From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 19 Feb 2007 14:11:14 +0000 (+0000)
Subject: Unified-Sub: lift_comm completed
X-Git-Tag: 0.4.95@7852~591
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=b0b69bb764af48d7f022477398f0beff5fb3f3f3;p=helm.git

Unified-Sub: lift_comm completed
---

diff --git a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma
index d8ee64fe5..b8a4e2a85 100644
--- a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma
+++ b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/defs.ma
@@ -22,12 +22,12 @@ include "datatypes/Term.ma".
 
 inductive Lift (l: Nat): Nat \to Term \to Term \to Prop \def
    | lift_sort   : \forall i,h. 
-                   Lift l i (leaf (sort h)) (leaf (sort h))
+                   Lift l i (sort h) (sort h)
    | lift_lref_gt: \forall i,j. i > j \to 
-                   Lift l i (leaf (lref j)) (leaf (lref j))
+                   Lift l i (lref j) (lref j)
    | lift_lref_le: \forall i,j1. i <= j1 \to
                    \forall j2. (l + j1  == j2) \to
-                   Lift l i (leaf (lref j1)) (leaf (lref j2))
+                   Lift l i (lref j1) (lref j2)
    | lift_bind   : \forall i,u1,u2. Lift l i u1 u2 \to
                    \forall t1,t2. Lift l (succ i) t1 t2 \to 
                    \forall r. Lift l i (intb r u1 t1) (intb r u2 t2)
diff --git a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma
index fd794fe9b..cfddffd3d 100644
--- a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma
+++ b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/fun.ma
@@ -17,18 +17,26 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Unified-Sub/Lift/fun".
 include "Lift/inv.ma".
 
 (* Functional properties ****************************************************)
-(*
+
 theorem lift_total: \forall l, t, i. \exists u. Lift l i t u.
  intros 2. elim t; clear t;
- [ 
-*)
+ [ auto
+ | lapply (nle_gt_or_le n i). decompose;
+   [ auto
+   | lapply (nplus_total l n). decompose. auto
+   ]
+ | lapply (H i1). lapply (H1 (succ i1)). decompose. auto
+ | lapply (H i1). lapply (H1 i1). decompose. auto  
+ ].
+qed.
+
 theorem lift_mono: \forall l,i,t,t1. Lift l i t t1 \to
                    \forall t2. Lift l i t t2 \to
                    t1 = t2.
  intros 5. elim H; clear H i t t1;
  [ lapply linear lift_inv_sort_1 to H1
  | lapply linear lift_inv_lref_1_gt to H2, H1
- | lapply linear lift_inv_lref_1_le to H3, H1, H2
+ | lapply linear lift_inv_lref_1_le_nplus to H3, H1, H2
  | lapply linear lift_inv_bind_1 to H5. decompose
  | lapply linear lift_inv_flat_1 to H5. decompose
  ]; subst; auto.
@@ -40,9 +48,9 @@ theorem lift_inj: \forall l,i,t1,t. Lift l i t1 t \to
  intros 5. elim H; clear H i t1 t;
  [ lapply linear lift_inv_sort_2 to H1
  | lapply linear lift_inv_lref_2_gt to H2, H1
- | lapply nle_nplus to H2 as H. (**)
+ | lapply nle_nplus to H2 as H.
    lapply linear nle_trans to H1, H as H0.
-   lapply lift_inv_lref_2_le to H3, H0, H2
+   lapply lift_inv_lref_2_le_nplus to H3, H0, H2
  | lapply linear lift_inv_bind_2 to H5. decompose
  | lapply linear lift_inv_flat_2 to H5. decompose
  ]; subst; auto.
diff --git a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
index d664e5f07..a4b218f44 100644
--- a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
+++ b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/inv.ma
@@ -19,8 +19,8 @@ include "Lift/defs.ma".
 (* Inversion properties *****************************************************)
 
 theorem lift_inv_sort_1: \forall l, i, h, x.
-                         Lift l i (leaf (sort h)) x \to
-                         x = leaf (sort h).
+                         Lift l i (sort h) x \to
+                         x = sort h.
  intros. inversion H; clear H; intros;
  [ auto
  | destruct H2
@@ -31,10 +31,10 @@ theorem lift_inv_sort_1: \forall l, i, h, x.
 qed.
 
