Definition Z_le_gt_dec : {x <= y} + {x > y}.Definition Z_gt_le_dec : {x > y} + {x <= y}.Definition Z_ge_lt_dec : {x >= y} + {x < y}.Definition Z_le_lt_eq_dec : x <= y -> {x < y} + {x = y}.

(* Specifications: *)
Lemma Zlt_cotrans : forall n m:Z, n < m -> forall p:Z, {n < p} + {p < m}.
Lemma Zlt_cotrans_pos : forall n m:Z, 0 < n + m -> {0 < n} + {0 < m}.
Lemma Zlt_cotrans_neg : forall n m:Z, n + m < 0 -> {n < 0} + {m < 0}.
Lemma not_Zeq_inf : forall n m:Z, n <> m -> {n < m} + {m < n}.
Lemma Z_dec : forall n m:Z, {n < m} + {n > m} + {n = m}.
Lemma Z_dec' : forall n m:Z, {n < m} + {m < n} + {n = m}.