(*****************************************************************)

Inductive k_deco_sound (k : kripke_tree) (i0 i1 : Int) 
(work : nf_list) (ds : disjs) (ni : nested_imps) (ai : atomic_imps)
(a : atoms) : Prop :=
    k_deco_sound_intro :
      Is_Monotone_kripke_tree k ->
      forces_ngamma work ds ni ai a k ->
      (forces_t k (Imp (Atom i0) (Atom i1)) -> False) ->
      k_deco_sound k i0 i1 work ds ni ai a.(*****************************************************************)

Inductive k_deco_sound (k : kripke_tree) (i0 i1 : Int) 
(work : nf_list) (ds : disjs) (ni : nested_imps) (ai : atomic_imps)
(a : atoms) : Prop :=
    k_deco_sound_intro :
      Is_Monotone_kripke_tree k ->
      forces_ngamma work ds ni ai a k ->
      (forces_t k (Imp (Atom i0) (Atom i1)) -> False) ->
      k_deco_sound k i0 i1 work ds ni ai a.Definition deco_sound (work : nf_list) (ds : disjs) 
  (ni : nested_imps) (ai : atomic_imps) (a : atoms) :=
  forall (k : kripke_tree) (i0 i1 : Int) (b : normal_form),
  In (Decorated (NImp i0 i1 b) k) ni -> k_deco_sound k i0 i1 work ds ni ai a.

Lemma deco_sound_shift_ni_x_ni_work : forall (x : nested_imp) (work : nf_list) (ds : disjs) (ni1 ni2 : nested_imps) (ai : atomic_imps) (a : atoms), deco_sound work ds (ni1 ++ x :: ni2) ai a -> deco_sound (NImp_NF (nested_imp2nimp x) :: work) ds (ni1 ++ ni2) ai a.