Require Import ADPUnif.
Require Import ADecomp.
Require Import ADuplicateSymb.
Require Import AGraph.
Require Import APolyInt_MA.
Require Import ATrs.
Require Import List.
Require Import LogicUtil.
Require Import MonotonePolynom.
Require Import Polynom.
Require Import SN.
Require Import VecUtil.

Open Scope nat_scope.
(* termination problem *)

Module M.
  Inductive symb : Type :=
  | _0_4 : symb
  | _1_5 : symb
  | _if_6 : symb
  | _minus__3 : symb
  | _plus__2 : symb
  | _sharp__1 : symb
  | and : symb
  | bs : symb
  | false : symb
  | ge : symb
  | l : symb
  | max : symb
  | min : symb
  | n : symb
  | not : symb
  | size : symb
  | true : symb
  | val : symb
  | wb : symb.
End M.

Lemma eq_symb_dec : forall f g : M.symb, {f=g}+{~f=g}.

Proof.
decide equality.
Defined.

Open Scope nat_scope.
Definition ar (s : M.symb) : nat :=
  match s with
  | M._0_4 => 1
  | M._1_5 => 1
  | M._if_6 => 3
  | M._minus__3 => 2
  | M._plus__2 => 2
  | M._sharp__1 => 0
  | M.and => 2
  | M.bs => 1
  | M.false => 0
  | M.ge => 2
  | M.l => 1
  | M.max => 1
  | M.min => 1
  | M.n => 3
  | M.not => 1
  | M.size => 1
  | M.true => 0
  | M.val => 1
  | M.wb => 1
  end.

Definition s0 := ASignature.mkSignature ar eq_symb_dec.
Definition s0_p := s0.
Definition V0 := @ATerm.Var s0.
Definition F0 := @ATerm.Fun s0.
Definition R0 := @ATrs.mkRule s0.

