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 :=
  | a : symb
  | a__b : symb
  | a__f : symb
  | b : symb
  | f : symb
  | mark : 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.a => 0
  | M.a__b => 0
  | M.a__f => 2
  | M.b => 0
  | M.f => 2
  | M.mark => 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 a := F0 M.a Vnil.
  Definition a__b := F0 M.a__b Vnil.
  Definition a__f x2 x1 := F0 M.a__f (Vcons x2 (Vcons x1 Vnil)).
  Definition b := F0 M.b Vnil.
  Definition f x2 x1 := F0 M.f (Vcons x2 (Vcons x1 Vnil)).
  Definition mark x1 := F0 M.mark (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.a__f (V0 0) (V0 0))
      (S0.a__f S0.a S0.b)
:: R0 S0.a__b
      S0.a
:: R0 (S0.mark (S0.f (V0 0) (V0 1)))
      (S0.a__f (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark S0.b)
      S0.a__b
:: R0 (S0.mark S0.a)
      S0.a
:: R0 (S0.a__f (V0 0) (V0 1))
      (S0.f (V0 0) (V0 1))
:: R0 S0.a__b
      S0.b
:: @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 ha := F1 (hd_symb s1_p M.a) Vnil.
  Definition a := F1 (int_symb s1_p M.a) Vnil.
  Definition ha__b := F1 (hd_symb s1_p M.a__b) Vnil.
  Definition a__b := F1 (int_symb s1_p M.a__b) Vnil.
  Definition ha__f x2 x1 := F1 (hd_symb s1_p M.a__f) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__f x2 x1 := F1 (int_symb s1_p M.a__f) (Vcons x2 (Vcons x1 Vnil)).
  Definition hb := F1 (hd_symb s1_p M.b) Vnil.
  Definition b := F1 (int_symb s1_p M.b) 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 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).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmark (S1.b))
         (S1.ha__b)
   :: nil)

:: (  R1 (S1.ha__f (V1 0) (V1 0))
         (S1.ha__f (S1.a) (S1.b))
   :: nil)

:: (  R1 (S1.hmark (S1.f (V1 0) (V1 1)))
         (S1.ha__f (S1.mark (V1 0)) (V1 1))
   :: nil)

:: (  R1 (S1.hmark (S1.f (V1 0) (V1 1)))
         (S1.hmark (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.a__f) =>
         nil
    | (int_symb M.a__f) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.b) =>
         nil
    | (int_symb M.b) =>
         nil
    | (hd_symb M.a__b) =>
         nil
    | (int_symb M.a__b) =>
         nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.f) =>
         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
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* 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.
termination_trivial.
Qed.