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
  | U12 : symb
  | U13 : symb
  | U14 : symb
  | U15 : symb
  | U16 : symb
  | U21 : symb
  | U22 : symb
  | U23 : symb
  | U31 : symb
  | U32 : symb
  | U41 : symb
  | U51 : symb
  | U52 : symb
  | U61 : symb
  | U62 : symb
  | U63 : symb
  | U64 : symb
  | _0_1 : symb
  | a__U11 : symb
  | a__U12 : symb
  | a__U13 : symb
  | a__U14 : symb
  | a__U15 : symb
  | a__U16 : symb
  | a__U21 : symb
  | a__U22 : symb
  | a__U23 : symb
  | a__U31 : symb
  | a__U32 : symb
  | a__U41 : symb
  | a__U51 : symb
  | a__U52 : symb
  | a__U61 : symb
  | a__U62 : symb
  | a__U63 : symb
  | a__U64 : symb
  | a__isNat : symb
  | a__isNatKind : symb
  | a__plus : symb
  | isNat : symb
  | isNatKind : symb
  | mark : symb
  | plus : symb
  | s : symb
  | tt : 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 => 3
  | M.U12 => 3
  | M.U13 => 3
  | M.U14 => 3
  | M.U15 => 2
  | M.U16 => 1
  | M.U21 => 2
  | M.U22 => 2
  | M.U23 => 1
  | M.U31 => 2
  | M.U32 => 1
  | M.U41 => 1
  | M.U51 => 2
  | M.U52 => 2
  | M.U61 => 3
  | M.U62 => 3
  | M.U63 => 3
  | M.U64 => 3
  | M._0_1 => 0
  | M.a__U11 => 3
  | M.a__U12 => 3
  | M.a__U13 => 3
  | M.a__U14 => 3
  | M.a__U15 => 2
  | M.a__U16 => 1
  | M.a__U21 => 2
  | M.a__U22 => 2
  | M.a__U23 => 1
  | M.a__U31 => 2
  | M.a__U32 => 1
  | M.a__U41 => 1
  | M.a__U51 => 2
  | M.a__U52 => 2
  | M.a__U61 => 3
  | M.a__U62 => 3
  | M.a__U63 => 3
  | M.a__U64 => 3
  | M.a__isNat => 1
  | M.a__isNatKind => 1
  | M.a__plus => 2
  | M.isNat => 1
  | M.isNatKind => 1
  | M.mark => 1
  | M.plus => 2
  | M.s => 1
  | M.tt => 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 x3 x2 x1 := F0 M.U11 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U12 x3 x2 x1 := F0 M.U12 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U13 x3 x2 x1 := F0 M.U13 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U14 x3 x2 x1 := F0 M.U14 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U15 x2 x1 := F0 M.U15 (Vcons x2 (Vcons x1 Vnil)).
  Definition U16 x1 := F0 M.U16 (Vcons x1 Vnil).
  Definition U21 x2 x1 := F0 M.U21 (Vcons x2 (Vcons x1 Vnil)).
  Definition U22 x2 x1 := F0 M.U22 (Vcons x2 (Vcons x1 Vnil)).
  Definition U23 x1 := F0 M.U23 (Vcons x1 Vnil).
  Definition U31 x2 x1 := F0 M.U31 (Vcons x2 (Vcons x1 Vnil)).
  Definition U32 x1 := F0 M.U32 (Vcons x1 Vnil).
  Definition U41 x1 := F0 M.U41 (Vcons x1 Vnil).
  Definition U51 x2 x1 := F0 M.U51 (Vcons x2 (Vcons x1 Vnil)).
  Definition U52 x2 x1 := F0 M.U52 (Vcons x2 (Vcons x1 Vnil)).
  Definition U61 x3 x2 x1 := F0 M.U61 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U62 x3 x2 x1 := F0 M.U62 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U63 x3 x2 x1 := F0 M.U63 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U64 x3 x2 x1 := F0 M.U64 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _0_1 := F0 M._0_1 Vnil.
  Definition a__U11 x3 x2 x1 := F0 M.a__U11 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U12 x3 x2 x1 := F0 M.a__U12 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U13 x3 x2 x1 := F0 M.a__U13 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U14 x3 x2 x1 := F0 M.a__U14 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U15 x2 x1 := F0 M.a__U15 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U16 x1 := F0 M.a__U16 (Vcons x1 Vnil).
  Definition a__U21 x2 x1 := F0 M.a__U21 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U22 x2 x1 := F0 M.a__U22 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U23 x1 := F0 M.a__U23 (Vcons x1 Vnil).
  Definition a__U31 x2 x1 := F0 M.a__U31 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U32 x1 := F0 M.a__U32 (Vcons x1 Vnil).
  Definition a__U41 x1 := F0 M.a__U41 (Vcons x1 Vnil).
  Definition a__U51 x2 x1 := F0 M.a__U51 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U52 x2 x1 := F0 M.a__U52 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U61 x3 x2 x1 := F0 M.a__U61 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U62 x3 x2 x1 := F0 M.a__U62 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U63 x3 x2 x1 := F0 M.a__U63 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U64 x3 x2 x1 := F0 M.a__U64 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__isNat x1 := F0 M.a__isNat (Vcons x1 Vnil).
  Definition a__isNatKind x1 := F0 M.a__isNatKind (Vcons x1 Vnil).
  Definition a__plus x2 x1 := F0 M.a__plus (Vcons x2 (Vcons x1 Vnil)).
  Definition isNat x1 := F0 M.isNat (Vcons x1 Vnil).
  Definition isNatKind x1 := F0 M.isNatKind (Vcons x1 Vnil).
  Definition mark x1 := F0 M.mark (Vcons x1 Vnil).
  Definition plus x2 x1 := F0 M.plus (Vcons x2 (Vcons x1 Vnil)).
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition tt := F0 M.tt Vnil.
End S0.

