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 :=
  | U11 : symb
  | U21 : symb
  | U22 : symb
  | U31 : symb
  | U41 : symb
  | U42 : symb
  | U51 : symb
  | U52 : symb
  | U61 : symb
  | U71 : symb
  | U72 : symb
  | U81 : symb
  | __ : symb
  | a : symb
  | activate : symb
  | e : symb
  | i : symb
  | isList : symb
  | isNeList : symb
  | isNePal : symb
  | isPal : symb
  | isQid : symb
  | n____ : symb
  | n__a : symb
  | n__e : symb
  | n__i : symb
  | n__nil : symb
  | n__o : symb
  | n__u : symb
  | nil : symb
  | o : symb
  | tt : symb
  | u : 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.U11 => 1
  | M.U21 => 2
  | M.U22 => 1
  | M.U31 => 1
  | M.U41 => 2
  | M.U42 => 1
  | M.U51 => 2
  | M.U52 => 1
  | M.U61 => 1
  | M.U71 => 2
  | M.U72 => 1
  | M.U81 => 1
  | M.__ => 2
  | M.a => 0
  | M.activate => 1
  | M.e => 0
  | M.i => 0
  | M.isList => 1
  | M.isNeList => 1
  | M.isNePal => 1
  | M.isPal => 1
  | M.isQid => 1
  | M.n____ => 2
  | M.n__a => 0
  | M.n__e => 0
  | M.n__i => 0
  | M.n__nil => 0
  | M.n__o => 0
  | M.n__u => 0
  | M.nil => 0
  | M.o => 0
  | M.tt => 0
  | M.u => 0
  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 U11 x1 := F0 M.U11 (Vcons x1 Vnil).
  Definition U21 x2 x1 := F0 M.U21 (Vcons x2 (Vcons x1 Vnil)).
  Definition U22 x1 := F0 M.U22 (Vcons x1 Vnil).
  Definition U31 x1 := F0 M.U31 (Vcons x1 Vnil).
  Definition U41 x2 x1 := F0 M.U41 (Vcons x2 (Vcons x1 Vnil)).
  Definition U42 x1 := F0 M.U42 (Vcons x1 Vnil).
  Definition U51 x2 x1 := F0 M.U51 (Vcons x2 (Vcons x1 Vnil)).
  Definition U52 x1 := F0 M.U52 (Vcons x1 Vnil).
  Definition U61 x1 := F0 M.U61 (Vcons x1 Vnil).
  Definition U71 x2 x1 := F0 M.U71 (Vcons x2 (Vcons x1 Vnil)).
  Definition U72 x1 := F0 M.U72 (Vcons x1 Vnil).
  Definition U81 x1 := F0 M.U81 (Vcons x1 Vnil).
  Definition __ x2 x1 := F0 M.__ (Vcons x2 (Vcons x1 Vnil)).
  Definition a := F0 M.a Vnil.
  Definition activate x1 := F0 M.activate (Vcons x1 Vnil).
  Definition e := F0 M.e Vnil.
  Definition i := F0 M.i Vnil.
  Definition isList x1 := F0 M.isList (Vcons x1 Vnil).
  Definition isNeList x1 := F0 M.isNeList (Vcons x1 Vnil).
  Definition isNePal x1 := F0 M.isNePal (Vcons x1 Vnil).
  Definition isPal x1 := F0 M.isPal (Vcons x1 Vnil).
  Definition isQid x1 := F0 M.isQid (Vcons x1 Vnil).
  Definition n____ x2 x1 := F0 M.n____ (Vcons x2 (Vcons x1 Vnil)).
  Definition n__a := F0 M.n__a Vnil.
  Definition n__e := F0 M.n__e Vnil.
  Definition n__i := F0 M.n__i Vnil.
  Definition n__nil := F0 M.n__nil Vnil.
  Definition n__o := F0 M.n__o Vnil.
  Definition n__u := F0 M.n__u Vnil.
  Definition nil := F0 M.nil Vnil.
  Definition o := F0 M.o Vnil.
  Definition tt := F0 M.tt Vnil.
  Definition u := F0 M.u Vnil.
End S0.

