Require Import ADPUnif.
Require Import ADecomp.
Require Import ADuplicateSymb.
Require Import AGraph.
Require Import APolyInt_MA.
Require Import ATrs.
Require Import List.
Require Import LogicUtil.
Require Import MonotonePolynom.
Require Import Polynom.
Require Import SN.
Require Import VecUtil.

Open Scope nat_scope.
(* termination problem *)

Module M.
  Inductive symb : Type :=
  | Cons : symb
  | False : symb
  | Nil : symb
  | Pair : symb
  | True : symb
  | append : symb
  | match_0 : symb
  | match_1 : symb
  | match_2 : symb
  | match_3 : symb
  | match_4 : symb
  | match_5 : symb
  | part : symb
  | quick : symb
  | test : 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.Cons => 2
  | M.False => 0
  | M.Nil => 0
  | M.Pair => 2
  | M.True => 0
  | M.append => 2
  | M.match_0 => 3
  | M.match_1 => 3
  | M.match_2 => 5
  | M.match_3 => 7
  | M.match_4 => 2
  | M.match_5 => 4
  | M.part => 2
  | M.quick => 1
  | M.test => 2
  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 Cons x2 x1 := F0 M.Cons (Vcons x2 (Vcons x1 Vnil)).
  Definition False := F0 M.False Vnil.
  Definition Nil := F0 M.Nil Vnil.
  Definition Pair x2 x1 := F0 M.Pair (Vcons x2 (Vcons x1 Vnil)).
  Definition True := F0 M.True Vnil.
  Definition append x2 x1 := F0 M.append (Vcons x2 (Vcons x1 Vnil)).
  Definition match_0 x3 x2 x1 := F0 M.match_0 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition match_1 x3 x2 x1 := F0 M.match_1 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition match_2 x5 x4 x3 x2 x1 := F0 M.match_2 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))).
  Definition match_3 x7 x6 x5 x4 x3 x2 x1 := F0 M.match_3 (Vcons x7 (Vcons x6 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))))).
  Definition match_4 x2 x1 := F0 M.match_4 (Vcons x2 (Vcons x1 Vnil)).
  Definition match_5 x4 x3 x2 x1 := F0 M.match_5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition part x2 x1 := F0 M.part (Vcons x2 (Vcons x1 Vnil)).
  Definition quick x1 := F0 M.quick (Vcons x1 Vnil).
  Definition test x2 x1 := F0 M.test (Vcons x2 (Vcons x1 Vnil)).
End S0.

