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
  | active : symb
  | e : symb
  | i : symb
  | isList : symb
  | isNeList : symb
  | isNePal : symb
  | isPal : symb
  | isQid : symb
  | mark : symb
  | nil : symb
  | o : symb
  | ok : symb
  | proper : symb
  | top : 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.active => 1
  | M.e => 0
  | M.i => 0
  | M.isList => 1
  | M.isNeList => 1
  | M.isNePal => 1
  | M.isPal => 1
  | M.isQid => 1
  | M.mark => 1
  | M.nil => 0
  | M.o => 0
  | M.ok => 1
  | M.proper => 1
  | M.top => 1
  | 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 active x1 := F0 M.active (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 mark x1 := F0 M.mark (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition o := F0 M.o Vnil.
  Definition ok x1 := F0 M.ok (Vcons x1 Vnil).
  Definition proper x1 := F0 M.proper (Vcons x1 Vnil).
  Definition top x1 := F0 M.top (Vcons x1 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.active (S0.__ (S0.__ (V0 0) (V0 1)) (V0 2)))
      (S0.mark (S0.__ (V0 0) (S0.__ (V0 1) (V0 2))))
:: R0 (S0.active (S0.__ (V0 0) S0.nil))
      (S0.mark (V0 0))
:: R0 (S0.active (S0.__ S0.nil (V0 0)))
      (S0.mark (V0 0))
:: R0 (S0.active (S0.U11 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U21 S0.tt (V0 0)))
      (S0.mark (S0.U22 (S0.isList (V0 0))))
:: R0 (S0.active (S0.U22 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U31 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U41 S0.tt (V0 0)))
      (S0.mark (S0.U42 (S0.isNeList (V0 0))))
:: R0 (S0.active (S0.U42 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U51 S0.tt (V0 0)))
      (S0.mark (S0.U52 (S0.isList (V0 0))))
:: R0 (S0.active (S0.U52 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U61 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U71 S0.tt (V0 0)))
      (S0.mark (S0.U72 (S0.isPal (V0 0))))
:: R0 (S0.active (S0.U72 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.U81 S0.tt))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isList (V0 0)))
      (S0.mark (S0.U11 (S0.isNeList (V0 0))))
:: R0 (S0.active (S0.isList S0.nil))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isList (S0.__ (V0 0) (V0 1))))
      (S0.mark (S0.U21 (S0.isList (V0 0)) (V0 1)))
:: R0 (S0.active (S0.isNeList (V0 0)))
      (S0.mark (S0.U31 (S0.isQid (V0 0))))
:: R0 (S0.active (S0.isNeList (S0.__ (V0 0) (V0 1))))
      (S0.mark (S0.U41 (S0.isList (V0 0)) (V0 1)))
:: R0 (S0.active (S0.isNeList (S0.__ (V0 0) (V0 1))))
      (S0.mark (S0.U51 (S0.isNeList (V0 0)) (V0 1)))
:: R0 (S0.active (S0.isNePal (V0 0)))
      (S0.mark (S0.U61 (S0.isQid (V0 0))))
:: R0 (S0.active (S0.isNePal (S0.__ (V0 0) (S0.__ (V0 1) (V0 0)))))
      (S0.mark (S0.U71 (S0.isQid (V0 0)) (V0 1)))
:: R0 (S0.active (S0.isPal (V0 0)))
      (S0.mark (S0.U81 (S0.isNePal (V0 0))))
:: R0 (S0.active (S0.isPal S0.nil))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isQid S0.a))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isQid S0.e))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isQid S0.i))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isQid S0.o))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.isQid S0.u))
