Definition singleton {A} k (x : A) : list (option A) := repeat None k ++ Some x :: nil.Definition gname := nat.Definition list_set {A} (m : list (option A)) k v : list (option A) :=
  firstn k m ++ repeat None (k - length m) ++ Some v :: skipn (S k) m.

Lemma ghost_fmap_singleton: forall f g k v, ghost_fmap f g (singleton k v) = singleton k (match v with (a, b) => (a, preds_fmap f g b) end).