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

Definition R :=
   R0 (S0.a__U11 S0.tt (V0 0) (V0 1))
      (S0.a__U12 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1))
:: R0 (S0.a__U12 S0.tt (V0 0) (V0 1))
      (S0.a__U13 (S0.a__isNatKind (V0 1)) (V0 0) (V0 1))
:: R0 (S0.a__U13 S0.tt (V0 0) (V0 1))
      (S0.a__U14 (S0.a__isNatKind (V0 1)) (V0 0) (V0 1))
:: R0 (S0.a__U14 S0.tt (V0 0) (V0 1))
      (S0.a__U15 (S0.a__isNat (V0 0)) (V0 1))
:: R0 (S0.a__U15 S0.tt (V0 0))
      (S0.a__U16 (S0.a__isNat (V0 0)))
:: R0 (S0.a__U16 S0.tt)
      S0.tt
:: R0 (S0.a__U21 S0.tt (V0 0))
      (S0.a__U22 (S0.a__isNatKind (V0 0)) (V0 0))
:: R0 (S0.a__U22 S0.tt (V0 0))
      (S0.a__U23 (S0.a__isNat (V0 0)))
:: R0 (S0.a__U23 S0.tt)
      S0.tt
:: R0 (S0.a__U31 S0.tt (V0 0))
      (S0.a__U32 (S0.a__isNatKind (V0 0)))
:: R0 (S0.a__U32 S0.tt)
      S0.tt
:: R0 (S0.a__U41 S0.tt)
      S0.tt
:: R0 (S0.a__U51 S0.tt (V0 0))
      (S0.a__U52 (S0.a__isNatKind (V0 0)) (V0 0))
:: R0 (S0.a__U52 S0.tt (V0 0))
      (S0.mark (V0 0))
:: R0 (S0.a__U61 S0.tt (V0 0) (V0 1))
      (S0.a__U62 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1))
:: R0 (S0.a__U62 S0.tt (V0 0) (V0 1))
      (S0.a__U63 (S0.a__isNat (V0 1)) (V0 0) (V0 1))
:: R0 (S0.a__U63 S0.tt (V0 0) (V0 1))
      (S0.a__U64 (S0.a__isNatKind (V0 1)) (V0 0) (V0 1))
:: R0 (S0.a__U64 S0.tt (V0 0) (V0 1))
      (S0.s (S0.a__plus (S0.mark (V0 1)) (S0.mark (V0 0))))
:: R0 (S0.a__isNat S0._0_1)
      S0.tt
:: R0 (S0.a__isNat (S0.plus (V0 0) (V0 1)))
      (S0.a__U11 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1))
:: R0 (S0.a__isNat (S0.s (V0 0)))
      (S0.a__U21 (S0.a__isNatKind (V0 0)) (V0 0))
:: R0 (S0.a__isNatKind S0._0_1)
      S0.tt
:: R0 (S0.a__isNatKind (S0.plus (V0 0) (V0 1)))
      (S0.a__U31 (S0.a__isNatKind (V0 0)) (V0 1))
:: R0 (S0.a__isNatKind (S0.s (V0 0)))
      (S0.a__U41 (S0.a__isNatKind (V0 0)))
:: R0 (S0.a__plus (V0 0) S0._0_1)
      (S0.a__U51 (S0.a__isNat (V0 0)) (V0 0))
:: R0 (S0.a__plus (V0 0) (S0.s (V0 1)))
      (S0.a__U61 (S0.a__isNat (V0 1)) (V0 1) (V0 0))
:: R0 (S0.mark (S0.U11 (V0 0) (V0 1) (V0 2)))
      (S0.a__U11 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.U12 (V0 0) (V0 1) (V0 2)))
      (S0.a__U12 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.isNatKind (V0 0)))
      (S0.a__isNatKind (V0 0))
:: R0 (S0.mark (S0.U13 (V0 0) (V0 1) (V0 2)))
      (S0.a__U13 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.U14 (V0 0) (V0 1) (V0 2)))
      (S0.a__U14 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.U15 (V0 0) (V0 1)))
      (S0.a__U15 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.isNat (V0 0)))
      (S0.a__isNat (V0 0))
:: R0 (S0.mark (S0.U16 (V0 0)))
      (S0.a__U16 (S0.mark (V0 0)))
:: R0 (S0.mark (S0.U21 (V0 0) (V0 1)))
      (S0.a__U21 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.U22 (V0 0) (V0 1)))
      (S0.a__U22 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.U23 (V0 0)))
      (S0.a__U23 (S0.mark (V0 0)))
:: R0 (S0.mark (S0.U31 (V0 0) (V0 1)))
      (S0.a__U31 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.U32 (V0 0)))
      (S0.a__U32 (S0.mark (V0 0)))
:: R0 (S0.mark (S0.U41 (V0 0)))
      (S0.a__U41 (S0.mark (V0 0)))
:: R0 (S0.mark (S0.U51 (V0 0) (V0 1)))
      (S0.a__U51 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.U52 (V0 0) (V0 1)))