Module S0.
  Definition _0_4 x1 := F0 M._0_4 (Vcons x1 Vnil).
  Definition _1_5 x1 := F0 M._1_5 (Vcons x1 Vnil).
  Definition _if_6 x3 x2 x1 := F0 M._if_6 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _minus__3 x2 x1 := F0 M._minus__3 (Vcons x2 (Vcons x1 Vnil)).
  Definition _plus__2 x2 x1 := F0 M._plus__2 (Vcons x2 (Vcons x1 Vnil)).
  Definition _sharp__1 := F0 M._sharp__1 Vnil.
  Definition and x2 x1 := F0 M.and (Vcons x2 (Vcons x1 Vnil)).
  Definition bs x1 := F0 M.bs (Vcons x1 Vnil).
  Definition false := F0 M.false Vnil.
  Definition ge x2 x1 := F0 M.ge (Vcons x2 (Vcons x1 Vnil)).
  Definition l x1 := F0 M.l (Vcons x1 Vnil).
  Definition max x1 := F0 M.max (Vcons x1 Vnil).
  Definition min x1 := F0 M.min (Vcons x1 Vnil).
  Definition n x3 x2 x1 := F0 M.n (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition not x1 := F0 M.not (Vcons x1 Vnil).
  Definition size x1 := F0 M.size (Vcons x1 Vnil).
  Definition true := F0 M.true Vnil.
  Definition val x1 := F0 M.val (Vcons x1 Vnil).
  Definition wb x1 := F0 M.wb (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0._0_4 S0._sharp__1)
      S0._sharp__1
:: R0 (S0._plus__2 (V0 0) S0._sharp__1)
      (V0 0)
:: R0 (S0._plus__2 S0._sharp__1 (V0 0))
      (V0 0)
:: R0 (S0._plus__2 (S0._0_4 (V0 0)) (S0._0_4 (V0 1)))
      (S0._0_4 (S0._plus__2 (V0 0) (V0 1)))
:: R0 (S0._plus__2 (S0._0_4 (V0 0)) (S0._1_5 (V0 1)))
      (S0._1_5 (S0._plus__2 (V0 0) (V0 1)))
:: R0 (S0._plus__2 (S0._1_5 (V0 0)) (S0._0_4 (V0 1)))
      (S0._1_5 (S0._plus__2 (V0 0) (V0 1)))
:: R0 (S0._plus__2 (S0._1_5 (V0 0)) (S0._1_5 (V0 1)))
      (S0._0_4 (S0._plus__2 (S0._plus__2 (V0 0) (V0 1)) (S0._1_5 S0._sharp__1)))
:: R0 (S0._plus__2 (V0 0) (S0._plus__2 (V0 1) (V0 2)))
      (S0._plus__2 (S0._plus__2 (V0 0) (V0 1)) (V0 2))
:: R0 (S0._minus__3 (V0 0) S0._sharp__1)
      (V0 0)
:: R0 (S0._minus__3 S0._sharp__1 (V0 0))
      S0._sharp__1
:: R0 (S0._minus__3 (S0._0_4 (V0 0)) (S0._0_4 (V0 1)))
      (S0._0_4 (S0._minus__3 (V0 0) (V0 1)))
:: R0 (S0._minus__3 (S0._0_4 (V0 0)) (S0._1_5 (V0 1)))
      (S0._1_5 (S0._minus__3 (S0._minus__3 (V0 0) (V0 1)) (S0._1_5 S0._sharp__1)))
:: R0 (S0._minus__3 (S0._1_5 (V0 0)) (S0._0_4 (V0 1)))
      (S0._1_5 (S0._minus__3 (V0 0) (V0 1)))
:: R0 (S0._minus__3 (S0._1_5 (V0 0)) (S0._1_5 (V0 1)))
      (S0._0_4 (S0._minus__3 (V0 0) (V0 1)))
:: R0 (S0.not S0.false)
      S0.true
:: R0 (S0.not S0.true)
      S0.false
:: R0 (S0.and (V0 0) S0.true)
      (V0 0)
:: R0 (S0.and (V0 0) S0.false)
      S0.false
:: R0 (S0._if_6 S0.true (V0 0) (V0 1))
      (V0 0)
:: R0 (S0._if_6 S0.false (V0 0) (V0 1))
      (V0 1)
:: R0 (S0.ge (S0._0_4 (V0 0)) (S0._0_4 (V0 1)))
      (S0.ge (V0 0) (V0 1))
:: R0 (S0.ge (S0._0_4 (V0 0)) (S0._1_5 (V0 1)))
      (S0.not (S0.ge (V0 1) (V0 0)))
:: R0 (S0.ge (S0._1_5 (V0 0)) (S0._0_4 (V0 1)))
      (S0.ge (V0 0) (V0 1))
:: R0 (S0.ge (S0._1_5 (V0 0)) (S0._1_5 (V0 1)))
      (S0.ge (V0 0) (V0 1))
:: R0 (S0.ge (V0 0) S0._sharp__1)
      S0.true
:: R0 (S0.ge S0._sharp__1 (S0._1_5 (V0 0)))
      S0.false
:: R0 (S0.ge S0._sharp__1 (S0._0_4 (V0 0)))
      (S0.ge S0._sharp__1 (V0 0))
:: R0 (S0.val (S0.l (V0 0)))
      (V0 0)
:: R0 (S0.val (S0.n (V0 0) (V0 1) (V0 2)))
      (V0 0)
:: R0 (S0.min (S0.l (V0 0)))
      (V0 0)
:: R0 (S0.min (S0.n (V0 0) (V0 1) (V0 2)))
      (S0.min (V0 1))
:: R0 (S0.max (S0.l (V0 0)))
      (V0 0)
:: R0 (S0.max (S0.n (V0 0) (V0 1) (V0 2)))
      (S0.max (V0 2))
:: R0 (S0.bs (S0.l (V0 0)))
      S0.true
:: R0 (S0.bs (S0.n (V0 0) (V0 1) (V0 2)))
      (S0.and (S0.and (S0.ge (V0 0) (S0.max (V0 1))) (S0.ge (S0.min (V0 2)) (V0 0))) (S0.and (S0.bs (V0 1)) (S0.bs (V0 2))))
:: R0 (S0.size (S0.l (V0 0)))
      (S0._1_5 S0._sharp__1)
:: R0 (S0.size (S0.n (V0 0) (V0 1) (V0 2)))
      (S0._plus__2 (S0._plus__2 (S0.size (V0 0)) (S0.size (V0 1))) (S0._1_5 S0._sharp__1))
:: R0 (S0.wb (S0.l (V0 0)))
      S0.true
:: R0 (S0.wb (S0.n (V0 0) (V0 1) (V0 2)))
      (S0.and (S0._if_6 (S0.ge (S0.size (V0 1)) (S0.size (V0 2))) (S0.ge (S0._1_5 S0._sharp__1) (S0._minus__3 (S0.size (V0 1)) (S0.size (V0 2)))) (S0.ge (S0._1_5 S0._sharp__1) (S0._minus__3 (S0.size (V0 2)) (S0.size (V0 1))))) (S0.and (S0.wb (V0 1)) (S0.wb (V0 2))))
:: @nil (@ATrs.rule s0).

