**** Experimental: ****

[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 da249902bc3a251b80c7c0e445a8785b39fb52e5..118cdd55eaaa3eb8e0f6b5eb35e1461d12dd8e1b 100644 (file)
--- a/helm/matita/library/Z/z.ma
+++ b/helm/matita/library/Z/z.ma
@@ -51,8 +51,8 @@ match z with
 
 theorem OZ_test_to_Prop :\forall z:Z.
 match OZ_test z with
-[true \Rightarrow eq Z z OZ 
-|false \Rightarrow \lnot (eq Z z OZ)].
+[true \Rightarrow z=OZ 
+|false \Rightarrow \lnot (z=OZ)].
 intros.elim z.
 simplify.reflexivity.
 simplify.intros.
@@ -87,14 +87,14 @@ 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 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 n.reflexivity.
@@ -127,7 +127,7 @@ definition Zplus :Z \to Z \to Z \def
 (*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. Zplus z OZ = z.
+theorem Zplus_z_OZ:  \forall z:Z. z+OZ = z.
 intro.elim z.
 simplify.reflexivity.
 simplify.reflexivity.
@@ -136,7 +136,7 @@ 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.
@@ -154,7 +154,7 @@ simplify. reflexivity.
 simplify.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.
@@ -162,7 +162,7 @@ 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 n.simplify.reflexivity.
@@ -171,7 +171,7 @@ 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.
@@ -184,12 +184,12 @@ 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.
@@ -200,7 +200,7 @@ 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.
@@ -211,7 +211,7 @@ 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.
@@ -230,15 +230,15 @@ 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 n2.simplify. reflexivity.
@@ -252,10 +252,10 @@ 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 n2.simplify. reflexivity.
@@ -269,10 +269,10 @@ 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 n2.simplify. reflexivity.
@@ -286,7 +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.
@@ -300,9 +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.
@@ -313,16 +313,16 @@ qed.
 
 
 theorem associative_Zplus: associative Z Zplus.
-change with \forall x,y,z:Z. eq Z (Zplus (Zplus x y) z) (Zplus x (Zplus y z)). 
+change with \forall x,y,z:Z. (x + y) + z = x + (y + z). 
 (* simplify. *)
 intros.elim x.simplify.reflexivity.
-elim n.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 n1).
 rewrite > Zplus_Zpred (neg n1).
-rewrite > Zplus_Zpred (Zplus (neg n1) y).
+rewrite > Zplus_Zpred ((neg n1)+y).
 apply eq_f.assumption.
 elim n.rewrite < Zsucc_Zplus_pos_O.
 rewrite < Zsucc_Zplus_pos_O.
@@ -330,11 +330,11 @@ rewrite > Zplus_Zsucc.
 reflexivity.
 rewrite > Zplus_Zsucc (pos n1).
 rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc (Zplus (pos n1) y).
+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 *)
@@ -344,7 +344,10 @@ definition Zopp : Z \to Z \def
 | (pos n) \Rightarrow (neg n)
 | (neg n) \Rightarrow (pos n) ].
 
-theorem Zplus_Zopp: \forall x:Z. (eq Z (Zplus x (Zopp x)) OZ).
+(*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.