X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Ftests%2Fmatch.ma;h=e36e684c2bba815d70f38d4c204e3310170b555b;hb=9f8a383035b272c628c555b728e84caf9229cd57;hp=f9b8a0fb6cc73bf8d848b2fdf2c366e8835ac8c0;hpb=65f34cf91a06b727d5387d92e70c875d15c88fd7;p=helm.git

diff --git a/helm/matita/tests/match.ma b/helm/matita/tests/match.ma
index f9b8a0fb6..e36e684c2 100644
--- a/helm/matita/tests/match.ma
+++ b/helm/matita/tests/match.ma
@@ -1,24 +1,41 @@
+(**************************************************************************)
+(*       ___	                                                            *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/tests/".
+
+
 inductive True: Prop \def
 I : True.
 
 inductive False: Prop \def .
 
 definition Not: Prop \to Prop \def
-\lambda A:Prop. (A \to False).
+\lambda A. (A \to False).
 
 theorem absurd : \forall A,C:Prop. A \to Not A \to C.
-intro.cut False.elim Hcut.apply H1.assumption.
+intros. elim (H1 H).
 qed.
 
 inductive And (A,B:Prop) : Prop \def
     conj : A \to B \to (And A B).
 
 theorem proj1: \forall A,B:Prop. (And A B) \to A.
-intro. elim H. assumption.
+intros. elim H. assumption.
 qed.
 
 theorem proj2: \forall A,B:Prop. (And A B) \to A.
-intro. elim H. assumption.
+intros. elim H. assumption.
 qed.
 
 inductive Or (A,B:Prop) : Prop \def
@@ -35,23 +52,23 @@ inductive eq (A:Type) (x:A) : A \to Prop \def
     refl_equal : eq A x x.
 
 theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y  \to eq A y x.
-intro. elim H. apply refl_equal.
+intros. elim H. apply refl_equal.
 qed.
 
 theorem trans_eq : \forall A:Type.
 \forall x,y,z:A. eq A x y  \to eq A y z \to eq A x z.
-intro.elim H1.assumption.
+intros.elim H1.assumption.
 qed.
 
 theorem f_equal: \forall  A,B:Type.\forall f:A\to B.
 \forall x,y:A. eq A x y \to eq B (f x) (f y).
-intro.elim H.apply refl_equal.
+intros.elim H.apply refl_equal.
 qed.
 
 theorem f_equal2: \forall  A,B,C:Type.\forall f:A\to B \to C.
 \forall x1,x2:A. \forall y1,y2:B.
 eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
-intro.elim H1.elim H.apply refl_equal.
+intros.elim H1.elim H.apply refl_equal.
 qed.
 
 inductive nat : Set \def
@@ -65,18 +82,18 @@ definition pred: nat \to nat \def
 
 theorem pred_Sn : \forall n:nat.
 (eq nat n (pred (S n))).
-intro.apply refl_equal.
+intros.apply refl_equal.
 qed.
 
 theorem injective_S : \forall n,m:nat. 
 (eq nat (S n) (S m)) \to (eq nat n m).
-intro.(elim (sym_eq ? ? ? (pred_Sn n))).(elim (sym_eq ? ? ? (pred_Sn m))).
+intros.(elim (sym_eq ? ? ? (pred_Sn n))).(elim (sym_eq ? ? ? (pred_Sn m))).
 apply f_equal. assumption.
 qed.
 
 theorem not_eq_S  : \forall n,m:nat. 
 Not (eq nat n m) \to Not (eq nat (S n) (S m)).
-intro. simplify.intro.
+intros. simplify.intros.
 apply H.apply injective_S.assumption.
 qed.
 
@@ -87,57 +104,51 @@ definition not_zero : nat \to Prop \def
   | (S p) \Rightarrow True ].
 
 theorem O_S : \forall n:nat. Not (eq nat O (S n)).
-intro.simplify.intro.
+intros.simplify.intros.
 cut (not_zero O).exact Hcut.elim (sym_eq ? ? ? H).
 exact I.
 qed.
 
 theorem n_Sn : \forall n:nat. Not (eq nat n (S n)).
-intro.elim n.apply O_S.apply not_eq_S.assumption.
+intros.elim n.apply O_S.apply not_eq_S.assumption.
 qed.
 
