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 :=
  | _0_2 : symb
  | _1_3 : symb
  | _2_4 : symb
  | _3_5 : symb
  | _4_6 : symb
  | _5_7 : symb
  | _6_8 : symb
  | _7_9 : symb
  | _8_10 : symb
  | _9_11 : symb
  | _plus__1 : symb
  | c : 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_2 => 0
  | M._1_3 => 0
  | M._2_4 => 0
  | M._3_5 => 0
  | M._4_6 => 0
  | M._5_7 => 0
  | M._6_8 => 0
  | M._7_9 => 0
  | M._8_10 => 0
  | M._9_11 => 0
  | M._plus__1 => 2
  | M.c => 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 _0_2 := F0 M._0_2 Vnil.
  Definition _1_3 := F0 M._1_3 Vnil.
  Definition _2_4 := F0 M._2_4 Vnil.
  Definition _3_5 := F0 M._3_5 Vnil.
  Definition _4_6 := F0 M._4_6 Vnil.
  Definition _5_7 := F0 M._5_7 Vnil.
  Definition _6_8 := F0 M._6_8 Vnil.
  Definition _7_9 := F0 M._7_9 Vnil.
  Definition _8_10 := F0 M._8_10 Vnil.
  Definition _9_11 := F0 M._9_11 Vnil.
  Definition _plus__1 x2 x1 := F0 M._plus__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition c x2 x1 := F0 M.c (Vcons x2 (Vcons x1 Vnil)).
End S0.

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

Definition R :=
   R0 (S0._plus__1 S0._0_2 S0._0_2)
      S0._0_2
:: R0 (S0._plus__1 S0._0_2 S0._1_3)
      S0._1_3
:: R0 (S0._plus__1 S0._0_2 S0._2_4)
      S0._2_4
:: R0 (S0._plus__1 S0._0_2 S0._3_5)
      S0._3_5
:: R0 (S0._plus__1 S0._0_2 S0._4_6)
      S0._4_6
:: R0 (S0._plus__1 S0._0_2 S0._5_7)
      S0._5_7
:: R0 (S0._plus__1 S0._0_2 S0._6_8)
      S0._6_8
:: R0 (S0._plus__1 S0._0_2 S0._7_9)
      S0._7_9
:: R0 (S0._plus__1 S0._0_2 S0._8_10)
      S0._8_10
:: R0 (S0._plus__1 S0._0_2 S0._9_11)
      S0._9_11
:: R0 (S0._plus__1 S0._1_3 S0._0_2)
      S0._1_3
:: R0 (S0._plus__1 S0._1_3 S0._1_3)
      S0._2_4
:: R0 (S0._plus__1 S0._1_3 S0._2_4)
      S0._3_5
:: R0 (S0._plus__1 S0._1_3 S0._3_5)
      S0._4_6
:: R0 (S0._plus__1 S0._1_3 S0._4_6)
      S0._5_7
:: R0 (S0._plus__1 S0._1_3 S0._5_7)
      S0._6_8
:: R0 (S0._plus__1 S0._1_3 S0._6_8)
      S0._7_9
:: R0 (S0._plus__1 S0._1_3 S0._7_9)
      S0._8_10
:: R0 (S0._plus__1 S0._1_3 S0._8_10)
      S0._9_11
:: R0 (S0._plus__1 S0._1_3 S0._9_11)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._2_4 S0._0_2)
      S0._2_4
:: R0 (S0._plus__1 S0._2_4 S0._1_3)
      S0._3_5
:: R0 (S0._plus__1 S0._2_4 S0._2_4)
      S0._4_6
:: R0 (S0._plus__1 S0._2_4 S0._3_5)
      S0._5_7
:: R0 (S0._plus__1 S0._2_4 S0._4_6)
      S0._6_8
:: R0 (S0._plus__1 S0._2_4 S0._5_7)
      S0._7_9
:: R0 (S0._plus__1 S0._2_4 S0._6_8)
      S0._8_10
:: R0 (S0._plus__1 S0._2_4 S0._7_9)
      S0._9_11
:: R0 (S0._plus__1 S0._2_4 S0._8_10)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._2_4 S0._9_11)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._3_5 S0._0_2)
      S0._3_5
:: R0 (S0._plus__1 S0._3_5 S0._1_3)
      S0._4_6
:: R0 (S0._plus__1 S0._3_5 S0._2_4)
      S0._5_7
:: R0 (S0._plus__1 S0._3_5 S0._3_5)
