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 :=
  | der : symb
  | din : symb
  | dout : symb
  | plus : symb
  | times : symb
  | u21 : symb
  | u22 : symb
  | u31 : symb
  | u32 : symb
  | u41 : symb
  | u42 : 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.der => 1
  | M.din => 1
  | M.dout => 1
  | M.plus => 2
  | M.times => 2
  | M.u21 => 3
  | M.u22 => 4
  | M.u31 => 3
  | M.u32 => 4
  | M.u41 => 2
  | M.u42 => 3
  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 der x1 := F0 M.der (Vcons x1 Vnil).
  Definition din x1 := F0 M.din (Vcons x1 Vnil).
  Definition dout x1 := F0 M.dout (Vcons x1 Vnil).
  Definition plus x2 x1 := F0 M.plus (Vcons x2 (Vcons x1 Vnil)).
  Definition times x2 x1 := F0 M.times (Vcons x2 (Vcons x1 Vnil)).
  Definition u21 x3 x2 x1 := F0 M.u21 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition u22 x4 x3 x2 x1 := F0 M.u22 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition u31 x3 x2 x1 := F0 M.u31 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition u32 x4 x3 x2 x1 := F0 M.u32 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition u41 x2 x1 := F0 M.u41 (Vcons x2 (Vcons x1 Vnil)).
  Definition u42 x3 x2 x1 := F0 M.u42 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
End S0.

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

Definition R :=
   R0 (S0.din (S0.der (S0.plus (V0 0) (V0 1))))
      (S0.u21 (S0.din (S0.der (V0 0))) (V0 0) (V0 1))
:: R0 (S0.u21 (S0.dout (V0 0)) (V0 1) (V0 2))
      (S0.u22 (S0.din (S0.der (V0 2))) (V0 1) (V0 2) (V0 0))
:: R0 (S0.u22 (S0.dout (V0 0)) (V0 1) (V0 2) (V0 3))
      (S0.dout (S0.plus (V0 3) (V0 0)))
:: R0 (S0.din (S0.der (S0.times (V0 0) (V0 1))))
      (S0.u31 (S0.din (S0.der (V0 0))) (V0 0) (V0 1))
:: R0 (S0.u31 (S0.dout (V0 0)) (V0 1) (V0 2))
      (S0.u32 (S0.din (S0.der (V0 2))) (V0 1) (V0 2) (V0 0))
:: R0 (S0.u32 (S0.dout (V0 0)) (V0 1) (V0 2) (V0 3))
      (S0.dout (S0.plus (S0.times (V0 1) (V0 0)) (S0.times (V0 2) (V0 3))))
:: R0 (S0.din (S0.der (S0.der (V0 0))))
      (S0.u41 (S0.din (S0.der (V0 0))) (V0 0))
:: R0 (S0.u41 (S0.dout (V0 0)) (V0 1))
      (S0.u42 (S0.din (S0.der (V0 0))) (V0 1) (V0 0))
:: R0 (S0.u42 (S0.dout (V0 0)) (V0 1) (V0 2))
      (S0.dout (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 hder x1 := F1 (hd_symb s1_p M.der) (Vcons x1 Vnil).
  Definition der x1 := F1 (int_symb s1_p M.der) (Vcons x1 Vnil).
  Definition hdin x1 := F1 (hd_symb s1_p M.din) (Vcons x1 Vnil).
  Definition din x1 := F1 (int_symb s1_p M.din) (Vcons x1 Vnil).
  Definition hdout x1 := F1 (hd_symb s1_p M.dout) (Vcons x1 Vnil).
  Definition dout x1 := F1 (int_symb s1_p M.dout) (Vcons x1 Vnil).
  Definition hplus x2 x1 := F1 (hd_symb s1_p M.plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition plus x2 x1 := F1 (int_symb s1_p M.plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition htimes x2 x1 := F1 (hd_symb s1_p M.times) (Vcons x2 (Vcons x1 Vnil)).
  Definition times x2 x1 := F1 (int_symb s1_p M.times) (Vcons x2 (Vcons x1 Vnil)).
  Definition hu21 x3 x2 x1 := F1 (hd_symb s1_p M.u21) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition u21 x3 x2 x1 := F1 (int_symb s1_p M.u21) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hu22 x4 x3 x2 x1 := F1 (hd_symb s1_p M.u22) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition u22 x4 x3 x2 x1 := F1 (int_symb s1_p M.u22) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hu31 x3 x2 x1 := F1 (hd_symb s1_p M.u31) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition u31 x3 x2 x1 := F1 (int_symb s1_p M.u31) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hu32 x4 x3 x2 x1 := F1 (hd_symb s1_p M.u32) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition u32 x4 x3 x2 x1 := F1 (int_symb s1_p M.u32) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hu41 x2 x1 := F1 (hd_symb s1_p M.u41) (Vcons x2 (Vcons x1 Vnil)).
  Definition u41 x2 x1 := F1 (int_symb s1_p M.u41) (Vcons x2 (Vcons x1 Vnil)).
  Definition hu42 x3 x2 x1 := F1 (hd_symb s1_p M.u42) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition u42 x3 x2 x1 := F1 (int_symb s1_p M.u42) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hu41 (S1.dout (V1 0)) (V1 1))
         (S1.hu42 (S1.din (S1.der (V1 0))) (V1 1) (V1 0))
   :: nil)

