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
  | f : 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.f => 2
  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 f x2 x1 := F0 M.f (Vcons x2 (Vcons x1 Vnil)).
End S0.

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

Definition R :=
   R0 (S0.f S0.a (S0.f (S0.f S0.a (V0 0)) S0.a))
      (S0.f (S0.f S0.a (S0.f S0.a (V0 0))) S0.a)
:: @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 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)).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hf (S1.a) (S1.f (S1.f (S1.a) (V1 0)) (S1.a)))
         (S1.hf (S1.f (S1.a) (S1.f (S1.a) (V1 0))) (S1.a))
   :: nil)

:: (  R1 (S1.hf (S1.a) (S1.f (S1.f (S1.a) (V1 0)) (S1.a)))
         (S1.hf (S1.a) (V1 0))
   :: R1 (S1.hf (S1.a) (S1.f (S1.f (S1.a) (V1 0)) (S1.a)))
         (S1.hf (S1.a) (S1.f (S1.a) (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.f) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         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.
right. PI1.prove_termination.
termination_trivial.
Qed.