      S0._6_8
:: R0 (S0._plus__1 S0._3_5 S0._4_6)
      S0._7_9
:: R0 (S0._plus__1 S0._3_5 S0._5_7)
      S0._8_10
:: R0 (S0._plus__1 S0._3_5 S0._6_8)
      S0._9_11
:: R0 (S0._plus__1 S0._3_5 S0._7_9)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._3_5 S0._8_10)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._3_5 S0._9_11)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._4_6 S0._0_2)
      S0._4_6
:: R0 (S0._plus__1 S0._4_6 S0._1_3)
      S0._5_7
:: R0 (S0._plus__1 S0._4_6 S0._2_4)
      S0._6_8
:: R0 (S0._plus__1 S0._4_6 S0._3_5)
      S0._7_9
:: R0 (S0._plus__1 S0._4_6 S0._4_6)
      S0._8_10
:: R0 (S0._plus__1 S0._4_6 S0._5_7)
      S0._9_11
:: R0 (S0._plus__1 S0._4_6 S0._6_8)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._4_6 S0._7_9)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._4_6 S0._8_10)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._4_6 S0._9_11)
      (S0.c S0._1_3 S0._3_5)
:: R0 (S0._plus__1 S0._5_7 S0._0_2)
      S0._5_7
:: R0 (S0._plus__1 S0._5_7 S0._1_3)
      S0._6_8
:: R0 (S0._plus__1 S0._5_7 S0._2_4)
      S0._7_9
:: R0 (S0._plus__1 S0._5_7 S0._3_5)
      S0._8_10
:: R0 (S0._plus__1 S0._5_7 S0._4_6)
      S0._9_11
:: R0 (S0._plus__1 S0._5_7 S0._5_7)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._5_7 S0._6_8)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._5_7 S0._7_9)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._5_7 S0._8_10)
      (S0.c S0._1_3 S0._3_5)
:: R0 (S0._plus__1 S0._5_7 S0._9_11)
      (S0.c S0._1_3 S0._4_6)
:: R0 (S0._plus__1 S0._6_8 S0._0_2)
      S0._6_8
:: R0 (S0._plus__1 S0._6_8 S0._1_3)
      S0._7_9
:: R0 (S0._plus__1 S0._6_8 S0._2_4)
      S0._8_10
:: R0 (S0._plus__1 S0._6_8 S0._3_5)
      S0._9_11
:: R0 (S0._plus__1 S0._6_8 S0._4_6)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._6_8 S0._5_7)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._6_8 S0._6_8)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._6_8 S0._7_9)
      (S0.c S0._1_3 S0._3_5)
:: R0 (S0._plus__1 S0._6_8 S0._8_10)
      (S0.c S0._1_3 S0._4_6)
:: R0 (S0._plus__1 S0._6_8 S0._9_11)
      (S0.c S0._1_3 S0._5_7)
:: R0 (S0._plus__1 S0._7_9 S0._0_2)
      S0._7_9
:: R0 (S0._plus__1 S0._7_9 S0._1_3)
      S0._8_10
:: R0 (S0._plus__1 S0._7_9 S0._2_4)
      S0._9_11
:: R0 (S0._plus__1 S0._7_9 S0._3_5)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._7_9 S0._4_6)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._7_9 S0._5_7)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._7_9 S0._6_8)
      (S0.c S0._1_3 S0._3_5)
:: R0 (S0._plus__1 S0._7_9 S0._7_9)
      (S0.c S0._1_3 S0._4_6)
:: R0 (S0._plus__1 S0._7_9 S0._8_10)
      (S0.c S0._1_3 S0._5_7)
:: R0 (S0._plus__1 S0._7_9 S0._9_11)
      (S0.c S0._1_3 S0._6_8)
:: R0 (S0._plus__1 S0._8_10 S0._0_2)
      S0._8_10
:: R0 (S0._plus__1 S0._8_10 S0._1_3)
      S0._9_11
