Inductive search_spec_aux (goal : form) (gamma : flist) 
(work : nf_list) (context : flist) : Set :=
  | derivable :
      Derivable context goal -> search_spec_aux goal gamma work context
  | refutable :
      forall k : kripke_tree,
      Is_Monotone_kripke_tree k ->
      forces_gamma gamma work k ->
      (forces_t k goal -> False) -> search_spec_aux goal gamma work context.Inductive search_spec_aux (goal : form) (gamma : flist) 
(work : nf_list) (context : flist) : Set :=
  | derivable :
      Derivable context goal -> search_spec_aux goal gamma work context
  | refutable :
      forall k : kripke_tree,
      Is_Monotone_kripke_tree k ->
      forces_gamma gamma work k ->
      (forces_t k goal -> False) -> search_spec_aux goal gamma work context.Inductive search_spec_aux (goal : form) (gamma : flist) 
(work : nf_list) (context : flist) : Set :=
  | derivable :
      Derivable context goal -> search_spec_aux goal gamma work context
  | refutable :
      forall k : kripke_tree,
      Is_Monotone_kripke_tree k ->
      forces_gamma gamma work k ->
      (forces_t k goal -> False) -> search_spec_aux goal gamma work context.Definition search_spec (goal : form) (gamma : flist) 
  (work : nf_list) (context : flist) (i : Int) :=
  below_form goal i ->
  below_list gamma i ->
  below_list context i ->
  sound gamma work context ->
  minimal gamma work context -> search_spec_aux goal gamma work context.

Lemma rule_atom_imp_atom_imp_c_gamma : forall (goal : form) (l : list Int) (a b : Int) (c : form) (gamma : flist) (work : nf_list) (context : flist) (j j1 : Int), Less j j1 -> search_spec goal (Imp (Atom j) c :: gamma) (nvimp l (NImp_NF (NImp a b (NAtom j))) :: work) (vimp l (Imp (Imp (Atom a) (Atom b)) (Atom j)) :: Imp (Atom j) c :: context) j1 -> search_spec goal (vimp l (Imp (Imp (Atom a) (Atom b)) c) :: gamma) work context j.