      (S0.mark S0.tt)
:: R0 (S0.active (S0.__ (V0 0) (V0 1)))
      (S0.__ (S0.active (V0 0)) (V0 1))
:: R0 (S0.active (S0.__ (V0 0) (V0 1)))
      (S0.__ (V0 0) (S0.active (V0 1)))
:: R0 (S0.active (S0.U11 (V0 0)))
      (S0.U11 (S0.active (V0 0)))
:: R0 (S0.active (S0.U21 (V0 0) (V0 1)))
      (S0.U21 (S0.active (V0 0)) (V0 1))
:: R0 (S0.active (S0.U22 (V0 0)))
      (S0.U22 (S0.active (V0 0)))
:: R0 (S0.active (S0.U31 (V0 0)))
      (S0.U31 (S0.active (V0 0)))
:: R0 (S0.active (S0.U41 (V0 0) (V0 1)))
      (S0.U41 (S0.active (V0 0)) (V0 1))
:: R0 (S0.active (S0.U42 (V0 0)))
      (S0.U42 (S0.active (V0 0)))
:: R0 (S0.active (S0.U51 (V0 0) (V0 1)))
      (S0.U51 (S0.active (V0 0)) (V0 1))
:: R0 (S0.active (S0.U52 (V0 0)))
      (S0.U52 (S0.active (V0 0)))
:: R0 (S0.active (S0.U61 (V0 0)))
      (S0.U61 (S0.active (V0 0)))
:: R0 (S0.active (S0.U71 (V0 0) (V0 1)))
      (S0.U71 (S0.active (V0 0)) (V0 1))
:: R0 (S0.active (S0.U72 (V0 0)))
      (S0.U72 (S0.active (V0 0)))
:: R0 (S0.active (S0.U81 (V0 0)))
      (S0.U81 (S0.active (V0 0)))
:: R0 (S0.__ (S0.mark (V0 0)) (V0 1))
      (S0.mark (S0.__ (V0 0) (V0 1)))
:: R0 (S0.__ (V0 0) (S0.mark (V0 1)))
      (S0.mark (S0.__ (V0 0) (V0 1)))
:: R0 (S0.U11 (S0.mark (V0 0)))
      (S0.mark (S0.U11 (V0 0)))
:: R0 (S0.U21 (S0.mark (V0 0)) (V0 1))
      (S0.mark (S0.U21 (V0 0) (V0 1)))
:: R0 (S0.U22 (S0.mark (V0 0)))
      (S0.mark (S0.U22 (V0 0)))
:: R0 (S0.U31 (S0.mark (V0 0)))
      (S0.mark (S0.U31 (V0 0)))
:: R0 (S0.U41 (S0.mark (V0 0)) (V0 1))
      (S0.mark (S0.U41 (V0 0) (V0 1)))
:: R0 (S0.U42 (S0.mark (V0 0)))
      (S0.mark (S0.U42 (V0 0)))
:: R0 (S0.U51 (S0.mark (V0 0)) (V0 1))
      (S0.mark (S0.U51 (V0 0) (V0 1)))
:: R0 (S0.U52 (S0.mark (V0 0)))
      (S0.mark (S0.U52 (V0 0)))
:: R0 (S0.U61 (S0.mark (V0 0)))
      (S0.mark (S0.U61 (V0 0)))
:: R0 (S0.U71 (S0.mark (V0 0)) (V0 1))
      (S0.mark (S0.U71 (V0 0) (V0 1)))
:: R0 (S0.U72 (S0.mark (V0 0)))
      (S0.mark (S0.U72 (V0 0)))
:: R0 (S0.U81 (S0.mark (V0 0)))
      (S0.mark (S0.U81 (V0 0)))
:: R0 (S0.proper (S0.__ (V0 0) (V0 1)))
      (S0.__ (S0.proper (V0 0)) (S0.proper (V0 1)))
:: R0 (S0.proper S0.nil)
      (S0.ok S0.nil)
:: R0 (S0.proper (S0.U11 (V0 0)))
      (S0.U11 (S0.proper (V0 0)))
:: R0 (S0.proper S0.tt)
      (S0.ok S0.tt)
:: R0 (S0.proper (S0.U21 (V0 0) (V0 1)))
      (S0.U21 (S0.proper (V0 0)) (S0.proper (V0 1)))
:: R0 (S0.proper (S0.U22 (V0 0)))
      (S0.U22 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.isList (V0 0)))
      (S0.isList (S0.proper (V0 0)))
:: R0 (S0.proper (S0.U31 (V0 0)))
      (S0.U31 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.U41 (V0 0) (V0 1)))
      (S0.U41 (S0.proper (V0 0)) (S0.proper (V0 1)))
:: R0 (S0.proper (S0.U42 (V0 0)))
      (S0.U42 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.isNeList (V0 0)))
      (S0.isNeList (S0.proper (V0 0)))
:: R0 (S0.proper (S0.U51 (V0 0) (V0 1)))
      (S0.U51 (S0.proper (V0 0)) (S0.proper (V0 1)))