 
-definition plus : nat \to nat \to nat \def
-let rec plus (n,m:nat) \def 
- match n:nat with 
+let rec plus n m \def 
+ match n with 
  [ O \Rightarrow m
- | (S p) \Rightarrow S (plus p m) ]
-in
-plus.
+ | (S p) \Rightarrow S (plus p m) ].
 
 theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
 qed.
 
 theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n  m)) (plus n (S m)).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
 qed.
 
 theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
-intro.elim n.simplify.apply plus_n_O.
+intros.elim n.simplify.apply plus_n_O.
 simplify.elim (sym_eq ? ? ? H).apply plus_n_Sm.
 qed.
 
 theorem assoc_plus: 
 \forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p)).
-intro.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
 qed.
 
-definition times : nat \to nat \to nat \def
-let rec times (n,m:nat) \def 
- match n:nat with 
+let rec times n m \def 
+ match n with 
  [ O \Rightarrow O
- | (S p) \Rightarrow (plus m (times p m)) ]
-in
-times.
+ | (S p) \Rightarrow (plus m (times p m)) ].
 
 theorem times_n_O: \forall n:nat. eq nat O (times n O).
-intro.elim n.simplify.apply refl_equal.simplify.assumption.
+intros.elim n.simplify.apply refl_equal.simplify.assumption.
 qed.
 
 theorem times_n_Sm : 
 \forall n,m:nat. eq nat (plus n (times n  m)) (times n (S m)).
-intro.elim n.simplify.apply refl_equal.
+intros.elim n.simplify.apply refl_equal.
 simplify.apply f_equal.elim H.
 apply trans_eq ? ? (plus (plus e m) (times e m)).apply sym_eq.
 apply assoc_plus.apply trans_eq ? ? (plus (plus m e) (times e m)).
@@ -147,25 +158,22 @@ qed.
 
 theorem sym_times : 
 \forall n,m:nat. eq nat (times n m) (times m n).
-intro.elim n.simplify.apply times_n_O.
+intros.elim n.simplify.apply times_n_O.
 simplify.elim (sym_eq ? ? ? H).apply times_n_Sm.
 qed.
 
-definition minus : nat \to nat \to nat \def
-let rec minus (n,m:nat) \def 
- [\lambda n:nat.nat] match n:nat with 
+let rec minus n m \def 
+ match n with 
  [ O \Rightarrow O
  | (S p) \Rightarrow 
-	[\lambda n:nat.nat] match m:nat with
+	match m with
 	[O \Rightarrow (S p)
-        | (S q) \Rightarrow minus p q ]]
-in
-minus.
+        | (S q) \Rightarrow minus p q ]].
 
 theorem nat_case :
 \forall n:nat.\forall P:nat \to Prop. 
 P O \to  (\forall m:nat. P (S m)) \to P n.
-intro.elim n.assumption.apply H1.
+intros.elim n.assumption.apply H1.
 qed.
 
 theorem nat_double_ind :
@@ -173,8 +181,8 @@ theorem nat_double_ind :
 (\forall n:nat. R O n) \to 
 (\forall n:nat. R (S n) O) \to 
 (\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
-intro.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
-apply nat_case m1.apply H1.intro.apply H2. apply H3.
+intros.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
+apply nat_case m1.apply H1.intros.apply H2. apply H3.
 qed.
 
 inductive bool : Set \def 
@@ -212,73 +220,71 @@ inductive le (n:nat) : nat \to Prop \def
   | le_S : \forall m:nat. le n m \to le n (S m).
 
 theorem trans_le: \forall n,m,p:nat. le n m \to le m p \to le n p.
-intro.
+intros.
 elim H1.assumption.
 apply le_S.assumption.
 qed.
 
 theorem le_n_S: \forall n,m:nat. le n m \to le (S n) (S m).
-intro.elim H.
+intros.elim H.
 apply le_n.apply le_S.assumption.
 qed.
 
