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_inv: forall k a b c, join (singleton k a) (singleton k b) c -> exists c', join a b c' /\ c = singleton k c'.