear p.
+   auto
+ | clear H. clear H1.
+   lapply eq_gen_O_S to H3 as H0. apply H0
+ ].
+qed.
+
+theorem add_gen_S_3: \forall p,q,r. add p q (S r) \to
+                     \exists s. p = S s \land add s q r \lor
+                                q = S s \land add p s r.
+ intros. inversion H; clear H; intros;
+ [ rewrite < H1. clear H1. clear p
+ | clear H1.
+   lapply eq_gen_S_S to H3 as H0. clear H3.
+   rewrite > H0. clear H0. clear r.
+ ]; apply ex_intro; [| auto || auto ] (**)
+qed.
+
+(* alternative proofs invoking add_gen_2 *)
+
+variant add_gen_O_3_alt: \forall p,q. add p q O \to p = O \land q = O.
+ intros 2. elim q; clear q; intros;
+ [ lapply add_gen_O_2 to H as H0. clear H.
+   rewrite > H0. clear H0. clear p.
+   auto
+ | clear H.
+   lapply add_gen_S_2 to H1 as H0. clear H1.
+   decompose H0.
+   lapply eq_gen_O_S to H1 as H0. apply H0
+ ].
+qed.
+
+variant add_gen_S_3_alt: \forall p,q,r. add p q (S r) \to
+                         \exists s. p = S s \land add s q r \lor
+                                    q = S s \land add p s r.
+ intros 2. elim q; clear q; intros;
+ [ lapply add_gen_O_2 to H as H0. clear H.
+   rewrite > H0. clear H0. clear p
+ | clear H.
+   lapply add_gen_S_2 to H1 as H0. clear H1.
+   decompose H0.
+   lapply eq_gen_S_S to H1 as H0. clear H1.
+   rewrite > H0. clear H0. clear r.
+ ]; apply ex_intro; [| auto || auto ]. (**)
+qed.