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_1 : symb
  | add : symb
  | app : symb
  | eq : symb
  | false : symb
  | if_min : symb
  | if_minsort : symb
  | if_rm : symb
  | le : symb
  | min : symb
  | minsort : symb
  | nil : symb
  | rm : symb
  | s : symb
  | true : 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_1 => 0
  | M.add => 2
  | M.app => 2
  | M.eq => 2
  | M.false => 0
  | M.if_min => 2
  | M.if_minsort => 3
  | M.if_rm => 3
  | M.le => 2
  | M.min => 1
  | M.minsort => 2
  | M.nil => 0
  | M.rm => 2
  | M.s => 1
  | M.true => 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 _0_1 := F0 M._0_1 Vnil.
  Definition add x2 x1 := F0 M.add (Vcons x2 (Vcons x1 Vnil)).
  Definition app x2 x1 := F0 M.app (Vcons x2 (Vcons x1 Vnil)).
  Definition eq x2 x1 := F0 M.eq (Vcons x2 (Vcons x1 Vnil)).
  Definition false := F0 M.false Vnil.
  Definition if_min x2 x1 := F0 M.if_min (Vcons x2 (Vcons x1 Vnil)).
  Definition if_minsort x3 x2 x1 := F0 M.if_minsort (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition if_rm x3 x2 x1 := F0 M.if_rm (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition le x2 x1 := F0 M.le (Vcons x2 (Vcons x1 Vnil)).
  Definition min x1 := F0 M.min (Vcons x1 Vnil).
  Definition minsort x2 x1 := F0 M.minsort (Vcons x2 (Vcons x1 Vnil)).
  Definition nil := F0 M.nil Vnil.
  Definition rm x2 x1 := F0 M.rm (Vcons x2 (Vcons x1 Vnil)).
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition true := F0 M.true Vnil.
End S0.

