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
  | _0_1 : symb
  | a__U11 : symb
  | a__and : symb
  | a__isNat : symb
  | a__isNatIList : symb
  | a__isNatList : symb
  | a__length : symb
  | a__zeros : symb
  | and : symb
  | cons : symb
  | isNat : symb
  | isNatIList : symb
  | isNatList : symb
  | length : symb
  | mark : symb
  | nil : symb
  | s : symb
  | tt : symb
  | zeros : 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 => 2
  | M._0_1 => 0
  | M.a__U11 => 2
  | M.a__and => 2
  | M.a__isNat => 1
  | M.a__isNatIList => 1
  | M.a__isNatList => 1
  | M.a__length => 1
  | M.a__zeros => 0
  | M.and => 2
  | M.cons => 2
  | M.isNat => 1
  | M.isNatIList => 1
  | M.isNatList => 1
  | M.length => 1
  | M.mark => 1
  | M.nil => 0
  | M.s => 1
  | M.tt => 0
  | M.zeros => 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 x2 x1 := F0 M.U11 (Vcons x2 (Vcons x1 Vnil)).
  Definition _0_1 := F0 M._0_1 Vnil.
  Definition a__U11 x2 x1 := F0 M.a__U11 (Vcons x2 (Vcons x1 Vnil)).
  Definition a__and x2 x1 := F0 M.a__and (Vcons x2 (Vcons x1 Vnil)).
  Definition a__isNat x1 := F0 M.a__isNat (Vcons x1 Vnil).
  Definition a__isNatIList x1 := F0 M.a__isNatIList (Vcons x1 Vnil).
  Definition a__isNatList x1 := F0 M.a__isNatList (Vcons x1 Vnil).
  Definition a__length x1 := F0 M.a__length (Vcons x1 Vnil).
  Definition a__zeros := F0 M.a__zeros Vnil.
  Definition and x2 x1 := F0 M.and (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition isNat x1 := F0 M.isNat (Vcons x1 Vnil).
  Definition isNatIList x1 := F0 M.isNatIList (Vcons x1 Vnil).
  Definition isNatList x1 := F0 M.isNatList (Vcons x1 Vnil).
  Definition length x1 := F0 M.length (Vcons x1 Vnil).
  Definition mark x1 := F0 M.mark (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition tt := F0 M.tt Vnil.
  Definition zeros := F0 M.zeros Vnil.
End S0.

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

Definition R :=
   R0 S0.a__zeros
      (S0.cons S0._0_1 S0.zeros)
:: R0 (S0.a__U11 S0.tt (V0 0))
      (S0.s (S0.a__length (S0.mark (V0 0))))
:: R0 (S0.a__and S0.tt (V0 0))
      (S0.mark (V0 0))
:: R0 (S0.a__isNat S0._0_1)
      S0.tt
:: R0 (S0.a__isNat (S0.length (V0 0)))
      (S0.a__isNatList (V0 0))
:: R0 (S0.a__isNat (S0.s (V0 0)))
      (S0.a__isNat (V0 0))
:: R0 (S0.a__isNatIList (V0 0))
      (S0.a__isNatList (V0 0))
:: R0 (S0.a__isNatIList S0.zeros)
      S0.tt
:: R0 (S0.a__isNatIList (S0.cons (V0 0) (V0 1)))
      (S0.a__and (S0.a__isNat (V0 0)) (S0.isNatIList (V0 1)))
:: R0 (S0.a__isNatList S0.nil)
      S0.tt
:: R0 (S0.a__isNatList (S0.cons (V0 0) (V0 1)))
      (S0.a__and (S0.a__isNat (V0 0)) (S0.isNatList (V0 1)))
:: R0 (S0.a__length S0.nil)
      S0._0_1
:: R0 (S0.a__length (S0.cons (V0 0) (V0 1)))
      (S0.a__U11 (S0.a__and (S0.a__isNatList (V0 1)) (S0.isNat (V0 0))) (V0 1))
:: R0 (S0.mark S0.zeros)
      S0.a__zeros
:: R0 (S0.mark (S0.U11 (V0 0) (V0 1)))
      (S0.a__U11 (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.length (V0 0)))
      (S0.a__length (S0.mark (V0 0)))
:: R0 (S0.mark (S0.and (V0 0) (V0 1)))
      (S0.a__and (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.isNat (V0 0)))
      (S0.a__isNat (V0 0))
:: R0 (S0.mark (S0.isNatList (V0 0)))
      (S0.a__isNatList (V0 0))
:: R0 (S0.mark (S0.isNatIList (V0 0)))
      (S0.a__isNatIList (V0 0))
:: R0 (S0.mark (S0.cons (V0 0) (V0 1)))
      (S0.cons (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark S0._0_1)
      S0._0_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.nil)
      S0.nil
:: R0 S0.a__zeros
      S0.zeros
:: R0 (S0.a__U11 (V0 0) (V0 1))
      (S0.U11 (V0 0) (V0 1))
:: R0 (S0.a__length (V0 0))
      (S0.length (V0 0))
:: R0 (S0.a__and (V0 0) (V0 1))
      (S0.and (V0 0) (V0 1))
:: R0 (S0.a__isNat (V0 0))
      (S0.isNat (V0 0))
:: R0 (S0.a__isNatList (V0 0))
      (S0.isNatList (V0 0))
:: R0 (S0.a__isNatIList (V0 0))
      (S0.isNatIList (V0 0))
:: @nil (@ATrs.rule s0).