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

Definition R :=
   R0 (S0.__ (S0.__ (V0 0) (V0 1)) (V0 2))
      (S0.__ (V0 0) (S0.__ (V0 1) (V0 2)))
:: R0 (S0.__ (V0 0) S0.nil)
      (V0 0)
:: R0 (S0.__ S0.nil (V0 0))
      (V0 0)
:: R0 (S0.U11 S0.tt)
      S0.tt
:: R0 (S0.U21 S0.tt (V0 0))
      (S0.U22 (S0.isList (S0.activate (V0 0))))
:: R0 (S0.U22 S0.tt)
      S0.tt
:: R0 (S0.U31 S0.tt)
      S0.tt
:: R0 (S0.U41 S0.tt (V0 0))
      (S0.U42 (S0.isNeList (S0.activate (V0 0))))
:: R0 (S0.U42 S0.tt)
      S0.tt
:: R0 (S0.U51 S0.tt (V0 0))
      (S0.U52 (S0.isList (S0.activate (V0 0))))
:: R0 (S0.U52 S0.tt)
      S0.tt
:: R0 (S0.U61 S0.tt)
      S0.tt
:: R0 (S0.U71 S0.tt (V0 0))
      (S0.U72 (S0.isPal (S0.activate (V0 0))))
:: R0 (S0.U72 S0.tt)
      S0.tt
:: R0 (S0.U81 S0.tt)
      S0.tt
:: R0 (S0.isList (V0 0))
      (S0.U11 (S0.isNeList (S0.activate (V0 0))))
:: R0 (S0.isList S0.n__nil)
      S0.tt
:: R0 (S0.isList (S0.n____ (V0 0) (V0 1)))
      (S0.U21 (S0.isList (S0.activate (V0 0))) (S0.activate (V0 1)))
:: R0 (S0.isNeList (V0 0))
      (S0.U31 (S0.isQid (S0.activate (V0 0))))
:: R0 (S0.isNeList (S0.n____ (V0 0) (V0 1)))
      (S0.U41 (S0.isList (S0.activate (V0 0))) (S0.activate (V0 1)))
:: R0 (S0.isNeList (S0.n____ (V0 0) (V0 1)))
      (S0.U51 (S0.isNeList (S0.activate (V0 0))) (S0.activate (V0 1)))
:: R0 (S0.isNePal (V0 0))
      (S0.U61 (S0.isQid (S0.activate (V0 0))))
:: R0 (S0.isNePal (S0.n____ (V0 0) (S0.__ (V0 1) (V0 0))))
      (S0.U71 (S0.isQid (S0.activate (V0 0))) (S0.activate (V0 1)))
:: R0 (S0.isPal (V0 0))
      (S0.U81 (S0.isNePal (S0.activate (V0 0))))
:: R0 (S0.isPal S0.n__nil)
      S0.tt
:: R0 (S0.isQid S0.n__a)
      S0.tt
:: R0 (S0.isQid S0.n__e)
      S0.tt
:: R0 (S0.isQid S0.n__i)
      S0.tt
:: R0 (S0.isQid S0.n__o)
      S0.tt
:: R0 (S0.isQid S0.n__u)
      S0.tt
:: R0 S0.nil
      S0.n__nil
:: R0 (S0.__ (V0 0) (V0 1))
      (S0.n____ (V0 0) (V0 1))
:: R0 S0.a
      S0.n__a
:: R0 S0.e
      S0.n__e
:: R0 S0.i
      S0.n__i
:: R0 S0.o
      S0.n__o
:: R0 S0.u
      S0.n__u
:: R0 (S0.activate S0.n__nil)
      S0.nil
:: R0 (S0.activate (S0.n____ (V0 0) (V0 1)))
      (S0.__ (V0 0) (V0 1))
:: R0 (S0.activate S0.n__a)
      S0.a
:: R0 (S0.activate S0.n__e)
