From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Fri, 1 Jul 2005 17:21:13 +0000 (+0000)
Subject: * match.ma removed (it is now splitted in several files in library/*.ma)
X-Git-Tag: PRE_GETTER_STORAGE~39
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=00f04989b1a7792a7cf34deed1a4305d290808d0;p=helm.git

* match.ma removed (it is now splitted in several files in library/*.ma)
* Makefile.tests removed (redundant with tests/Makefile)
* make tests and make tests.opt now call the same targets in library and tests
---

diff --git a/helm/matita/Makefile.tests b/helm/matita/Makefile.tests
deleted file mode 100644
index cab67026a..000000000
--- a/helm/matita/Makefile.tests
+++ /dev/null
@@ -1,40 +0,0 @@
-TODO=\
-	tests/apply.moo\
-	tests/auto.moo\
-	tests/baseuri.moo\
-	tests/coercions.moo\
-	tests/comments.moo\
-	tests/fguidi.moo\
-	tests/first.moo\
-	tests/fix_betareduction.moo\
-	tests/inversion.moo\
-	tests/letrec.moo\
-	tests/match.moo\
-	tests/match_inference.moo\
-	tests/mysql_escaping.moo\
-	tests/record.moo\
-	tests/replace.moo\
-	tests/rewrite.moo\
-	tests/second.moo\
-	tests/simpl.moo\
-	tests/test2.moo\
-	tests/test3.moo\
-	tests/test4.moo\
-	tests/third.moo
-
-DEPEND_NAME=.depend.moo
-
-all: $(TODO)
-
-clean:
-	rm -f $(DEPEND_NAME) $(TODO) 
-	./matitaclean all
-
-%.moo:%.ma
-	[ ! -e $@ ] || ./matitaclean $< 
-	./matitac $< || ./matitaclean $<
-
-$(DEPEND_NAME): $(TODO:%.moo=%.ma)
-	./matitadep $^ > $@
-
-include $(DEPEND_NAME)
diff --git a/helm/matita/matita.ml b/helm/matita/matita.ml
index 6e2fe2526..93441ed83 100644
--- a/helm/matita/matita.ml
+++ b/helm/matita/matita.ml
@@ -35,8 +35,8 @@ let _ =
   Helm_registry.load_from "matita.conf.xml";  (* read conf *)
   Http_getter.init ();
   MetadataTypes.ownerize_tables (Helm_registry.get "matita.owner");
-  MatitaDb.clean_owner_environment ();
-  MatitaDb.create_owner_environment ();
+(*  MatitaDb.clean_owner_environment ();
+  MatitaDb.create_owner_environment (); *)
   GtkMain.Rc.add_default_file BuildTimeConf.gtkrc_file; (* loads gtk rc *)
   ignore (GMain.Main.init ());
 
