-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.
+eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (neg (pred (times (S n) (S (pred (times (S n1) (S n2))))))).
+rewrite < S_pred ? ?.rewrite < S_pred ? ?.rewrite < assoc_times.reflexivity.
+apply lt_O_times_S_S.
+apply lt_O_times_S_S.
+change with
+eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (pos (pred (times (S n) (S (pred (times (S n1) (S n2))))))).
+rewrite < S_pred ? ?.rewrite < S_pred ? ?.rewrite < assoc_times.reflexivity.
+apply lt_O_times_S_S.apply lt_O_times_S_S.
+elim z.simplify.reflexivity.
+change with
+eq Z (pos (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (pos(pred (times (S n) (S (pred (times (S n1) (S n2))))))).
+rewrite < S_pred ? ?.rewrite < S_pred ? ?.rewrite < assoc_times.reflexivity.
+apply lt_O_times_S_S.apply lt_O_times_S_S.
+change with
+eq Z (neg (pred (times (S (pred (times (S n) (S n1)))) (S n2))))
+ (neg (pred (times (S n) (S (pred (times (S n1) (S n2))))))).
+rewrite < S_pred ? ?.rewrite < S_pred ? ?.rewrite < assoc_times.reflexivity.
+apply lt_O_times_S_S.
+apply lt_O_times_S_S.