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 singleton_join_gen: forall k a c (m: ghost) (Hjoin: join (Some a) (nth k m None) (Some c)), join (singleton k a) m (list_set m k c).