Definition rel := ATrs.red_mod E R.

(* symbol marking *)

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

Module S1.
  Definition h_0_4 x1 := F1 (hd_symb s1_p M._0_4) (Vcons x1 Vnil).
  Definition _0_4 x1 := F1 (int_symb s1_p M._0_4) (Vcons x1 Vnil).
  Definition h_1_5 x1 := F1 (hd_symb s1_p M._1_5) (Vcons x1 Vnil).
  Definition _1_5 x1 := F1 (int_symb s1_p M._1_5) (Vcons x1 Vnil).
  Definition h_if_6 x3 x2 x1 := F1 (hd_symb s1_p M._if_6) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _if_6 x3 x2 x1 := F1 (int_symb s1_p M._if_6) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition h_minus__3 x2 x1 := F1 (hd_symb s1_p M._minus__3) (Vcons x2 (Vcons x1 Vnil)).
  Definition _minus__3 x2 x1 := F1 (int_symb s1_p M._minus__3) (Vcons x2 (Vcons x1 Vnil)).
  Definition h_plus__2 x2 x1 := F1 (hd_symb s1_p M._plus__2) (Vcons x2 (Vcons x1 Vnil)).
  Definition _plus__2 x2 x1 := F1 (int_symb s1_p M._plus__2) (Vcons x2 (Vcons x1 Vnil)).
  Definition h_sharp__1 := F1 (hd_symb s1_p M._sharp__1) Vnil.
  Definition _sharp__1 := F1 (int_symb s1_p M._sharp__1) 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 hbs x1 := F1 (hd_symb s1_p M.bs) (Vcons x1 Vnil).
  Definition bs x1 := F1 (int_symb s1_p M.bs) (Vcons x1 Vnil).
  Definition hfalse := F1 (hd_symb s1_p M.false) Vnil.
  Definition false := F1 (int_symb s1_p M.false) Vnil.
  Definition hge x2 x1 := F1 (hd_symb s1_p M.ge) (Vcons x2 (Vcons x1 Vnil)).
  Definition ge x2 x1 := F1 (int_symb s1_p M.ge) (Vcons x2 (Vcons x1 Vnil)).
  Definition hl x1 := F1 (hd_symb s1_p M.l) (Vcons x1 Vnil).
  Definition l x1 := F1 (int_symb s1_p M.l) (Vcons x1 Vnil).
  Definition hmax x1 := F1 (hd_symb s1_p M.max) (Vcons x1 Vnil).
  Definition max x1 := F1 (int_symb s1_p M.max) (Vcons x1 Vnil).
  Definition hmin x1 := F1 (hd_symb s1_p M.min) (Vcons x1 Vnil).
  Definition min x1 := F1 (int_symb s1_p M.min) (Vcons x1 Vnil).
  Definition hn x3 x2 x1 := F1 (hd_symb s1_p M.n) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition n x3 x2 x1 := F1 (int_symb s1_p M.n) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hnot x1 := F1 (hd_symb s1_p M.not) (Vcons x1 Vnil).
  Definition not x1 := F1 (int_symb s1_p M.not) (Vcons x1 Vnil).
  Definition hsize x1 := F1 (hd_symb s1_p M.size) (Vcons x1 Vnil).
  Definition size x1 := F1 (int_symb s1_p M.size) (Vcons x1 Vnil).
  Definition htrue := F1 (hd_symb s1_p M.true) Vnil.
  Definition true := F1 (int_symb s1_p M.true) Vnil.
  Definition hval x1 := F1 (hd_symb s1_p M.val) (Vcons x1 Vnil).
  Definition val x1 := F1 (int_symb s1_p M.val) (Vcons x1 Vnil).
  Definition hwb x1 := F1 (hd_symb s1_p M.wb) (Vcons x1 Vnil).
  Definition wb x1 := F1 (int_symb s1_p M.wb) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hand (S1.wb (V1 1)) (S1.wb (V1 2)))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.h_if_6 (S1.ge (S1.size (V1 1)) (S1.size (V1 2))) (S1.ge (S1._1_5 (S1._sharp__1)) (S1._minus__3 (S1.size (V1 1)) (S1.size (V1 2)))) (S1.ge (S1._1_5 (S1._sharp__1)) (S1._minus__3 (S1.size (V1 2)) (S1.size (V1 1)))))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hand (S1._if_6 (S1.ge (S1.size (V1 1)) (S1.size (V1 2))) (S1.ge (S1._1_5 (S1._sharp__1)) (S1._minus__3 (S1.size (V1 1)) (S1.size (V1 2)))) (S1.ge (S1._1_5 (S1._sharp__1)) (S1._minus__3 (S1.size (V1 2)) (S1.size (V1 1))))) (S1.and (S1.wb (V1 1)) (S1.wb (V1 2))))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hand (S1.bs (V1 1)) (S1.bs (V1 2)))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hand (S1.ge (V1 0) (S1.max (V1 1))) (S1.ge (S1.min (V1 2)) (V1 0)))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hand (S1.and (S1.ge (V1 0) (S1.max (V1 1))) (S1.ge (S1.min (V1 2)) (V1 0))) (S1.and (S1.bs (V1 1)) (S1.bs (V1 2))))
   :: nil)

