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

Open Scope nat_scope.
(* termination problem *)

Module M.
  Inductive symb : Type :=
  | U11 : symb
  | U12 : symb
  | U21 : symb
  | U31 : symb
  | U41 : symb
  | U42 : symb
  | _0_1 : symb
  | activate : symb
  | isNat : symb
  | n__0 : symb
  | n__plus : symb
  | n__s : symb
  | plus : symb
  | s : symb
  | tt : symb.
End M.

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

Proof.
decide equality.
Defined.

Open Scope nat_scope.
Definition ar (s : M.symb) : nat :=
  match s with
  | M.U11 => 2
  | M.U12 => 1
  | M.U21 => 1
  | M.U31 => 2
  | M.U41 => 3
  | M.U42 => 3
  | M._0_1 => 0
  | M.activate => 1
  | M.isNat => 1
  | M.n__0 => 0
  | M.n__plus => 2
  | M.n__s => 1
  | M.plus => 2
  | M.s => 1
  | M.tt => 0
  end.

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

Module S0.
  Definition U11 x2 x1 := F0 M.U11 (Vcons x2 (Vcons x1 Vnil)).
  Definition U12 x1 := F0 M.U12 (Vcons x1 Vnil).
  Definition U21 x1 := F0 M.U21 (Vcons x1 Vnil).
  Definition U31 x2 x1 := F0 M.U31 (Vcons x2 (Vcons x1 Vnil)).
  Definition U41 x3 x2 x1 := F0 M.U41 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U42 x3 x2 x1 := F0 M.U42 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _0_1 := F0 M._0_1 Vnil.
  Definition activate x1 := F0 M.activate (Vcons x1 Vnil).
  Definition isNat x1 := F0 M.isNat (Vcons x1 Vnil).
  Definition n__0 := F0 M.n__0 Vnil.
  Definition n__plus x2 x1 := F0 M.n__plus (Vcons x2 (Vcons x1 Vnil)).
  Definition n__s x1 := F0 M.n__s (Vcons x1 Vnil).
  Definition plus x2 x1 := F0 M.plus (Vcons x2 (Vcons x1 Vnil)).
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition tt := F0 M.tt Vnil.
End S0.

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

Definition R :=
   R0 (S0.U11 S0.tt (V0 0))
      (S0.U12 (S0.isNat (S0.activate (V0 0))))
:: R0 (S0.U12 S0.tt)
      S0.tt
:: R0 (S0.U21 S0.tt)
      S0.tt
:: R0 (S0.U31 S0.tt (V0 0))
      (S0.activate (V0 0))
:: R0 (S0.U41 S0.tt (V0 0) (V0 1))
      (S0.U42 (S0.isNat (S0.activate (V0 1))) (S0.activate (V0 0)) (S0.activate (V0 1)))
:: R0 (S0.U42 S0.tt (V0 0) (V0 1))
      (S0.s (S0.plus (S0.activate (V0 1)) (S0.activate (V0 0))))
:: R0 (S0.isNat S0.n__0)
      S0.tt
:: R0 (S0.isNat (S0.n__plus (V0 0) (V0 1)))
      (S0.U11 (S0.isNat (S0.activate (V0 0))) (S0.activate (V0 1)))
:: R0 (S0.isNat (S0.n__s (V0 0)))
      (S0.U21 (S0.isNat (S0.activate (V0 0))))
:: R0 (S0.plus (V0 0) S0._0_1)
      (S0.U31 (S0.isNat (V0 0)) (V0 0))
:: R0 (S0.plus (V0 0) (S0.s (V0 1)))
      (S0.U41 (S0.isNat (V0 1)) (V0 1) (V0 0))
:: R0 S0._0_1
      S0.n__0
:: R0 (S0.plus (V0 0) (V0 1))
      (S0.n__plus (V0 0) (V0 1))
:: R0 (S0.s (V0 0))
      (S0.n__s (V0 0))
:: R0 (S0.activate S0.n__0)
      S0._0_1
:: R0 (S0.activate (S0.n__plus (V0 0) (V0 1)))
      (S0.plus (S0.activate (V0 0)) (S0.activate (V0 1)))
:: R0 (S0.activate (S0.n__s (V0 0)))
      (S0.s (S0.activate (V0 0)))
:: R0 (S0.activate (V0 0))
      (V0 0)
:: @nil (@ATrs.rule s0).