      (S0.a__U52 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.U61 (V0 0) (V0 1) (V0 2)))
      (S0.a__U61 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.U62 (V0 0) (V0 1) (V0 2)))
      (S0.a__U62 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.U63 (V0 0) (V0 1) (V0 2)))
      (S0.a__U63 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.U64 (V0 0) (V0 1) (V0 2)))
      (S0.a__U64 (S0.mark (V0 0)) (V0 1) (V0 2))
:: R0 (S0.mark (S0.plus (V0 0) (V0 1)))
      (S0.a__plus (S0.mark (V0 0)) (S0.mark (V0 1)))
:: R0 (S0.mark S0.tt)
      S0.tt
:: R0 (S0.mark (S0.s (V0 0)))
      (S0.s (S0.mark (V0 0)))
:: R0 (S0.mark S0._0_1)
      S0._0_1
:: R0 (S0.a__U11 (V0 0) (V0 1) (V0 2))
      (S0.U11 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__U12 (V0 0) (V0 1) (V0 2))
      (S0.U12 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__isNatKind (V0 0))
      (S0.isNatKind (V0 0))
:: R0 (S0.a__U13 (V0 0) (V0 1) (V0 2))
      (S0.U13 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__U14 (V0 0) (V0 1) (V0 2))
      (S0.U14 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__U15 (V0 0) (V0 1))
      (S0.U15 (V0 0) (V0 1))
:: R0 (S0.a__isNat (V0 0))
      (S0.isNat (V0 0))
:: R0 (S0.a__U16 (V0 0))
      (S0.U16 (V0 0))
:: R0 (S0.a__U21 (V0 0) (V0 1))
      (S0.U21 (V0 0) (V0 1))
:: R0 (S0.a__U22 (V0 0) (V0 1))
      (S0.U22 (V0 0) (V0 1))
:: R0 (S0.a__U23 (V0 0))
      (S0.U23 (V0 0))
:: R0 (S0.a__U31 (V0 0) (V0 1))
      (S0.U31 (V0 0) (V0 1))
:: R0 (S0.a__U32 (V0 0))
      (S0.U32 (V0 0))
:: R0 (S0.a__U41 (V0 0))
      (S0.U41 (V0 0))
:: R0 (S0.a__U51 (V0 0) (V0 1))
      (S0.U51 (V0 0) (V0 1))
:: R0 (S0.a__U52 (V0 0) (V0 1))
      (S0.U52 (V0 0) (V0 1))
:: R0 (S0.a__U61 (V0 0) (V0 1) (V0 2))
      (S0.U61 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__U62 (V0 0) (V0 1) (V0 2))
      (S0.U62 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__U63 (V0 0) (V0 1) (V0 2))
      (S0.U63 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__U64 (V0 0) (V0 1) (V0 2))
      (S0.U64 (V0 0) (V0 1) (V0 2))
:: R0 (S0.a__plus (V0 0) (V0 1))
      (S0.plus (V0 0) (V0 1))
:: @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 x3 x2 x1 := F1 (hd_symb s1_p M.U11) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U11 x3 x2 x1 := F1 (int_symb s1_p M.U11) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU12 x3 x2 x1 := F1 (hd_symb s1_p M.U12) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U12 x3 x2 x1 := F1 (int_symb s1_p M.U12) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU13 x3 x2 x1 := F1 (hd_symb s1_p M.U13) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U13 x3 x2 x1 := F1 (int_symb s1_p M.U13) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU14 x3 x2 x1 := F1 (hd_symb s1_p M.U14) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U14 x3 x2 x1 := F1 (int_symb s1_p M.U14) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU15 x2 x1 := F1 (hd_symb s1_p M.U15) (Vcons x2 (Vcons x1 Vnil)).
  Definition U15 x2 x1 := F1 (int_symb s1_p M.U15) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU16 x1 := F1 (hd_symb s1_p M.U16) (Vcons x1 Vnil).
  Definition U16 x1 := F1 (int_symb s1_p M.U16) (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 x2 x1 := F1 (hd_symb s1_p M.U22) (Vcons x2 (Vcons x1 Vnil)).