:: R0 (S0.proper (S0.U52 (V0 0)))
      (S0.U52 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.U61 (V0 0)))
      (S0.U61 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.U71 (V0 0) (V0 1)))
      (S0.U71 (S0.proper (V0 0)) (S0.proper (V0 1)))
:: R0 (S0.proper (S0.U72 (V0 0)))
      (S0.U72 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.isPal (V0 0)))
      (S0.isPal (S0.proper (V0 0)))
:: R0 (S0.proper (S0.U81 (V0 0)))
      (S0.U81 (S0.proper (V0 0)))
:: R0 (S0.proper (S0.isQid (V0 0)))
      (S0.isQid (S0.proper (V0 0)))
:: R0 (S0.proper (S0.isNePal (V0 0)))
      (S0.isNePal (S0.proper (V0 0)))
:: R0 (S0.proper S0.a)
      (S0.ok S0.a)
:: R0 (S0.proper S0.e)
      (S0.ok S0.e)
:: R0 (S0.proper S0.i)
      (S0.ok S0.i)
:: R0 (S0.proper S0.o)
      (S0.ok S0.o)
:: R0 (S0.proper S0.u)
      (S0.ok S0.u)
:: R0 (S0.__ (S0.ok (V0 0)) (S0.ok (V0 1)))
      (S0.ok (S0.__ (V0 0) (V0 1)))
:: R0 (S0.U11 (S0.ok (V0 0)))
      (S0.ok (S0.U11 (V0 0)))
:: R0 (S0.U21 (S0.ok (V0 0)) (S0.ok (V0 1)))
      (S0.ok (S0.U21 (V0 0) (V0 1)))
:: R0 (S0.U22 (S0.ok (V0 0)))
      (S0.ok (S0.U22 (V0 0)))
:: R0 (S0.isList (S0.ok (V0 0)))
      (S0.ok (S0.isList (V0 0)))
:: R0 (S0.U31 (S0.ok (V0 0)))
      (S0.ok (S0.U31 (V0 0)))
:: R0 (S0.U41 (S0.ok (V0 0)) (S0.ok (V0 1)))
      (S0.ok (S0.U41 (V0 0) (V0 1)))
:: R0 (S0.U42 (S0.ok (V0 0)))
      (S0.ok (S0.U42 (V0 0)))
:: R0 (S0.isNeList (S0.ok (V0 0)))
      (S0.ok (S0.isNeList (V0 0)))
:: R0 (S0.U51 (S0.ok (V0 0)) (S0.ok (V0 1)))
      (S0.ok (S0.U51 (V0 0) (V0 1)))
:: R0 (S0.U52 (S0.ok (V0 0)))
      (S0.ok (S0.U52 (V0 0)))
:: R0 (S0.U61 (S0.ok (V0 0)))
      (S0.ok (S0.U61 (V0 0)))
:: R0 (S0.U71 (S0.ok (V0 0)) (S0.ok (V0 1)))
      (S0.ok (S0.U71 (V0 0) (V0 1)))
:: R0 (S0.U72 (S0.ok (V0 0)))
      (S0.ok (S0.U72 (V0 0)))
:: R0 (S0.isPal (S0.ok (V0 0)))
      (S0.ok (S0.isPal (V0 0)))
:: R0 (S0.U81 (S0.ok (V0 0)))
      (S0.ok (S0.U81 (V0 0)))
:: R0 (S0.isQid (S0.ok (V0 0)))
      (S0.ok (S0.isQid (V0 0)))
:: R0 (S0.isNePal (S0.ok (V0 0)))
      (S0.ok (S0.isNePal (V0 0)))
:: R0 (S0.top (S0.mark (V0 0)))
      (S0.top (S0.proper (V0 0)))
:: R0 (S0.top (S0.ok (V0 0)))
      (S0.top (S0.active (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 hactive x1 := F1 (hd_symb s1_p M.active) (Vcons x1 Vnil).
  Definition active x1 := F1 (int_symb s1_p M.active) (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 hmark x1 := F1 (hd_symb s1_p M.mark) (Vcons x1 Vnil).
  Definition mark x1 := F1 (int_symb s1_p M.mark) (Vcons x1 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 hok x1 := F1 (hd_symb s1_p M.ok) (Vcons x1 Vnil).
  Definition ok x1 := F1 (int_symb s1_p M.ok) (Vcons x1 Vnil).
  Definition hproper x1 := F1 (hd_symb s1_p M.proper) (Vcons x1 Vnil).
  Definition proper x1 := F1 (int_symb s1_p M.proper) (Vcons x1 Vnil).
  Definition htop x1 := F1 (hd_symb s1_p M.top) (Vcons x1 Vnil).
  Definition top x1 := F1 (int_symb s1_p M.top) (Vcons x1 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.hisNePal (S1.ok (V1 0)))
         (S1.hisNePal (V1 0))
   :: nil)