 theorem le_O_n : \forall n:nat. le O n.
-intro.elim n.apply le_n.apply le_S. assumption.
+intros.elim n.apply le_n.apply le_S. assumption.
 qed.
 
 theorem le_n_Sn : \forall n:nat. le n (S n).
-intro. apply le_S.apply le_n.
+intros. apply le_S.apply le_n.
 qed.
 
 theorem le_pred_n : \forall n:nat. le (pred n) n.
-intro.elim n.simplify.apply le_n.simplify.
+intros.elim n.simplify.apply le_n.simplify.
 apply le_n_Sn.
 qed.
 
 theorem not_zero_le : \forall n,m:nat. (le (S n) m ) \to not_zero m.
-intro.elim H.exact I.exact I.
+intros.elim H.exact I.exact I.
 qed.
 
 theorem le_Sn_O: \forall n:nat. Not (le (S n) O).
-intro.simplify.intro.apply not_zero_le ? O H.
+intros.simplify.intros.apply not_zero_le ? O H.
 qed.
 
 theorem le_n_O_eq : \forall n:nat. (le n O) \to (eq nat O n).
-intro.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
+intros.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
 elim n.apply refl_equal.apply False_ind.apply  (le_Sn_O ? H2).
 qed.
 
 theorem le_S_n : \forall n,m:nat. le (S n) (S m) \to le n m.
-intro.cut le (pred (S n)) (pred (S m)).exact Hcut.
+intros.cut le (pred (S n)) (pred (S m)).exact Hcut.
 elim H.apply le_n.apply trans_le ? (pred x).assumption.
 apply le_pred_n.
 qed.
 
 theorem le_Sn_n : \forall n:nat. Not (le (S n) n).
-intro.elim n.apply le_Sn_O.simplify.intro.
+intros.elim n.apply le_Sn_O.simplify.intros.
 cut le (S e) e.apply H.assumption.apply le_S_n.assumption.
 qed.
 
 theorem le_antisym : \forall n,m:nat. (le n m) \to (le m n) \to (eq nat n m).
-intro.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
+intros.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
 apply nat_double_ind (\lambda n,m.((le n m) \to (le m n) \to eq nat n m)).
-intro.whd.intro.
+intros.whd.intros.
 apply le_n_O_eq.assumption.
-intro.whd.intro.apply sym_eq.apply le_n_O_eq.assumption.
-intro.whd.intro.apply f_equal.apply H2.
+intros.whd.intros.apply sym_eq.apply le_n_O_eq.assumption.
+intros.whd.intros.apply f_equal.apply H2.
 apply le_S_n.assumption.
 apply le_S_n.assumption.
 qed.
 
-definition leb : nat \to nat \to bool \def
-let rec leb (n,m:nat) \def 
-   [\lambda n:nat.bool] match n:nat with 
+let rec leb n m \def 
+    match n with 
     [ O \Rightarrow true
     | (S p) \Rightarrow
-	[\lambda n:nat.bool] match m:nat with 
+	match m with 
         [ O \Rightarrow false
-	| (S q) \Rightarrow leb p q]]
-in leb.
+	| (S q) \Rightarrow leb p q]].
 