      S0.e
:: R0 (S0.activate S0.n__i)
      S0.i
:: R0 (S0.activate S0.n__o)
      S0.o
:: R0 (S0.activate S0.n__u)
      S0.u
:: R0 (S0.activate (V0 0))
      (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 hU11 x1 := F1 (hd_symb s1_p M.U11) (Vcons x1 Vnil).
  Definition U11 x1 := F1 (int_symb s1_p M.U11) (Vcons x1 Vnil).
  Definition hU21 x2 x1 := F1 (hd_symb s1_p M.U21) (Vcons x2 (Vcons x1 Vnil)).
  Definition U21 x2 x1 := F1 (int_symb s1_p M.U21) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU22 x1 := F1 (hd_symb s1_p M.U22) (Vcons x1 Vnil).
  Definition U22 x1 := F1 (int_symb s1_p M.U22) (Vcons x1 Vnil).
  Definition hU31 x1 := F1 (hd_symb s1_p M.U31) (Vcons x1 Vnil).
  Definition U31 x1 := F1 (int_symb s1_p M.U31) (Vcons x1 Vnil).
  Definition hU41 x2 x1 := F1 (hd_symb s1_p M.U41) (Vcons x2 (Vcons x1 Vnil)).
  Definition U41 x2 x1 := F1 (int_symb s1_p M.U41) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU42 x1 := F1 (hd_symb s1_p M.U42) (Vcons x1 Vnil).
  Definition U42 x1 := F1 (int_symb s1_p M.U42) (Vcons x1 Vnil).
  Definition hU51 x2 x1 := F1 (hd_symb s1_p M.U51) (Vcons x2 (Vcons x1 Vnil)).
  Definition U51 x2 x1 := F1 (int_symb s1_p M.U51) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU52 x1 := F1 (hd_symb s1_p M.U52) (Vcons x1 Vnil).
  Definition U52 x1 := F1 (int_symb s1_p M.U52) (Vcons x1 Vnil).
  Definition hU61 x1 := F1 (hd_symb s1_p M.U61) (Vcons x1 Vnil).
  Definition U61 x1 := F1 (int_symb s1_p M.U61) (Vcons x1 Vnil).
  Definition hU71 x2 x1 := F1 (hd_symb s1_p M.U71) (Vcons x2 (Vcons x1 Vnil)).
  Definition U71 x2 x1 := F1 (int_symb s1_p M.U71) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU72 x1 := F1 (hd_symb s1_p M.U72) (Vcons x1 Vnil).
  Definition U72 x1 := F1 (int_symb s1_p M.U72) (Vcons x1 Vnil).
  Definition hU81 x1 := F1 (hd_symb s1_p M.U81) (Vcons x1 Vnil).
  Definition U81 x1 := F1 (int_symb s1_p M.U81) (Vcons x1 Vnil).
  Definition h__ x2 x1 := F1 (hd_symb s1_p M.__) (Vcons x2 (Vcons x1 Vnil)).
  Definition __ x2 x1 := F1 (int_symb s1_p M.__) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha := F1 (hd_symb s1_p M.a) Vnil.
  Definition a := F1 (int_symb s1_p M.a) Vnil.
  Definition hactivate x1 := F1 (hd_symb s1_p M.activate) (Vcons x1 Vnil).
