(* File: NDeco_Sound.v  (last edited on 27/10/2000) (c) Klaus Weich  *)
Require Export NMinimal.
(*****************************************************************)
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.
(*****************************************************************)
intros c work ds ni ai a complete.
unfold deco_sound in |- *.
intros k i0 i1 j in_k.
elim (complete k i0 i1 j in_k); clear complete in_k.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_cons_work_tail with c; assumption.
Qed.
intros work i j ds ni ai a complete.
unfold deco_sound in |- *.
intros k a0 a1 b in_k.
elim (complete k a0 a1 b in_k); clear complete in_k.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_cons_ds_tail with i j; assumption.
Qed.
intros work ds x ni ai a complete.
unfold deco_sound in |- *.
intros k a0 a1 b in_k.
elim (complete k a0 a1 b); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_cons_ni_tail with x; try assumption.
right; assumption.
Qed.
(*****************************************************************)
intros i j work ds ni ai a complete.
unfold deco_sound in |- *.
intros k a0 a1 b in_k.
elim (complete k a0 a1 b in_k); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_ds_work; assumption.
Qed.
intros i j work ds ni ai a complete.
unfold deco_sound in |- *.
intros k a0 a1 b in_k.
elim (complete k a0 a1 b in_k); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_work_ds; assumption.
Qed.
intros x work ds ni ai a complete.
unfold deco_sound in |- *.
intros k a0 a1 b in_k.
elim (complete k a0 a1 b); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_ni_work; assumption.
right; assumption.
Qed.
intros x work ds ni ai a complete.
unfold deco_sound in |- *.
intros k i0 i1 j in_k.
elim (complete k i0 i1 j); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_work_ni; assumption.
inversion_clear in_k.
discriminate H.
assumption.
Qed.
intros i b work ds ni ai ai' a notlookup equiv_ins complete.
unfold deco_sound in |- *.
intros k a0 a1 b0 in_k.
elim (complete k a0 a1 b0 in_k); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_ai_work_new with ai'; assumption.
Qed.
intros i b work ds ni bs ai ai' a lookup equiv_ins complete.
unfold deco_sound in |- *.
intros k a0 a1 b0 in_k.
elim (complete k a0 a1 b0 in_k); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_ai_work_old with bs ai'; assumption.
Qed.
intros i b work ds ni ai ai' a equiv_ins complete.
unfold deco_sound in |- *.
intros k a0 a1 b0 in_k.
elim (complete k a0 a1 b0 in_k); clear complete.
intros k_is_mon k_forces_ngamma k_notforces_i0i1.
apply k_deco_sound_intro; try assumption.
apply forces_ngamma_shift_work_ai with i b ai; assumption.
Qed.

Lemma deco_sound_shift_a_work :
forall (i : Int) (work : nf_list) (ds : disjs) (ni : nested_imps)
(ai : atomic_imps) (a a' : atoms),
EQUIV_INS unit i (fun _ : unit => tt) tt a a' ->
deco_sound work ds ni ai a' -> deco_sound (NAtom i :: work) ds ni ai a.