  Definition U22 x2 x1 := F1 (int_symb s1_p M.U22) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU23 x1 := F1 (hd_symb s1_p M.U23) (Vcons x1 Vnil).
  Definition U23 x1 := F1 (int_symb s1_p M.U23) (Vcons x1 Vnil).
  Definition hU31 x2 x1 := F1 (hd_symb s1_p M.U31) (Vcons x2 (Vcons x1 Vnil)).
  Definition U31 x2 x1 := F1 (int_symb s1_p M.U31) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU32 x1 := F1 (hd_symb s1_p M.U32) (Vcons x1 Vnil).
  Definition U32 x1 := F1 (int_symb s1_p M.U32) (Vcons x1 Vnil).
  Definition hU41 x1 := F1 (hd_symb s1_p M.U41) (Vcons x1 Vnil).
  Definition U41 x1 := F1 (int_symb s1_p M.U41) (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 x2 x1 := F1 (hd_symb s1_p M.U52) (Vcons x2 (Vcons x1 Vnil)).
  Definition U52 x2 x1 := F1 (int_symb s1_p M.U52) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU61 x3 x2 x1 := F1 (hd_symb s1_p M.U61) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U61 x3 x2 x1 := F1 (int_symb s1_p M.U61) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU62 x3 x2 x1 := F1 (hd_symb s1_p M.U62) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U62 x3 x2 x1 := F1 (int_symb s1_p M.U62) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU63 x3 x2 x1 := F1 (hd_symb s1_p M.U63) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U63 x3 x2 x1 := F1 (int_symb s1_p M.U63) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hU64 x3 x2 x1 := F1 (hd_symb s1_p M.U64) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U64 x3 x2 x1 := F1 (int_symb s1_p M.U64) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition h_0_1 := F1 (hd_symb s1_p M._0_1) Vnil.
  Definition _0_1 := F1 (int_symb s1_p M._0_1) Vnil.
  Definition ha__U11 x3 x2 x1 := F1 (hd_symb s1_p M.a__U11) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U11 x3 x2 x1 := F1 (int_symb s1_p M.a__U11) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U12 x3 x2 x1 := F1 (hd_symb s1_p M.a__U12) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U12 x3 x2 x1 := F1 (int_symb s1_p M.a__U12) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U13 x3 x2 x1 := F1 (hd_symb s1_p M.a__U13) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U13 x3 x2 x1 := F1 (int_symb s1_p M.a__U13) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U14 x3 x2 x1 := F1 (hd_symb s1_p M.a__U14) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U14 x3 x2 x1 := F1 (int_symb s1_p M.a__U14) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U15 x2 x1 := F1 (hd_symb s1_p M.a__U15) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U15 x2 x1 := F1 (int_symb s1_p M.a__U15) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__U16 x1 := F1 (hd_symb s1_p M.a__U16) (Vcons x1 Vnil).
  Definition a__U16 x1 := F1 (int_symb s1_p M.a__U16) (Vcons x1 Vnil).
  Definition ha__U21 x2 x1 := F1 (hd_symb s1_p M.a__U21) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U21 x2 x1 := F1 (int_symb s1_p M.a__U21) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__U22 x2 x1 := F1 (hd_symb s1_p M.a__U22) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U22 x2 x1 := F1 (int_symb s1_p M.a__U22) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__U23 x1 := F1 (hd_symb s1_p M.a__U23) (Vcons x1 Vnil).
  Definition a__U23 x1 := F1 (int_symb s1_p M.a__U23) (Vcons x1 Vnil).