:: (  R1 (S1.hisQid (S1.ok (V1 0)))
         (S1.hisQid (V1 0))
   :: nil)

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

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

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

:: (  R1 (S1.hproper (S1.isNePal (V1 0)))
         (S1.hisNePal (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hproper (S1.isQid (V1 0)))
         (S1.hisQid (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hproper (S1.isPal (V1 0)))
         (S1.hisPal (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hproper (S1.isNeList (V1 0)))
         (S1.hisNeList (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hproper (S1.isList (V1 0)))
         (S1.hisList (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU81 (S1.ok (V1 0)))
         (S1.hU81 (V1 0))
   :: R1 (S1.hU81 (S1.mark (V1 0)))
         (S1.hU81 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U81 (V1 0)))
         (S1.hU81 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU72 (S1.ok (V1 0)))
         (S1.hU72 (V1 0))
   :: R1 (S1.hU72 (S1.mark (V1 0)))
         (S1.hU72 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U72 (V1 0)))
         (S1.hU72 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU71 (S1.ok (V1 0)) (S1.ok (V1 1)))
         (S1.hU71 (V1 0) (V1 1))
   :: R1 (S1.hU71 (S1.mark (V1 0)) (V1 1))
         (S1.hU71 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hproper (S1.U71 (V1 0) (V1 1)))
         (S1.hU71 (S1.proper (V1 0)) (S1.proper (V1 1)))
   :: nil)

:: (  R1 (S1.hU61 (S1.ok (V1 0)))
         (S1.hU61 (V1 0))
   :: R1 (S1.hU61 (S1.mark (V1 0)))
         (S1.hU61 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U61 (V1 0)))
         (S1.hU61 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU52 (S1.ok (V1 0)))
         (S1.hU52 (V1 0))
   :: R1 (S1.hU52 (S1.mark (V1 0)))
         (S1.hU52 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U52 (V1 0)))
         (S1.hU52 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU51 (S1.ok (V1 0)) (S1.ok (V1 1)))
         (S1.hU51 (V1 0) (V1 1))
   :: R1 (S1.hU51 (S1.mark (V1 0)) (V1 1))
         (S1.hU51 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hproper (S1.U51 (V1 0) (V1 1)))
         (S1.hU51 (S1.proper (V1 0)) (S1.proper (V1 1)))
   :: nil)

:: (  R1 (S1.hU42 (S1.ok (V1 0)))
         (S1.hU42 (V1 0))
   :: R1 (S1.hU42 (S1.mark (V1 0)))
         (S1.hU42 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U42 (V1 0)))
         (S1.hU42 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU41 (S1.ok (V1 0)) (S1.ok (V1 1)))
         (S1.hU41 (V1 0) (V1 1))
   :: R1 (S1.hU41 (S1.mark (V1 0)) (V1 1))
         (S1.hU41 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hproper (S1.U41 (V1 0) (V1 1)))
         (S1.hU41 (S1.proper (V1 0)) (S1.proper (V1 1)))
   :: nil)

:: (  R1 (S1.hU31 (S1.ok (V1 0)))
         (S1.hU31 (V1 0))
   :: R1 (S1.hU31 (S1.mark (V1 0)))
         (S1.hU31 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U31 (V1 0)))
         (S1.hU31 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU22 (S1.ok (V1 0)))
         (S1.hU22 (V1 0))
   :: R1 (S1.hU22 (S1.mark (V1 0)))
         (S1.hU22 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U22 (V1 0)))
         (S1.hU22 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hU21 (S1.ok (V1 0)) (S1.ok (V1 1)))
         (S1.hU21 (V1 0) (V1 1))
   :: R1 (S1.hU21 (S1.mark (V1 0)) (V1 1))
         (S1.hU21 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hproper (S1.U21 (V1 0) (V1 1)))
         (S1.hU21 (S1.proper (V1 0)) (S1.proper (V1 1)))
   :: nil)