:: (  R1 (S1.hmax (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hmax (V1 2))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hmax (V1 1))
   :: nil)

:: (  R1 (S1.hmin (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hmin (V1 1))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hmin (V1 2))
   :: nil)

:: (  R1 (S1.hge (S1._0_4 (V1 0)) (S1._1_5 (V1 1)))
         (S1.hnot (S1.ge (V1 1) (V1 0)))
   :: nil)

:: (  R1 (S1.hge (S1._sharp__1) (S1._0_4 (V1 0)))
         (S1.hge (S1._sharp__1) (V1 0))
   :: nil)

:: (  R1 (S1.hge (S1._0_4 (V1 0)) (S1._1_5 (V1 1)))
         (S1.hge (V1 1) (V1 0))
   :: R1 (S1.hge (S1._0_4 (V1 0)) (S1._0_4 (V1 1)))
         (S1.hge (V1 0) (V1 1))
   :: R1 (S1.hge (S1._1_5 (V1 0)) (S1._0_4 (V1 1)))
         (S1.hge (V1 0) (V1 1))
   :: R1 (S1.hge (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.hge (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hge (S1._1_5 (S1._sharp__1)) (S1._minus__3 (S1.size (V1 2)) (S1.size (V1 1))))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hge (S1._1_5 (S1._sharp__1)) (S1._minus__3 (S1.size (V1 1)) (S1.size (V1 2))))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hge (S1.size (V1 1)) (S1.size (V1 2)))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hge (S1.min (V1 2)) (V1 0))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hge (V1 0) (S1.max (V1 1)))
   :: nil)

:: (  R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hbs (V1 2))
   :: R1 (S1.hbs (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hbs (V1 1))
   :: nil)

:: (  R1 (S1.h_minus__3 (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_0_4 (S1._minus__3 (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.h_minus__3 (S1._0_4 (V1 0)) (S1._0_4 (V1 1)))
         (S1.h_0_4 (S1._minus__3 (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.h_minus__3 (S1._0_4 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_minus__3 (S1._minus__3 (V1 0) (V1 1)) (S1._1_5 (S1._sharp__1)))
   :: R1 (S1.h_minus__3 (S1._0_4 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_minus__3 (V1 0) (V1 1))
   :: R1 (S1.h_minus__3 (S1._0_4 (V1 0)) (S1._0_4 (V1 1)))
         (S1.h_minus__3 (V1 0) (V1 1))
   :: R1 (S1.h_minus__3 (S1._1_5 (V1 0)) (S1._0_4 (V1 1)))
         (S1.h_minus__3 (V1 0) (V1 1))
   :: R1 (S1.h_minus__3 (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_minus__3 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.h_minus__3 (S1.size (V1 2)) (S1.size (V1 1)))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.h_minus__3 (S1.size (V1 1)) (S1.size (V1 2)))
   :: nil)

:: (  R1 (S1.h_plus__2 (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_0_4 (S1._plus__2 (S1._plus__2 (V1 0) (V1 1)) (S1._1_5 (S1._sharp__1))))
   :: nil)

