new naming

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/NPlus.ma b/matita/contribs/RELATIONAL-ARITHMETICS/NPlus.ma
new file mode 100644 (file)
index 0000000..8531c76
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/NPlus.ma
@@ -0,0 +1,23 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/NPlus".
+
+include "logic/equality.ma".
+
+include "Nat.ma".
+
+inductive NPlus (p:Nat): Nat \to Nat \to Prop \def
+   | NPlus_zero_2: NPlus p zero p
+   | NPlus_succ_2: \forall q, r. NPlus p q r \to NPlus p (succ q) (succ r).

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/NPlus_fwd.ma b/matita/contribs/RELATIONAL-ARITHMETICS/NPlus_fwd.ma
new file mode 100644 (file)
index 0000000..bb73113
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/NPlus_fwd.ma
@@ -0,0 +1,128 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/NPlus_fwd".
+
+include "Nat_fwd.ma".
+include "NPlus.ma".
+
+(* primitive generation lemmas proved by elimination and inversion *)
+
+theorem NPlus_gen_zero_1: \forall q,r. NPlus zero q r \to q = r.
+ intros. elim H; clear H q r; intros;
+ [ reflexivity
+ | clear H1. auto
+ ].
+qed.
+
+theorem NPlus_gen_succ_1: \forall p,q,r. NPlus (succ p) q r \to 
+                          \exists s. r = (succ s) \land NPlus p q s.
+ intros. elim H; clear H q r; intros;
+ [
+ | clear H1.
+   decompose.
+   rewrite > H1. clear H1 n2
+ ]; apply ex_intro; [| auto || auto ]. (**)
+qed.
+
+theorem NPlus_gen_zero_2: \forall p,r. NPlus p zero r \to p = r.
+ intros. inversion H; clear H; intros;
+ [ auto
+ | clear H H1.
+   lapply eq_gen_zero_succ to H2 as H0. apply H0
+ ].
+qed.
+
+theorem NPlus_gen_succ_2: \forall p,q,r. NPlus p (succ q) r \to 
+                          \exists s. r = (succ s) \land NPlus p q s.
+ intros. inversion H; clear H; intros;
+ [ lapply eq_gen_succ_zero to H as H0. apply H0
+ | clear H1 H3 r.
+   lapply linear eq_gen_succ_succ to H2 as H0.
+   rewrite > H0. clear H0 q.
+   apply ex_intro; [| auto ] (**)
+ ].
+qed.
+
+theorem NPlus_gen_zero_3: \forall p,q. NPlus p q zero \to p = zero \land q = zero.
+ intros. inversion H; clear H; intros;
+ [ rewrite < H1. clear H1 p.
+   auto
+ | clear H H1.
+   lapply eq_gen_zero_succ to H3 as H0. apply H0
+ ].
+qed.
+
+theorem NPlus_gen_succ_3: \forall p,q,r. NPlus p q (succ r) \to
+                          \exists s. p = succ s \land NPlus s q r \lor
+                                     q = succ s \land NPlus p s r.
+ intros. inversion H; clear H; intros;
+ [ rewrite < H1. clear H1 p
+ | clear H1.
+   lapply linear eq_gen_succ_succ to H3 as H0.
+   rewrite > H0. clear H0 r.
+ ]; apply ex_intro; [| auto || auto ] (**)
+qed.
+(*
+(* alternative proofs invoking NPlus_gen_2 *)
+
+variant NPlus_gen_zero_3_alt: \forall p,q. NPlus p q zero \to p = zero \land q = zero.
+ intros 2. elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p.
+   auto
+ | clear H.
+   lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   lapply linear eq_gen_zero_succ to H1 as H0. apply H0
+ ].
+qed.
+
+variant NPlus_gen_succ_3_alt: \forall p,q,r. NPlus p q (succ r) \to
+                              \exists s. p = succ s \land NPlus s q r \lor
+                                         q = succ s \land NPlus p s r.
+ intros 2. elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p
+ | clear H.
+   lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   lapply linear eq_gen_succ_succ to H1 as H0.
+   rewrite > H0. clear H0 r.
+ ]; apply ex_intro; [| auto || auto ]. (**)
+qed.
+*)
+(* other simplification lemmas *)
+
+theorem NPlus_gen_eq_2_3: \forall p,q. NPlus p q q \to p = zero.
+ intros 2. elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   lapply linear eq_gen_succ_succ to H2 as H0.
+   rewrite < H0 in H3. clear H0 a
+ ]; auto.
+qed.
+
+theorem NPlus_gen_eq_1_3: \forall p,q. NPlus p q p \to q = zero.
+ intros 1. elim p; clear p; intros;
+ [ lapply linear NPlus_gen_zero_1 to H as H0.
+   rewrite > H0. clear H0 q
+ | lapply linear NPlus_gen_succ_1 to H1 as H0.
+   decompose.
+   lapply linear eq_gen_succ_succ to H2 as H0.
+   rewrite < H0 in H3. clear H0 a
+ ]; auto.
+qed.