  Definition ha__U31 x2 x1 := F1 (hd_symb s1_p M.a__U31) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U31 x2 x1 := F1 (int_symb s1_p M.a__U31) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__U32 x1 := F1 (hd_symb s1_p M.a__U32) (Vcons x1 Vnil).
  Definition a__U32 x1 := F1 (int_symb s1_p M.a__U32) (Vcons x1 Vnil).
  Definition ha__U41 x1 := F1 (hd_symb s1_p M.a__U41) (Vcons x1 Vnil).
  Definition a__U41 x1 := F1 (int_symb s1_p M.a__U41) (Vcons x1 Vnil).
  Definition ha__U51 x2 x1 := F1 (hd_symb s1_p M.a__U51) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U51 x2 x1 := F1 (int_symb s1_p M.a__U51) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__U52 x2 x1 := F1 (hd_symb s1_p M.a__U52) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U52 x2 x1 := F1 (int_symb s1_p M.a__U52) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__U61 x3 x2 x1 := F1 (hd_symb s1_p M.a__U61) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U61 x3 x2 x1 := F1 (int_symb s1_p M.a__U61) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U62 x3 x2 x1 := F1 (hd_symb s1_p M.a__U62) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U62 x3 x2 x1 := F1 (int_symb s1_p M.a__U62) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U63 x3 x2 x1 := F1 (hd_symb s1_p M.a__U63) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U63 x3 x2 x1 := F1 (int_symb s1_p M.a__U63) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__U64 x3 x2 x1 := F1 (hd_symb s1_p M.a__U64) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a__U64 x3 x2 x1 := F1 (int_symb s1_p M.a__U64) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ha__isNat x1 := F1 (hd_symb s1_p M.a__isNat) (Vcons x1 Vnil).
  Definition a__isNat x1 := F1 (int_symb s1_p M.a__isNat) (Vcons x1 Vnil).
  Definition ha__isNatKind x1 := F1 (hd_symb s1_p M.a__isNatKind) (Vcons x1 Vnil).
  Definition a__isNatKind x1 := F1 (int_symb s1_p M.a__isNatKind) (Vcons x1 Vnil).
  Definition ha__plus x2 x1 := F1 (hd_symb s1_p M.a__plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__plus x2 x1 := F1 (int_symb s1_p M.a__plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition hisNat x1 := F1 (hd_symb s1_p M.isNat) (Vcons x1 Vnil).
  Definition isNat x1 := F1 (int_symb s1_p M.isNat) (Vcons x1 Vnil).
  Definition hisNatKind x1 := F1 (hd_symb s1_p M.isNatKind) (Vcons x1 Vnil).
  Definition isNatKind x1 := F1 (int_symb s1_p M.isNatKind) (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 hplus x2 x1 := F1 (hd_symb s1_p M.plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition plus x2 x1 := F1 (int_symb s1_p M.plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition hs x1 := F1 (hd_symb s1_p M.s) (Vcons x1 Vnil).
  Definition s x1 := F1 (int_symb s1_p M.s) (Vcons x1 Vnil).
  Definition htt := F1 (hd_symb s1_p M.tt) Vnil.
  Definition tt := F1 (int_symb s1_p M.tt) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmark (S1.U41 (V1 0)))
         (S1.ha__U41 (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.hmark (S1.U32 (V1 0)))
         (S1.ha__U32 (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.hmark (S1.U23 (V1 0)))
         (S1.ha__U23 (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.hmark (S1.U16 (V1 0)))
         (S1.ha__U16 (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.ha__isNatKind (S1.s (V1 0)))
         (S1.ha__U41 (S1.a__isNatKind (V1 0)))
   :: nil)

