-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 n2.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 n2.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 n2.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)).
-(* simplify. *)
-intros.elim x.simplify.reflexivity.
-elim n.rewrite < (Zpred_Zplus_neg_O (Zplus 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).
-apply eq_f.assumption.
-elim n.rewrite < Zsucc_Zplus_pos_O.
-rewrite < Zsucc_Zplus_pos_O.
-rewrite > Zplus_Zsucc.
-reflexivity.
-rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc (pos n1).
-rewrite > Zplus_Zsucc (Zplus (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))
-\def associative_Zplus.
-
-(* Zopp *)
-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.