 theorem lift_inv_lref_1: \forall l, i, j1, x.
-                         Lift l i (leaf (lref j1)) x \to
-                         (i > j1 \land x = leaf (lref j1)) \lor
+                         Lift l i (lref j1) x \to
+                         (i > j1 \land x = lref j1) \lor
                          (i <= j1 \land 
-                          \exists j2. (l + j1 == j2) \land x = leaf (lref j2)
+                          \exists j2. (l + j1 == j2) \land x = lref j2
                          ).
  intros. inversion H; clear H; intros;
  [ destruct H1
@@ -76,8 +76,8 @@ theorem lift_inv_flat_1: \forall l, i, r, u1, t1, x.
 qed.
 
 theorem lift_inv_sort_2: \forall l, i, x, h.
-                         Lift l i x (leaf (sort h)) \to
-                         x = leaf (sort h).
+                         Lift l i x (sort h) \to
+                         x = sort h.
  intros. inversion H; clear H; intros;
  [ auto
  | destruct H3
@@ -88,10 +88,10 @@ theorem lift_inv_sort_2: \forall l, i, x, h.
 qed.
 
 theorem lift_inv_lref_2: \forall l, i, x, j2.
-                         Lift l i x (leaf (lref j2)) \to
-                         (i > j2 \land x = leaf (lref j2)) \lor
+                         Lift l i x (lref j2) \to
+                         (i > j2 \land x = lref j2) \lor
                          (i <= j2 \land 
-                          \exists j1. (l + j1 == j2) \land x = leaf (lref j1)
+                          \exists j1. (l + j1 == j2) \land x = lref j1
                          ).
  intros. inversion H; clear H; intros;
  [ destruct H2
@@ -135,20 +135,30 @@ qed.
 (* Corollaries of inversion properties ***************************************)
 
 theorem lift_inv_lref_1_gt: \forall l, i, j1, x.
-                            Lift l i (leaf (lref j1)) x \to
-                            i > j1 \to x = leaf (lref j1).
- intros 5.
+                            Lift l i (lref j1) x \to
+                            i > j1 \to x = lref j1.
+ intros.
  lapply linear lift_inv_lref_1 to H. decompose; subst;
  [ auto
  | lapply linear nle_false to H2, H1. decompose
  ].
- qed.
+qed.
 
 theorem lift_inv_lref_1_le: \forall l, i, j1, x.
-                            Lift l i (leaf (lref j1)) x \to
-                            i <= j1 \to \forall j2. (l + j1 == j2) \to
-                            x = leaf (lref j2).
- intros 5.
+                            Lift l i (lref j1) x \to i <= j1 \to 
+                            \exists j2. (l + j1 == j2) \land x = lref j2.
+ intros.
+ lapply linear lift_inv_lref_1 to H. decompose; subst;
+ [ lapply linear nle_false to H1, H2. decompose
+ | auto
+ ].
+qed.
+ 
+theorem lift_inv_lref_1_le_nplus: \forall l, i, j1, x.
+                                 Lift l i (lref j1) x \to
+                                 i <= j1 \to \forall j2. (l + j1 == j2) \to
+                                 x = lref j2.
+ intros.
  lapply linear lift_inv_lref_1 to H. decompose; subst;
  [ lapply linear nle_false to H1, H3. decompose
  | lapply linear nplus_mono to H2, H4. subst. auto
@@ -156,9 +166,9 @@ theorem lift_inv_lref_1_le: \forall l, i, j1, x.
 qed.
 