:: (  R1 (S1.hU11 (S1.ok (V1 0)))
         (S1.hU11 (V1 0))
   :: R1 (S1.hU11 (S1.mark (V1 0)))
         (S1.hU11 (V1 0))
   :: nil)

:: (  R1 (S1.hproper (S1.U11 (V1 0)))
         (S1.hU11 (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.h__ (V1 0) (S1.mark (V1 1)))
         (S1.h__ (V1 0) (V1 1))
   :: R1 (S1.h__ (S1.mark (V1 0)) (V1 1))
         (S1.h__ (V1 0) (V1 1))
   :: R1 (S1.h__ (S1.ok (V1 0)) (S1.ok (V1 1)))
         (S1.h__ (V1 0) (V1 1))
   :: nil)

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

:: (  R1 (S1.hproper (S1.__ (V1 0) (V1 1)))
         (S1.hproper (V1 1))
   :: R1 (S1.hproper (S1.__ (V1 0) (V1 1)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U11 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U21 (V1 0) (V1 1)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U21 (V1 0) (V1 1)))
         (S1.hproper (V1 1))
   :: R1 (S1.hproper (S1.U22 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.isList (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U31 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U41 (V1 0) (V1 1)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U41 (V1 0) (V1 1)))
         (S1.hproper (V1 1))
   :: R1 (S1.hproper (S1.U42 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.isNeList (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U51 (V1 0) (V1 1)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U51 (V1 0) (V1 1)))
         (S1.hproper (V1 1))
   :: R1 (S1.hproper (S1.U52 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U61 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U71 (V1 0) (V1 1)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U71 (V1 0) (V1 1)))
         (S1.hproper (V1 1))
   :: R1 (S1.hproper (S1.U72 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.isPal (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.U81 (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.isQid (V1 0)))
         (S1.hproper (V1 0))
   :: R1 (S1.hproper (S1.isNePal (V1 0)))
         (S1.hproper (V1 0))
   :: nil)

:: (  R1 (S1.htop (S1.mark (V1 0)))
         (S1.hproper (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.U81 (V1 0)))
         (S1.hU81 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U72 (V1 0)))
         (S1.hU72 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U71 (V1 0) (V1 1)))
         (S1.hU71 (S1.active (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.U61 (V1 0)))
         (S1.hU61 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U52 (V1 0)))
         (S1.hU52 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U51 (V1 0) (V1 1)))
         (S1.hU51 (S1.active (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.U42 (V1 0)))
         (S1.hU42 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U41 (V1 0) (V1 1)))
         (S1.hU41 (S1.active (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.U31 (V1 0)))
         (S1.hU31 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U22 (V1 0)))
         (S1.hU22 (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U21 (V1 0) (V1 1)))
         (S1.hU21 (S1.active (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.U11 (V1 0)))
         (S1.hU11 (S1.active (V1 0)))
   :: nil)

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

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

:: (  R1 (S1.hactive (S1.isPal (V1 0)))
         (S1.hisNePal (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isPal (V1 0)))
         (S1.hU81 (S1.isNePal (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.isNePal (S1.__ (V1 0) (S1.__ (V1 1) (V1 0)))))
         (S1.hisQid (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isNePal (S1.__ (V1 0) (S1.__ (V1 1) (V1 0)))))
         (S1.hU71 (S1.isQid (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.isNePal (V1 0)))
         (S1.hisQid (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isNePal (V1 0)))
         (S1.hU61 (S1.isQid (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.isNeList (S1.__ (V1 0) (V1 1))))
         (S1.hisNeList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isNeList (S1.__ (V1 0) (V1 1))))
         (S1.hU51 (S1.isNeList (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.isNeList (S1.__ (V1 0) (V1 1))))
         (S1.hisList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isNeList (S1.__ (V1 0) (V1 1))))
         (S1.hU41 (S1.isList (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.isNeList (V1 0)))
         (S1.hisQid (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isNeList (V1 0)))
         (S1.hU31 (S1.isQid (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.isList (S1.__ (V1 0) (V1 1))))
         (S1.hisList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isList (S1.__ (V1 0) (V1 1))))
         (S1.hU21 (S1.isList (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hactive (S1.isList (V1 0)))
         (S1.hisNeList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.isList (V1 0)))
         (S1.hU11 (S1.isNeList (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U71 (S1.tt) (V1 0)))
         (S1.hisPal (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.U71 (S1.tt) (V1 0)))
         (S1.hU72 (S1.isPal (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U51 (S1.tt) (V1 0)))
         (S1.hisList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.U51 (S1.tt) (V1 0)))
         (S1.hU52 (S1.isList (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U41 (S1.tt) (V1 0)))
         (S1.hisNeList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.U41 (S1.tt) (V1 0)))
         (S1.hU42 (S1.isNeList (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.U21 (S1.tt) (V1 0)))
         (S1.hisList (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.U21 (S1.tt) (V1 0)))
         (S1.hU22 (S1.isList (V1 0)))
   :: nil)