:: (  R1 (S1.hu31 (S1.dout (V1 0)) (V1 1) (V1 2))
         (S1.hu32 (S1.din (S1.der (V1 2))) (V1 1) (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hu21 (S1.dout (V1 0)) (V1 1) (V1 2))
         (S1.hu22 (S1.din (S1.der (V1 2))) (V1 1) (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hu21 (S1.dout (V1 0)) (V1 1) (V1 2))
         (S1.hdin (S1.der (V1 2)))
   :: R1 (S1.hdin (S1.der (S1.plus (V1 0) (V1 1))))
         (S1.hu21 (S1.din (S1.der (V1 0))) (V1 0) (V1 1))
   :: R1 (S1.hdin (S1.der (S1.plus (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hu31 (S1.din (S1.der (V1 0))) (V1 0) (V1 1))
   :: R1 (S1.hu31 (S1.dout (V1 0)) (V1 1) (V1 2))
         (S1.hdin (S1.der (V1 2)))
   :: R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hu41 (S1.din (S1.der (V1 0))) (V1 0))
   :: R1 (S1.hu41 (S1.dout (V1 0)) (V1 1))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hdin (S1.der (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.din) =>
         nil
    | (int_symb M.din) =>
         nil
    | (hd_symb M.der) =>
         nil
    | (int_symb M.der) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         nil
    | (hd_symb M.u21) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (int_symb M.u21) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.dout) =>
         nil
    | (int_symb M.dout) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.u22) =>
         nil
    | (int_symb M.u22) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.times) =>
         nil
    | (int_symb M.times) =>
         nil
    | (hd_symb M.u31) =>
         nil
    | (int_symb M.u31) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.u32) =>
         nil
    | (int_symb M.u32) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.u41) =>
         nil
    | (int_symb M.u41) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u42) =>
         nil
    | (int_symb M.u42) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 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.hdin (S1.der (S1.plus (V1 0) (V1 1))))
         (S1.hu21 (S1.din (S1.der (V1 0))) (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hu31 (S1.din (S1.der (V1 0))) (V1 0) (V1 1))
   :: R1 (S1.hu31 (S1.dout (V1 0)) (V1 1) (V1 2))
         (S1.hdin (S1.der (V1 2)))
   :: R1 (S1.hdin (S1.der (S1.plus (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hu41 (S1.din (S1.der (V1 0))) (V1 0))
   :: R1 (S1.hu41 (S1.dout (V1 0)) (V1 1))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hdin (S1.der (V1 0)))
   :: 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.din) =>
         nil
    | (int_symb M.din) =>
         nil
    | (hd_symb M.der) =>
         nil
    | (int_symb M.der) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u21) =>
         nil
    | (int_symb M.u21) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.dout) =>
         nil
    | (int_symb M.dout) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.u22) =>
         nil
    | (int_symb M.u22) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.times) =>
         nil
    | (int_symb M.times) =>
         nil
    | (hd_symb M.u31) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (int_symb M.u31) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.u32) =>
         nil
    | (int_symb M.u32) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.u41) =>
         nil
    | (int_symb M.u41) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u42) =>
         nil
    | (int_symb M.u42) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS2.