 theorem lift_inv_lref_2_gt: \forall l, i, x, j2.
-                            Lift l i x (leaf (lref j2)) \to
-                            i > j2 \to x = leaf (lref j2).
- intros 5.
+                            Lift l i x (lref j2) \to
+                            i > j2 \to x = lref j2.
+ intros.
  lapply linear lift_inv_lref_2 to H. decompose; subst;
  [ auto
  | lapply linear nle_false to H2, H1. decompose
@@ -166,10 +176,20 @@ theorem lift_inv_lref_2_gt: \forall l, i, x, j2.
  qed.
 
 theorem lift_inv_lref_2_le: \forall l, i, x, j2.
-                            Lift l i x (leaf (lref j2)) \to
-                            i <= j2 \to \forall j1. (l + j1 == j2) \to
-                            x = leaf (lref j1).
- intros 5.
+                            Lift l i x (lref j2) \to i <= j2 \to 
+                            \exists j1. (l + j1 == j2) \land x = lref j1.
+ intros.
+ lapply linear lift_inv_lref_2 to H. decompose; subst;
+ [ lapply linear nle_false to H1, H2. decompose
+ | auto
+ ].
+qed.
+
+theorem lift_inv_lref_2_le_nplus: \forall l, i, x, j2.
+                                  Lift l i x (lref j2) \to
+                                  i <= j2 \to \forall j1. (l + j1 == j2) \to
+                                  x = lref j1.
+ intros.
  lapply linear lift_inv_lref_2 to H. decompose; subst;
  [ lapply linear nle_false to H1, H3. decompose
  | lapply linear nplus_inj_2 to H2, H4. subst. auto
diff --git a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma
index a5c566dab..34964a5b1 100644
--- a/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma
+++ b/matita/contribs/LAMBDA-TYPES/Unified-Sub/Lift/props.ma
@@ -16,108 +16,64 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Unified-Sub/Lift/props".
 
 include "Lift/fun.ma".
 
-theorem lift_comp_ge_1: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
-                        \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
-                        \forall u2. Lift l2 i2 t2 u2 \to
-                        i1 >= i2 \to \forall i. (l2 + i1 == i) \to
-                        Lift l1 i u1 u2.
+(* NOTE: this holds because of nplus_comm_1 *)
+theorem lift_comp: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+                   \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
+                   \forall x. Lift l2 i2 t2 x \to
+                   \forall i,y. Lift l1 i u1 y \to
+                   i1 >= i2 \to (l2 + i1 == i) \to x = y.
  intros 5. elim H; clear H i1 t1 t2;
- [ lapply lift_conf to H1, H2. clear H2. subst.
-   lapply linear lift_inv_sort_1 to H1.
-   subst. auto
- | lapply lift_conf to H2, H3. clear H3. subst.
+ [ lapply lift_mono to H1, H2. clear H2. subst.
+   lapply linear lift_inv_sort_1 to H1. subst.
+   lapply linear lift_inv_sort_1 to H3. subst. auto
+ | lapply lift_mono to H2, H3. clear H3. subst.
    lapply linear lift_inv_lref_1 to H2.
-   decompose; subst; clear H2 H4 i2;
-   [ lapply linear nle_nplus to H5 as H0. clear l2. (**)
-     lapply linear nle_trans to H1, H0.
-     auto
-   | lapply nle_nplus_comp_lt_2 to H3, H5; auto.
-   ]
- | lapply linear lift_inv_lref_1 to H3.
-   decompose; subst;
-   [ clear H2 H4 H6 n3 l2.
-     lapply linear nle_trans to H3, H5 as H0.
-     lapply linear nle_false to H1, H0. decompose
-   | lapply linear lift_inv_lref_1 to H4.
-     decompose; subst;
-     [ clear H1 H5 H6 H7 n1.
-       lapply linear nle_nplus to H2 as H0. (**)
-       lapply linear nle_trans to H3, H0 as H2.
-       lapply linear nle_false to H2, H4. decompose
-     | clear H3 H4 H5.
-       lapply nle_nplus_comp to H6, H7; auto.
-     ]
-   ]
+   decompose; subst; clear H2 H5;
+   lapply linear lift_inv_lref_1_gt to H4; subst; auto width = 4
+ | lapply lift_inv_lref_1_le to H3; [ 2: auto ]. clear H3.
+   lapply lift_inv_lref_1_le to H4; [ 2: auto ]. clear H4.
+   decompose. subst. clear H6 i2.
+   lapply lift_inv_lref_1_le to H5; [ 2: auto depth = 4 width = 4 ]. 
+   decompose. subst. clear H5 H1 H7 i. auto depth = 4
  | clear H1 H3.
    lapply linear lift_inv_bind_1 to H5.
-   lapply linear lift_inv_bind_1 to H6.
-   decompose. subst. auto width = 4
+   lapply linear lift_inv_bind_1 to H6. decompose. subst.
+   lapply linear lift_inv_bind_1 to H7. decompose. subst.
+   auto width = 5
  | clear H1 H3.
    lapply linear lift_inv_flat_1 to H5.
-   lapply linear lift_inv_flat_1 to H6.
-   decompose. subst. auto width = 4
+   lapply linear lift_inv_flat_1 to H6. decompose. subst.
+   lapply linear lift_inv_flat_1 to H7. decompose. subst.
+   auto width = 5
  ].
 qed.
 