 theorem le_dec: \forall n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m)).
 intros.
@@ -288,5 +294,341 @@ simplify.intros.apply le_O_n.
 simplify.exact le_Sn_O.
 intros 2.simplify.elim (leb n1 m1).
 simplify.apply le_n_S.apply H.
-simplify.intro.apply H.apply le_S_n.assumption.
+simplify.intros.apply H.apply le_S_n.assumption.
+qed.
+
+inductive Z : Set \def
+  OZ : Z
+| pos : nat \to Z
+| neg : nat \to Z.
+
+definition Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow  OZ 
+| (S n)\Rightarrow  pos n].
+
+coercion Z_of_nat.
+
+definition neg_Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow  OZ 
+| (S n)\Rightarrow  neg n].
+
+definition absZ \def
+\lambda z.
+ match z with 
+[ OZ \Rightarrow O
+| (pos n) \Rightarrow n
+| (neg n) \Rightarrow n].
+
+definition OZ_testb \def
+\lambda z.
+match z with 
+[ OZ \Rightarrow true
+| (pos n) \Rightarrow false
+| (neg n) \Rightarrow false].
+
+theorem OZ_discr :
+\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)). 
+intros.elim z.simplify.apply refl_equal.
+simplify.intros.
+cut match neg e with 
+[ OZ \Rightarrow True 
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+simplify.intros.
+cut match pos e with 
+[ OZ \Rightarrow True 
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+qed.
+
+definition Zsucc \def
+\lambda z. match z with
+[ OZ \Rightarrow pos O
+| (pos n) \Rightarrow pos (S n)
+| (neg n) \Rightarrow 
+	  match n with
+	  [ O \Rightarrow OZ
+	  | (S p) \Rightarrow neg p]].
+
+definition Zpred \def
+\lambda z. match z with
+[ OZ \Rightarrow neg O
+| (pos n) \Rightarrow 
+	  match n with
+	  [ O \Rightarrow OZ
+	  | (S p) \Rightarrow pos p]
+| (neg n) \Rightarrow neg (S n)].
+
+theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
+intros.elim z.apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.	    
+apply refl_equal.
+qed.
+
+theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
+intros.elim z.apply refl_equal.
+apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.
+qed.
+
+inductive compare :Set \def
+| LT : compare
+| EQ : compare
+| GT : compare.
+
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow 
+    match m with 
+      [ O \Rightarrow EQ
+      | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow 
+    match m with 
+      [ O \Rightarrow GT
+      | (S q) \Rightarrow nat_compare p q]].
+
+definition compare_invert: compare \to compare \def
+  \lambda c.
+    match c with
+      [ LT \Rightarrow GT
+      | EQ \Rightarrow EQ
+      | GT \Rightarrow LT ].
+
+theorem nat_compare_invert: \forall n,m:nat. 
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros. 
+apply nat_double_ind (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intro.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intros.simplify.elim H.apply refl_equal.
+qed.
+
+let rec Zplus x y : Z \def
+  match x with
+    [ OZ \Rightarrow y
+    | (pos m) \Rightarrow
+        match y with
+         [ OZ \Rightarrow x
+         | (pos n) \Rightarrow (pos (S (plus m n)))
+         | (neg n) \Rightarrow 
+              match nat_compare m n with
+                [ LT \Rightarrow (neg (pred (minus n m)))
+                | EQ \Rightarrow OZ
+                | GT \Rightarrow (pos (pred (minus m n)))]]
+    | (neg m) \Rightarrow
+        match y with
+         [ OZ \Rightarrow x
+         | (pos n) \Rightarrow 
+              match nat_compare m n with
+                [ LT \Rightarrow (pos (pred (minus n m)))
+                | EQ \Rightarrow OZ
+                | GT \Rightarrow (neg (pred (minus m n)))]     
+         | (neg n) \Rightarrow (neg (S (plus m n)))]].
+         
+theorem Zplus_z_O:  \forall z:Z. eq Z (Zplus z OZ) z.
+intro.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
+intros.elim x.simplify.elim (sym_eq ? ? ? (Zplus_z_O y)).apply refl_equal.
+elim y.simplify.reflexivity.
+simplify.elim (sym_plus e e1).apply refl_equal.
+simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
+simplify.elim nat_compare e1 e.simplify.apply refl_equal.
+simplify. apply refl_equal.
+simplify. apply refl_equal.