:: (  R1 (S1.ha__U31 (S1.tt) (V1 0))
         (S1.ha__U32 (S1.a__isNatKind (V1 0)))
   :: nil)

:: (  R1 (S1.ha__U31 (S1.tt) (V1 0))
         (S1.ha__isNatKind (V1 0))
   :: R1 (S1.ha__isNatKind (S1.plus (V1 0) (V1 1)))
         (S1.ha__U31 (S1.a__isNatKind (V1 0)) (V1 1))
   :: R1 (S1.ha__isNatKind (S1.plus (V1 0) (V1 1)))
         (S1.ha__isNatKind (V1 0))
   :: R1 (S1.ha__isNatKind (S1.s (V1 0)))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.hmark (S1.isNatKind (V1 0)))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.ha__isNat (S1.s (V1 0)))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.ha__isNat (S1.plus (V1 0) (V1 1)))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.ha__U63 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNatKind (V1 1))
   :: nil)

:: (  R1 (S1.ha__U61 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.ha__U51 (S1.tt) (V1 0))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.hmark (S1.U31 (V1 0) (V1 1)))
         (S1.ha__U31 (S1.mark (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.ha__U22 (S1.tt) (V1 0))
         (S1.ha__U23 (S1.a__isNat (V1 0)))
   :: nil)

:: (  R1 (S1.ha__U21 (S1.tt) (V1 0))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.ha__U15 (S1.tt) (V1 0))
         (S1.ha__U16 (S1.a__isNat (V1 0)))
   :: nil)

:: (  R1 (S1.ha__U13 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNatKind (V1 1))
   :: nil)

:: (  R1 (S1.ha__U12 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNatKind (V1 1))
   :: nil)

:: (  R1 (S1.ha__U11 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNatKind (V1 0))
   :: nil)

:: (  R1 (S1.ha__U12 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U13 (S1.a__isNatKind (V1 1)) (V1 0) (V1 1))
   :: R1 (S1.ha__U13 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U14 (S1.a__isNatKind (V1 1)) (V1 0) (V1 1))
   :: R1 (S1.ha__U14 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U15 (S1.a__isNat (V1 0)) (V1 1))
   :: R1 (S1.ha__U15 (S1.tt) (V1 0))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.ha__isNat (S1.plus (V1 0) (V1 1)))
         (S1.ha__U11 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1))
   :: R1 (S1.ha__U11 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U12 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1))
   :: R1 (S1.ha__isNat (S1.s (V1 0)))
         (S1.ha__U21 (S1.a__isNatKind (V1 0)) (V1 0))
   :: R1 (S1.ha__U21 (S1.tt) (V1 0))
         (S1.ha__U22 (S1.a__isNatKind (V1 0)) (V1 0))
   :: R1 (S1.ha__U22 (S1.tt) (V1 0))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.ha__U14 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNat (V1 0))
   :: nil)

:: (  R1 (S1.hmark (S1.U11 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U11 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hmark (S1.U22 (V1 0) (V1 1)))
         (S1.ha__U22 (S1.mark (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hmark (S1.U21 (V1 0) (V1 1)))
         (S1.ha__U21 (S1.mark (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hmark (S1.isNat (V1 0)))
         (S1.ha__isNat (V1 0))
   :: nil)

:: (  R1 (S1.ha__plus (V1 0) (S1.s (V1 1)))
         (S1.ha__isNat (V1 1))
   :: nil)

:: (  R1 (S1.ha__plus (V1 0) (S1._0_1))
         (S1.ha__isNat (V1 0))
   :: nil)

:: (  R1 (S1.ha__U62 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNat (V1 1))
   :: nil)

