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 :=
  | a : symb
  | f : symb
  | g1 : symb
  | g2 : symb
  | h : 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.a => 0
  | M.f => 5
  | M.g1 => 4
  | M.g2 => 4
  | M.h => 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 a := F0 M.a Vnil.
  Definition f x5 x4 x3 x2 x1 := F0 M.f (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))).
  Definition g1 x4 x3 x2 x1 := F0 M.g1 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition g2 x4 x3 x2 x1 := F0 M.g2 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition h x2 x1 := F0 M.h (Vcons x2 (Vcons x1 Vnil)).
End S0.

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

Definition R :=
   R0 (S0.f (V0 0) (V0 1) (V0 2) (V0 2) S0.a)
      (S0.g1 (V0 0) (V0 0) (V0 1) (V0 2))
:: R0 (S0.f (V0 0) (V0 1) (V0 2) S0.a S0.a)
      (S0.g1 (V0 1) (V0 0) (V0 0) (V0 2))
:: R0 (S0.f (V0 0) (V0 1) S0.a S0.a (V0 2))
      (S0.g2 (V0 0) (V0 1) (V0 1) (V0 2))
:: R0 (S0.f (V0 0) (V0 1) S0.a (V0 2) (V0 2))
      (S0.g2 (V0 1) (V0 1) (V0 0) (V0 2))
:: R0 (S0.g1 (V0 0) (V0 0) (V0 2) S0.a)
      (S0.h (V0 0) (V0 2))
:: R0 (S0.g1 (V0 0) (V0 1) (V0 1) S0.a)
      (S0.h (V0 1) (V0 0))
:: R0 (S0.g2 (V0 0) (V0 1) (V0 1) S0.a)
      (S0.h (V0 0) (V0 1))
:: R0 (S0.g2 (V0 0) (V0 0) (V0 2) S0.a)
      (S0.h (V0 2) (V0 0))
:: R0 (S0.h (V0 0) (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 ha := F1 (hd_symb s1_p M.a) Vnil.
  Definition a := F1 (int_symb s1_p M.a) Vnil.
  Definition hf x5 x4 x3 x2 x1 := F1 (hd_symb s1_p M.f) (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))).
  Definition f x5 x4 x3 x2 x1 := F1 (int_symb s1_p M.f) (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))).
  Definition hg1 x4 x3 x2 x1 := F1 (hd_symb s1_p M.g1) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition g1 x4 x3 x2 x1 := F1 (int_symb s1_p M.g1) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hg2 x4 x3 x2 x1 := F1 (hd_symb s1_p M.g2) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition g2 x4 x3 x2 x1 := F1 (int_symb s1_p M.g2) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hh x2 x1 := F1 (hd_symb s1_p M.h) (Vcons x2 (Vcons x1 Vnil)).
  Definition h x2 x1 := F1 (int_symb s1_p M.h) (Vcons x2 (Vcons x1 Vnil)).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hg2 (V1 0) (V1 0) (V1 2) (S1.a))
         (S1.hh (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hg2 (V1 0) (V1 1) (V1 1) (S1.a))
         (S1.hh (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hg1 (V1 0) (V1 1) (V1 1) (S1.a))
         (S1.hh (V1 1) (V1 0))
   :: nil)

:: (  R1 (S1.hg1 (V1 0) (V1 0) (V1 2) (S1.a))
         (S1.hh (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hf (V1 0) (V1 1) (S1.a) (V1 2) (V1 2))
         (S1.hg2 (V1 1) (V1 1) (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hf (V1 0) (V1 1) (S1.a) (S1.a) (V1 2))
         (S1.hg2 (V1 0) (V1 1) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hf (V1 0) (V1 1) (V1 2) (S1.a) (S1.a))
         (S1.hg1 (V1 1) (V1 0) (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hf (V1 0) (V1 1) (V1 2) (V1 2) (S1.a))
         (S1.hg1 (V1 0) (V1 0) (V1 1) (V1 2))
   :: 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.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.