+(**************************************************************************)
+(* ___ *)
+(* ||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/Z/times".
+
+include "Z/z.ma".
+
+definition Ztimes :Z \to Z \to Z \def
+\lambda x,y.
+ match x with
+ [ OZ \Rightarrow OZ
+ | (pos m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow OZ
+ | (pos n) \Rightarrow (pos (pred (times (S m) (S n))))
+ | (neg n) \Rightarrow (neg (pred (times (S m) (S n))))]
+ | (neg m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow OZ
+ | (pos n) \Rightarrow (neg (pred (times (S m) (S n))))
+ | (neg n) \Rightarrow (pos (pred (times (S m) (S n))))]].
+
+theorem Ztimes_z_OZ: \forall z:Z. eq Z (Ztimes z OZ) OZ.
+intro.elim z.
+simplify.reflexivity.
+simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+(* da spostare in nat/order *)
+theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
+intro.elim n.apply False_ind.exact not_le_Sn_O O H.
+apply eq_f.apply pred_Sn.
+qed.
+
+(*
+theorem symmetric_Ztimes : symmetric Z Ztimes.
+change with \forall x,y:Z. eq Z (Ztimes x y) (Ztimes y x).
+intros.elim x.rewrite > Ztimes_z_OZ.reflexivity.
+elim y.simplify.reflexivity.
+change with eq Z (pos (pred (times (S e1) (S e)))) (pos (pred (times (S e) (S e1)))).
+rewrite < sym_times.reflexivity.
+change with eq Z (neg (pred (times (S e1) (S e2)))) (neg (pred (times (S e2) (S e1)))).
+rewrite < sym_times.reflexivity.
+elim y.simplify.reflexivity.
+change with eq Z (neg (pred (times (S e2) (S e1)))) (neg (pred (times (S e1) (S e2)))).
+rewrite < sym_times.reflexivity.
+change with eq Z (pos (pred (times (S e2) (S e)))) (pos (pred (times (S e) (S e2)))).
+rewrite < sym_times.reflexivity.
+qed.
+
+variant sym_Ztimes : \forall x,y:Z. eq Z (Ztimes x y) (Ztimes y x)
+\def symmetric_Ztimes.
+
+theorem associative_Ztimes: associative Z Ztimes.
+change with \forall x,y,z:Z.eq Z (Ztimes (Ztimes x y) z) (Ztimes x (Ztimes y z)).
+intros.
+elim x.simplify.reflexivity.
+elim y.simplify.reflexivity.
+elim z.simplify.reflexivity.
+change with
+eq Z (neg (pred (times (S (pred (times (S e1) (S e)))) (S e2))))
+ (neg (pred (times (S e1) (S (pred (times (S e) (S e2))))))).
+rewrite < S_pred_S.
+
+theorem Zpred_Zplus_neg_O : \forall z:Z. eq Z (Zpred z) (Zplus (neg O) z).
+intros.elim z.
+simplify.reflexivity.
+simplify.reflexivity.
+elim e2.simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem Zsucc_Zplus_pos_O : \forall z:Z. eq Z (Zsucc z) (Zplus (pos O) z).
+intros.elim z.
+simplify.reflexivity.
+elim e1.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))).
+intros.
+elim n.elim m.
+simplify.reflexivity.
+simplify.reflexivity.
+elim m.
+simplify.
+rewrite < plus_n_O.reflexivity.
+simplify.
+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))).
+intros.reflexivity.
+qed.
+
+theorem Zplus_neg_pos :
+\forall n,m. eq Z (Zplus (neg n) (pos m)) (Zplus (Zsucc (neg n)) (Zpred (pos m))).
+intros.
+elim n.elim m.
+simplify.reflexivity.
+simplify.reflexivity.
+elim m.
+simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem Zplus_neg_neg:
+\forall n,m. eq Z (Zplus (neg n) (neg m)) (Zplus (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.
+qed.
+
+theorem Zplus_Zsucc_Zpred:
+\forall x,y. eq Z (Zplus x y) (Zplus (Zsucc x) (Zpred y)).
+intros.
+elim x. elim y.
+simplify.reflexivity.
+simplify.reflexivity.
+rewrite < Zsucc_Zplus_pos_O.
+rewrite > Zsucc_Zpred.reflexivity.
+elim y.rewrite < sym_Zplus.rewrite < sym_Zplus (Zpred OZ).
+rewrite < Zpred_Zplus_neg_O.
+rewrite > Zpred_Zsucc.
+simplify.reflexivity.
+rewrite < Zplus_neg_neg.reflexivity.
+apply Zplus_neg_pos.
+elim y.simplify.reflexivity.
+apply Zplus_pos_neg.
+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))).
+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))).
+intros.
+apply nat_elim2
+(\lambda n,m. eq Z (Zplus (Zsucc (pos n)) (neg m)) (Zsucc (Zplus (pos n) (neg m)))).intro.
+intros.elim n1.
+simplify. reflexivity.
+elim e1.simplify. reflexivity.
+simplify. reflexivity.
+intros. elim n1.
+simplify. reflexivity.
+simplify.reflexivity.
+intros.
+rewrite < (Zplus_pos_neg ? m1).
+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))).
+intros.
+apply nat_elim2
+(\lambda n,m. eq Z (Zplus (Zsucc (neg n)) (neg m)) (Zsucc (Zplus (neg n) (neg m)))).intro.
+intros.elim n1.
+simplify. reflexivity.
+elim e1.simplify. reflexivity.
+simplify. reflexivity.
+intros. elim n1.
+simplify. reflexivity.
+simplify.reflexivity.
+intros.
+rewrite < (Zplus_neg_neg ? m1).
+reflexivity.
+qed.
+
+theorem Zplus_Zsucc_neg_pos:
+\forall n,m. eq Z (Zplus (Zsucc (neg n)) (pos m)) (Zsucc (Zplus (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)))).
+intros.elim n1.
+simplify. reflexivity.
+elim e1.simplify. reflexivity.
+simplify. reflexivity.
+intros. elim n1.
+simplify. reflexivity.
+simplify.reflexivity.
+intros.
+rewrite < H.
+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)).
+intros.elim x.elim y.
+simplify. reflexivity.
+rewrite < Zsucc_Zplus_pos_O.reflexivity.
+simplify.reflexivity.
+elim y.rewrite < sym_Zplus.rewrite < sym_Zplus OZ.simplify.reflexivity.
+apply Zplus_Zsucc_neg_neg.
+apply Zplus_Zsucc_neg_pos.
+elim y.
+rewrite < sym_Zplus OZ.reflexivity.
+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)).
+intros.
+cut eq Z (Zpred (Zplus x y)) (Zpred (Zplus (Zsucc (Zpred x)) y)).
+rewrite > Hcut.
+rewrite > Zplus_Zsucc.
+rewrite > Zpred_Zsucc.
+reflexivity.
+rewrite > Zsucc_Zpred.
+reflexivity.
+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)).
+
+intros.elim x.simplify.reflexivity.
+elim e1.rewrite < (Zpred_Zplus_neg_O (Zplus 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).
+apply eq_f.assumption.
+elim e2.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).
+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))
+\def associative_Zplus.
+
+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) ].
+
+theorem Zplus_Zopp: \forall x:Z. (eq Z (Zplus x (Zopp 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.
+*)