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

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

:: (  R1 (S1.hactive (S1.__ (V1 0) (V1 1)))
         (S1.hactive (V1 1))
   :: R1 (S1.hactive (S1.__ (V1 0) (V1 1)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U11 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U21 (V1 0) (V1 1)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U22 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U31 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U41 (V1 0) (V1 1)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U42 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U51 (V1 0) (V1 1)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U52 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U61 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U71 (V1 0) (V1 1)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U72 (V1 0)))
         (S1.hactive (V1 0))
   :: R1 (S1.hactive (S1.U81 (V1 0)))
         (S1.hactive (V1 0))
   :: nil)

:: (  R1 (S1.htop (S1.ok (V1 0)))
         (S1.hactive (V1 0))
   :: nil)

:: (  R1 (S1.htop (S1.ok (V1 0)))
         (S1.htop (S1.active (V1 0)))
   :: R1 (S1.htop (S1.mark (V1 0)))
         (S1.htop (S1.proper (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.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         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.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         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.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isPal) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

(* 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.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS4.

Module PI4 := PolyInt PIS4.

(* 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.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* polynomial interpretation 6 *)

Module PIS6 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U81) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS6.

Module PI6 := PolyInt PIS6.

(* polynomial interpretation 7 *)

Module PIS7 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS7.

Module PI7 := PolyInt PIS7.

(* polynomial interpretation 8 *)

Module PIS8 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS8.

Module PI8 := PolyInt PIS8.

(* polynomial interpretation 9 *)

Module PIS9 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS9.

Module PI9 := PolyInt PIS9.

(* polynomial interpretation 10 *)

Module PIS10 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS10.

Module PI10 := PolyInt PIS10.

(* polynomial interpretation 11 *)

Module PIS11 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS11.

Module PI11 := PolyInt PIS11.

(* polynomial interpretation 12 *)

Module PIS12 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS12.

Module PI12 := PolyInt PIS12.

(* polynomial interpretation 13 *)

Module PIS13 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS13.

Module PI13 := PolyInt PIS13.

(* polynomial interpretation 14 *)

Module PIS14 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS14.

Module PI14 := PolyInt PIS14.

(* polynomial interpretation 15 *)

Module PIS15 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS15.

Module PI15 := PolyInt PIS15.

(* polynomial interpretation 16 *)

Module PIS16 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS16.

Module PI16 := PolyInt PIS16.

(* polynomial interpretation 17 *)

Module PIS17 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS17.

Module PI17 := PolyInt PIS17.

(* polynomial interpretation 18 *)

Module PIS18 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS18.

Module PI18 := PolyInt PIS18.

(* polynomial interpretation 19 *)

Module PIS19 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS19.

Module PI19 := PolyInt PIS19.

(* polynomial interpretation 20 *)

Module PIS20 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS20.

Module PI20 := PolyInt PIS20.

(* polynomial interpretation 21 *)

Module PIS21 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS21.

Module PI21 := PolyInt PIS21.

(* polynomial interpretation 22 *)

Module PIS22 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS22.

Module PI22 := PolyInt PIS22.

(* polynomial interpretation 23 *)

Module PIS23 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS23.

Module PI23 := PolyInt PIS23.

(* polynomial interpretation 24 *)

Module PIS24 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS24.

Module PI24 := PolyInt PIS24.

(* polynomial interpretation 25 *)

Module PIS25 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS25.

Module PI25 := PolyInt PIS25.

(* polynomial interpretation 26 *)

Module PIS26 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: 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) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS26.

Module PI26 := PolyInt PIS26.

(* polynomial interpretation 27 *)

Module PIS27 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS27.

Module PI27 := PolyInt PIS27.

(* polynomial interpretation 28 *)

Module PIS28 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 0 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) =>
         nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS28.

