show that there are infinitely many locations

theories/heap_lang/locations.v

@@ -11,6 +11,20 @@ Instance loc_inhabited : Inhabited loc := populate {|loc_car := 0 |}.
 Instance loc_countable : Countable loc.
 Proof. by apply (inj_countable' loc_car (λ i, {| loc_car := i |})); intros []. Qed.
 
+Program Definition loc_infinite : Infinite loc := {|
+  infinite_fresh l := {| loc_car := infinite_fresh (loc_car <$> l) |}
+|}.
+Next Obligation.
+  intros xs Hfresh. unfold fresh in Hfresh.
+  eapply (infinite_is_fresh (loc_car <$> xs)).
+  eapply elem_of_list_fmap. eexists. split; last done.
+  reflexivity.
+Qed.
+Next Obligation.
+  intros xs ys Hperm. unfold fresh. f_equal.
+  eapply infinite_fresh_Permutation. rewrite Hperm //.
+Qed.
+
 Definition loc_add (l : loc) (off : Z) : loc :=
   {| loc_car := loc_car l + off|}.