:: (  R1 (S1.hmark (S1.U15 (V1 0) (V1 1)))
         (S1.ha__U15 (S1.mark (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hmark (S1.U14 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U14 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hmark (S1.U13 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U13 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hmark (S1.U12 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U12 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hmark (S1.U11 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U12 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U13 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U14 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U15 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U16 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U21 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U22 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U23 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U31 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U32 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U41 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U51 (V1 0) (V1 1)))
         (S1.ha__U51 (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.ha__U51 (S1.tt) (V1 0))
         (S1.ha__U52 (S1.a__isNatKind (V1 0)) (V1 0))
   :: R1 (S1.ha__U52 (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U51 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U52 (V1 0) (V1 1)))
         (S1.ha__U52 (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.hmark (S1.U52 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U61 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U61 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: R1 (S1.ha__U61 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U62 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1))
   :: R1 (S1.ha__U62 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U63 (S1.a__isNat (V1 1)) (V1 0) (V1 1))
   :: R1 (S1.ha__U63 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U64 (S1.a__isNatKind (V1 1)) (V1 0) (V1 1))
   :: R1 (S1.ha__U64 (S1.tt) (V1 0) (V1 1))
         (S1.ha__plus (S1.mark (V1 1)) (S1.mark (V1 0)))
   :: R1 (S1.ha__plus (V1 0) (S1._0_1))
         (S1.ha__U51 (S1.a__isNat (V1 0)) (V1 0))
   :: R1 (S1.ha__plus (V1 0) (S1.s (V1 1)))
         (S1.ha__U61 (S1.a__isNat (V1 1)) (V1 1) (V1 0))
   :: R1 (S1.ha__U64 (S1.tt) (V1 0) (V1 1))
         (S1.hmark (V1 1))
   :: R1 (S1.hmark (S1.U61 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U62 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U62 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: R1 (S1.hmark (S1.U62 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U63 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U63 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: R1 (S1.hmark (S1.U63 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U64 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U64 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: R1 (S1.ha__U64 (S1.tt) (V1 0) (V1 1))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U64 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.plus (V1 0) (V1 1)))
         (S1.ha__plus (S1.mark (V1 0)) (S1.mark (V1 1)))
   :: R1 (S1.hmark (S1.plus (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.plus (V1 0) (V1 1)))
         (S1.hmark (V1 1))
   :: R1 (S1.hmark (S1.s (V1 0)))
         (S1.hmark (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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         nil
    | (hd_symb M.a__isNatKind) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         nil
    | (hd_symb M.a__U31) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U31) =>
         nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (2%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) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* polynomial interpretation 2 *)

Module PIS2 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         nil
    | (hd_symb M.a__isNatKind) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         nil
    | (hd_symb M.a__U31) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U31) =>
         nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         nil
    end.

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

End PIS2.

Module PI2 := PolyInt PIS2.

(* graph decomposition 2 *)

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

   (  R1 (S1.ha__isNatKind (S1.plus (V1 0) (V1 1)))
         (S1.ha__U31 (S1.a__isNatKind (V1 0)) (V1 1))
   :: nil)

:: nil.

(* 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.a__U11) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U12) =>
         nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U13) =>
         nil
    | (hd_symb M.a__U14) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U14) =>
         nil
    | (hd_symb M.a__U15) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (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 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         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) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

(* 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.a__U11) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U12) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U13) =>
         nil
    | (hd_symb M.a__U14) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U14) =>
         nil
    | (hd_symb M.a__U15) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (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._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (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.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         nil
    end.

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

End PIS4.

Module PI4 := PolyInt PIS4.

(* graph decomposition 3 *)

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

   (  R1 (S1.ha__U14 (S1.tt) (V1 0) (V1 1))
         (S1.ha__isNat (V1 0))
   :: nil)

:: (  R1 (S1.ha__U11 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U12 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.ha__U15 (S1.tt) (V1 0))
         (S1.ha__isNat (V1 0))
   :: nil)

:: (  R1 (S1.ha__U14 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U15 (S1.a__isNat (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.ha__U13 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U14 (S1.a__isNatKind (V1 1)) (V1 0) (V1 1))
   :: nil)

:: nil.

(* polynomial interpretation 5 *)

Module PIS5 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U51) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U52) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U62) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U63) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.a__U64) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__plus) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__plus) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         (3%Z, (Vcons 1 Vnil))
      :: 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) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* graph decomposition 4 *)

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

   (  R1 (S1.hmark (S1.U62 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U62 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.ha__plus (V1 0) (S1._0_1))
         (S1.ha__U51 (S1.a__isNat (V1 0)) (V1 0))
   :: nil)

:: (  R1 (S1.ha__U61 (S1.tt) (V1 0) (V1 1))
         (S1.ha__U62 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.ha__plus (V1 0) (S1.s (V1 1)))
         (S1.ha__U61 (S1.a__isNat (V1 1)) (V1 1) (V1 0))
   :: nil)

:: (  R1 (S1.hmark (S1.plus (V1 0) (V1 1)))
         (S1.ha__plus (S1.mark (V1 0)) (S1.mark (V1 1)))
   :: nil)