:: (  R1 (S1.h_plus__2 (S1._0_4 (V1 0)) (S1._0_4 (V1 1)))
         (S1.h_0_4 (S1._plus__2 (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.h_plus__2 (S1._0_4 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_plus__2 (V1 0) (V1 1))
   :: R1 (S1.h_plus__2 (S1._0_4 (V1 0)) (S1._0_4 (V1 1)))
         (S1.h_plus__2 (V1 0) (V1 1))
   :: R1 (S1.h_plus__2 (S1._1_5 (V1 0)) (S1._0_4 (V1 1)))
         (S1.h_plus__2 (V1 0) (V1 1))
   :: R1 (S1.h_plus__2 (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_plus__2 (S1._plus__2 (V1 0) (V1 1)) (S1._1_5 (S1._sharp__1)))
   :: R1 (S1.h_plus__2 (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_plus__2 (V1 0) (V1 1))
   :: R1 (S1.h_plus__2 (V1 0) (S1._plus__2 (V1 1) (V1 2)))
         (S1.h_plus__2 (S1._plus__2 (V1 0) (V1 1)) (V1 2))
   :: R1 (S1.h_plus__2 (V1 0) (S1._plus__2 (V1 1) (V1 2)))
         (S1.h_plus__2 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hsize (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.h_plus__2 (S1.size (V1 0)) (S1.size (V1 1)))
   :: nil)

:: (  R1 (S1.hsize (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.h_plus__2 (S1._plus__2 (S1.size (V1 0)) (S1.size (V1 1))) (S1._1_5 (S1._sharp__1)))
   :: nil)

:: (  R1 (S1.hsize (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hsize (V1 1))
   :: R1 (S1.hsize (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hsize (V1 0))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hsize (V1 2))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hsize (V1 1))
   :: nil)

:: (  R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hwb (V1 2))
   :: R1 (S1.hwb (S1.n (V1 0) (V1 1) (V1 2)))
         (S1.hwb (V1 1))
   :: 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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (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))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.max) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         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
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    end.

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

End PIS4.

Module PI4 := PolyInt PIS4.

(* polynomial interpretation 5 *)

Module PIS5 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M._0_4) =>
         nil
    | (int_symb M._0_4) =>
         nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (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.ge) =>
         nil
    | (int_symb M.ge) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.bs) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (2%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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (2%Z, (Vcons 0 Vnil))
      :: 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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%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.min) =>
         nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         nil
    end.

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

End PIS7.

Module PI7 := PolyInt PIS7.

(* graph decomposition 2 *)

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

   (  R1 (S1.h_plus__2 (S1._1_5 (V1 0)) (S1._1_5 (V1 1)))
         (S1.h_plus__2 (S1._plus__2 (V1 0) (V1 1)) (S1._1_5 (S1._sharp__1)))
   :: nil)

:: (  R1 (S1.h_plus__2 (V1 0) (S1._plus__2 (V1 1) (V1 2)))
         (S1.h_plus__2 (V1 0) (V1 1))
   :: R1 (S1.h_plus__2 (V1 0) (S1._plus__2 (V1 1) (V1 2)))
         (S1.h_plus__2 (S1._plus__2 (V1 0) (V1 1)) (V1 2))
   :: 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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (1%Z, (Vcons 0 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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M._plus__2) =>
         (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._1_5) =>
         nil
    | (int_symb M._1_5) =>
         nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.ge) =>
         nil
    | (int_symb M.ge) =>
         nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.size) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.size) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         nil
    | (int_symb M.wb) =>
         (1%Z, (Vcons 0 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._0_4) =>
         nil
    | (int_symb M._0_4) =>
         nil
    | (hd_symb M._sharp__1) =>
         nil
    | (int_symb M._sharp__1) =>
         nil
    | (hd_symb M._plus__2) =>
         nil
    | (int_symb M._plus__2) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._1_5) =>
         nil
    | (int_symb M._1_5) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._minus__3) =>
         nil
    | (int_symb M._minus__3) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_6) =>
         nil
    | (int_symb M._if_6) =>
         (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.ge) =>
         nil
    | (int_symb M.ge) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.val) =>
         nil
    | (int_symb M.val) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.l) =>
         nil
    | (int_symb M.l) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.n) =>
         nil
    | (int_symb M.n) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.bs) =>
         nil
    | (int_symb M.bs) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.size) =>
         nil
    | (int_symb M.size) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.wb) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.wb) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    end.

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

End PIS11.

Module PI11 := PolyInt PIS11.

(* 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.
termination_trivial.
left. co_scc.
right. PI2.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI3.prove_termination.
termination_trivial.
right. PI4.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI5.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI6.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
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) cs2; subst D; subst R.
dpg_unif_N_correct.
right. PI8.prove_termination.
termination_trivial.
right. PI9.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI10.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI11.prove_termination.
termination_trivial.
Qed.