Require Import ADPUnif.
Require Import ADecomp.
Require Import ADuplicateSymb.
Require Import AGraph.
Require Import ATrs.
Require Import List.
Require Import LogicUtil.
Require Import SN.
Require Import VecUtil.

Open Scope nat_scope.
(* termination problem *)

Module M.
  Inductive symb : Type :=
  | _0_1 : symb
  | activate : symb
  | cons : symb
  | n__zeros : symb
  | tail : symb
  | zeros : symb.
End M.

Lemma eq_symb_dec : forall f g : M.symb, {f=g}+{~f=g}.

Proof.
decide equality.
Defined.

Open Scope nat_scope.
Definition ar (s : M.symb) : nat :=
  match s with
  | M._0_1 => 0
  | M.activate => 1
  | M.cons => 2
  | M.n__zeros => 0
  | M.tail => 1
  | M.zeros => 0
  end.

Definition s0 := ASignature.mkSignature ar eq_symb_dec.
Definition s0_p := s0.
Definition V0 := @ATerm.Var s0.
Definition F0 := @ATerm.Fun s0.
Definition R0 := @ATrs.mkRule s0.

Module S0.
  Definition _0_1 := F0 M._0_1 Vnil.
  Definition activate x1 := F0 M.activate (Vcons x1 Vnil).
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition n__zeros := F0 M.n__zeros Vnil.
  Definition tail x1 := F0 M.tail (Vcons x1 Vnil).
  Definition zeros := F0 M.zeros Vnil.
End S0.

Definition E :=
   @nil (@ATrs.rule s0).

Definition R :=
   R0 S0.zeros
      (S0.cons S0._0_1 S0.n__zeros)
:: R0 (S0.tail (S0.cons (V0 0) (V0 1)))
      (S0.activate (V0 1))
:: R0 S0.zeros
      S0.n__zeros
:: R0 (S0.activate S0.n__zeros)
      S0.zeros
:: R0 (S0.activate (V0 0))
      (V0 0)
:: @nil (@ATrs.rule s0).

Definition rel := ATrs.red_mod E R.

(* symbol marking *)

Definition s1 := dup_sig s0.
Definition s1_p := s0.
Definition V1 := @ATerm.Var s1.
Definition F1 := @ATerm.Fun s1.
Definition R1 := @ATrs.mkRule s1.

Module S1.
  Definition h_0_1 := F1 (hd_symb s1_p M._0_1) Vnil.
  Definition _0_1 := F1 (int_symb s1_p M._0_1) Vnil.
  Definition hactivate x1 := F1 (hd_symb s1_p M.activate) (Vcons x1 Vnil).
  Definition activate x1 := F1 (int_symb s1_p M.activate) (Vcons x1 Vnil).
  Definition hcons x2 x1 := F1 (hd_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F1 (int_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition hn__zeros := F1 (hd_symb s1_p M.n__zeros) Vnil.
  Definition n__zeros := F1 (int_symb s1_p M.n__zeros) Vnil.
  Definition htail x1 := F1 (hd_symb s1_p M.tail) (Vcons x1 Vnil).
  Definition tail x1 := F1 (int_symb s1_p M.tail) (Vcons x1 Vnil).
  Definition hzeros := F1 (hd_symb s1_p M.zeros) Vnil.
  Definition zeros := F1 (int_symb s1_p M.zeros) Vnil.
End S1.

(* graph decomposition 1 *)

Definition cs1 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hactivate (S1.n__zeros))
         (S1.hzeros)
   :: nil)

:: (  R1 (S1.htail (S1.cons (V1 0) (V1 1)))
         (S1.hactivate (V1 1))
   :: nil)

:: nil.

(* termination proof *)

Lemma termination : WF rel.

Proof.
unfold rel.
dp_trans.
mark.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs1; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
Qed.