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 :=
  | app : symb
  | cons : symb
  | from : symb
  | nil : symb
  | prefix : symb
  | zWadr : 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.app => 2
  | M.cons => 1
  | M.from => 1
  | M.nil => 0
  | M.prefix => 1
  | M.zWadr => 2
  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 app x2 x1 := F0 M.app (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x1 := F0 M.cons (Vcons x1 Vnil).
  Definition from x1 := F0 M.from (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition prefix x1 := F0 M.prefix (Vcons x1 Vnil).
  Definition zWadr x2 x1 := F0 M.zWadr (Vcons x2 (Vcons x1 Vnil)).
End S0.

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

Definition R :=
   R0 (S0.app S0.nil (V0 0))
      (V0 0)
:: R0 (S0.app (S0.cons (V0 0)) (V0 1))
      (S0.cons (V0 0))
:: R0 (S0.from (V0 0))
      (S0.cons (V0 0))
:: R0 (S0.zWadr S0.nil (V0 0))
      S0.nil
:: R0 (S0.zWadr (V0 0) S0.nil)
      S0.nil
:: R0 (S0.zWadr (S0.cons (V0 0)) (S0.cons (V0 1)))
      (S0.cons (S0.app (V0 1) (S0.cons (V0 0))))
:: R0 (S0.prefix (V0 0))
      (S0.cons S0.nil)
:: @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 happ x2 x1 := F1 (hd_symb s1_p M.app) (Vcons x2 (Vcons x1 Vnil)).
  Definition app x2 x1 := F1 (int_symb s1_p M.app) (Vcons x2 (Vcons x1 Vnil)).
  Definition hcons x1 := F1 (hd_symb s1_p M.cons) (Vcons x1 Vnil).
  Definition cons x1 := F1 (int_symb s1_p M.cons) (Vcons x1 Vnil).
  Definition hfrom x1 := F1 (hd_symb s1_p M.from) (Vcons x1 Vnil).
  Definition from x1 := F1 (int_symb s1_p M.from) (Vcons x1 Vnil).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hprefix x1 := F1 (hd_symb s1_p M.prefix) (Vcons x1 Vnil).
  Definition prefix x1 := F1 (int_symb s1_p M.prefix) (Vcons x1 Vnil).
  Definition hzWadr x2 x1 := F1 (hd_symb s1_p M.zWadr) (Vcons x2 (Vcons x1 Vnil)).
  Definition zWadr x2 x1 := F1 (int_symb s1_p M.zWadr) (Vcons x2 (Vcons x1 Vnil)).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hzWadr (S1.cons (V1 0)) (S1.cons (V1 1)))
         (S1.happ (V1 1) (S1.cons (V1 0)))
   :: 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.
Qed.