Definition Ropp_Ropp_IZR := opp_IZR.

(* Specifications: *)
Lemma minus_IZR : forall n m:Z, IZR (n - m) = IZR n - IZR m.
Lemma Z_R_minus : forall n m:Z, IZR n - IZR m = IZR (n - m).
Lemma lt_0_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z.
Lemma lt_IZR : forall n m:Z, IZR n < IZR m -> (n < m)%Z.
Lemma eq_IZR_R0 : forall n:Z, IZR n = 0 -> n = 0%Z.
Lemma eq_IZR : forall n m:Z, IZR n = IZR m -> n = m.
Lemma not_0_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0.
Lemma le_0_IZR : forall n:Z, 0 <= IZR n -> (0 <= n)%Z.
Lemma le_IZR : forall n m:Z, IZR n <= IZR m -> (n <= m)%Z.
Lemma le_IZR_R1 : forall n:Z, IZR n <= 1 -> (n <= 1)%Z.
Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m.
Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m.
Lemma IZR_lt : forall n m:Z, (n < m)%Z -> IZR n < IZR m.
Lemma eq_IZR_contrapositive : forall n m:Z, n <> m -> IZR n <> IZR m.
Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z.
Lemma one_IZR_r_R1 : forall r (n m:Z), r < IZR n <= r + 1 -> r < IZR m <= r + 1 -> n = m.
Lemma INR_unbounded : forall A, exists n, INR n > A.
Lemma INR_archimed : forall eps A : R, eps > 0 -> exists n : nat, (INR n) * eps > A.
Lemma R_rm : ring_morph 0%R 1%R Rplus Rmult Rminus Ropp eq 0%Z 1%Z Zplus Zmult Zminus Z.opp Zeq_bool IZR.
Lemma Zeq_bool_IZR : forall x y:Z, IZR x = IZR y -> Zeq_bool x y = true.