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

Definition R :=
   R0 (S0.eq S0._0_1 S0._0_1)
      S0.true
:: R0 (S0.eq S0._0_1 (S0.s (V0 0)))
      S0.false
:: R0 (S0.eq (S0.s (V0 0)) S0._0_1)
      S0.false
:: R0 (S0.eq (S0.s (V0 0)) (S0.s (V0 1)))
      (S0.eq (V0 0) (V0 1))
:: R0 (S0.le S0._0_1 (V0 0))
      S0.true
:: R0 (S0.le (S0.s (V0 0)) S0._0_1)
      S0.false
:: R0 (S0.le (S0.s (V0 0)) (S0.s (V0 1)))
      (S0.le (V0 0) (V0 1))
:: R0 (S0.app S0.nil (V0 0))
      (V0 0)
:: R0 (S0.app (S0.add (V0 0) (V0 1)) (V0 2))
      (S0.add (V0 0) (S0.app (V0 1) (V0 2)))
:: R0 (S0.min (S0.add (V0 0) S0.nil))
      (V0 0)
:: R0 (S0.min (S0.add (V0 0) (S0.add (V0 1) (V0 2))))
      (S0.if_min (S0.le (V0 0) (V0 1)) (S0.add (V0 0) (S0.add (V0 1) (V0 2))))
:: R0 (S0.if_min S0.true (S0.add (V0 0) (S0.add (V0 1) (V0 2))))
      (S0.min (S0.add (V0 0) (V0 2)))
:: R0 (S0.if_min S0.false (S0.add (V0 0) (S0.add (V0 1) (V0 2))))
      (S0.min (S0.add (V0 1) (V0 2)))
:: R0 (S0.rm (V0 0) S0.nil)
      S0.nil
:: R0 (S0.rm (V0 0) (S0.add (V0 1) (V0 2)))
      (S0.if_rm (S0.eq (V0 0) (V0 1)) (V0 0) (S0.add (V0 1) (V0 2)))
:: R0 (S0.if_rm S0.true (V0 0) (S0.add (V0 1) (V0 2)))
      (S0.rm (V0 0) (V0 2))
:: R0 (S0.if_rm S0.false (V0 0) (S0.add (V0 1) (V0 2)))
      (S0.add (V0 1) (S0.rm (V0 0) (V0 2)))
:: R0 (S0.minsort S0.nil S0.nil)
      S0.nil
:: R0 (S0.minsort (S0.add (V0 0) (V0 1)) (V0 2))
      (S0.if_minsort (S0.eq (V0 0) (S0.min (S0.add (V0 0) (V0 1)))) (S0.add (V0 0) (V0 1)) (V0 2))
:: R0 (S0.if_minsort S0.true (S0.add (V0 0) (V0 1)) (V0 2))
      (S0.add (V0 0) (S0.minsort (S0.app (S0.rm (V0 0) (V0 1)) (V0 2)) S0.nil))
:: R0 (S0.if_minsort S0.false (S0.add (V0 0) (V0 1)) (V0 2))
      (S0.minsort (V0 1) (S0.add (V0 0) (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_1 := F1 (hd_symb s1_p M._0_1) Vnil.
  Definition _0_1 := F1 (int_symb s1_p M._0_1) Vnil.
  Definition hadd x2 x1 := F1 (hd_symb s1_p M.add) (Vcons x2 (Vcons x1 Vnil)).
  Definition add x2 x1 := F1 (int_symb s1_p M.add) (Vcons x2 (Vcons x1 Vnil)).
  Definition happ x2 x1 := F1 (hd_symb s1_p M.app) (Vcons x2 (Vcons x1 Vnil)).
  Definition app x2 x1 := F1 (int_symb s1_p M.app) (Vcons x2 (Vcons x1 Vnil)).
  Definition heq x2 x1 := F1 (hd_symb s1_p M.eq) (Vcons x2 (Vcons x1 Vnil)).
  Definition eq x2 x1 := F1 (int_symb s1_p M.eq) (Vcons x2 (Vcons x1 Vnil)).
  Definition hfalse := F1 (hd_symb s1_p M.false) Vnil.
  Definition false := F1 (int_symb s1_p M.false) Vnil.
  Definition hif_min x2 x1 := F1 (hd_symb s1_p M.if_min) (Vcons x2 (Vcons x1 Vnil)).
  Definition if_min x2 x1 := F1 (int_symb s1_p M.if_min) (Vcons x2 (Vcons x1 Vnil)).
  Definition hif_minsort x3 x2 x1 := F1 (hd_symb s1_p M.if_minsort) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition if_minsort x3 x2 x1 := F1 (int_symb s1_p M.if_minsort) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hif_rm x3 x2 x1 := F1 (hd_symb s1_p M.if_rm) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition if_rm x3 x2 x1 := F1 (int_symb s1_p M.if_rm) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hle x2 x1 := F1 (hd_symb s1_p M.le) (Vcons x2 (Vcons x1 Vnil)).
  Definition le x2 x1 := F1 (int_symb s1_p M.le) (Vcons x2 (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 hminsort x2 x1 := F1 (hd_symb s1_p M.minsort) (Vcons x2 (Vcons x1 Vnil)).
  Definition minsort x2 x1 := F1 (int_symb s1_p M.minsort) (Vcons x2 (Vcons x1 Vnil)).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hrm x2 x1 := F1 (hd_symb s1_p M.rm) (Vcons x2 (Vcons x1 Vnil)).
  Definition rm x2 x1 := F1 (int_symb s1_p M.rm) (Vcons x2 (Vcons x1 Vnil)).
  Definition hs x1 := F1 (hd_symb s1_p M.s) (Vcons x1 Vnil).
  Definition s x1 := F1 (int_symb s1_p M.s) (Vcons x1 Vnil).
  Definition htrue := F1 (hd_symb s1_p M.true) Vnil.
  Definition true := F1 (int_symb s1_p M.true) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.happ (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.happ (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hif_minsort (S1.true) (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.happ (S1.rm (V1 0) (V1 1)) (V1 2))
   :: nil)

:: (  R1 (S1.hle (S1.s (V1 0)) (S1.s (V1 1)))
         (S1.hle (V1 0) (V1 1))
   :: nil)

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

:: (  R1 (S1.hmin (S1.add (V1 0) (S1.add (V1 1) (V1 2))))
         (S1.hif_min (S1.le (V1 0) (V1 1)) (S1.add (V1 0) (S1.add (V1 1) (V1 2))))
   :: R1 (S1.hif_min (S1.true) (S1.add (V1 0) (S1.add (V1 1) (V1 2))))
         (S1.hmin (S1.add (V1 0) (V1 2)))
   :: R1 (S1.hif_min (S1.false) (S1.add (V1 0) (S1.add (V1 1) (V1 2))))
         (S1.hmin (S1.add (V1 1) (V1 2)))
   :: nil)