  Definition activate x1 := F1 (int_symb s1_p M.activate) (Vcons x1 Vnil).
  Definition he := F1 (hd_symb s1_p M.e) Vnil.
  Definition e := F1 (int_symb s1_p M.e) Vnil.
  Definition hi := F1 (hd_symb s1_p M.i) Vnil.
  Definition i := F1 (int_symb s1_p M.i) Vnil.
  Definition hisList x1 := F1 (hd_symb s1_p M.isList) (Vcons x1 Vnil).
  Definition isList x1 := F1 (int_symb s1_p M.isList) (Vcons x1 Vnil).
  Definition hisNeList x1 := F1 (hd_symb s1_p M.isNeList) (Vcons x1 Vnil).
  Definition isNeList x1 := F1 (int_symb s1_p M.isNeList) (Vcons x1 Vnil).
  Definition hisNePal x1 := F1 (hd_symb s1_p M.isNePal) (Vcons x1 Vnil).
  Definition isNePal x1 := F1 (int_symb s1_p M.isNePal) (Vcons x1 Vnil).
  Definition hisPal x1 := F1 (hd_symb s1_p M.isPal) (Vcons x1 Vnil).
  Definition isPal x1 := F1 (int_symb s1_p M.isPal) (Vcons x1 Vnil).
  Definition hisQid x1 := F1 (hd_symb s1_p M.isQid) (Vcons x1 Vnil).
  Definition isQid x1 := F1 (int_symb s1_p M.isQid) (Vcons x1 Vnil).
  Definition hn____ x2 x1 := F1 (hd_symb s1_p M.n____) (Vcons x2 (Vcons x1 Vnil)).
  Definition n____ x2 x1 := F1 (int_symb s1_p M.n____) (Vcons x2 (Vcons x1 Vnil)).
  Definition hn__a := F1 (hd_symb s1_p M.n__a) Vnil.
  Definition n__a := F1 (int_symb s1_p M.n__a) Vnil.
  Definition hn__e := F1 (hd_symb s1_p M.n__e) Vnil.
  Definition n__e := F1 (int_symb s1_p M.n__e) Vnil.
  Definition hn__i := F1 (hd_symb s1_p M.n__i) Vnil.
  Definition n__i := F1 (int_symb s1_p M.n__i) Vnil.
  Definition hn__nil := F1 (hd_symb s1_p M.n__nil) Vnil.
  Definition n__nil := F1 (int_symb s1_p M.n__nil) Vnil.
  Definition hn__o := F1 (hd_symb s1_p M.n__o) Vnil.
  Definition n__o := F1 (int_symb s1_p M.n__o) Vnil.
  Definition hn__u := F1 (hd_symb s1_p M.n__u) Vnil.
  Definition n__u := F1 (int_symb s1_p M.n__u) Vnil.
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition ho := F1 (hd_symb s1_p M.o) Vnil.
  Definition o := F1 (int_symb s1_p M.o) Vnil.
  Definition htt := F1 (hd_symb s1_p M.tt) Vnil.
  Definition tt := F1 (int_symb s1_p M.tt) Vnil.
  Definition hu := F1 (hd_symb s1_p M.u) Vnil.
  Definition u := F1 (int_symb s1_p M.u) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hactivate (S1.n__u))
         (S1.hu)
   :: nil)

