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 :=
  | _1_2 : symb
  | _dot__1 : symb
  | i : 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._1_2 => 0
  | M._dot__1 => 2
  | M.i => 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 _1_2 := F0 M._1_2 Vnil.
  Definition _dot__1 x2 x1 := F0 M._dot__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition i x1 := F0 M.i (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0._dot__1 S0._1_2 (V0 0))
      (V0 0)
:: R0 (S0._dot__1 (V0 0) S0._1_2)
      (V0 0)
:: R0 (S0._dot__1 (S0.i (V0 0)) (V0 0))
      S0._1_2
:: R0 (S0._dot__1 (V0 0) (S0.i (V0 0)))
      S0._1_2
:: R0 (S0.i S0._1_2)
      S0._1_2
:: R0 (S0.i (S0.i (V0 0)))
      (V0 0)
:: R0 (S0._dot__1 (S0.i (V0 0)) (S0._dot__1 (V0 0) (V0 2)))
      (V0 2)
:: R0 (S0._dot__1 (V0 0) (S0._dot__1 (S0.i (V0 0)) (V0 2)))
      (V0 2)
:: @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 h_1_2 := F1 (hd_symb s1_p M._1_2) Vnil.
  Definition _1_2 := F1 (int_symb s1_p M._1_2) Vnil.
  Definition h_dot__1 x2 x1 := F1 (hd_symb s1_p M._dot__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _dot__1 x2 x1 := F1 (int_symb s1_p M._dot__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition hi x1 := F1 (hd_symb s1_p M.i) (Vcons x1 Vnil).
  Definition i x1 := F1 (int_symb s1_p M.i) (Vcons x1 Vnil).
End S1.

(* termination proof *)

Lemma termination : WF rel.

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