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 :=
  | _if_1 : symb
  | activate : symb
  | c : symb
  | f : symb
  | false : symb
  | n__f : 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._if_1 => 3
  | M.activate => 1
  | M.c => 0
  | M.f => 1
  | M.false => 0
  | M.n__f => 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 _if_1 x3 x2 x1 := F0 M._if_1 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition activate x1 := F0 M.activate (Vcons x1 Vnil).
  Definition c := F0 M.c Vnil.
  Definition f x1 := F0 M.f (Vcons x1 Vnil).
  Definition false := F0 M.false Vnil.
  Definition n__f x1 := F0 M.n__f (Vcons x1 Vnil).
  Definition true := F0 M.true Vnil.
End S0.

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

Definition R :=
   R0 (S0.f (V0 0))
      (S0._if_1 (V0 0) S0.c (S0.n__f S0.true))
:: R0 (S0._if_1 S0.true (V0 0) (V0 1))
      (V0 0)
:: R0 (S0._if_1 S0.false (V0 0) (V0 1))
      (S0.activate (V0 1))
:: R0 (S0.f (V0 0))
      (S0.n__f (V0 0))
:: R0 (S0.activate (S0.n__f (V0 0)))
      (S0.f (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 h_if_1 x3 x2 x1 := F1 (hd_symb s1_p M._if_1) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _if_1 x3 x2 x1 := F1 (int_symb s1_p M._if_1) (Vcons x3 (Vcons x2 (Vcons x1 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 hc := F1 (hd_symb s1_p M.c) Vnil.
  Definition c := F1 (int_symb s1_p M.c) Vnil.
  Definition hf x1 := F1 (hd_symb s1_p M.f) (Vcons x1 Vnil).
  Definition f x1 := F1 (int_symb s1_p M.f) (Vcons x1 Vnil).
  Definition hfalse := F1 (hd_symb s1_p M.false) Vnil.
  Definition false := F1 (int_symb s1_p M.false) Vnil.
  Definition hn__f x1 := F1 (hd_symb s1_p M.n__f) (Vcons x1 Vnil).
  Definition n__f x1 := F1 (int_symb s1_p M.n__f) (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.

(* 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) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.f) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._if_1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M._if_1) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         nil
    | (hd_symb M.n__f) =>
         nil
    | (int_symb M.n__f) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.activate) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.activate) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* graph decomposition 1 *)

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

   (  R1 (S1.h_if_1 (S1.false) (V1 0) (V1 1))
         (S1.hactivate (V1 1))
   :: nil)

:: nil.

(* termination proof *)

Lemma termination : WF rel.

Proof.
unfold rel.
dp_trans.
mark.
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) cs1; subst D; subst R.
dpg_unif_N_correct.
left. co_scc.
Qed.