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+Require Import Coq.ZArith.ZArith Coq.Classes.RelationClasses.
+
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma ge_refl x : x >= x.
+  Proof. rewrite !Z.ge_le_iff; reflexivity. Qed.
+  Lemma ge_trans n m p : n >= m -> m >= p -> n >= p.
+  Proof. rewrite !Z.ge_le_iff; eauto using Z.le_trans. Qed.
+
+  Global Instance ge_preorder : PreOrder Z.ge.
+  Proof. constructor; hnf; [ apply ge_refl | apply ge_trans ]. Defined.
+End Z.