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

:: (  R1 (S1.heq (S1.s (V1 0)) (S1.s (V1 1)))
         (S1.heq (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hminsort (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.heq (V1 0) (S1.min (S1.add (V1 0) (V1 1))))
   :: nil)

:: (  R1 (S1.hrm (V1 0) (S1.add (V1 1) (V1 2)))
         (S1.heq (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hrm (V1 0) (S1.add (V1 1) (V1 2)))
         (S1.hif_rm (S1.eq (V1 0) (V1 1)) (V1 0) (S1.add (V1 1) (V1 2)))
   :: R1 (S1.hif_rm (S1.true) (V1 0) (S1.add (V1 1) (V1 2)))
         (S1.hrm (V1 0) (V1 2))
   :: R1 (S1.hif_rm (S1.false) (V1 0) (S1.add (V1 1) (V1 2)))
         (S1.hrm (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hif_minsort (S1.true) (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.hrm (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hif_minsort (S1.true) (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.hminsort (S1.app (S1.rm (V1 0) (V1 1)) (V1 2)) (S1.nil))
   :: R1 (S1.hminsort (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.hif_minsort (S1.eq (V1 0) (S1.min (S1.add (V1 0) (V1 1)))) (S1.add (V1 0) (V1 1)) (V1 2))
   :: R1 (S1.hif_minsort (S1.false) (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.hminsort (V1 1) (S1.add (V1 0) (V1 2)))
   :: 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.eq) =>
         nil
    | (int_symb M.eq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.le) =>
         nil
    | (int_symb M.le) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (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.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         nil
    | (int_symb M.if_min) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         nil
    | (int_symb M.minsort) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         nil
    | (int_symb M.if_minsort) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* polynomial interpretation 2 *)

Module PIS2 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.le) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.le) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         nil
    | (int_symb M.if_min) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         nil
    | (int_symb M.minsort) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         nil
    | (int_symb M.if_minsort) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End 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.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.le) =>
         nil
    | (int_symb M.le) =>
         nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.if_min) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         nil
    | (int_symb M.minsort) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         nil
    | (int_symb M.if_minsort) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

(* polynomial interpretation 4 *)

Module PIS4 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.eq) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.eq) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.le) =>
         nil
    | (int_symb M.le) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         nil
    | (int_symb M.if_min) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         nil
    | (int_symb M.minsort) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         nil
    | (int_symb M.if_minsort) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End 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.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.le) =>
         nil
    | (int_symb M.le) =>
         nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (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.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         nil
    | (int_symb M.if_min) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         nil
    | (int_symb M.minsort) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         nil
    | (int_symb M.if_minsort) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS5.

Module PI5 := PolyInt PIS5.

(* graph decomposition 2 *)

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

   (  R1 (S1.hrm (V1 0) (S1.add (V1 1) (V1 2)))
         (S1.hif_rm (S1.eq (V1 0) (V1 1)) (V1 0) (S1.add (V1 1) (V1 2)))
   :: nil)

:: nil.

(* polynomial interpretation 6 *)

Module PIS6 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.le) =>
         nil
    | (int_symb M.le) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         nil
    | (int_symb M.if_min) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.minsort) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.if_minsort) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End 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.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.le) =>
         nil
    | (int_symb M.le) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.add) =>
         nil
    | (int_symb M.add) =>
         (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.min) =>
         nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.if_min) =>
         nil
    | (int_symb M.if_min) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.rm) =>
         nil
    | (int_symb M.rm) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_rm) =>
         nil
    | (int_symb M.if_rm) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.minsort) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.minsort) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.if_minsort) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (int_symb M.if_minsort) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    end.

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

End PIS7.

Module PI7 := PolyInt PIS7.

(* graph decomposition 3 *)

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

   (  R1 (S1.hif_minsort (S1.false) (S1.add (V1 0) (V1 1)) (V1 2))
         (S1.hminsort (V1 1) (S1.add (V1 0) (V1 2)))
   :: 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.
right. PI1.prove_termination.
termination_trivial.
left. co_scc.
right. PI2.prove_termination.
termination_trivial.
left. co_scc.
right. PI3.prove_termination.
termination_trivial.
left. co_scc.
right. PI4.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI5.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs2; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
right. PI6.prove_termination.
PI7.prove_termination.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs3; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
Qed.