:: R0 (S0._plus__1 S0._8_10 S0._2_4)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._8_10 S0._3_5)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._8_10 S0._4_6)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._8_10 S0._5_7)
      (S0.c S0._1_3 S0._3_5)
:: R0 (S0._plus__1 S0._8_10 S0._6_8)
      (S0.c S0._1_3 S0._4_6)
:: R0 (S0._plus__1 S0._8_10 S0._7_9)
      (S0.c S0._1_3 S0._5_7)
:: R0 (S0._plus__1 S0._8_10 S0._8_10)
      (S0.c S0._1_3 S0._6_8)
:: R0 (S0._plus__1 S0._8_10 S0._9_11)
      (S0.c S0._1_3 S0._7_9)
:: R0 (S0._plus__1 S0._9_11 S0._0_2)
      S0._9_11
:: R0 (S0._plus__1 S0._9_11 S0._1_3)
      (S0.c S0._1_3 S0._0_2)
:: R0 (S0._plus__1 S0._9_11 S0._2_4)
      (S0.c S0._1_3 S0._1_3)
:: R0 (S0._plus__1 S0._9_11 S0._3_5)
      (S0.c S0._1_3 S0._2_4)
:: R0 (S0._plus__1 S0._9_11 S0._4_6)
      (S0.c S0._1_3 S0._3_5)
:: R0 (S0._plus__1 S0._9_11 S0._5_7)
      (S0.c S0._1_3 S0._4_6)
:: R0 (S0._plus__1 S0._9_11 S0._6_8)
      (S0.c S0._1_3 S0._5_7)
:: R0 (S0._plus__1 S0._9_11 S0._7_9)
      (S0.c S0._1_3 S0._6_8)
:: R0 (S0._plus__1 S0._9_11 S0._8_10)
      (S0.c S0._1_3 S0._7_9)
:: R0 (S0._plus__1 S0._9_11 S0._9_11)
      (S0.c S0._1_3 S0._8_10)
:: R0 (S0._plus__1 (V0 0) (S0.c (V0 1) (V0 2)))
      (S0.c (V0 1) (S0._plus__1 (V0 0) (V0 2)))
:: R0 (S0._plus__1 (S0.c (V0 0) (V0 1)) (V0 2))
      (S0.c (V0 0) (S0._plus__1 (V0 1) (V0 2)))
:: R0 (S0.c S0._0_2 (V0 0))
      (V0 0)
:: R0 (S0.c (V0 0) (S0.c (V0 1) (V0 2)))
      (S0.c (S0._plus__1 (V0 0) (V0 1)) (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_0_2 := F1 (hd_symb s1_p M._0_2) Vnil.
  Definition _0_2 := F1 (int_symb s1_p M._0_2) Vnil.
  Definition h_1_3 := F1 (hd_symb s1_p M._1_3) Vnil.
  Definition _1_3 := F1 (int_symb s1_p M._1_3) Vnil.
  Definition h_2_4 := F1 (hd_symb s1_p M._2_4) Vnil.
  Definition _2_4 := F1 (int_symb s1_p M._2_4) Vnil.
  Definition h_3_5 := F1 (hd_symb s1_p M._3_5) Vnil.
  Definition _3_5 := F1 (int_symb s1_p M._3_5) Vnil.
  Definition h_4_6 := F1 (hd_symb s1_p M._4_6) Vnil.
  Definition _4_6 := F1 (int_symb s1_p M._4_6) Vnil.
  Definition h_5_7 := F1 (hd_symb s1_p M._5_7) Vnil.
  Definition _5_7 := F1 (int_symb s1_p M._5_7) Vnil.
  Definition h_6_8 := F1 (hd_symb s1_p M._6_8) Vnil.
  Definition _6_8 := F1 (int_symb s1_p M._6_8) Vnil.
  Definition h_7_9 := F1 (hd_symb s1_p M._7_9) Vnil.
  Definition _7_9 := F1 (int_symb s1_p M._7_9) Vnil.
  Definition h_8_10 := F1 (hd_symb s1_p M._8_10) Vnil.
  Definition _8_10 := F1 (int_symb s1_p M._8_10) Vnil.
  Definition h_9_11 := F1 (hd_symb s1_p M._9_11) Vnil.
  Definition _9_11 := F1 (int_symb s1_p M._9_11) Vnil.
  Definition h_plus__1 x2 x1 := F1 (hd_symb s1_p M._plus__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _plus__1 x2 x1 := F1 (int_symb s1_p M._plus__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition hc x2 x1 := F1 (hd_symb s1_p M.c) (Vcons x2 (Vcons x1 Vnil)).