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/NPlus_props.ma b/matita/contribs/RELATIONAL-ARITHMETICS/NPlus_props.ma
new file mode 100644 (file)
index 0000000..de0de1b
--- /dev/null
+++ b/matita/contribs/RELATIONAL-ARITHMETICS/NPlus_props.ma
@@ -0,0 +1,107 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/NPlus_props".
+
+include "NPlus_fwd.ma".
+
+theorem NPlus_zero_1: \forall q. NPlus 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).
+ intros 2. elim q; clear q;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   rewrite > H2. clear H2 r
+ ]; auto.
+qed.
+
+theorem NPlus_sym: \forall p,q,r. NPlus p q r \to NPlus q p r.
+ intros 2. elim q; clear q;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   rewrite > H2. clear H2 r
+ ]; auto.
+qed.
+
+theorem NPlus_shift_succ_sx: \forall p,q,r. 
+                             NPlus p (succ q) r \to NPlus (succ p) q r.
+ intros.
+ lapply linear NPlus_gen_succ_2 to H as H0.
+ decompose.
+ rewrite > H1. clear H1 r.
+ auto.
+qed.
+
+theorem NPlus_shift_succ_dx: \forall p,q,r. 
+                             NPlus (succ p) q r \to NPlus p (succ q) r.
+ intros.
+ lapply linear NPlus_gen_succ_1 to H as H0.
+ decompose.
+ rewrite > H1. clear H1 r.
+ 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.
+ intros 2; elim q1; clear q1; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+   lapply linear NPlus_gen_succ_1 to H2 as H0.
+   decompose.
+   rewrite > H2. clear H2 r2.
+   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.
+ intros 2; elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0. clear H0 p1
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+   lapply linear NPlus_gen_succ_2 to H2 as H0.
+   decompose.
+   rewrite > H2. clear H2 r2.
+   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.
+ intros 2. elim q; clear q; intros;
+ [ lapply linear NPlus_gen_zero_2 to H as H0.
+   rewrite > H0 in H1. clear H0 p
+ | lapply linear NPlus_gen_succ_2 to H1 as H0.
+   decompose.
+   rewrite > H3. clear H3 r1.
+   lapply linear NPlus_gen_succ_2 to H2 as H0.
+   decompose.
+   rewrite > H2. clear H2 r2.
+ ]; auto.
+qed.

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/Plus.ma b/matita/contribs/RELATIONAL-ARITHMETICS/Plus.ma
deleted file mode 100644 (file)
index 5b86bf7..0000000
--- a/matita/contribs/RELATIONAL-ARITHMETICS/Plus.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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/Plus".
-
-include "logic/equality.ma".
-
-include "Nat.ma".
-
-inductive Plus (p:Nat): Nat \to Nat \to Prop \def
-   | Plus_zero_2: Plus p zero p
-   | Plus_succ_2: \forall q, r. Plus p q r \to Plus p (succ q) (succ r).

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/Plus_fwd.ma b/matita/contribs/RELATIONAL-ARITHMETICS/Plus_fwd.ma
deleted file mode 100644 (file)
index 8f3e20a..0000000
--- a/matita/contribs/RELATIONAL-ARITHMETICS/Plus_fwd.ma
+++ /dev/null
@@ -1,128 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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/Plus_fwd".
-
-include "Nat_fwd.ma".
-include "Plus.ma".
-
-(* primitive generation lemmas proved by elimination and inversion *)
-
-theorem Plus_gen_zero_1: \forall q,r. Plus zero q r \to q = r.
- intros. elim H; clear H q r; intros;
- [ reflexivity
- | clear H1. auto
- ].
-qed.
-
-theorem Plus_gen_succ_1: \forall p,q,r. Plus (succ p) q r \to 
-                         \exists s. r = (succ s) \land Plus p q s.
- intros. elim H; clear H q r; intros;
- [
- | clear H1.
-   decompose.
-   rewrite > H1. clear H1 n2
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-
-theorem Plus_gen_zero_2: \forall p,r. Plus p zero r \to p = r.
- intros. inversion H; clear H; intros;
- [ auto
- | clear H H1.
-   lapply eq_gen_zero_succ to H2 as H0. apply H0
- ].
-qed.
-
-theorem Plus_gen_succ_2: \forall p,q,r. Plus p (succ q) r \to 
-                         \exists s. r = (succ s) \land Plus p q s.
- intros. inversion H; clear H; intros;
- [ lapply eq_gen_succ_zero to H as H0. apply H0
- | clear H1 H3 r.
-   lapply linear eq_gen_succ_succ to H2 as H0.
-   rewrite > H0. clear H0 q.
-   apply ex_intro; [| auto ] (**)
- ].
-qed.
-
-theorem Plus_gen_zero_3: \forall p,q. Plus p q zero \to p = zero \land q = zero.
- intros. inversion H; clear H; intros;
- [ rewrite < H1. clear H1 p.
-   auto
- | clear H H1.
-   lapply eq_gen_zero_succ to H3 as H0. apply H0
- ].
-qed.
-
-theorem Plus_gen_succ_3: \forall p,q,r. Plus p q (succ r) \to
-                     \exists s. p = succ s \land Plus s q r \lor
-                                q = succ s \land Plus p s r.
- intros. inversion H; clear H; intros;
- [ rewrite < H1. clear H1 p
- | clear H1.
-   lapply linear eq_gen_succ_succ to H3 as H0.
-   rewrite > H0. clear H0 r.
- ]; apply ex_intro; [| auto || auto ] (**)
-qed.
-(*
-(* alternative proofs invoking Plus_gen_2 *)
-
-variant Plus_gen_zero_3_alt: \forall p,q. Plus p q zero \to p = zero \land q = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p.
-   auto
- | clear H.
-   lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   lapply linear eq_gen_zero_succ to H1 as H0. apply H0
- ].
-qed.
-
-variant Plus_gen_succ_3_alt: \forall p,q,r. Plus p q (succ r) \to
-                             \exists s. p = succ s \land Plus s q r \lor
-                                        q = succ s \land Plus p s r.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
- | clear H.
-   lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   lapply linear eq_gen_succ_succ to H1 as H0.
-   rewrite > H0. clear H0 r.
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-*)
-(* other simplification lemmas *)
-
-theorem Plus_gen_eq_2_3: \forall p,q. Plus p q q \to p = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   lapply linear eq_gen_succ_succ to H2 as H0.
-   rewrite < H0 in H3. clear H0 a
- ]; auto.
-qed.
-
-theorem Plus_gen_eq_1_3: \forall p,q. Plus p q p \to q = zero.
- intros 1. elim p; clear p; intros;
- [ lapply linear Plus_gen_zero_1 to H as H0.
-   rewrite > H0. clear H0 q
- | lapply linear Plus_gen_succ_1 to H1 as H0.
-   decompose.
-   lapply linear eq_gen_succ_succ to H2 as H0.
-   rewrite < H0 in H3. clear H0 a
- ]; auto.
-qed.