Definition rel := ATrs.red_mod E R.

(* symbol marking *)

Definition s1 := dup_sig s0.
Definition s1_p := s0.
Definition V1 := @ATerm.Var s1.
Definition F1 := @ATerm.Fun s1.
Definition R1 := @ATrs.mkRule s1.

Module S1.
  Definition hU11 x2 x1 := F1 (hd_symb s1_p M.U11) (Vcons x2 (Vcons x1 Vnil)).
  Definition U11 x2 x1 := F1 (int_symb s1_p M.U11) (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 x2 x1 := F1 (hd_symb s1_p M.a__U11) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__U11 x2 x1 := F1 (int_symb s1_p M.a__U11) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__and x2 x1 := F1 (hd_symb s1_p M.a__and) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__and x2 x1 := F1 (int_symb s1_p M.a__and) (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__isNatIList x1 := F1 (hd_symb s1_p M.a__isNatIList) (Vcons x1 Vnil).
  Definition a__isNatIList x1 := F1 (int_symb s1_p M.a__isNatIList) (Vcons x1 Vnil).
  Definition ha__isNatList x1 := F1 (hd_symb s1_p M.a__isNatList) (Vcons x1 Vnil).
  Definition a__isNatList x1 := F1 (int_symb s1_p M.a__isNatList) (Vcons x1 Vnil).
  Definition ha__length x1 := F1 (hd_symb s1_p M.a__length) (Vcons x1 Vnil).
  Definition a__length x1 := F1 (int_symb s1_p M.a__length) (Vcons x1 Vnil).
  Definition ha__zeros := F1 (hd_symb s1_p M.a__zeros) Vnil.
  Definition a__zeros := F1 (int_symb s1_p M.a__zeros) Vnil.
  Definition hand x2 x1 := F1 (hd_symb s1_p M.and) (Vcons x2 (Vcons x1 Vnil)).
  Definition and x2 x1 := F1 (int_symb s1_p M.and) (Vcons x2 (Vcons x1 Vnil)).
  Definition hcons x2 x1 := F1 (hd_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F1 (int_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition 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 hisNatIList x1 := F1 (hd_symb s1_p M.isNatIList) (Vcons x1 Vnil).
  Definition isNatIList x1 := F1 (int_symb s1_p M.isNatIList) (Vcons x1 Vnil).
  Definition hisNatList x1 := F1 (hd_symb s1_p M.isNatList) (Vcons x1 Vnil).
  Definition isNatList x1 := F1 (int_symb s1_p M.isNatList) (Vcons x1 Vnil).
  Definition hlength x1 := F1 (hd_symb s1_p M.length) (Vcons x1 Vnil).
  Definition length x1 := F1 (int_symb s1_p M.length) (Vcons x1 Vnil).
  Definition hmark x1 := F1 (hd_symb s1_p M.mark) (Vcons x1 Vnil).
  Definition mark x1 := F1 (int_symb s1_p M.mark) (Vcons x1 Vnil).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition 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.
  Definition hzeros := F1 (hd_symb s1_p M.zeros) Vnil.
  Definition zeros := F1 (int_symb s1_p M.zeros) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmark (S1.zeros))
         (S1.ha__zeros)
   :: nil)

:: (  R1 (S1.ha__length (S1.cons (V1 0) (V1 1)))
         (S1.ha__U11 (S1.a__and (S1.a__isNatList (V1 1)) (S1.isNat (V1 0))) (V1 1))
   :: R1 (S1.ha__U11 (S1.tt) (V1 0))
         (S1.ha__length (S1.mark (V1 0)))
   :: R1 (S1.ha__length (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNatList (V1 1)) (S1.isNat (V1 0)))
   :: R1 (S1.ha__and (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U11 (V1 0) (V1 1)))
         (S1.ha__U11 (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.ha__U11 (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.U11 (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.length (V1 0)))
         (S1.ha__length (S1.mark (V1 0)))
   :: R1 (S1.ha__length (S1.cons (V1 0) (V1 1)))
         (S1.ha__isNatList (V1 1))
   :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatList (V1 1)))
   :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.ha__isNat (S1.length (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.ha__isNat (S1.s (V1 0)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.hmark (S1.length (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.ha__and (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.isNat (V1 0)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.hmark (S1.isNatList (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.hmark (S1.isNatIList (V1 0)))
         (S1.ha__isNatIList (V1 0))
   :: R1 (S1.ha__isNatIList (V1 0))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatIList (V1 1)))
   :: R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.hmark (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (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__and) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (2%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) =>
         nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (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.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* graph decomposition 2 *)

