ready for 0.1.1 release

[helm.git] / helm / matita / library / Z / z.ma

diff --git a/helm/matita/library/Z/z.ma b/helm/matita/library/Z/z.ma
index 11dd836f667b6ed884753f80ec88ac86e731e8d7..cdc0e44d7c28f29fb01681a6d2b52a139a2955d0 100644 (file)
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/Z/".
+set "baseuri" "cic:/matita/Z/z".
 
 include "nat/compare.ma".
 include "nat/minus.ma".
@@ -51,18 +51,18 @@ match z with
 
 theorem OZ_test_to_Prop :\forall z:Z.
 match OZ_test z with
-[true \Rightarrow eq Z z OZ 
-|false \Rightarrow Not (eq Z z OZ)].
+[true \Rightarrow z=OZ 
+|false \Rightarrow \lnot (z=OZ)].
 intros.elim z.
 simplify.reflexivity.
 simplify.intros.
-cut match neg e1 with 
+cut match neg n with 
 [ OZ \Rightarrow True 
 | (pos n) \Rightarrow False
 | (neg n) \Rightarrow False].
 apply Hcut.rewrite > H.simplify.exact I.
 simplify.intros.
-cut match pos e2 with 
+cut match pos n with 
 [ OZ \Rightarrow True 
 | (pos n) \Rightarrow False
 | (neg n) \Rightarrow False].
@@ -87,17 +87,17 @@ definition Zpred \def
          | (S p) \Rightarrow pos p]
 | (neg n) \Rightarrow neg (S n)].
 
-theorem Zpred_Zsucc: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
+theorem Zpred_Zsucc: \forall z:Z. Zpred (Zsucc z) = z.
 intros.elim z.reflexivity.
-elim e1.reflexivity.
+elim n.reflexivity.
 reflexivity.
 reflexivity.
 qed.
 
-theorem Zsucc_Zpred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
+theorem Zsucc_Zpred: \forall z:Z. Zsucc (Zpred z) = z.
 intros.elim z.reflexivity.
 reflexivity.
-elim e2.reflexivity.
+elim n.reflexivity.
 reflexivity.
 qed.
 
@@ -108,23 +108,26 @@ definition Zplus :Z \to Z \to Z \def
     | (pos m) \Rightarrow
         match y with
          [ OZ \Rightarrow x
-         | (pos n) \Rightarrow (pos (S (plus m n)))
+         | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
          | (neg n) \Rightarrow 
               match nat_compare m n with
-                [ LT \Rightarrow (neg (pred (minus n m)))
+                [ LT \Rightarrow (neg (pred (n-m)))
                 | EQ \Rightarrow OZ
-                | GT \Rightarrow (pos (pred (minus m n)))]]
+                | GT \Rightarrow (pos (pred (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)))
+                [ LT \Rightarrow (pos (pred (n-m)))
                 | EQ \Rightarrow OZ
-                | GT \Rightarrow (neg (pred (minus m n)))]     
-         | (neg n) \Rightarrow (neg (S (plus m n)))]].
+                | GT \Rightarrow (neg (pred (m-n)))]     
+         | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))]].
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer plus" 'plus x y = (cic:/matita/Z/z/Zplus.con x y).
          
-theorem Zplus_z_OZ:  \forall z:Z. eq Z (Zplus z OZ) z.
+theorem Zplus_z_OZ:  \forall z:Z. z+OZ = z.
 intro.elim z.
 simplify.reflexivity.
 simplify.reflexivity.
@@ -133,11 +136,11 @@ qed.
 
 (* theorem symmetric_Zplus: symmetric Z Zplus. *)
 
-theorem sym_Zplus : \forall x,y:Z. eq Z (Zplus x y) (Zplus y x).
+theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
 intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
 elim y.simplify.reflexivity.
 simplify.
-rewrite < sym_plus.reflexivity.
+rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
 simplify.
 rewrite > nat_compare_n_m_m_n.
 simplify.elim nat_compare ? ?.simplify.reflexivity.
@@ -148,45 +151,45 @@ simplify.rewrite > nat_compare_n_m_m_n.
 simplify.elim nat_compare ? ?.simplify.reflexivity.
 simplify. reflexivity.
 simplify. reflexivity.
-simplify.rewrite < sym_plus.reflexivity.
+simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
 qed.
 