-theorem lift_comp_ge_2: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
-                        \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
-                        \forall i,u2. Lift l2 i t2 u2 \to
-                        i2 >= i1 \to (l1 + i2 == i) \to
-                        Lift l1 i1 u1 u2.
- intros 5. elim H; clear H i1 t1 t2;
- [ lapply linear lift_inv_sort_1 to H1.
-   lapply linear lift_inv_sort_1 to H2.
-   subst. auto
- | lapply linear lift_inv_lref_1 to H2.
-   lapply linear lift_inv_lref_1 to H3.
-   decompose; subst; (* clear H2 H4 i2; *)
-   [ clear H H3 H4 H5. auto
-   | clear H1 H4 H7.
-     lapply linear nle_nplus to H5 as H0. (**)
-     lapply linear nle_trans to H3, H0 as H2.
-     lapply nle_false to H, H2. decompose
-   | clear H H5 H6.
-     lapply linear nle_trans to H4, H3 as H.
-     lapply nle_false to H, H1. decompose
-   | clear H H2 H5 H7. 
-     lapply linear nle_trans to H4, H3 as H.
-     lapply nle_false to H, H1. decompose
-   ]
-(* 
- | lapply linear lift_inv_lref_1 to H3.
-   decompose; subst;
-   [ clear H2 H4 H6 n3 l2.
-     lapply linear nle_trans to H3, H5 as H0.
-     lapply linear nle_false to H1, H0. decompose
-   | lapply linear lift_inv_lref_1 to H4.
-     decompose; subst;
-     [ clear H1 H5 H6 H7 n1.
-       lapply linear nle_nplus to H2 as H0. (**)
-       lapply linear nle_trans to H3, H0 as H2.
-       lapply linear nle_false to H2, H4. decompose
-     | clear H3 H4 H5.
-       lapply nle_nplus_comp to H6, H7; auto.
-     ]
-   ]
- | clear H1 H3.
-   lapply linear lift_inv_bind_1 to H5.
-   lapply linear lift_inv_bind_1 to H6.
-   decompose. subst. auto width = 4
- | clear H1 H3.
-   lapply linear lift_inv_flat_1 to H5.
-   lapply linear lift_inv_flat_1 to H6.
-   decompose. subst. auto width = 4
- ].
+theorem lift_comp_rew_dx: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+                          \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
+                          \forall u2. Lift l2 i2 t2 u2 \to
+                          i1 >= i2 \to \forall i. (l2 + i1 == i) \to
+                          Lift l1 i u1 u2.
+ intros.
+ lapply (lift_total l1 u1 i). decompose.
+ lapply lift_comp to H, H1, H2, H5, H3, H4. subst. auto.
 qed.
 
