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 :=
  | X : symb
  | _match_1 : symb
  | active : symb
  | c : symb
  | check : symb
  | f : symb
  | found : symb
  | mark : symb
  | ok : symb
  | proper : symb
  | start : symb
  | top : 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.X => 0
  | M._match_1 => 2
  | M.active => 1
  | M.c => 0
  | M.check => 1
  | M.f => 1
  | M.found => 1
  | M.mark => 1
  | M.ok => 1
  | M.proper => 1
  | M.start => 1
  | M.top => 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 X := F0 M.X Vnil.
  Definition _match_1 x2 x1 := F0 M._match_1 (Vcons x2 (Vcons x1 Vnil)).
  Definition active x1 := F0 M.active (Vcons x1 Vnil).
  Definition c := F0 M.c Vnil.
  Definition check x1 := F0 M.check (Vcons x1 Vnil).
  Definition f x1 := F0 M.f (Vcons x1 Vnil).
  Definition found x1 := F0 M.found (Vcons x1 Vnil).
  Definition mark x1 := F0 M.mark (Vcons x1 Vnil).
  Definition ok x1 := F0 M.ok (Vcons x1 Vnil).
  Definition proper x1 := F0 M.proper (Vcons x1 Vnil).
  Definition start x1 := F0 M.start (Vcons x1 Vnil).
  Definition top x1 := F0 M.top (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.active (S0.f (V0 0)))
      (S0.mark (V0 0))
:: R0 (S0.top (S0.active S0.c))
      (S0.top (S0.mark S0.c))
:: R0 (S0.top (S0.mark (V0 0)))
      (S0.top (S0.check (V0 0)))
:: R0 (S0.check (S0.f (V0 0)))
      (S0.f (S0.check (V0 0)))
:: R0 (S0.check (V0 0))
      (S0.start (S0._match_1 (S0.f S0.X) (V0 0)))
:: R0 (S0._match_1 (S0.f (V0 0)) (S0.f (V0 1)))
      (S0.f (S0._match_1 (V0 0) (V0 1)))
:: R0 (S0._match_1 S0.X (V0 0))
      (S0.proper (V0 0))
:: R0 (S0.proper S0.c)
      (S0.ok S0.c)
:: R0 (S0.proper (S0.f (V0 0)))
      (S0.f (S0.proper (V0 0)))
:: R0 (S0.f (S0.ok (V0 0)))
      (S0.ok (S0.f (V0 0)))
:: R0 (S0.start (S0.ok (V0 0)))
      (S0.found (V0 0))
:: R0 (S0.f (S0.found (V0 0)))
      (S0.found (S0.f (V0 0)))
:: R0 (S0.top (S0.found (V0 0)))
      (S0.top (S0.active (V0 0)))
:: R0 (S0.active (S0.f (V0 0)))
      (S0.f (S0.active (V0 0)))
:: R0 (S0.f (S0.mark (V0 0)))
      (S0.mark (S0.f (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 hX := F1 (hd_symb s1_p M.X) Vnil.
  Definition X := F1 (int_symb s1_p M.X) Vnil.
  Definition h_match_1 x2 x1 := F1 (hd_symb s1_p M._match_1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _match_1 x2 x1 := F1 (int_symb s1_p M._match_1) (Vcons x2 (Vcons x1 Vnil)).
  Definition hactive x1 := F1 (hd_symb s1_p M.active) (Vcons x1 Vnil).
  Definition active x1 := F1 (int_symb s1_p M.active) (Vcons x1 Vnil).
  Definition hc := F1 (hd_symb s1_p M.c) Vnil.
  Definition c := F1 (int_symb s1_p M.c) Vnil.
  Definition hcheck x1 := F1 (hd_symb s1_p M.check) (Vcons x1 Vnil).
  Definition check x1 := F1 (int_symb s1_p M.check) (Vcons x1 Vnil).
  Definition hf x1 := F1 (hd_symb s1_p M.f) (Vcons x1 Vnil).
  Definition f x1 := F1 (int_symb s1_p M.f) (Vcons x1 Vnil).
  Definition hfound x1 := F1 (hd_symb s1_p M.found) (Vcons x1 Vnil).
  Definition found x1 := F1 (int_symb s1_p M.found) (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 hok x1 := F1 (hd_symb s1_p M.ok) (Vcons x1 Vnil).
  Definition ok x1 := F1 (int_symb s1_p M.ok) (Vcons x1 Vnil).
  Definition hproper x1 := F1 (hd_symb s1_p M.proper) (Vcons x1 Vnil).
  Definition proper x1 := F1 (int_symb s1_p M.proper) (Vcons x1 Vnil).
  Definition hstart x1 := F1 (hd_symb s1_p M.start) (Vcons x1 Vnil).
  Definition start x1 := F1 (int_symb s1_p M.start) (Vcons x1 Vnil).
  Definition htop x1 := F1 (hd_symb s1_p M.top) (Vcons x1 Vnil).
  Definition top x1 := F1 (int_symb s1_p M.top) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hf (S1.found (V1 0)))
         (S1.hf (V1 0))
   :: R1 (S1.hf (S1.ok (V1 0)))
         (S1.hf (V1 0))
   :: R1 (S1.hf (S1.mark (V1 0)))
         (S1.hf (V1 0))
   :: nil)