-theorem Zpred_Zplus_neg_O : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
 intros.elim z.
 simplify.reflexivity.
 simplify.reflexivity.
-elim e2.simplify.reflexivity.
+elim n.simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
-theorem Zsucc_Zplus_pos_O : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
 intros.elim z.
 simplify.reflexivity.
-elim e1.simplify.reflexivity.
+elim n.simplify.reflexivity.
 simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
 theorem Zplus_pos_pos:
-\forall n,m. eq Z (Zplus (pos n) (pos m)) (Zplus (Zsucc (pos n)) (Zpred (pos m))).
+\forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
 intros.
 elim n.elim m.
 simplify.reflexivity.
 simplify.reflexivity.
 elim m.
-simplify.
+simplify.rewrite < plus_n_Sm.
 rewrite < plus_n_O.reflexivity.
-simplify.
+simplify.rewrite < plus_n_Sm.
 rewrite < plus_n_Sm.reflexivity.
 qed.
 
 theorem Zplus_pos_neg:
-\forall n,m. eq Z (Zplus (pos n) (neg m)) (Zplus (Zsucc (pos n)) (Zpred (neg m))).
+\forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
 intros.reflexivity.
 qed.
 
 theorem Zplus_neg_pos :
-\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
+\forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
 intros.
 elim n.elim m.
 simplify.reflexivity.
@@ -197,18 +200,18 @@ simplify.reflexivity.
 qed.
 
 theorem Zplus_neg_neg:
-\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (Zsucc (neg n)) (Zpred (neg m))).
+\forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)).
 intros.
 elim n.elim m.
 simplify.reflexivity.
 simplify.reflexivity.
 elim m.
-simplify.rewrite < plus_n_Sm.reflexivity.
+simplify.rewrite > plus_n_Sm.reflexivity.
 simplify.rewrite > plus_n_Sm.reflexivity.
 qed.
 
 theorem Zplus_Zsucc_Zpred:
-\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
+\forall x,y. x+y = (Zsucc x)+(Zpred y).
 intros.
 elim x. elim y.
 simplify.reflexivity.
@@ -227,18 +230,18 @@ apply Zplus_pos_pos.
 qed.
 
 theorem Zplus_Zsucc_pos_pos : 
-\forall n,m. eq Z (Zplus (Zsucc (pos n)) (pos m)) (Zsucc (Zplus (pos n) (pos m))).
+\forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
 intros.reflexivity.
 qed.
 
 theorem Zplus_Zsucc_pos_neg: 
-\forall n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m))).
+\forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
 intros.
 apply nat_elim2
-(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
+(\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m)))).intro.
 intros.elim n1.
 simplify. reflexivity.
-elim e1.simplify. reflexivity.
+elim n2.simplify. reflexivity.
 simplify. reflexivity.
 intros. elim n1.
 simplify. reflexivity.
@@ -249,13 +252,13 @@ elim H.reflexivity.
 qed.
 
 theorem Zplus_Zsucc_neg_neg : 
-\forall n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m))).
+\forall n,m. (Zsucc (neg n))+(neg m) = Zsucc ((neg n)+(neg m)).
 intros.
 apply nat_elim2
-(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
+(\lambda n,m. ((Zsucc (neg n))+(neg m)) = Zsucc ((neg n)+(neg m))).intro.
 intros.elim n1.
 simplify. reflexivity.
-elim e1.simplify. reflexivity.
+elim n2.simplify. reflexivity.
 simplify. reflexivity.
 intros. elim n1.
 simplify. reflexivity.
@@ -266,13 +269,13 @@ reflexivity.
 qed.
 
 theorem Zplus_Zsucc_neg_pos: 
-\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m))).
+\forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
 intros.
 apply nat_elim2
-(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (neg n) (pos m)))).
+(\lambda n,m. (Zsucc (neg n))+(pos m) = Zsucc ((neg n)+(pos m))).
 intros.elim n1.
 simplify. reflexivity.
-elim e1.simplify. reflexivity.
+elim n2.simplify. reflexivity.
 simplify. reflexivity.
 intros. elim n1.
 simplify. reflexivity.
@@ -283,7 +286,7 @@ rewrite < (Zplus_neg_pos ? (S m1)).
 reflexivity.
 qed.
 