diff --git a/matita/contribs/RELATIONAL-ARITHMETICS/Plus_props.ma b/matita/contribs/RELATIONAL-ARITHMETICS/Plus_props.ma
deleted file mode 100644 (file)
index b260839..0000000
--- a/matita/contribs/RELATIONAL-ARITHMETICS/Plus_props.ma
+++ /dev/null
@@ -1,107 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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-ARITHMETICsucc/Plus_props".
-
-include "Plus_fwd.ma".
-
-theorem Plus_zero_1: \forall q. Plus zero q q.
- intros. elim q; clear q; auto.
-qed.
-
-theorem Plus_succ_1: \forall p,q,r. Plus p q r \to Plus (succ p) q (succ r).
- intros 2. elim q; clear q;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H2. clear H2 r
- ]; auto.
-qed.
-
-theorem Plus_sym: \forall p,q,r. Plus p q r \to Plus q p r.
- intros 2. elim q; clear q;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H2. clear H2 r
- ]; auto.
-qed.
-
-theorem Plus_shift_succ_sx: \forall p,q,r. 
-                            Plus p (succ q) r \to Plus (succ p) q r.
- intros.
- lapply linear Plus_gen_succ_2 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
-qed.
-
-theorem Plus_shift_succ_dx: \forall p,q,r. 
-                            Plus (succ p) q r \to Plus p (succ q) r.
- intros.
- lapply linear Plus_gen_succ_1 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
-qed.
-
-theorem Plus_trans_1: \forall p,q1,r1. Plus p q1 r1 \to 
-                      \forall q2,r2. Plus r1 q2 r2 \to
-                      \exists q. Plus q1 q2 q \land Plus p q r2.
- intros 2; elim q1; clear q1; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
-   lapply linear Plus_gen_succ_1 to H2 as H0.
-   decompose.
-   rewrite > H2. clear H2 r2.
-   lapply linear H to H4, H3 as H0.
-   decompose.
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-
-theorem Plus_trans_2: \forall p1,q,r1. Plus p1 q r1 \to 
-                      \forall p2,r2. Plus p2 r1 r2 \to
-                      \exists p. Plus p1 p2 p \land Plus p q r2.
- intros 2; elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0. clear H0 p1
- | lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H3. rewrite > H3 in H2. clear H3 r1.
-   lapply linear Plus_gen_succ_2 to H2 as H0.
-   decompose.
-   rewrite > H2. clear H2 r2.
-   lapply linear H to H4, H3 as H0.
-   decompose.
- ]; apply ex_intro; [| auto || auto ]. (**)
-qed.
-
-theorem Plus_conf: \forall p,q,r1. Plus p q r1 \to 
-                   \forall r2. Plus p q r2 \to r1 = r2.
- intros 2. elim q; clear q; intros;
- [ lapply linear Plus_gen_zero_2 to H as H0.
-   rewrite > H0 in H1. clear H0 p
- | lapply linear Plus_gen_succ_2 to H1 as H0.
-   decompose.
-   rewrite > H3. clear H3 r1.
-   lapply linear Plus_gen_succ_2 to H2 as H0.
-   decompose.
-   rewrite > H2. clear H2 r2.
- ]; auto.
-qed.