-
+theorem lift_comp_rew_sx: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+                          \forall l2,i2,u1. Lift l2 i2 t1 u1 \to
+                          \forall i,u2. Lift l2 i t2 u2 \to
+                          i2 >= i1 \to (l1 + i2 == i) \to
+                          Lift l1 i1 u1 u2.
+ intros.
+ lapply (lift_total l1 u1 i1). decompose.
+ lapply lift_comp to H1, H, H5, H2, H3, H4. subst. auto.
+qed.
 (*
 theorem lift_trans_le: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
                        \forall l2,i2,z. Lift l2 i2 t2 t3 \to
 		       i1 <= i2 \to 
 		       \forall i. \to i2 <= i \to (l1 + i1 == i) \to
 		       \forall l. (l1 + l2 == l) \to Lift l i1 t1 t3.
-*)
+
 axiom lift_conf_back_ge: \forall l1,i1,u1,u2. Lift l1 i1 u1 u2 \to
 	                 \forall l2,i,t2. Lift l2 i t2 u2 \to
 	                 \forall i2. i2 >= i1 \to (l1 + i2 == i) \to
diff --git a/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma b/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma
index ba37390d9..8c2fb6ec6 100644
--- a/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma
+++ b/matita/contribs/LAMBDA-TYPES/Unified-Sub/datatypes/Term.ma
@@ -18,7 +18,6 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Unified-Sub/datatypes/Term".
    - Naming policy:
      - natural numbers      : sorts h k, local references i j, lengths l o
      - boolean values       : p q
-     - generic leaf items   : m n
      - generic binding items: r s 
      - generic flat items   : r s
      - generic head items   : m n
@@ -27,11 +26,6 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Unified-Sub/datatypes/Term".
 
 include "preamble.ma".
 
-inductive Leaf: Set \def
-   | sort: Nat \to Leaf
-   | lref: Nat \to Leaf
-.
-
 inductive Bind: Set \def
    | abbr: Bind
    | abst: Bind
@@ -52,7 +46,8 @@ inductive IntF: Set \def
 .
 
 inductive Term: Set \def
-   | leaf: Leaf \to Term
+   | sort: Nat  \to Term
+   | lref: Nat  \to Term
    | intb: IntB \to Term \to Term \to Term
    | intf: IntF \to Term \to Term \to Term
 .
diff --git a/matita/contribs/RELATIONAL/NLE/inv.ma b/matita/contribs/RELATIONAL/NLE/inv.ma
index e42eecde3..67a18d820 100644
--- a/matita/contribs/RELATIONAL/NLE/inv.ma
+++ b/matita/contribs/RELATIONAL/NLE/inv.ma
@@ -18,9 +18,9 @@ include "NPlus/inv.ma".
 include "NLE/defs.ma".
 
 theorem nle_inv_succ_1: \forall x,y. x < y \to 
-                         \exists z. y = succ z \land x <= z.
+                        \exists z. y = succ z \land x <= z.
  intros. elim H.
- lapply linear nplus_gen_succ_2 to H1.
+ lapply linear nplus_inv_succ_2 to H1.
  decompose. subst. auto depth = 4.
 qed.
 
diff --git a/matita/contribs/RELATIONAL/NLE/props.ma b/matita/contribs/RELATIONAL/NLE/props.ma
index eccc2f9db..a6d0ca7a8 100644
--- a/matita/contribs/RELATIONAL/NLE/props.ma
+++ b/matita/contribs/RELATIONAL/NLE/props.ma
@@ -65,7 +65,7 @@ theorem nle_lt_or_eq: \forall y,x. x <= y \to x < y \lor x = y.
  ].
 qed.
 