:: (  R1 (S1.hactive (S1.f (V1 0)))
         (S1.hf (S1.active (V1 0)))
   :: nil)

:: (  R1 (S1.hactive (S1.f (V1 0)))
         (S1.hactive (V1 0))
   :: nil)

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

:: (  R1 (S1.hproper (S1.f (V1 0)))
         (S1.hf (S1.proper (V1 0)))
   :: nil)

:: (  R1 (S1.hproper (S1.f (V1 0)))
         (S1.hproper (V1 0))
   :: nil)

:: (  R1 (S1.h_match_1 (S1.X) (V1 0))
         (S1.hproper (V1 0))
   :: nil)

:: (  R1 (S1.h_match_1 (S1.f (V1 0)) (S1.f (V1 1)))
         (S1.hf (S1._match_1 (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.h_match_1 (S1.f (V1 0)) (S1.f (V1 1)))
         (S1.h_match_1 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hcheck (V1 0))
         (S1.hf (S1.X))
   :: nil)

:: (  R1 (S1.hcheck (V1 0))
         (S1.h_match_1 (S1.f (S1.X)) (V1 0))
   :: nil)

:: (  R1 (S1.hcheck (V1 0))
         (S1.hstart (S1._match_1 (S1.f (S1.X)) (V1 0)))
   :: nil)

:: (  R1 (S1.hcheck (S1.f (V1 0)))
         (S1.hf (S1.check (V1 0)))
   :: nil)

:: (  R1 (S1.hcheck (S1.f (V1 0)))
         (S1.hcheck (V1 0))
   :: nil)

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

:: (  R1 (S1.htop (S1.mark (V1 0)))
         (S1.htop (S1.check (V1 0)))
   :: R1 (S1.htop (S1.active (S1.c)))
         (S1.htop (S1.mark (S1.c)))
   :: R1 (S1.htop (S1.found (V1 0)))
         (S1.htop (S1.active (V1 0)))
   :: nil)

:: nil.

(* polynomial interpretation 1 *)

Module PIS1 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.f) =>
         (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))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* polynomial interpretation 2 *)

Module PIS2 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%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.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

(* polynomial interpretation 4 *)

Module PIS4 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         nil
    end.

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

End PIS4.

Module PI4 := PolyInt PIS4.

(* polynomial interpretation 5 *)

Module PIS5 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         nil
    | (hd_symb M._match_1) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M._match_1) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* polynomial interpretation 6 *)

Module PIS6 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.check) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.check) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         nil
    end.

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

End PIS6.

Module PI6 := PolyInt PIS6.

(* polynomial interpretation 7 *)

Module PIS7 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         nil
    | (hd_symb M.top) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    end.

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

End PIS7.

Module PI7 := PolyInt PIS7.

(* polynomial interpretation 8 *)

Module PIS8 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%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.top) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         (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 PIS8.

Module PI8 := PolyInt PIS8.

(* polynomial interpretation 9 *)

Module PIS9 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.active) =>
         nil
    | (int_symb M.active) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.top) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.top) =>
         nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         nil
    | (hd_symb M.check) =>
         nil
    | (int_symb M.check) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.start) =>
         nil
    | (int_symb M.start) =>
         nil
    | (hd_symb M._match_1) =>
         nil
    | (int_symb M._match_1) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.X) =>
         nil
    | (int_symb M.X) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.proper) =>
         nil
    | (int_symb M.proper) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.found) =>
         nil
    | (int_symb M.found) =>
         nil
    end.

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

End PIS9.

Module PI9 := PolyInt PIS9.

(* termination proof *)

Lemma termination : WF rel.

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