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 :=
  | f : symb
  | g : symb
  | p : symb
  | q : 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.f => 1
  | M.g => 1
  | M.p => 1
  | M.q => 1
  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 f x1 := F0 M.f (Vcons x1 Vnil).
  Definition g x1 := F0 M.g (Vcons x1 Vnil).
  Definition p x1 := F0 M.p (Vcons x1 Vnil).
  Definition q x1 := F0 M.q (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.p (S0.f (S0.f (V0 0))))
      (S0.q (S0.f (S0.g (V0 0))))
:: R0 (S0.p (S0.g (S0.g (V0 0))))
      (S0.q (S0.g (S0.f (V0 0))))
:: R0 (S0.q (S0.f (S0.f (V0 0))))
      (S0.p (S0.f (S0.g (V0 0))))
:: R0 (S0.q (S0.g (S0.g (V0 0))))
      (S0.p (S0.g (S0.f (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 hf x1 := F1 (hd_symb s1_p M.f) (Vcons x1 Vnil).
  Definition f x1 := F1 (int_symb s1_p M.f) (Vcons x1 Vnil).
  Definition hg x1 := F1 (hd_symb s1_p M.g) (Vcons x1 Vnil).
  Definition g x1 := F1 (int_symb s1_p M.g) (Vcons x1 Vnil).
  Definition hp x1 := F1 (hd_symb s1_p M.p) (Vcons x1 Vnil).
  Definition p x1 := F1 (int_symb s1_p M.p) (Vcons x1 Vnil).
  Definition hq x1 := F1 (hd_symb s1_p M.q) (Vcons x1 Vnil).
  Definition q x1 := F1 (int_symb s1_p M.q) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hq (S1.g (S1.g (V1 0))))
         (S1.hp (S1.g (S1.f (V1 0))))
   :: nil)

:: (  R1 (S1.hq (S1.f (S1.f (V1 0))))
         (S1.hp (S1.f (S1.g (V1 0))))
   :: nil)

:: (  R1 (S1.hp (S1.g (S1.g (V1 0))))
         (S1.hq (S1.g (S1.f (V1 0))))
   :: nil)

:: (  R1 (S1.hp (S1.f (S1.f (V1 0))))
         (S1.hq (S1.f (S1.g (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.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.