  Definition c x2 x1 := F1 (int_symb s1_p M.c) (Vcons x2 (Vcons x1 Vnil)).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.h_plus__1 (S1._9_11) (S1._9_11))
         (S1.hc (S1._1_3) (S1._8_10))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._8_10))
         (S1.hc (S1._1_3) (S1._7_9))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._7_9))
         (S1.hc (S1._1_3) (S1._6_8))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._6_8))
         (S1.hc (S1._1_3) (S1._5_7))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._5_7))
         (S1.hc (S1._1_3) (S1._4_6))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._4_6))
         (S1.hc (S1._1_3) (S1._3_5))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._3_5))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._2_4))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._9_11) (S1._1_3))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._9_11))
         (S1.hc (S1._1_3) (S1._7_9))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._8_10))
         (S1.hc (S1._1_3) (S1._6_8))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._7_9))
         (S1.hc (S1._1_3) (S1._5_7))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._6_8))
         (S1.hc (S1._1_3) (S1._4_6))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._5_7))
         (S1.hc (S1._1_3) (S1._3_5))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._4_6))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._3_5))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._8_10) (S1._2_4))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._9_11))
         (S1.hc (S1._1_3) (S1._6_8))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._8_10))
         (S1.hc (S1._1_3) (S1._5_7))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._7_9))
         (S1.hc (S1._1_3) (S1._4_6))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._6_8))
         (S1.hc (S1._1_3) (S1._3_5))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._5_7))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._4_6))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._7_9) (S1._3_5))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._6_8) (S1._9_11))
         (S1.hc (S1._1_3) (S1._5_7))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._6_8) (S1._8_10))
         (S1.hc (S1._1_3) (S1._4_6))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._6_8) (S1._7_9))
         (S1.hc (S1._1_3) (S1._3_5))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._6_8) (S1._6_8))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._6_8) (S1._5_7))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._6_8) (S1._4_6))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._5_7) (S1._9_11))
         (S1.hc (S1._1_3) (S1._4_6))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._5_7) (S1._8_10))
         (S1.hc (S1._1_3) (S1._3_5))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._5_7) (S1._7_9))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._5_7) (S1._6_8))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._5_7) (S1._5_7))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._4_6) (S1._9_11))
         (S1.hc (S1._1_3) (S1._3_5))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._4_6) (S1._8_10))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._4_6) (S1._7_9))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._4_6) (S1._6_8))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._3_5) (S1._9_11))
         (S1.hc (S1._1_3) (S1._2_4))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._3_5) (S1._8_10))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._3_5) (S1._7_9))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._2_4) (S1._9_11))
         (S1.hc (S1._1_3) (S1._1_3))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._2_4) (S1._8_10))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (S1._1_3) (S1._9_11))
         (S1.hc (S1._1_3) (S1._0_2))
   :: nil)

:: (  R1 (S1.h_plus__1 (V1 0) (S1.c (V1 1) (V1 2)))
         (S1.hc (V1 1) (S1._plus__1 (V1 0) (V1 2)))
   :: R1 (S1.hc (V1 0) (S1.c (V1 1) (V1 2)))
         (S1.hc (S1._plus__1 (V1 0) (V1 1)) (V1 2))
   :: R1 (S1.hc (V1 0) (S1.c (V1 1) (V1 2)))
         (S1.h_plus__1 (V1 0) (V1 1))
   :: R1 (S1.h_plus__1 (V1 0) (S1.c (V1 1) (V1 2)))
         (S1.h_plus__1 (V1 0) (V1 2))
   :: R1 (S1.h_plus__1 (S1.c (V1 0) (V1 1)) (V1 2))
         (S1.hc (V1 0) (S1._plus__1 (V1 1) (V1 2)))
   :: R1 (S1.h_plus__1 (S1.c (V1 0) (V1 1)) (V1 2))
         (S1.h_plus__1 (V1 1) (V1 2))
   :: 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._plus__1) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M._plus__1) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_2) =>
         nil
    | (int_symb M._0_2) =>
         nil
    | (hd_symb M._1_3) =>
         nil
    | (int_symb M._1_3) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M._2_4) =>
         nil
    | (int_symb M._2_4) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M._3_5) =>
         nil
    | (int_symb M._3_5) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M._4_6) =>
         nil
    | (int_symb M._4_6) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M._5_7) =>
         nil
    | (int_symb M._5_7) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M._6_8) =>
         nil
    | (int_symb M._6_8) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M._7_9) =>
         nil
    | (int_symb M._7_9) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M._8_10) =>
         nil
    | (int_symb M._8_10) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M._9_11) =>
         nil
    | (int_symb M._9_11) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.c) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.c) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (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.

(* 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.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI1.prove_termination.
termination_trivial.
Qed.