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.
| (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.
| EQ \Rightarrow OZ
| GT \Rightarrow (neg (pred (minus m n)))]
| (neg n) \Rightarrow (neg (S (plus m 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.
(* 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 *)
| (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.