-theorem nat_dec_lt_ge: \forall x,y. x < y \lor y <= x.
+theorem nle_gt_or_le: \forall x,y. y > x \lor y <= x.
  intros 2; elim y; clear y;
  [ auto
  | decompose;
diff --git a/matita/contribs/RELATIONAL/NPlus/fun.ma b/matita/contribs/RELATIONAL/NPlus/fun.ma
index d2d686c51..4b31c570f 100644
--- a/matita/contribs/RELATIONAL/NPlus/fun.ma
+++ b/matita/contribs/RELATIONAL/NPlus/fun.ma
@@ -26,15 +26,15 @@ qed.
 theorem nplus_mono: \forall p,q,r1. (p + q == r1) \to 
                     \forall r2. (p + q == r2) \to r1 = r2.
  intros 4. elim H; clear H q r1;
- [ lapply linear nplus_gen_zero_2 to H1
- | lapply linear nplus_gen_succ_2 to H3. decompose
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2 to H3. decompose
  ]; subst; auto.
 qed.
 
 theorem nplus_inj_1: \forall p1, q, r. (p1 + q == r) \to
                      \forall p2. (p2 + q == r) \to p2 = p1.
  intros 4. elim H; clear H q r;
- [ lapply linear nplus_gen_zero_2 to H1
- | lapply linear nplus_gen_succ_2_3 to H3
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2_3 to H3
  ]; auto.
 qed.
diff --git a/matita/contribs/RELATIONAL/NPlus/inv.ma b/matita/contribs/RELATIONAL/NPlus/inv.ma
index 68043c84a..99bf3a9ee 100644
--- a/matita/contribs/RELATIONAL/NPlus/inv.ma
+++ b/matita/contribs/RELATIONAL/NPlus/inv.ma
@@ -16,119 +16,78 @@ set "baseuri" "cic:/matita/RELATIONAL/NPlus/inv".
 
 include "NPlus/defs.ma".
 
-(* primitive generation lemmas proved by elimination and inversion *)
+(* Inversion lemmas *********************************************************)
 
-theorem nplus_gen_zero_1: \forall q,r. (zero + q == r) \to q = r.
- intros. elim H; clear H q r; intros;
- [ reflexivity
- | clear H1. auto new timeout=30
- ].
+theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
+ intros. elim H; clear H q r; auto.
 qed.
 
-theorem nplus_gen_succ_1: \forall p,q,r. ((succ p) + q == r) \to 
+theorem nplus_inv_succ_1: \forall p,q,r. ((succ p) + q == r) \to 
                           \exists s. r = (succ s) \land p + q == s.
  intros. elim H; clear H q r; intros;
- [
- | clear H1.
-   decompose.
-   subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
+ [ auto depth = 4
+ | clear H1. decompose. subst. auto depth = 4
+ ]
 qed.
 
-theorem nplus_gen_zero_2: \forall p,r. (p + zero == r) \to p = r.
+theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
  intros. inversion H; clear H; intros;
- [ auto new timeout=30
- | clear H H1.
-   destruct H2.
+ [ auto.
+ | clear H H1. destruct H2.
  ].
 qed.
 
-theorem nplus_gen_succ_2: \forall p,q,r. (p + (succ q) == r) \to 
+theorem nplus_inv_succ_2: \forall p,q,r. (p + (succ q) == r) \to 
                           \exists s. r = (succ s) \land p + q == s.
  intros. inversion H; clear H; intros;
  [ destruct H.
- | clear H1 H3 r.
-   destruct H2; clear H2.
-   subst.
-   apply ex_intro; [| auto new timeout=30 ] (**)
+ | clear H1 H3 r. destruct H2; clear H2. subst. auto depth = 4.
  ].
 qed.
 