Module PI28 := PolyInt PIS28.

(* polynomial interpretation 29 *)

Module PIS29 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.o) =>
         nil
    | (int_symb M.o) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS29.

Module PI29 := PolyInt PIS29.

(* polynomial interpretation 30 *)

Module PIS30 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         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) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         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
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS30.

Module PI30 := PolyInt PIS30.

(* polynomial interpretation 31 *)

Module PIS31 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: 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)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         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) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS31.

Module PI31 := PolyInt PIS31.

(* polynomial interpretation 32 *)

Module PIS32 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS32.

Module PI32 := PolyInt PIS32.

(* polynomial interpretation 33 *)

Module PIS33 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (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))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         nil
    | (hd_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS33.

Module PI33 := PolyInt PIS33.

(* polynomial interpretation 34 *)

Module PIS34 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (2%Z, (Vcons 1 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 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%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) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (3%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) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (3%Z, (Vcons 0 Vnil))
      :: 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 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         nil
    | (hd_symb M.isPal) =>
         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) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS34.

Module PI34 := PolyInt PIS34.

(* polynomial interpretation 35 *)

Module PIS35 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         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) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (2%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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.proper) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS35.

Module PI35 := PolyInt PIS35.

(* polynomial interpretation 36 *)

Module PIS36 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%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) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: 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) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         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) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS36.

Module PI36 := PolyInt PIS36.

(* polynomial interpretation 37 *)

Module PIS37 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.active) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (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.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (1%Z, Vnil)
      :: 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
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS37.

Module PI37 := PolyInt PIS37.

(* polynomial interpretation 38 *)

Module PIS38 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (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) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 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) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS38.

Module PI38 := PolyInt PIS38.

(* polynomial interpretation 39 *)

Module PIS39 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.__) =>
         nil
    | (int_symb M.__) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         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))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isList) =>
         nil
    | (int_symb M.isList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNeList) =>
         nil
    | (int_symb M.isNeList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U71) =>
         nil
    | (int_symb M.U71) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U72) =>
         nil
    | (int_symb M.U72) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isPal) =>
         nil
    | (int_symb M.isPal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U81) =>
         nil
    | (int_symb M.U81) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isQid) =>
         nil
    | (int_symb M.isQid) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNePal) =>
         nil
    | (int_symb M.isNePal) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.e) =>
         nil
    | (int_symb M.e) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (2%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) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.top) =>
         nil
    end.

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

End PIS39.

Module PI39 := PolyInt PIS39.

(* 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.
right. PI1.prove_termination.
termination_trivial.
right. PI2.prove_termination.
termination_trivial.
right. PI3.prove_termination.
termination_trivial.
right. PI4.prove_termination.
termination_trivial.
right. PI5.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI6.prove_termination.
PI7.prove_termination.
termination_trivial.
left. co_scc.
right. PI8.prove_termination.
PI9.prove_termination.
termination_trivial.
left. co_scc.
right. PI10.prove_termination.
PI11.prove_termination.
termination_trivial.
left. co_scc.
right. PI12.prove_termination.
PI13.prove_termination.
termination_trivial.
left. co_scc.
right. PI14.prove_termination.
PI15.prove_termination.
termination_trivial.
left. co_scc.
right. PI16.prove_termination.
PI17.prove_termination.
termination_trivial.
left. co_scc.
right. PI18.prove_termination.
PI19.prove_termination.
termination_trivial.
left. co_scc.
right. PI20.prove_termination.
PI21.prove_termination.
termination_trivial.
left. co_scc.
right. PI22.prove_termination.
PI23.prove_termination.
termination_trivial.
left. co_scc.
right. PI24.prove_termination.
PI25.prove_termination.
termination_trivial.
left. co_scc.
right. PI26.prove_termination.
PI27.prove_termination.
termination_trivial.
left. co_scc.
right. PI28.prove_termination.
PI29.prove_termination.
termination_trivial.
left. co_scc.
right. PI30.prove_termination.
PI31.prove_termination.
PI32.prove_termination.
termination_trivial.
left. co_scc.
right. PI33.prove_termination.
PI34.prove_termination.
PI35.prove_termination.
PI36.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.
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. PI37.prove_termination.
termination_trivial.
left. co_scc.
right. PI38.prove_termination.
PI39.prove_termination.
termination_trivial.
Qed.