diff --git a/helm/matita/tests/match.ma b/helm/matita/tests/match.ma
deleted file mode 100644
index f006a7045..000000000
--- a/helm/matita/tests/match.ma
+++ /dev/null
@@ -1,634 +0,0 @@
-(**************************************************************************)
-(*       ___	                                                            *)
-(*      ||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/match/".
-
-
-inductive True: Prop \def
-I : True.
-
-inductive False: Prop \def .
-
-definition Not: Prop \to Prop \def
-\lambda A. (A \to False).
-
-theorem absurd : \forall A,C:Prop. A \to Not A \to C.
-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.
-intros. elim H. assumption.
-qed.
-
-theorem proj2: \forall A,B:Prop. (And A B) \to A.
-intros. elim H. assumption.
-qed.
-
-inductive Or (A,B:Prop) : Prop \def
-     or_introl : A \to (Or A B)
-   | or_intror : B \to (Or A B).
-
-inductive ex (A:Type) (P:A \to Prop) : Prop \def
-    ex_intro: \forall x:A. P x \to ex A P.
-
-inductive ex2 (A:Type) (P,Q:A \to Prop) : Prop \def
-    ex_intro2: \forall x:A. P x \to Q x \to ex2 A P Q.
-
-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.
-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.
-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).
-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).
-intros.elim H1.elim H.apply refl_equal.
-qed.
-
-inductive nat : Set \def
-  | O : nat
-  | S : nat \to nat.
-
-definition pred: nat \to nat \def
-\lambda n:nat. match n with
-[ O \Rightarrow  O
-| (S u) \Rightarrow u ].
-
-theorem pred_Sn : \forall n:nat.
-(eq nat n (pred (S n))).
-intros.apply refl_equal.
-qed.
-
-theorem injective_S : \forall n,m:nat. 
-(eq nat (S n) (S m)) \to (eq nat n 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)).
-intros. simplify.intros.
-apply H.apply injective_S.assumption.
-qed.
-
-definition not_zero : nat \to Prop \def
-\lambda n: nat.
-  match n with
-  [ O \Rightarrow False
-  | (S p) \Rightarrow True ].
-
-theorem O_S : \forall n:nat. Not (eq nat O (S n)).
-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)).
-intros.elim n.apply O_S.apply not_eq_S.assumption.
-qed.
-
-
-let rec plus n m \def 
- match n with 
- [ O \Rightarrow m
- | (S p) \Rightarrow S (plus p m) ].
-
-theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
-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)).
-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).
-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)).
-intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
-qed.
-
-let rec times n m \def 
- match n with 
- [ O \Rightarrow O
- | (S p) \Rightarrow (plus m (times p m)) ].
-
-theorem times_n_O: \forall n:nat. eq nat O (times n O).
-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)).
-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)).
-apply f_equal2.
-apply sym_plus.apply refl_equal.apply assoc_plus.
-qed.
-
-theorem sym_times : 
-\forall n,m:nat. eq nat (times n m) (times m n).
-intros.elim n.simplify.apply times_n_O.
-simplify.elim (sym_eq ? ? ? H).apply times_n_Sm.
-qed.
-
-let rec minus n m \def 
- match n with 
- [ O \Rightarrow O
- | (S p) \Rightarrow 
-	match m with
-	[O \Rightarrow (S p)
-        | (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.
-intros.elim n.assumption.apply H1.
-qed.
-
-theorem nat_double_ind :
-\forall R:nat \to nat \to Prop.
-(\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.
-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 
-  | true : bool
-  | false : bool.
-
-definition notn : bool \to bool \def
-\lambda b:bool. 
- match b with 
- [ true \Rightarrow false
- | false \Rightarrow true ].
-
-definition andb : bool \to bool \to bool\def
-\lambda b1,b2:bool. 
- match b1 with 
- [ true \Rightarrow 
-	match b2 with [true \Rightarrow true | false \Rightarrow false]
- | false \Rightarrow false ].
-
-definition orb : bool \to bool \to bool\def
-\lambda b1,b2:bool. 
- match b1 with 
- [ true \Rightarrow 
-	match b2 with [true \Rightarrow true | false \Rightarrow false]
- | false \Rightarrow false ].
-
-definition if_then_else : bool \to Prop \to Prop \to Prop \def 
-\lambda b:bool.\lambda P,Q:Prop.
-match b with
-[ true \Rightarrow P
-| false  \Rightarrow Q].
-
-inductive le (n:nat) : nat \to Prop \def
-  | le_n : le n n
-  | 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.
-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).
-intros.elim H.
-apply le_n.apply le_S.assumption.
-qed.
-
-theorem le_O_n : \forall n:nat. le O n.
-intros.elim n.apply le_n.apply le_S. assumption.
-qed.
-
-theorem le_n_Sn : \forall n:nat. le n (S n).
-intros. apply le_S.apply le_n.
-qed.
-
-theorem le_pred_n : \forall n:nat. le (pred n) n.
-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.
-intros.elim H.exact I.exact I.
-qed.
-
-theorem le_Sn_O: \forall n:nat. Not (le (S n) O).
-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).
-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.
-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).
-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).
-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)).
-intros.whd.intros.
-apply le_n_O_eq.assumption.
-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.
-
-let rec leb n m \def 
-    match n with 
-    [ O \Rightarrow true
-    | (S p) \Rightarrow
-	match m with 
-        [ O \Rightarrow false
-	| (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.
-apply (nat_double_ind 
-(\lambda n,m:nat.if_then_else (leb n m) (le n m) (Not (le n m))) ? ? ? n m).
-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.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.
-
-
-