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 :=
  | _0_1 : symb
  | cons : symb
  | first : symb
  | from : symb
  | nil : symb
  | s : 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._0_1 => 0
  | M.cons => 1
  | M.first => 2
  | M.from => 1
  | M.nil => 0
  | M.s => 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 _0_1 := F0 M._0_1 Vnil.
  Definition cons x1 := F0 M.cons (Vcons x1 Vnil).
  Definition first x2 x1 := F0 M.first (Vcons x2 (Vcons x1 Vnil)).
  Definition from x1 := F0 M.from (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.first S0._0_1 (V0 0))
      S0.nil
:: R0 (S0.first (S0.s (V0 0)) (S0.cons (V0 1)))
      (S0.cons (V0 1))
:: R0 (S0.from (V0 0))
      (S0.cons (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 h_0_1 := F1 (hd_symb s1_p M._0_1) Vnil.
  Definition _0_1 := F1 (int_symb s1_p M._0_1) Vnil.
  Definition hcons x1 := F1 (hd_symb s1_p M.cons) (Vcons x1 Vnil).
  Definition cons x1 := F1 (int_symb s1_p M.cons) (Vcons x1 Vnil).
  Definition hfirst x2 x1 := F1 (hd_symb s1_p M.first) (Vcons x2 (Vcons x1 Vnil)).
  Definition first x2 x1 := F1 (int_symb s1_p M.first) (Vcons x2 (Vcons x1 Vnil)).
  Definition hfrom x1 := F1 (hd_symb s1_p M.from) (Vcons x1 Vnil).
  Definition from x1 := F1 (int_symb s1_p M.from) (Vcons x1 Vnil).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hs x1 := F1 (hd_symb s1_p M.s) (Vcons x1 Vnil).
  Definition s x1 := F1 (int_symb s1_p M.s) (Vcons x1 Vnil).
End S1.

(* termination proof *)

Lemma termination : WF rel.

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