+elim y.simplify.reflexivity.
+simplify.elim (sym_eq ? ? ?(nat_compare_invert e e1)).
+simplify.elim nat_compare e1 e.simplify.apply refl_equal.
+simplify. apply refl_equal.
+simplify. apply refl_equal.
+simplify.elim (sym_plus e1 e).apply refl_equal.
+qed.
+
+theorem Zpred_neg : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+intros.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim e.simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zsucc_pos : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+intros.elim z.
+simplify.apply refl_equal.
+elim e.simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_pp :
+\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.elim (plus_n_O ?).apply refl_equal.
+simplify.elim (plus_n_Sm ? ?).apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_pn :
+\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
+intros.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_np :
+\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred_nn:
+\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
+intros.
+elim n.elim m.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim m.
+simplify.elim (plus_n_Sm ? ?).apply refl_equal.
+simplify.elim (sym_eq ? ? ? (plus_n_Sm ? ?)).apply refl_equal.
+qed.
+
+theorem Zplus_succ_pred:
+\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
+intros.
+elim x. elim y.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+elim (Zsucc_pos ?).elim (sym_eq ? ? ? (Zsucc_pred ?)).apply refl_equal.
+elim y.elim sym_Zplus ? ?.elim sym_Zplus (Zpred OZ) ?.
+elim (Zpred_neg ?).elim (sym_eq ? ? ? (Zpred_succ ?)).
+simplify.apply refl_equal.
+apply Zplus_succ_pred_nn.
+apply Zplus_succ_pred_np.
+elim y.simplify.apply refl_equal.
+apply Zplus_succ_pred_pn.
+apply Zplus_succ_pred_pp.
+qed.
+
+theorem Zsucc_plus_pp : 
+\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
+intros.apply refl_equal.
+qed.
+
+theorem Zsucc_plus_pn : 
+\forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim (Zplus_succ_pred_pn ? m1).
+elim H.apply refl_equal.
 qed.
+
+theorem Zsucc_plus_nn : 
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim (Zplus_succ_pred_nn ? m1).
+apply refl_equal.
+qed.
+
+theorem Zsucc_plus_np : 
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
+intros.
+apply nat_double_ind
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
+intros.elim n1.
+simplify. apply refl_equal.
+elim e.simplify. apply refl_equal.
+simplify. apply refl_equal.
+intros. elim n1.
+simplify. apply refl_equal.
+simplify.apply refl_equal.
+intros.
+elim H.
+elim (Zplus_succ_pred_np ? (S m1)).
+apply refl_equal.
+qed.
+
+
+theorem Zsucc_plus : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
+intros.elim x.elim y.
+simplify. apply refl_equal.
+elim (Zsucc_pos ?).apply refl_equal.
+simplify.apply refl_equal.
+elim y.elim sym_Zplus ? ?.elim sym_Zplus OZ ?.simplify.apply refl_equal.
+apply Zsucc_plus_nn.
+apply Zsucc_plus_np.
+elim y.
+elim (sym_Zplus OZ ?).apply refl_equal.
+apply Zsucc_plus_pn.
+apply Zsucc_plus_pp.
+qed.
+
+theorem Zpred_plus : \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
+intros.
+cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
+elim (sym_eq ? ? ? Hcut).
+elim (sym_eq ? ? ? (Zsucc_plus ? ?)).
+elim (sym_eq ? ? ? (Zpred_succ ?)).
+apply refl_equal.
+elim (sym_eq ? ? ? (Zsucc_pred ?)).
+apply refl_equal.
+qed.
+
+theorem assoc_Zplus : 
+\forall x,y,z:Z. eq Z (Zplus x (Zplus y z)) (Zplus (Zplus x y) z).
+intros.elim x.simplify.apply refl_equal.
+elim e.elim (Zpred_neg (Zplus y z)).
+elim (Zpred_neg y).
+elim (Zpred_plus ? ?).
+apply refl_equal.
+elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
+elim (sym_eq ? ? ? (Zpred_plus (neg e1) ?)).
+elim (sym_eq ? ? ? (Zpred_plus (Zplus (neg e1) y) ?)).
+apply f_equal.assumption.
+elim e.elim (Zsucc_pos ?).
+elim (Zsucc_pos ?).
+apply (sym_eq ? ? ? (Zsucc_plus ? ?)) .
+elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
+elim (sym_eq ? ? ? (Zsucc_plus (pos e1) ?)).
+elim (sym_eq ? ? ? (Zsucc_plus (Zplus (pos e1) y) ?)).
+apply f_equal.assumption.
+qed.
+
+
+