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

Definition R :=
   R0 (S0.test (V0 0) (V0 1))
      S0.True
:: R0 (S0.test (V0 0) (V0 1))
      S0.False
:: R0 (S0.append (V0 0) (V0 1))
      (S0.match_0 (V0 0) (V0 1) (V0 0))
:: R0 (S0.match_0 (V0 0) (V0 1) S0.Nil)
      (V0 1)
:: R0 (S0.match_0 (V0 0) (V0 1) (S0.Cons (V0 2) (V0 3)))
      (S0.Cons (V0 2) (S0.append (V0 3) (V0 1)))
:: R0 (S0.part (V0 0) (V0 1))
      (S0.match_1 (V0 0) (V0 1) (V0 1))
:: R0 (S0.match_1 (V0 0) (V0 1) S0.Nil)
      (S0.Pair S0.Nil S0.Nil)
:: R0 (S0.match_1 (V0 0) (V0 1) (S0.Cons (V0 2) (V0 3)))
      (S0.match_2 (V0 2) (V0 3) (V0 0) (V0 1) (S0.part (V0 0) (V0 3)))
:: R0 (S0.match_2 (V0 0) (V0 1) (V0 2) (V0 3) (S0.Pair (V0 4) (V0 5)))
      (S0.match_3 (V0 4) (V0 5) (V0 0) (V0 1) (V0 2) (V0 3) (S0.test (V0 2) (V0 0)))
:: R0 (S0.match_3 (V0 0) (V0 1) (V0 2) (V0 3) (V0 4) (V0 5) S0.False)
      (S0.Pair (S0.Cons (V0 2) (V0 0)) (V0 1))
:: R0 (S0.match_3 (V0 0) (V0 1) (V0 2) (V0 3) (V0 4) (V0 5) S0.True)
      (S0.Pair (V0 0) (S0.Cons (V0 2) (V0 1)))
:: R0 (S0.quick (V0 0))
      (S0.match_4 (V0 0) (V0 0))
:: R0 (S0.match_4 (V0 0) S0.Nil)
      S0.Nil
:: R0 (S0.match_4 (V0 0) (S0.Cons (V0 1) (V0 2)))
      (S0.match_5 (V0 1) (V0 2) (V0 0) (S0.part (V0 1) (V0 2)))
:: R0 (S0.match_5 (V0 0) (V0 1) (V0 2) (S0.Pair (V0 3) (V0 4)))
      (S0.append (S0.quick (V0 3)) (S0.Cons (V0 0) (S0.quick (V0 4))))
:: @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 hCons x2 x1 := F1 (hd_symb s1_p M.Cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition Cons x2 x1 := F1 (int_symb s1_p M.Cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition hFalse := F1 (hd_symb s1_p M.False) Vnil.
  Definition False := F1 (int_symb s1_p M.False) Vnil.
  Definition hNil := F1 (hd_symb s1_p M.Nil) Vnil.
  Definition Nil := F1 (int_symb s1_p M.Nil) Vnil.
  Definition hPair x2 x1 := F1 (hd_symb s1_p M.Pair) (Vcons x2 (Vcons x1 Vnil)).
  Definition Pair x2 x1 := F1 (int_symb s1_p M.Pair) (Vcons x2 (Vcons x1 Vnil)).
  Definition hTrue := F1 (hd_symb s1_p M.True) Vnil.
  Definition True := F1 (int_symb s1_p M.True) Vnil.
  Definition happend x2 x1 := F1 (hd_symb s1_p M.append) (Vcons x2 (Vcons x1 Vnil)).
  Definition append x2 x1 := F1 (int_symb s1_p M.append) (Vcons x2 (Vcons x1 Vnil)).
  Definition hmatch_0 x3 x2 x1 := F1 (hd_symb s1_p M.match_0) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition match_0 x3 x2 x1 := F1 (int_symb s1_p M.match_0) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hmatch_1 x3 x2 x1 := F1 (hd_symb s1_p M.match_1) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition match_1 x3 x2 x1 := F1 (int_symb s1_p M.match_1) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hmatch_2 x5 x4 x3 x2 x1 := F1 (hd_symb s1_p M.match_2) (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))).
  Definition match_2 x5 x4 x3 x2 x1 := F1 (int_symb s1_p M.match_2) (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))).
  Definition hmatch_3 x7 x6 x5 x4 x3 x2 x1 := F1 (hd_symb s1_p M.match_3) (Vcons x7 (Vcons x6 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))))).
  Definition match_3 x7 x6 x5 x4 x3 x2 x1 := F1 (int_symb s1_p M.match_3) (Vcons x7 (Vcons x6 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))))))).
  Definition hmatch_4 x2 x1 := F1 (hd_symb s1_p M.match_4) (Vcons x2 (Vcons x1 Vnil)).
  Definition match_4 x2 x1 := F1 (int_symb s1_p M.match_4) (Vcons x2 (Vcons x1 Vnil)).
  Definition hmatch_5 x4 x3 x2 x1 := F1 (hd_symb s1_p M.match_5) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition match_5 x4 x3 x2 x1 := F1 (int_symb s1_p M.match_5) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hpart x2 x1 := F1 (hd_symb s1_p M.part) (Vcons x2 (Vcons x1 Vnil)).
  Definition part x2 x1 := F1 (int_symb s1_p M.part) (Vcons x2 (Vcons x1 Vnil)).
  Definition hquick x1 := F1 (hd_symb s1_p M.quick) (Vcons x1 Vnil).
  Definition quick x1 := F1 (int_symb s1_p M.quick) (Vcons x1 Vnil).
  Definition htest x2 x1 := F1 (hd_symb s1_p M.test) (Vcons x2 (Vcons x1 Vnil)).
  Definition test x2 x1 := F1 (int_symb s1_p M.test) (Vcons x2 (Vcons x1 Vnil)).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmatch_2 (V1 0) (V1 1) (V1 2) (V1 3) (S1.Pair (V1 4) (V1 5)))
         (S1.htest (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hmatch_2 (V1 0) (V1 1) (V1 2) (V1 3) (S1.Pair (V1 4) (V1 5)))
         (S1.hmatch_3 (V1 4) (V1 5) (V1 0) (V1 1) (V1 2) (V1 3) (S1.test (V1 2) (V1 0)))
   :: nil)

:: (  R1 (S1.hmatch_1 (V1 0) (V1 1) (S1.Cons (V1 2) (V1 3)))
         (S1.hmatch_2 (V1 2) (V1 3) (V1 0) (V1 1) (S1.part (V1 0) (V1 3)))
   :: nil)

:: (  R1 (S1.hmatch_1 (V1 0) (V1 1) (S1.Cons (V1 2) (V1 3)))
         (S1.hpart (V1 0) (V1 3))
   :: R1 (S1.hpart (V1 0) (V1 1))
         (S1.hmatch_1 (V1 0) (V1 1) (V1 1))
   :: nil)

:: (  R1 (S1.hmatch_4 (V1 0) (S1.Cons (V1 1) (V1 2)))
         (S1.hpart (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hmatch_0 (V1 0) (V1 1) (S1.Cons (V1 2) (V1 3)))
         (S1.happend (V1 3) (V1 1))
   :: R1 (S1.happend (V1 0) (V1 1))
         (S1.hmatch_0 (V1 0) (V1 1) (V1 0))
   :: nil)

