End N.
Bind Scope N_scope with N.t N.
(** Exportation of notations *)
Number Notation N N.of_num_uint N.to_num_uint : N_scope.
Infix "+" := N.add : N_scope.
Infix "-" := N.sub : N_scope.
Infix "*" := N.mul : N_scope.
Infix "^" := N.pow : N_scope.
Infix "?=" := N.compare (at level 70, no associativity) : N_scope.
Infix "<=" := N.le : N_scope.
Infix "<" := N.lt : N_scope.
Infix ">=" := N.ge : N_scope.
Infix ">" := N.gt : N_scope.
Notation "x <= y <= z" := (x <= y /\ y <= z) : N_scope.
Notation "x <= y < z" := (x <= y /\ y < z) : N_scope.
Notation "x < y < z" := (x < y /\ y < z) : N_scope.
Notation "x < y <= z" := (x < y /\ y <= z) : N_scope.
Infix "=?" := N.eqb (at level 70, no associativity) : N_scope.
Infix "<=?" := N.leb (at level 70, no associativity) : N_scope.
Infix "<?" := N.ltb (at level 70, no associativity) : N_scope.
Infix "/" := N.div : N_scope.
Infix "mod" := N.modulo (at level 40, no associativity) : N_scope.
Notation "( p | q )" := (N.divide p q) (at level 0) : N_scope.
(** Compatibility notations *)
Notation N_rect := N_rect (only parsing).
Notation N_rec := N_rec (only parsing).
Notation N_ind := N_ind (only parsing).
Notation N0 := N0 (only parsing).
Notation Npos := N.pos (only parsing).
Notation Ndouble_plus_one := N.succ_double (only parsing).
Notation Nplus := N.add (only parsing).
Notation Nminus := N.sub (only parsing).
Notation Nmult := N.mul (only parsing).
Notation nat_of_N := N.to_nat (only parsing).
Notation N_of_nat := N.of_nat (only parsing).
Notation Nrect := N.peano_rect (only parsing).
Notation Nrect_base := N.peano_rect_base (only parsing).
Notation Nrect_step := N.peano_rect_succ (only parsing).
Notation Nind := N.peano_ind (only parsing).
Notation Nrec := N.peano_rec (only parsing).
Notation Nrec_base := N.peano_rec_base (only parsing).
Notation Nrec_succ := N.peano_rec_succ (only parsing).
Notation Npred_minus := N.pred_sub (only parsing).
Notation Ppred_N_spec := N.pos_pred_spec (only parsing).
Notation Ppred_Nsucc := N.pos_pred_succ (only parsing).
Notation Nplus_0_l := N.add_0_l (only parsing).
Notation Nplus_0_r := N.add_0_r (only parsing).
Notation Nplus_comm := N.add_comm (only parsing).
Notation Nplus_assoc := N.add_assoc (only parsing).
Notation Nplus_succ := N.add_succ_l (only parsing).
Notation Nsucc_0 := N.succ_0_discr (only parsing).
Notation Nminus_N0_Nle := N.sub_0_le (only parsing).
Notation Nminus_0_r := N.sub_0_r (only parsing).
Notation Nminus_succ_r:= N.sub_succ_r (only parsing).
Notation Nmult_0_l := N.mul_0_l (only parsing).
Notation Nmult_1_l := N.mul_1_l (only parsing).
Notation Nmult_1_r := N.mul_1_r (only parsing).
Notation Nmult_comm := N.mul_comm (only parsing).
Notation Nmult_assoc := N.mul_assoc (only parsing).
Notation Nmult_plus_distr_r := N.mul_add_distr_r (only parsing).
Notation Nle_0 := N.le_0_l (only parsing).
Notation Ncompare_Eq_eq := N.compare_eq (only parsing).
Notation Ncompare_eq_correct := N.compare_eq_iff (only parsing).
Notation Nle_lteq := N.lt_eq_cases (only parsing).
Notation Ncompare_0 := N.compare_0_r (only parsing).
Notation Ndouble_div2 := N.div2_double (only parsing).
Notation Ndouble_plus_one_div2 := N.div2_succ_double (only parsing).
Notation Ndouble_plus_one_inj := N.succ_double_inj (only parsing).
Notation Nlt_not_eq := N.lt_neq (only parsing).
Notation Ngt_Nlt := N.gt_lt (only parsing).
(** More complex compatibility facts, expressed as lemmas (to preserve scopes for instance) *)

(* Specifications: *)
Lemma Nplus_reg_l n m p : n + m = n + p -> m = p.
Proof (proj1 (N.add_cancel_l m p n)).
Lemma Nmult_Sn_m n m : N.succ n * m = m + n * m.
Proof (eq_trans (N.mul_succ_l n m) (N.add_comm _ _)).
Lemma Nmult_plus_distr_l n m p : p * (n + m) = p * n + p * m.
Proof (N.mul_add_distr_l p n m).
Lemma Nmult_reg_r n m p : p <> 0 -> n * p = m * p -> n = m.
Proof (fun H => proj1 (N.mul_cancel_r n m p H)).
Lemma Ncompare_antisym n m : CompOpp (n ?= m) = (m ?= n).
Proof (eq_sym (N.compare_antisym n m)).