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
  | f : symb
  | g : symb
  | h : symb
  | h1 : symb
  | h2 : symb
  | i : symb
  | j : symb
  | k : symb
  | s : 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.f => 2
  | M.g => 1
  | M.h => 1
  | M.h1 => 2
  | M.h2 => 3
  | M.i => 1
  | M.j => 2
  | M.k => 1
  | M.s => 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_1 := F0 M._0_1 Vnil.
  Definition f x2 x1 := F0 M.f (Vcons x2 (Vcons x1 Vnil)).
  Definition g x1 := F0 M.g (Vcons x1 Vnil).
  Definition h x1 := F0 M.h (Vcons x1 Vnil).
  Definition h1 x2 x1 := F0 M.h1 (Vcons x2 (Vcons x1 Vnil)).
  Definition h2 x3 x2 x1 := F0 M.h2 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition i x1 := F0 M.i (Vcons x1 Vnil).
  Definition j x2 x1 := F0 M.j (Vcons x2 (Vcons x1 Vnil)).
  Definition k x1 := F0 M.k (Vcons x1 Vnil).
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.f (S0.j (V0 0) (V0 1)) (V0 1))
      (S0.g (S0.f (V0 0) (S0.k (V0 1))))
:: R0 (S0.f (V0 0) (S0.h1 (V0 1) (V0 2)))
      (S0.h2 S0._0_1 (V0 0) (S0.h1 (V0 1) (V0 2)))
:: R0 (S0.g (S0.h2 (V0 0) (V0 1) (S0.h1 (V0 2) (V0 3))))
      (S0.h2 (S0.s (V0 0)) (V0 1) (S0.h1 (V0 2) (V0 3)))
:: R0 (S0.h2 (V0 0) (S0.j (V0 1) (S0.h1 (V0 2) (V0 3))) (S0.h1 (V0 2) (V0 3)))
      (S0.h2 (S0.s (V0 0)) (V0 1) (S0.h1 (S0.s (V0 2)) (V0 3)))
:: R0 (S0.i (S0.f (V0 0) (S0.h (V0 1))))
      (V0 1)
:: R0 (S0.i (S0.h2 (S0.s (V0 0)) (V0 1) (S0.h1 (V0 0) (V0 3))))
      (V0 3)
:: R0 (S0.k (S0.h (V0 0)))
      (S0.h1 S0._0_1 (V0 0))
:: R0 (S0.k (S0.h1 (V0 0) (V0 1)))
      (S0.h1 (S0.s (V0 0)) (V0 1))
:: @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 hf x2 x1 := F1 (hd_symb s1_p M.f) (Vcons x2 (Vcons x1 Vnil)).
  Definition f x2 x1 := F1 (int_symb s1_p M.f) (Vcons x2 (Vcons x1 Vnil)).
  Definition hg x1 := F1 (hd_symb s1_p M.g) (Vcons x1 Vnil).
  Definition g x1 := F1 (int_symb s1_p M.g) (Vcons x1 Vnil).
  Definition hh x1 := F1 (hd_symb s1_p M.h) (Vcons x1 Vnil).
  Definition h x1 := F1 (int_symb s1_p M.h) (Vcons x1 Vnil).
  Definition hh1 x2 x1 := F1 (hd_symb s1_p M.h1) (Vcons x2 (Vcons x1 Vnil)).
  Definition h1 x2 x1 := F1 (int_symb s1_p M.h1) (Vcons x2 (Vcons x1 Vnil)).
  Definition hh2 x3 x2 x1 := F1 (hd_symb s1_p M.h2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition h2 x3 x2 x1 := F1 (int_symb s1_p M.h2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hi x1 := F1 (hd_symb s1_p M.i) (Vcons x1 Vnil).
  Definition i x1 := F1 (int_symb s1_p M.i) (Vcons x1 Vnil).
  Definition hj x2 x1 := F1 (hd_symb s1_p M.j) (Vcons x2 (Vcons x1 Vnil)).
  Definition j x2 x1 := F1 (int_symb s1_p M.j) (Vcons x2 (Vcons x1 Vnil)).
  Definition hk x1 := F1 (hd_symb s1_p M.k) (Vcons x1 Vnil).
  Definition k x1 := F1 (int_symb s1_p M.k) (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).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hh2 (V1 0) (S1.j (V1 1) (S1.h1 (V1 2) (V1 3))) (S1.h1 (V1 2) (V1 3)))
         (S1.hh2 (S1.s (V1 0)) (V1 1) (S1.h1 (S1.s (V1 2)) (V1 3)))
   :: nil)

:: (  R1 (S1.hg (S1.h2 (V1 0) (V1 1) (S1.h1 (V1 2) (V1 3))))
         (S1.hh2 (S1.s (V1 0)) (V1 1) (S1.h1 (V1 2) (V1 3)))
   :: nil)

:: (  R1 (S1.hf (V1 0) (S1.h1 (V1 1) (V1 2)))
         (S1.hh2 (S1._0_1) (V1 0) (S1.h1 (V1 1) (V1 2)))
   :: nil)

:: (  R1 (S1.hf (S1.j (V1 0) (V1 1)) (V1 1))
         (S1.hk (V1 1))
   :: nil)

:: (  R1 (S1.hf (S1.j (V1 0) (V1 1)) (V1 1))
         (S1.hg (S1.f (V1 0) (S1.k (V1 1))))
   :: nil)

:: (  R1 (S1.hf (S1.j (V1 0) (V1 1)) (V1 1))
         (S1.hf (V1 0) (S1.k (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.f) =>
         nil
    | (int_symb M.f) =>
         (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.j) =>
         nil
    | (int_symb M.j) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.g) =>
         nil
    | (int_symb M.g) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.k) =>
         nil
    | (int_symb M.k) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.h1) =>
         nil
    | (int_symb M.h1) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.h2) =>
         (2%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
    | (int_symb M.h2) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.h) =>
         nil
    | (int_symb M.h) =>
         (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 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.f) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.f) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.j) =>
         nil
    | (int_symb M.j) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.g) =>
         nil
    | (int_symb M.g) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.k) =>
         nil
    | (int_symb M.k) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.h1) =>
         nil
    | (int_symb M.h1) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.h2) =>
         nil
    | (int_symb M.h2) =>
         (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
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.i) =>
         nil
    | (int_symb M.i) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.h) =>
         nil
    | (int_symb M.h) =>
         (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.

(* 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.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI2.prove_termination.
termination_trivial.
Qed.