:: (  R1 (S1.hactivate (S1.n__o))
         (S1.ho)
   :: nil)

:: (  R1 (S1.hactivate (S1.n__i))
         (S1.hi)
   :: nil)

:: (  R1 (S1.hactivate (S1.n__e))
         (S1.he)
   :: nil)

:: (  R1 (S1.hactivate (S1.n__a))
         (S1.ha)
   :: nil)

:: (  R1 (S1.hactivate (S1.n__nil))
         (S1.hnil)
   :: nil)

:: (  R1 (S1.hisPal (V1 0))
         (S1.hU81 (S1.isNePal (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0))))
         (S1.hisQid (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hisNePal (V1 0))
         (S1.hisQid (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hisNePal (V1 0))
         (S1.hU61 (S1.isQid (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hisNeList (V1 0))
         (S1.hisQid (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hisNeList (V1 0))
         (S1.hU31 (S1.isQid (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hisList (V1 0))
         (S1.hU11 (S1.isNeList (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU71 (S1.tt) (V1 0))
         (S1.hU72 (S1.isPal (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU51 (S1.tt) (V1 0))
         (S1.hU52 (S1.isList (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU41 (S1.tt) (V1 0))
         (S1.hU42 (S1.isNeList (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU21 (S1.tt) (V1 0))
         (S1.hU22 (S1.isList (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.h__ (S1.__ (V1 0) (V1 1)) (V1 2))
         (S1.h__ (V1 1) (V1 2))
   :: R1 (S1.h__ (S1.__ (V1 0) (V1 1)) (V1 2))
         (S1.h__ (V1 0) (S1.__ (V1 1) (V1 2)))
   :: nil)

:: (  R1 (S1.hactivate (S1.n____ (V1 0) (V1 1)))
         (S1.h__ (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hisPal (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0))))
         (S1.hactivate (V1 1))
   :: nil)

:: (  R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0))))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hisNePal (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hactivate (V1 1))
   :: nil)

:: (  R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hisNeList (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hisList (S1.n____ (V1 0) (V1 1)))
         (S1.hactivate (V1 1))
   :: nil)

:: (  R1 (S1.hisList (S1.n____ (V1 0) (V1 1)))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hisList (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hU71 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hU71 (S1.tt) (V1 0))
         (S1.hisPal (S1.activate (V1 0)))
   :: R1 (S1.hisPal (V1 0))
         (S1.hisNePal (S1.activate (V1 0)))
   :: R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0))))
         (S1.hU71 (S1.isQid (S1.activate (V1 0))) (S1.activate (V1 1)))
   :: nil)

:: (  R1 (S1.hU51 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hU41 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hU21 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: (  R1 (S1.hU21 (S1.tt) (V1 0))
         (S1.hisList (S1.activate (V1 0)))
   :: R1 (S1.hisList (V1 0))
         (S1.hisNeList (S1.activate (V1 0)))
   :: R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hU41 (S1.isList (S1.activate (V1 0))) (S1.activate (V1 1)))
   :: R1 (S1.hU41 (S1.tt) (V1 0))
         (S1.hisNeList (S1.activate (V1 0)))
   :: R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hisList (S1.activate (V1 0)))
   :: R1 (S1.hisList (S1.n____ (V1 0) (V1 1)))
         (S1.hU21 (S1.isList (S1.activate (V1 0))) (S1.activate (V1 1)))
   :: R1 (S1.hisList (S1.n____ (V1 0) (V1 1)))
         (S1.hisList (S1.activate (V1 0)))
   :: R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hU51 (S1.isNeList (S1.activate (V1 0))) (S1.activate (V1 1)))
   :: R1 (S1.hU51 (S1.tt) (V1 0))
         (S1.hisList (S1.activate (V1 0)))
   :: R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hisNeList (S1.activate (V1 0)))
   :: 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.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.__) =>
         (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.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.activate) =>
         nil
    | (int_symb M.activate) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         nil
    | (hd_symb M.n__nil) =>
         nil
    | (int_symb M.n__nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.n____) =>
         nil
    | (int_symb M.n____) =>
         (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.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n__a) =>
         nil
    | (int_symb M.n__a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n__e) =>
         nil
    | (int_symb M.n__e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n__i) =>
         nil
    | (int_symb M.n__i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n__o) =>
         nil
    | (int_symb M.n__o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n__u) =>
         nil
    | (int_symb M.n__u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, 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.__) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.__) =>
         (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.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.activate) =>
         nil
    | (int_symb M.activate) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         nil
    | (hd_symb M.n__nil) =>
         nil
    | (int_symb M.n__nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n____) =>
         nil
    | (int_symb M.n____) =>
         (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.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n__a) =>
         nil
    | (int_symb M.n__a) =>
         nil
    | (hd_symb M.n__e) =>
         nil
    | (int_symb M.n__e) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.n__i) =>
         nil
    | (int_symb M.n__i) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.n__o) =>
         nil
    | (int_symb M.n__o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n__u) =>
         nil
    | (int_symb M.n__u) =>
         nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, 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.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.activate) =>
         nil
    | (int_symb M.activate) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         nil
    | (hd_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U71) =>
         nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         nil
    | (hd_symb M.isPal) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isPal) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         nil
    | (hd_symb M.n__nil) =>
         nil
    | (int_symb M.n__nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n____) =>
         nil
    | (int_symb M.n____) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNePal) =>
         nil
    | (hd_symb M.n__a) =>
         nil
    | (int_symb M.n__a) =>
         nil
    | (hd_symb M.n__e) =>
         nil
    | (int_symb M.n__e) =>
         nil
    | (hd_symb M.n__i) =>
         nil
    | (int_symb M.n__i) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.n__o) =>
         nil
    | (int_symb M.n__o) =>
         nil
    | (hd_symb M.n__u) =>
         nil
    | (int_symb M.n__u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, 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.hisPal (V1 0))
         (S1.hisNePal (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hU71 (S1.tt) (V1 0))
         (S1.hisPal (S1.activate (V1 0)))
   :: nil)

