(** Compatibility Notations *)

Notation Zdouble_plus_one := Z.succ_double (only parsing).Notation Zdouble_minus_one := Z.pred_double (only parsing).Notation ZPminus := Z.pos_sub (only parsing).Notation Zplus := Z.add (only parsing).(* Slightly incompatible *)
Notation Zminus := Z.sub (only parsing).Notation Zmult := Z.mul (only parsing).Notation Z_of_nat := Z.of_nat (only parsing).Notation Z_of_N := Z.of_N (only parsing).Notation Zind := Z.peano_ind (only parsing).Notation Zplus_0_l := Z.add_0_l (only parsing).Notation Zplus_0_r := Z.add_0_r (only parsing).Notation Zplus_comm := Z.add_comm (only parsing).Notation Zopp_plus_distr := Z.opp_add_distr (only parsing).Notation Zplus_opp_r := Z.add_opp_diag_r (only parsing).Notation Zplus_opp_l := Z.add_opp_diag_l (only parsing).Notation Zplus_assoc := Z.add_assoc (only parsing).Notation Zplus_permute := Z.add_shuffle3 (only parsing).Notation Zplus_reg_l := Z.add_reg_l (only parsing).Notation Zplus_succ_l := Z.add_succ_l (only parsing).Notation Zplus_succ_comm := Z.add_succ_comm (only parsing).Notation Zsucc_discr := Z.neq_succ_diag_r (only parsing).Notation Zsucc'_discr := Z.neq_succ_diag_r (only parsing).Notation Zminus_0_r := Z.sub_0_r (only parsing).Notation Zminus_diag := Z.sub_diag (only parsing).Notation Zminus_plus_distr := Z.sub_add_distr (only parsing).Notation Zminus_succ_r := Z.sub_succ_r (only parsing).Notation Zminus_plus := Z.add_simpl_l (only parsing).Notation Zmult_0_l := Z.mul_0_l (only parsing).Notation Zmult_0_r := Z.mul_0_r (only parsing).Notation Zmult_1_l := Z.mul_1_l (only parsing).Notation Zmult_1_r := Z.mul_1_r (only parsing).Notation Zmult_comm := Z.mul_comm (only parsing).Notation Zmult_assoc := Z.mul_assoc (only parsing).Notation Zmult_permute := Z.mul_shuffle3 (only parsing).Notation Zmult_1_inversion_l := Z.mul_eq_1 (only parsing).Notation Zdouble_mult := Z.double_spec (only parsing).Notation Zdouble_plus_one_mult := Z.succ_double_spec (only parsing).Notation Zopp_mult_distr_l_reverse := Z.mul_opp_l (only parsing).Notation Zmult_opp_opp := Z.mul_opp_opp (only parsing).Notation Zmult_opp_comm := Z.mul_opp_comm (only parsing).Notation Zopp_eq_mult_neg_1 := Z.opp_eq_mul_m1 (only parsing).Notation Zmult_plus_distr_r := Z.mul_add_distr_l (only parsing).Notation Zmult_plus_distr_l := Z.mul_add_distr_r (only parsing).Notation Zmult_minus_distr_r := Z.mul_sub_distr_r (only parsing).Notation Zmult_reg_l := Z.mul_reg_l (only parsing).Notation Zmult_reg_r := Z.mul_reg_r (only parsing).Notation Zmult_succ_l := Z.mul_succ_l (only parsing).Notation Zmult_succ_r := Z.mul_succ_r (only parsing).Notation Zpos_xI := Pos2Z.inj_xI (only parsing).Notation Zpos_xO := Pos2Z.inj_xO (only parsing).Notation Zneg_xI := Pos2Z.neg_xI (only parsing).Notation Zneg_xO := Pos2Z.neg_xO (only parsing).Notation Zopp_neg := Pos2Z.opp_neg (only parsing).Notation Zpos_succ_morphism := Pos2Z.inj_succ (only parsing).Notation Zpos_mult_morphism := Pos2Z.inj_mul (only parsing).Notation Zpos_minus_morphism := Pos2Z.inj_sub (only parsing).Notation Zpos_eq_rev := Pos2Z.inj (only parsing).Notation Zpos_plus_distr := Pos2Z.inj_add (only parsing).Notation Zneg_plus_distr := Pos2Z.add_neg_neg (only parsing).Notation Z := Z (only parsing).Notation Z_rect := Z_rect (only parsing).Notation Z_rec := Z_rec (only parsing).Notation Z_ind := Z_ind (only parsing).Notation Z0 := Z0 (only parsing).Notation Zpos := Zpos (only parsing).Notation Zneg := Zneg (only parsing).(** Compatibility lemmas. These could be notations,
    but scope information would be lost.
*)

Notation SYM1 lem := (fun n => eq_sym (lem n)).Notation SYM2 lem := (fun n m => eq_sym (lem n m)).Notation SYM3 lem := (fun n m p => eq_sym (lem n m p)).

(* Specifications: *)
Lemma Zplus_assoc_reverse : forall n m p, n+m+p = n+(m+p).
Lemma Zplus_succ_r_reverse : forall n m, Z.succ (n+m) = n+Z.succ m.