Definition vlist := list (list Int * form).Fixpoint vlist2list (gamma : vlist) : flist :=
  match gamma with
  | nil => fnil
  | (l, a) :: gamma => vimp l a :: vlist2list gamma
  end.Fixpoint vlist2hlist (gamma : vlist) : flist :=
  match gamma with
  | nil => fnil
  | (l, a) :: gamma => a :: vlist2hlist gamma
  end.Definition search_atom_invariant (n : nat) :=
  forall (goal : Int) (gamma : vlist) (work : nf_list) 
    (context : flist) (j : Int),
  weight_gamma (vlist2hlist gamma) < n ->
  search_spec (Atom goal) (vlist2list gamma) work context j.

Lemma search_atom_aux : forall n : nat, search_atom_invariant n.