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
  | b : symb
  | c : symb
  | d : symb
  | e : symb
  | u : symb
  | v : 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 => 1
  | M.b => 1
  | M.c => 1
  | M.d => 1
  | M.e => 1
  | M.u => 1
  | M.v => 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 a x1 := F0 M.a (Vcons x1 Vnil).
  Definition b x1 := F0 M.b (Vcons x1 Vnil).
  Definition c x1 := F0 M.c (Vcons x1 Vnil).
  Definition d x1 := F0 M.d (Vcons x1 Vnil).
  Definition e x1 := F0 M.e (Vcons x1 Vnil).
  Definition u x1 := F0 M.u (Vcons x1 Vnil).
  Definition v x1 := F0 M.v (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.d (V0 0))
      (S0.e (S0.u (V0 0)))
:: R0 (S0.d (S0.u (V0 0)))
      (S0.c (V0 0))
:: R0 (S0.c (S0.u (V0 0)))
      (S0.b (V0 0))
:: R0 (S0.v (S0.e (V0 0)))
      (V0 0)
:: R0 (S0.b (S0.u (V0 0)))
      (S0.a (S0.e (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 x1 := F1 (hd_symb s1_p M.a) (Vcons x1 Vnil).
  Definition a x1 := F1 (int_symb s1_p M.a) (Vcons x1 Vnil).
  Definition hb x1 := F1 (hd_symb s1_p M.b) (Vcons x1 Vnil).
  Definition b x1 := F1 (int_symb s1_p M.b) (Vcons x1 Vnil).
  Definition hc x1 := F1 (hd_symb s1_p M.c) (Vcons x1 Vnil).
  Definition c x1 := F1 (int_symb s1_p M.c) (Vcons x1 Vnil).
  Definition hd x1 := F1 (hd_symb s1_p M.d) (Vcons x1 Vnil).
  Definition d x1 := F1 (int_symb s1_p M.d) (Vcons x1 Vnil).
  Definition he x1 := F1 (hd_symb s1_p M.e) (Vcons x1 Vnil).
  Definition e x1 := F1 (int_symb s1_p M.e) (Vcons x1 Vnil).
  Definition hu x1 := F1 (hd_symb s1_p M.u) (Vcons x1 Vnil).
  Definition u x1 := F1 (int_symb s1_p M.u) (Vcons x1 Vnil).
  Definition hv x1 := F1 (hd_symb s1_p M.v) (Vcons x1 Vnil).
  Definition v x1 := F1 (int_symb s1_p M.v) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hc (S1.u (V1 0)))
         (S1.hb (V1 0))
   :: nil)

:: (  R1 (S1.hd (S1.u (V1 0)))
         (S1.hc (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.
Qed.