(** For these one, used in omega, a Definition is necessary *)

Definition inj_eq := (f_equal Z.of_nat).Definition inj_le n m := proj1 (Nat2Z.inj_le n m).Definition inj_lt n m := proj1 (Nat2Z.inj_lt n m).Definition inj_ge n m := proj1 (Nat2Z.inj_ge n m).Definition inj_gt n m := proj1 (Nat2Z.inj_gt n m).(** For the others, a Notation is fine *)

Notation inj_0 := Nat2Z.inj_0 (only parsing).Notation inj_S := Nat2Z.inj_succ (only parsing).Notation inj_compare := Nat2Z.inj_compare (only parsing).Notation inj_eq_rev := Nat2Z.inj (only parsing).Notation inj_eq_iff := (fun n m => iff_sym (Nat2Z.inj_iff n m)) (only parsing).Notation inj_le_iff := Nat2Z.inj_le (only parsing).Notation inj_lt_iff := Nat2Z.inj_lt (only parsing).Notation inj_ge_iff := Nat2Z.inj_ge (only parsing).Notation inj_gt_iff := Nat2Z.inj_gt (only parsing).Notation inj_le_rev := (fun n m => proj2 (Nat2Z.inj_le n m)) (only parsing).Notation inj_lt_rev := (fun n m => proj2 (Nat2Z.inj_lt n m)) (only parsing).Notation inj_ge_rev := (fun n m => proj2 (Nat2Z.inj_ge n m)) (only parsing).Notation inj_gt_rev := (fun n m => proj2 (Nat2Z.inj_gt n m)) (only parsing).Notation inj_plus := Nat2Z.inj_add (only parsing).Notation inj_mult := Nat2Z.inj_mul (only parsing).Notation inj_minus1 := Nat2Z.inj_sub (only parsing).Notation inj_minus := Nat2Z.inj_sub_max (only parsing).Notation inj_min := Nat2Z.inj_min (only parsing).Notation inj_max := Nat2Z.inj_max (only parsing).Notation inj_div := Nat2Z.inj_div (only parsing).Notation inj_mod := Nat2Z.inj_mod (only parsing).Notation inj_pow := Nat2Z.inj_pow (only parsing).Notation Z_of_nat_of_P := positive_nat_Z (only parsing).Notation Zpos_eq_Z_of_nat_o_nat_of_P :=
  (fun p => eq_sym (positive_nat_Z p)) (only parsing).Notation Z_of_nat_of_N := N_nat_Z (only parsing).Notation Z_of_N_of_nat := nat_N_Z (only parsing).Notation Z_of_N_eq := (f_equal Z.of_N) (only parsing).Notation Z_of_N_eq_rev := N2Z.inj (only parsing).Notation Z_of_N_eq_iff := (fun n m => iff_sym (N2Z.inj_iff n m)) (only parsing).Notation Z_of_N_compare := N2Z.inj_compare (only parsing).Notation Z_of_N_le_iff := N2Z.inj_le (only parsing).Notation Z_of_N_lt_iff := N2Z.inj_lt (only parsing).Notation Z_of_N_ge_iff := N2Z.inj_ge (only parsing).Notation Z_of_N_gt_iff := N2Z.inj_gt (only parsing).Notation Z_of_N_le := (fun n m => proj1 (N2Z.inj_le n m)) (only parsing).Notation Z_of_N_lt := (fun n m => proj1 (N2Z.inj_lt n m)) (only parsing).Notation Z_of_N_ge := (fun n m => proj1 (N2Z.inj_ge n m)) (only parsing).Notation Z_of_N_gt := (fun n m => proj1 (N2Z.inj_gt n m)) (only parsing).Notation Z_of_N_le_rev := (fun n m => proj2 (N2Z.inj_le n m)) (only parsing).Notation Z_of_N_lt_rev := (fun n m => proj2 (N2Z.inj_lt n m)) (only parsing).Notation Z_of_N_ge_rev := (fun n m => proj2 (N2Z.inj_ge n m)) (only parsing).Notation Z_of_N_gt_rev := (fun n m => proj2 (N2Z.inj_gt n m)) (only parsing).Notation Z_of_N_pos := N2Z.inj_pos (only parsing).Notation Z_of_N_abs := N2Z.inj_abs_N (only parsing).Notation Z_of_N_le_0 := N2Z.is_nonneg (only parsing).Notation Z_of_N_plus := N2Z.inj_add (only parsing).Notation Z_of_N_mult := N2Z.inj_mul (only parsing).Notation Z_of_N_minus := N2Z.inj_sub_max (only parsing).Notation Z_of_N_succ := N2Z.inj_succ (only parsing).Notation Z_of_N_min := N2Z.inj_min (only parsing).Notation Z_of_N_max := N2Z.inj_max (only parsing).Notation Zabs_of_N := Zabs2N.id (only parsing).Notation Zabs_N_succ_abs := Zabs2N.inj_succ_abs (only parsing).Notation Zabs_N_succ := Zabs2N.inj_succ (only parsing).Notation Zabs_N_plus_abs := Zabs2N.inj_add_abs (only parsing).Notation Zabs_N_plus := Zabs2N.inj_add (only parsing).Notation Zabs_N_mult_abs := Zabs2N.inj_mul_abs (only parsing).Notation Zabs_N_mult := Zabs2N.inj_mul (only parsing).

Theorem inj_minus2 : forall n m:nat, (m > n)%nat -> Z.of_nat (n - m) = 0.