:: (  R1 (S1.hmatch_5 (V1 0) (V1 1) (V1 2) (S1.Pair (V1 3) (V1 4)))
         (S1.happend (S1.quick (V1 3)) (S1.Cons (V1 0) (S1.quick (V1 4))))
   :: nil)

:: (  R1 (S1.hmatch_5 (V1 0) (V1 1) (V1 2) (S1.Pair (V1 3) (V1 4)))
         (S1.hquick (V1 3))
   :: R1 (S1.hquick (V1 0))
         (S1.hmatch_4 (V1 0) (V1 0))
   :: R1 (S1.hmatch_4 (V1 0) (S1.Cons (V1 1) (V1 2)))
         (S1.hmatch_5 (V1 1) (V1 2) (V1 0) (S1.part (V1 1) (V1 2)))
   :: R1 (S1.hmatch_5 (V1 0) (V1 1) (V1 2) (S1.Pair (V1 3) (V1 4)))
         (S1.hquick (V1 4))
   :: nil)

:: nil.

(* polynomial interpretation 1 *)

Module PIS1 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.test) =>
         nil
    | (int_symb M.test) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.True) =>
         nil
    | (int_symb M.True) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.False) =>
         nil
    | (int_symb M.False) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_0) =>
         nil
    | (int_symb M.match_0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.Nil) =>
         nil
    | (int_symb M.Nil) =>
         nil
    | (hd_symb M.Cons) =>
         nil
    | (int_symb M.Cons) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.part) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.part) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_1) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.match_1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.Pair) =>
         nil
    | (int_symb M.Pair) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_2) =>
         nil
    | (int_symb M.match_2) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))))
      :: nil
    | (hd_symb M.match_3) =>
         nil
    | (int_symb M.match_3) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))))))
      :: nil
    | (hd_symb M.quick) =>
         nil
    | (int_symb M.quick) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.match_4) =>
         nil
    | (int_symb M.match_4) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_5) =>
         nil
    | (int_symb M.match_5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS1.

Module PI1 := PolyInt PIS1.

(* polynomial interpretation 2 *)

Module PIS2 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.test) =>
         nil
    | (int_symb M.test) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.True) =>
         nil
    | (int_symb M.True) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.False) =>
         nil
    | (int_symb M.False) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.append) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_0) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.match_0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.Nil) =>
         nil
    | (int_symb M.Nil) =>
         nil
    | (hd_symb M.Cons) =>
         nil
    | (int_symb M.Cons) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.part) =>
         nil
    | (int_symb M.part) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_1) =>
         nil
    | (int_symb M.match_1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.Pair) =>
         nil
    | (int_symb M.Pair) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_2) =>
         nil
    | (int_symb M.match_2) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))))
      :: nil
    | (hd_symb M.match_3) =>
         nil
    | (int_symb M.match_3) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))))))
      :: nil
    | (hd_symb M.quick) =>
         nil
    | (int_symb M.quick) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.match_4) =>
         nil
    | (int_symb M.match_4) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_5) =>
         nil
    | (int_symb M.match_5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS2.

Module PI2 := PolyInt PIS2.

(* polynomial interpretation 3 *)

Module PIS3 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.test) =>
         nil
    | (int_symb M.test) =>
         nil
    | (hd_symb M.True) =>
         nil
    | (int_symb M.True) =>
         nil
    | (hd_symb M.False) =>
         nil
    | (int_symb M.False) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_0) =>
         nil
    | (int_symb M.match_0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.Nil) =>
         nil
    | (int_symb M.Nil) =>
         nil
    | (hd_symb M.Cons) =>
         nil
    | (int_symb M.Cons) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.part) =>
         nil
    | (int_symb M.part) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_1) =>
         nil
    | (int_symb M.match_1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.Pair) =>
         nil
    | (int_symb M.Pair) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_2) =>
         nil
    | (int_symb M.match_2) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))))
      :: nil
    | (hd_symb M.match_3) =>
         nil
    | (int_symb M.match_3) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))))))
      :: nil
    | (hd_symb M.quick) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.quick) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.match_4) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.match_4) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.match_5) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (int_symb M.match_5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS3.

Module PI3 := PolyInt PIS3.

(* graph decomposition 2 *)

Definition cs2 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hquick (V1 0))
         (S1.hmatch_4 (V1 0) (V1 0))
   :: nil)

:: (  R1 (S1.hmatch_5 (V1 0) (V1 1) (V1 2) (S1.Pair (V1 3) (V1 4)))
         (S1.hquick (V1 4))
   :: nil)

:: (  R1 (S1.hmatch_5 (V1 0) (V1 1) (V1 2) (S1.Pair (V1 3) (V1 4)))
         (S1.hquick (V1 3))
   :: 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.
right. PI1.prove_termination.
termination_trivial.
left. co_scc.
right. PI2.prove_termination.
termination_trivial.
left. co_scc.
right. PI3.prove_termination.
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) cs2; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.