Definition rel := ATrs.red_mod E R.

(* symbol marking *)

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

Module S1.
  Definition hU11 x2 x1 := F1 (hd_symb s1_p M.U11) (Vcons x2 (Vcons x1 Vnil)).
  Definition U11 x2 x1 := F1 (int_symb s1_p M.U11) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU12 x1 := F1 (hd_symb s1_p M.U12) (Vcons x1 Vnil).
  Definition U12 x1 := F1 (int_symb s1_p M.U12) (Vcons x1 Vnil).
  Definition hU21 x1 := F1 (hd_symb s1_p M.U21) (Vcons x1 Vnil).
  Definition U21 x1 := F1 (int_symb s1_p M.U21) (Vcons x1 Vnil).
  Definition hU31 x2 x1 := F1 (hd_symb s1_p M.U31) (Vcons x2 (Vcons x1 Vnil)).
  Definition U31 x2 x1 := F1 (int_symb s1_p M.U31) (Vcons x2 (Vcons x1 Vnil)).
  Definition hU41 x3 x2 x1 := F1 (hd_symb s1_p M.U41) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition U41 x3 x2 x1 := F1 (int_symb s1_p M.U41) (Vcons x3 (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))).
  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 hactivate x1 := F1 (hd_symb s1_p M.activate) (Vcons x1 Vnil).
  Definition activate x1 := F1 (int_symb s1_p M.activate) (Vcons x1 Vnil).
  Definition hisNat x1 := F1 (hd_symb s1_p M.isNat) (Vcons x1 Vnil).
  Definition isNat x1 := F1 (int_symb s1_p M.isNat) (Vcons x1 Vnil).
  Definition hn__0 := F1 (hd_symb s1_p M.n__0) Vnil.
  Definition n__0 := F1 (int_symb s1_p M.n__0) Vnil.
  Definition hn__plus x2 x1 := F1 (hd_symb s1_p M.n__plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition n__plus x2 x1 := F1 (int_symb s1_p M.n__plus) (Vcons x2 (Vcons x1 Vnil)).
  Definition hn__s x1 := F1 (hd_symb s1_p M.n__s) (Vcons x1 Vnil).
  Definition n__s x1 := F1 (int_symb s1_p M.n__s) (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 hs x1 := F1 (hd_symb s1_p M.s) (Vcons x1 Vnil).
  Definition s x1 := F1 (int_symb s1_p M.s) (Vcons x1 Vnil).
  Definition htt := F1 (hd_symb s1_p M.tt) Vnil.
  Definition tt := F1 (int_symb s1_p M.tt) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hactivate (S1.n__s (V1 0)))
         (S1.hs (S1.activate (V1 0)))
   :: nil)

:: (  R1 (S1.hactivate (S1.n__0))
         (S1.h_0_1)
   :: nil)

:: (  R1 (S1.hisNat (S1.n__s (V1 0)))
         (S1.hU21 (S1.isNat (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU42 (S1.tt) (V1 0) (V1 1))
         (S1.hs (S1.plus (S1.activate (V1 1)) (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU11 (S1.tt) (V1 0))
         (S1.hU12 (S1.isNat (S1.activate (V1 0))))
   :: nil)

