Definition heap : Type := fpm nat nat.
Instance Join_heap: Join heap := Join_fpm (Join_discrete _).
Instance Perm_heap: Perm_alg heap := Perm_fpm _ (Perm_discrete nat).
Definition mapsto  (x y: nat) : pred heap :=
fun h => forall z, if eq_dec z x then lookup_fpm h z = Some y else lookup_fpm h z = None.
Inductive lis' : nat -> pred heap :=
| lis_nil : forall p w, (!!(p=0) && emp)%pred w -> lis' p w
| lis_cons: forall p w, (!!(p<>0) && EX r:nat, mapsto p r * lis' r)%pred w -> lis' p w.
Definition lisfun (Q: nat -> pred heap) (p: nat)  : pred heap:=
(!!(p=0) && emp || !!(p<>0) && EX r:nat, mapsto p r * Q r)%pred.

Lemma covariant_lisfun: covariant lisfun.