-theorem nplus_gen_zero_3: \forall p,q. (p + q == zero) \to 
+theorem nplus_inv_zero_3: \forall p,q. (p + q == zero) \to 
                           p = zero \land q = zero.
  intros. inversion H; clear H; intros;
- [ subst. auto new timeout=30
- | clear H H1.
-   destruct H3.
+ [ subst. auto
+ | clear H H1. destruct H3.
  ].
 qed.
 
-theorem nplus_gen_succ_3: \forall p,q,r. (p + q == (succ r)) \to
+theorem nplus_inv_succ_3: \forall p,q,r. (p + q == (succ r)) \to
                           \exists s. p = succ s \land (s + q == r) \lor
                                      q = succ s \land p + s == r.
  intros. inversion H; clear H; intros;
- [ subst.
- | clear H1.
-   destruct H3. clear H3.
-   subst.
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ] (**)
+ [ subst
+ | clear H1. destruct H3. clear H3. subst.
+ ]; auto depth = 4.
 qed.
-(*
-(* alternative proofs invoking nplus_gen_2 *)
 
-variant nplus_gen_zero_3_alt: \forall p,q. (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.
-   subst. auto new timeout=30
- | 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.
+(* Corollaries to inversion lemmas ******************************************)
 
-variant nplus_gen_succ_3_alt: \forall p,q,r. (p + q == (succ r)) \to
-                              \exists s. p = succ s \land (s + q == r) \lor
-                                         q = succ s \land p + s == r.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
-   subst
- | clear H.
-   lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose.
-   lapply linear eq_gen_succ_succ to H1 as H0.
-   subst
- ]; apply ex_intro; [| auto new timeout=30 || auto new timeout=30 ]. (**)
+theorem nplus_inv_succ_2_3: \forall p,q,r.
+                            (p + (succ q) == (succ r)) \to p + q == r.
+ intros. 
+ lapply linear nplus_inv_succ_2 to H. decompose. subst.
+ destruct H1. clear H1. subst. auto.
 qed.
-*)
-(* other simplification lemmas *)
 
-theorem nplus_gen_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
- intros 2. elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
-   subst
- | lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose.
-   destruct H2. clear H2.
-   subst
- ]; auto new timeout=30.
+theorem nplus_inv_succ_1_3: \forall p,q,r.
+                            ((succ p) + q == (succ r)) \to p + q == r.
+ intros. 
+ lapply linear nplus_inv_succ_1 to H. decompose. subst.
+ destruct H1. clear H1. subst. auto.
 qed.
 
-theorem nplus_gen_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
- intros 1. elim p; clear p; intros;
- [ lapply linear nplus_gen_zero_1 to H as H0.
-   subst
- | lapply linear nplus_gen_succ_1 to H1 as H0.
-   decompose.
-   destruct H2. clear H2.
-   subst
- ]; auto new timeout=30.
+theorem nplus_inv_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
+ intros 2. elim q; clear q;
+ [ lapply linear nplus_inv_zero_2 to H
+ | lapply linear nplus_inv_succ_2_3 to H1
+ ]; auto.
 qed.
 
-theorem nplus_gen_succ_2_3: \forall p,q,r.
-                            (p + (succ q) == (succ r)) \to p + q == r.
- intros. 
- lapply linear nplus_gen_succ_2 to H. decompose. subst.
- destruct H1. clear H1. subst. auto.
+theorem nplus_inv_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
+ intros 1. elim p; clear p;
+ [ lapply linear nplus_inv_zero_1 to H
+ | lapply linear nplus_inv_succ_1_3 to H1.
+ ]; auto.
 qed.
diff --git a/matita/contribs/RELATIONAL/NPlus/monoid.ma b/matita/contribs/RELATIONAL/NPlus/monoid.ma
index 804485e54..a7d56eeb3 100644
--- a/matita/contribs/RELATIONAL/NPlus/monoid.ma
+++ b/matita/contribs/RELATIONAL/NPlus/monoid.ma
@@ -27,6 +27,14 @@ theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
  intros. elim H; clear H q r; auto.
 qed.
 