:: (  R1 (S1.hU11 (S1.tt) (V1 0))
         (S1.hisNat (S1.activate (V1 0)))
   :: R1 (S1.hisNat (S1.n__plus (V1 0) (V1 1)))
         (S1.hU11 (S1.isNat (S1.activate (V1 0))) (S1.activate (V1 1)))
   :: R1 (S1.hU11 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: R1 (S1.hactivate (S1.n__plus (V1 0) (V1 1)))
         (S1.hplus (S1.activate (V1 0)) (S1.activate (V1 1)))
   :: R1 (S1.hplus (V1 0) (S1._0_1))
         (S1.hU31 (S1.isNat (V1 0)) (V1 0))
   :: R1 (S1.hU31 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: R1 (S1.hactivate (S1.n__plus (V1 0) (V1 1)))
         (S1.hactivate (V1 0))
   :: R1 (S1.hactivate (S1.n__plus (V1 0) (V1 1)))
         (S1.hactivate (V1 1))
   :: R1 (S1.hactivate (S1.n__s (V1 0)))
         (S1.hactivate (V1 0))
   :: R1 (S1.hplus (V1 0) (S1._0_1))
         (S1.hisNat (V1 0))
   :: R1 (S1.hisNat (S1.n__plus (V1 0) (V1 1)))
         (S1.hisNat (S1.activate (V1 0)))
   :: R1 (S1.hisNat (S1.n__plus (V1 0) (V1 1)))
         (S1.hactivate (V1 0))
   :: R1 (S1.hisNat (S1.n__plus (V1 0) (V1 1)))
         (S1.hactivate (V1 1))
   :: R1 (S1.hisNat (S1.n__s (V1 0)))
         (S1.hisNat (S1.activate (V1 0)))
   :: R1 (S1.hisNat (S1.n__s (V1 0)))
         (S1.hactivate (V1 0))
   :: R1 (S1.hplus (V1 0) (S1.s (V1 1)))
         (S1.hU41 (S1.isNat (V1 1)) (V1 1) (V1 0))
   :: R1 (S1.hU41 (S1.tt) (V1 0) (V1 1))
         (S1.hU42 (S1.isNat (S1.activate (V1 1))) (S1.activate (V1 0)) (S1.activate (V1 1)))
   :: R1 (S1.hU42 (S1.tt) (V1 0) (V1 1))
         (S1.hplus (S1.activate (V1 1)) (S1.activate (V1 0)))
   :: R1 (S1.hplus (V1 0) (S1.s (V1 1)))
         (S1.hisNat (V1 1))
   :: R1 (S1.hU42 (S1.tt) (V1 0) (V1 1))
         (S1.hactivate (V1 1))
   :: R1 (S1.hU42 (S1.tt) (V1 0) (V1 1))
         (S1.hactivate (V1 0))
   :: R1 (S1.hU41 (S1.tt) (V1 0) (V1 1))
         (S1.hisNat (S1.activate (V1 1)))
   :: R1 (S1.hU41 (S1.tt) (V1 0) (V1 1))
         (S1.hactivate (V1 1))
   :: R1 (S1.hU41 (S1.tt) (V1 0) (V1 1))
         (S1.hactivate (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.U11) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNat) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.activate) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.activate) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U31) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U31) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U41) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.U41) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.U42) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.U42) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.n__0) =>
         nil
    | (int_symb M.n__0) =>
         nil
    | (hd_symb M.n__plus) =>
         nil
    | (int_symb M.n__plus) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.n__s) =>
         nil
    | (int_symb M.n__s) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         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.U11) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U11) =>
         nil
    | (hd_symb M.tt) =>
         nil
    | (int_symb M.tt) =>
         nil
    | (hd_symb M.U12) =>
         nil
    | (int_symb M.U12) =>
         nil
    | (hd_symb M.isNat) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.isNat) =>
         nil
    | (hd_symb M.activate) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.activate) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.U21) =>
         nil
    | (int_symb M.U21) =>
         nil
    | (hd_symb M.U31) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.U31) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.U41) =>
         nil
    | (int_symb M.U41) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.U42) =>
         nil
    | (int_symb M.U42) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.plus) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.plus) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.n__0) =>
         nil
    | (int_symb M.n__0) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.n__plus) =>
         nil
    | (int_symb M.n__plus) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.n__s) =>
         nil
    | (int_symb M.n__s) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         (2%Z, Vnil)
      :: nil
    end.

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

End PIS2.

Module PI2 := PolyInt PIS2.

(* graph decomposition 2 *)

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

   (  R1 (S1.hU31 (S1.tt) (V1 0))
         (S1.hactivate (V1 0))
   :: nil)

:: nil.

(* termination proof *)

Lemma termination : WF rel.

Proof.
unfold rel.
dp_trans.
mark.
let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R;
graph_decomp (dpg_unif_N 100 R D) cs1; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI1.prove_termination.
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) cs2; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
Qed.