-theorem Zplus_Zsucc : \forall x,y:Z. eq Z (Zplus (Zsucc x) y) (Zsucc (Zplus x y)).
+theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
 intros.elim x.elim y.
 simplify. reflexivity.
 rewrite < Zsucc_Zplus_pos_O.reflexivity.
@@ -297,9 +300,9 @@ apply Zplus_Zsucc_pos_neg.
 apply Zplus_Zsucc_pos_pos.
 qed.
 
-theorem Zplus_Zpred: \forall x,y:Z. eq Z (Zplus (Zpred x) y) (Zpred (Zplus x y)).
+theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
 intros.
-cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
+cut Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y).
 rewrite > Hcut.
 rewrite > Zplus_Zsucc.
 rewrite > Zpred_Zsucc.
@@ -310,28 +313,48 @@ qed.
 
 
 theorem associative_Zplus: associative Z Zplus.
-change with (\forall x,y,z. eq ?(Zplus (Zplus x y) z) (Zplus x (Zplus y z))).
-(* simplify. *) 
+change with \forall x,y,z:Z. (x + y) + z = x + (y + z). 
+(* simplify. *)
 intros.elim x.simplify.reflexivity.
-elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus y z)).
+elim n.rewrite < (Zpred_Zplus_neg_O (y+z)).
 rewrite < (Zpred_Zplus_neg_O y).
 rewrite < Zplus_Zpred.
 reflexivity.
-rewrite > Zplus_Zpred (neg e).
-rewrite > Zplus_Zpred (neg e).
-rewrite > Zplus_Zpred (Zplus (neg e) y).
+rewrite > Zplus_Zpred (neg n1).
+rewrite > Zplus_Zpred (neg n1).
+rewrite > Zplus_Zpred ((neg n1)+y).
 apply eq_f.assumption.
-elim e2.rewrite < Zsucc_Zplus_pos_O.
+elim n.rewrite < Zsucc_Zplus_pos_O.
 rewrite < Zsucc_Zplus_pos_O.
 rewrite > Zplus_Zsucc.
 reflexivity.
-rewrite > Zplus_Zsucc (pos e1).
-rewrite > Zplus_Zsucc (pos e1).
-rewrite > Zplus_Zsucc (Zplus (pos e1) y).
+rewrite > Zplus_Zsucc (pos n1).
+rewrite > Zplus_Zsucc (pos n1).
+rewrite > Zplus_Zsucc ((pos n1)+y).
 apply eq_f.assumption.
 qed.
 
-variant assoc_Zplus : 
-\forall x,y,z:Z. eq Z (Zplus (Zplus x y) z) (Zplus x (Zplus y z))
+variant assoc_Zplus : \forall x,y,z:Z.  (x+y)+z = x+(y+z)
 \def associative_Zplus.
 
+(* Zopp *)
+definition Zopp : Z \to Z \def
+\lambda x:Z. match x with
+[ OZ \Rightarrow OZ
+| (pos n) \Rightarrow (neg n)
+| (neg n) \Rightarrow (pos n) ].
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "integer unary minus" 'uminus x = (cic:/matita/Z/z/Zopp.con x).
+
+theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
+intro.elim x.
+apply refl_eq.
+simplify.
+rewrite > nat_compare_n_n.
+simplify.apply refl_eq.
+simplify.
+rewrite > nat_compare_n_n.
+simplify.apply refl_eq.
+qed.
+