[ O \Rightarrow OZ
| (S n)\Rightarrow pos n].
-coercion Z_of_nat.
+coercion cic:/matita/Z/z/Z_of_nat.con.
definition neg_Z_of_nat \def
\lambda n. match n with
|false \Rightarrow z \neq OZ].
intros.elim z.
simplify.reflexivity.
-simplify.intros (H).
+simplify. unfold Not. intros (H).
discriminate H.
-simplify.intros (H).
+simplify. unfold Not. intros (H).
discriminate H.
qed.
(* discrimination *)
theorem injective_pos: injective nat Z pos.
-simplify.
+unfold injective.
intros.
change with (abs (pos x) = abs (pos y)).
apply eq_f.assumption.
\def injective_pos.
theorem injective_neg: injective nat Z neg.
-simplify.
+unfold injective.
intros.
change with (abs (neg x) = abs (neg y)).
apply eq_f.assumption.
\def injective_neg.
theorem not_eq_OZ_pos: \forall n:nat. OZ \neq pos n.
-simplify.intros (n H).
+unfold Not.intros (n H).
discriminate H.
qed.
theorem not_eq_OZ_neg :\forall n:nat. OZ \neq neg n.
-simplify.intros (n H).
+unfold Not.intros (n H).
discriminate H.
qed.
theorem not_eq_pos_neg :\forall n,m:nat. pos n \neq neg m.
-simplify.intros (n m H).
+unfold Not.intros (n m H).
discriminate H.
qed.
theorem decidable_eq_Z : \forall x,y:Z. decidable (x=y).
-intros.simplify.
+intros.unfold decidable.
elim x.
(* goal: x=OZ *)
elim y.
(* goal: x=pos *)
elim y.
(* goal: x=pos y=OZ *)
- right.intro.
+ right.unfold Not.intro.
apply (not_eq_OZ_pos n). symmetry. assumption.
(* goal: x=pos y=pos *)
elim (decidable_eq_nat n n1:((n=n1) \lor ((n=n1) \to False))).
left.apply eq_f.assumption.
- right.intros (H_inj).apply H. injection H_inj. assumption.
+ right.unfold Not.intros (H_inj).apply H. injection H_inj. assumption.
(* goal: x=pos y=neg *)
- right.intro.apply (not_eq_pos_neg n n1). assumption.
+ right.unfold Not.intro.apply (not_eq_pos_neg n n1). assumption.
(* goal: x=neg *)
elim y.
(* goal: x=neg y=OZ *)
- right.intro.
+ right.unfold Not.intro.
apply (not_eq_OZ_neg n). symmetry. assumption.
(* goal: x=neg y=pos *)
- right. intro. apply (not_eq_pos_neg n1 n). symmetry. assumption.
+ right. unfold Not.intro. apply (not_eq_pos_neg n1 n). symmetry. assumption.
(* goal: x=neg y=neg *)
elim (decidable_eq_nat n n1:((n=n1) \lor ((n=n1) \to False))).
left.apply eq_f.assumption.
- right.intro.apply H.apply injective_neg.assumption.
+ right.unfold Not.intro.apply H.apply injective_neg.assumption.
qed.
(* end discrimination *)