Module PI2 := PolyInt PIS2.

(* graph decomposition 3 *)

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

   (  R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hu31 (S1.din (S1.der (V1 0))) (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.plus (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hu41 (S1.din (S1.der (V1 0))) (V1 0))
   :: R1 (S1.hu41 (S1.dout (V1 0)) (V1 1))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hdin (S1.der (V1 0)))
   :: nil)

:: nil.

(* polynomial interpretation 3 *)

Module PIS3 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.din) =>
         nil
    | (int_symb M.din) =>
         nil
    | (hd_symb M.der) =>
         nil
    | (int_symb M.der) =>
         nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         nil
    | (hd_symb M.u21) =>
         nil
    | (int_symb M.u21) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.dout) =>
         nil
    | (int_symb M.dout) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.u22) =>
         nil
    | (int_symb M.u22) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.times) =>
         nil
    | (int_symb M.times) =>
         nil
    | (hd_symb M.u31) =>
         nil
    | (int_symb M.u31) =>
         nil
    | (hd_symb M.u32) =>
         nil
    | (int_symb M.u32) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.u41) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.u41) =>
         nil
    | (hd_symb M.u42) =>
         nil
    | (int_symb M.u42) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

(* graph decomposition 4 *)

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

   (  R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hu41 (S1.din (S1.der (V1 0))) (V1 0))
   :: nil)

:: (  R1 (S1.hdin (S1.der (S1.plus (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.times (V1 0) (V1 1))))
         (S1.hdin (S1.der (V1 0)))
   :: R1 (S1.hdin (S1.der (S1.der (V1 0))))
         (S1.hdin (S1.der (V1 0)))
   :: nil)

:: nil.

(* 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.din) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.din) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.der) =>
         nil
    | (int_symb M.der) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u21) =>
         nil
    | (int_symb M.u21) =>
         nil
    | (hd_symb M.dout) =>
         nil
    | (int_symb M.dout) =>
         nil
    | (hd_symb M.u22) =>
         nil
    | (int_symb M.u22) =>
         nil
    | (hd_symb M.times) =>
         nil
    | (int_symb M.times) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u31) =>
         nil
    | (int_symb M.u31) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.u32) =>
         nil
    | (int_symb M.u32) =>
         nil
    | (hd_symb M.u41) =>
         nil
    | (int_symb M.u41) =>
         nil
    | (hd_symb M.u42) =>
         nil
    | (int_symb M.u42) =>
         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.din) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.din) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.der) =>
         nil
    | (int_symb M.der) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.plus) =>
         nil
    | (int_symb M.plus) =>
         nil
    | (hd_symb M.u21) =>
         nil
    | (int_symb M.u21) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.dout) =>
         nil
    | (int_symb M.dout) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.u22) =>
         nil
    | (int_symb M.u22) =>
         nil
    | (hd_symb M.times) =>
         nil
    | (int_symb M.times) =>
         nil
    | (hd_symb M.u31) =>
         nil
    | (int_symb M.u31) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.u32) =>
         nil
    | (int_symb M.u32) =>
         nil
    | (hd_symb M.u41) =>
         nil
    | (int_symb M.u41) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u42) =>
         nil
    | (int_symb M.u42) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* 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.
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.
left. co_scc.
right. PI2.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs3; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
right. PI3.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs4; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
right. PI4.prove_termination.
PI5.prove_termination.
termination_trivial.
Qed.