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
  | c : symb
  | f : 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.c => 0
  | M.f => 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 := F0 M.a Vnil.
  Definition c := F0 M.c Vnil.
  Definition f x1 := F0 M.f (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.f (S0.f S0.a))
      S0.c
:: @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 hc := F1 (hd_symb s1_p M.c) Vnil.
  Definition c := F1 (int_symb s1_p M.c) Vnil.
  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).
End S1.

(* termination proof *)

Lemma termination : WF rel.

Proof.
unfold rel.
dp_trans.
mark.
termination_trivial.
Qed.