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

   (  R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.ha__and (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.ha__and (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.isNat (V1 0)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.ha__isNat (S1.length (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatList (V1 1)))
   :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.ha__isNat (S1.s (V1 0)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.hmark (S1.isNatList (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.hmark (S1.isNatIList (V1 0)))
         (S1.ha__isNatIList (V1 0))
   :: R1 (S1.ha__isNatIList (V1 0))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatIList (V1 1)))
   :: R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1)))
         (S1.ha__isNat (V1 0))
   :: R1 (S1.hmark (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.s (V1 0)))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__U11 (S1.tt) (V1 0))
         (S1.ha__length (S1.mark (V1 0)))
   :: R1 (S1.ha__length (S1.cons (V1 0) (V1 1)))
         (S1.ha__U11 (S1.a__and (S1.a__isNatList (V1 1)) (S1.isNat (V1 0))) (V1 1))
   :: nil)

:: nil.

(* 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         (2%Z, Vnil)
      :: nil
    | (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.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         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__and) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (1%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) =>
         nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End PIS2.

Module PI2 := PolyInt PIS2.

(* polynomial interpretation 3 *)

Module PIS3 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.a__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (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.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (2%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) =>
         nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         nil
    | (hd_symb M.a__isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (int_symb M.a__isNatIList) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%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.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNat) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNatList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNatIList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: 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__isNat (S1.s (V1 0)))
         (S1.ha__isNat (V1 0))
   :: nil)

:: (  R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__isNat (V1 0))
   :: nil)

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

:: (  R1 (S1.ha__and (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.ha__and (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.isNatList (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatList (V1 1)))
   :: R1 (S1.hmark (S1.isNatIList (V1 0)))
         (S1.ha__isNatIList (V1 0))
   :: R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatIList (V1 1)))
   :: R1 (S1.hmark (S1.s (V1 0)))
         (S1.hmark (V1 0))
   :: nil)

:: nil.

(* polynomial interpretation 5 *)

Module PIS5 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.a__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (1%Z, 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__length) =>
         nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         nil
    | (int_symb M.a__and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__isNat) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNat) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* polynomial interpretation 6 *)

Module PIS6 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.a__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         (1%Z, Vnil)
      :: nil
    | (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.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (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) =>
         nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         nil
    | (hd_symb M.a__isNatIList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNatIList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End PIS6.

Module PI6 := PolyInt PIS6.

(* graph decomposition 4 *)

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

   (  R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.ha__and (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.ha__and (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.isNatList (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatList (V1 1)))
   :: R1 (S1.hmark (S1.s (V1 0)))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1)))
         (S1.ha__and (S1.a__isNat (V1 0)) (S1.isNatIList (V1 1)))
   :: nil)

:: nil.

(* 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (1%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.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End PIS7.

Module PI7 := PolyInt PIS7.

(* graph decomposition 5 *)

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

   (  R1 (S1.hmark (S1.isNatList (V1 0)))
         (S1.ha__isNatList (V1 0))
   :: nil)

:: (  R1 (S1.ha__and (S1.tt) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.ha__and (S1.mark (V1 0)) (V1 1))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.s (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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__and) =>
         (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) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (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.

(* graph decomposition 6 *)

Definition cs6 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hmark (S1.s (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.and (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: nil)

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

:: nil.

(* 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         nil
    | (int_symb M.a__and) =>
         (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__isNat) =>
         nil
    | (int_symb M.a__isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         nil
    | (int_symb M.a__U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         (1%Z, 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__length) =>
         nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         nil
    | (int_symb M.a__and) =>
         (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 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End 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__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__U11) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__U11) =>
         (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.tt) =>
         nil
    | (int_symb M.tt) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__and) =>
         nil
    | (int_symb M.a__and) =>
         (1%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 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.length) =>
         nil
    | (int_symb M.length) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatList) =>
         nil
    | (int_symb M.a__isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__isNatIList) =>
         nil
    | (int_symb M.a__isNatIList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNatIList) =>
         nil
    | (int_symb M.isNatIList) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.isNatList) =>
         nil
    | (int_symb M.isNatList) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.isNat) =>
         nil
    | (int_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U11) =>
         nil
    | (int_symb M.U11) =>
         (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.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    end.

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

End PIS11.

Module PI11 := PolyInt PIS11.

(* graph decomposition 7 *)

Definition cs7 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.ha__U11 (S1.tt) (V1 0))
         (S1.ha__length (S1.mark (V1 0)))
   :: nil)

:: nil.

(* termination proof *)

Lemma termination : WF rel.

Proof.
unfold rel.
dp_trans.
mark.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs1; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
right. PI1.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.
right. PI2.prove_termination.
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.
right. PI5.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI6.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.
right. 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.
left. co_scc.
right. PI8.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) cs6; subst D; subst R.
dpg_unif_N_correct.
right. PI9.prove_termination.
PI10.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI11.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) cs7; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
Qed.