+theorem nplus_comm: \forall p, q, x. (p + q == x) \to
+                    \forall y. (q + p == y) \to x = y.
+ intros 4; elim H; clear H q x;
+ [ lapply linear nplus_inv_zero_1 to H1
+ | lapply linear nplus_inv_succ_1 to H3. decompose
+ ]; subst; auto.
+qed.
+
 theorem nplus_comm_rew: \forall p,q,r. (p + q == r) \to q + p == r.
  intros. elim H; clear H q r; auto.
 qed.
@@ -40,13 +48,26 @@ qed.
 
 (* Corollaries of nonoidal properties ***************************************)
 
+theorem nplus_comm_1: \forall p1, q, r1. (p1 + q == r1) \to
+                      \forall p2, r2. (p2 + q == r2) \to
+                      \forall x. (p2 + r1 == x) \to 
+                      \forall y. (p1 + r2 == y) \to
+                      x = y.
+ intros 4. elim H; clear H q r1;
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2 to H3.
+   lapply linear nplus_inv_succ_2 to H4. decompose. subst.
+   lapply linear nplus_inv_succ_2 to H5. decompose
+ ]; subst; auto.
+qed.
+
 theorem nplus_comm_1_rew: \forall p1,q,r1. (p1 + q == r1) \to
                           \forall p2,r2. (p2 + q == r2) \to
                           \forall s. (p1 + r2 == s) \to (p2 + r1 == s).
  intros 4. elim H; clear H q r1;
- [ lapply linear nplus_gen_zero_2 to H1. subst
- | lapply linear nplus_gen_succ_2 to H3. decompose. subst.
-   lapply linear nplus_gen_succ_2 to H4. decompose. subst
+ [ lapply linear nplus_inv_zero_2 to H1. subst
+ | lapply linear nplus_inv_succ_2 to H3. decompose. subst.
+   lapply linear nplus_inv_succ_2 to H4. decompose. subst
  ]; auto.
 qed.
 
@@ -54,14 +75,14 @@ qed.
 theorem nplus_shift_succ_sx: \forall p,q,r. 
                              (p + (succ q) == r) \to (succ p) + q == r.
  intros.
- lapply linear nplus_gen_succ_2 to H as H0.
+ lapply linear nplus_inv_succ_2 to H as H0.
  decompose. subst. auto new timeout=100.
 qed.
 
 theorem nplus_shift_succ_dx: \forall p,q,r. 
                              ((succ p) + q == r) \to p + (succ q) == r.
  intros.
- lapply linear nplus_gen_succ_1 to H as H0.
+ lapply linear nplus_inv_succ_1 to H as H0.
  decompose. subst. auto new timeout=100.
 qed.
 
@@ -69,11 +90,11 @@ theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
                        \forall q2,r2. (r1 + q2 == r2) \to
                        \exists q. (q1 + q2 == q) \land p + q == r2.
  intros 2; elim q1; clear q1; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
    subst.
- | lapply linear nplus_gen_succ_2 to H1 as H0.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
    decompose. subst.
-   lapply linear nplus_gen_succ_1 to H2 as H0.
+   lapply linear nplus_inv_succ_1 to H2 as H0.
    decompose. subst.
    lapply linear H to H4, H3 as H0.
    decompose.
@@ -84,15 +105,14 @@ theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
                        \forall p2,r2. (p2 + r1 == r2) \to
                        \exists p. (p1 + p2 == p) \land p + q == r2.
  intros 2; elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
    subst
- | lapply linear nplus_gen_succ_2 to H1 as H0.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
    decompose. subst.
-   lapply linear nplus_gen_succ_2 to H2 as H0.
+   lapply linear nplus_inv_succ_2 to H2 as H0.
    decompose. subst.
    lapply linear H to H4, H3 as H0.
    decompose.
  ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
 qed.
 *)
-