:: nil.

(* polynomial interpretation 4 *)

Module PIS4 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.isList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.activate) =>
         nil
    | (int_symb M.activate) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.n__nil) =>
         nil
    | (int_symb M.n__nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n____) =>
         nil
    | (int_symb M.n____) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n__a) =>
         nil
    | (int_symb M.n__a) =>
         nil
    | (hd_symb M.n__e) =>
         nil
    | (int_symb M.n__e) =>
         nil
    | (hd_symb M.n__i) =>
         nil
    | (int_symb M.n__i) =>
         nil
    | (hd_symb M.n__o) =>
         nil
    | (int_symb M.n__o) =>
         nil
    | (hd_symb M.n__u) =>
         nil
    | (int_symb M.n__u) =>
         nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    end.

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

End PIS4.

Module PI4 := PolyInt PIS4.

(* graph decomposition 3 *)

Definition cs3 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1)))
         (S1.hU51 (S1.isNeList (S1.activate (V1 0))) (S1.activate (V1 1)))
   :: R1 (S1.hU51 (S1.tt) (V1 0))
         (S1.hisList (S1.activate (V1 0)))
   :: R1 (S1.hisList (V1 0))
         (S1.hisNeList (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hU21 (S1.tt) (V1 0))
         (S1.hisList (S1.activate (V1 0)))
   :: nil)

:: nil.

(* polynomial interpretation 5 *)

Module PIS5 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (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.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.activate) =>
         nil
    | (int_symb M.activate) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         nil
    | (hd_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         nil
    | (hd_symb M.n__nil) =>
         nil
    | (int_symb M.n__nil) =>
         nil
    | (hd_symb M.n____) =>
         nil
    | (int_symb M.n____) =>
         (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.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         nil
    | (hd_symb M.n__a) =>
         nil
    | (int_symb M.n__a) =>
         nil
    | (hd_symb M.n__e) =>
         nil
    | (int_symb M.n__e) =>
         nil
    | (hd_symb M.n__i) =>
         nil
    | (int_symb M.n__i) =>
         nil
    | (hd_symb M.n__o) =>
         nil
    | (int_symb M.n__o) =>
         nil
    | (hd_symb M.n__u) =>
         nil
    | (int_symb M.n__u) =>
         nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* graph decomposition 4 *)

Definition cs4 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hisList (V1 0))
         (S1.hisNeList (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hU51 (S1.tt) (V1 0))
         (S1.hisList (S1.activate (V1 0)))
   :: 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.
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.
PI2.prove_termination.
termination_trivial.
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. 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.
left. co_scc.
left. co_scc.
right. PI4.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) cs3; subst D; subst R.
dpg_unif_N_correct.
right. PI5.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) cs4; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.