:: (  R1 (S1.ha__U64 (S1.tt) (V1 0) (V1 1))
         (S1.ha__plus (S1.mark (V1 1)) (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.hmark (S1.U61 (V1 0) (V1 1) (V1 2)))
         (S1.ha__U61 (S1.mark (V1 0)) (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hmark (S1.U51 (V1 0) (V1 1)))
         (S1.ha__U51 (S1.mark (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hmark (S1.U12 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U11 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U13 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U14 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U15 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U16 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U21 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U22 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U23 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U31 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U32 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U41 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U52 (V1 0) (V1 1)))
         (S1.ha__U52 (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.ha__U52 (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U52 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.plus (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.plus (V1 0) (V1 1)))
         (S1.hmark (V1 1))
   :: nil)

:: (  R1 (S1.ha__U64 (S1.tt) (V1 0) (V1 1))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__U64 (S1.tt) (V1 0) (V1 1))
         (S1.hmark (V1 1))
   :: nil)

:: nil.

(* 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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U52) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (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._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (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.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 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) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U52) =>
         (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.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         nil
    end.

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

End PIS7.

Module PI7 := PolyInt PIS7.

(* graph decomposition 5 *)

Definition cs5 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hmark (S1.U11 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U12 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U13 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U14 (V1 0) (V1 1) (V1 2)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U15 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U16 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U21 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U22 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U23 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U31 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U32 (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U41 (V1 0)))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__U52 (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: nil)

:: nil.

(* 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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (2%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) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: 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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (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.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (2%Z, (Vcons 0 Vnil))
      :: (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) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: 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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (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._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (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.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U32) =>
         nil
    | (int_symb M.U32) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         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.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.a__U12) =>
         nil
    | (int_symb M.a__U12) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__isNatKind) =>
         nil
    | (int_symb M.a__isNatKind) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U13) =>
         nil
    | (int_symb M.a__U13) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U14) =>
         nil
    | (int_symb M.a__U14) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.a__U15) =>
         nil
    | (int_symb M.a__U15) =>
         nil
    | (hd_symb M.a__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U16) =>
         nil
    | (int_symb M.a__U16) =>
         nil
    | (hd_symb M.a__U21) =>
         nil
    | (int_symb M.a__U21) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U22) =>
         nil
    | (int_symb M.a__U22) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.a__U23) =>
         nil
    | (int_symb M.a__U23) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.a__U31) =>
         nil
    | (int_symb M.a__U31) =>
         (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.a__U32) =>
         nil
    | (int_symb M.a__U32) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U41) =>
         nil
    | (int_symb M.a__U41) =>
         nil
    | (hd_symb M.a__U51) =>
         nil
    | (int_symb M.a__U51) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__U52) =>
         nil
    | (int_symb M.a__U52) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__U61) =>
         nil
    | (int_symb M.a__U61) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U62) =>
         nil
    | (int_symb M.a__U62) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U63) =>
         nil
    | (int_symb M.a__U63) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.a__U64) =>
         nil
    | (int_symb M.a__U64) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.a__plus) =>
         nil
    | (int_symb M.a__plus) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (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.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.isNatKind) =>
         nil
    | (int_symb M.isNatKind) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U13) =>
         nil
    | (int_symb M.U13) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U14) =>
         nil
    | (int_symb M.U14) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U15) =>
         nil
    | (int_symb M.U15) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U16) =>
         nil
    | (int_symb M.U16) =>
         nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U22) =>
         nil
    | (int_symb M.U22) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.U23) =>
         nil
    | (int_symb M.U23) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.U31) =>
         nil
    | (int_symb M.U31) =>
         (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.U32) =>
         nil
    | (int_symb M.U32) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         nil
    | (hd_symb M.U51) =>
         nil
    | (int_symb M.U51) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U52) =>
         nil
    | (int_symb M.U52) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U61) =>
         nil
    | (int_symb M.U61) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U62) =>
         nil
    | (int_symb M.U62) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U63) =>
         nil
    | (int_symb M.U63) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U64) =>
         nil
    | (int_symb M.U64) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS12.

Module PI12 := PolyInt PIS12.

(* termination proof *)

Lemma termination : WF rel.

Proof.
unfold rel.
dp_trans.
mark.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs1; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI1.prove_termination.
PI2.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs2; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI3.prove_termination.
PI4.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs3; subst D; subst R.
dpg_unif_N_correct.
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. PI5.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs4; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI6.prove_termination.
PI7.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs5; subst D; subst R.
dpg_unif_N_correct.
right. PI8.prove_termination.
PI9.prove_termination.
PI10.prove_termination.
PI11.prove_termination.
PI12.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.