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 :=
  | append : symb
  | cons : symb
  | false : symb
  | hd : symb
  | ifappend : symb
  | is_empty : symb
  | nil : symb
  | tl : 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.append => 2
  | M.cons => 2
  | M.false => 0
  | M.hd => 1
  | M.ifappend => 3
  | M.is_empty => 1
  | M.nil => 0
  | M.tl => 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 append x2 x1 := F0 M.append (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition false := F0 M.false Vnil.
  Definition hd x1 := F0 M.hd (Vcons x1 Vnil).
  Definition ifappend x3 x2 x1 := F0 M.ifappend (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition is_empty x1 := F0 M.is_empty (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition tl x1 := F0 M.tl (Vcons x1 Vnil).
  Definition true := F0 M.true Vnil.
End S0.

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

Definition R :=
   R0 (S0.is_empty S0.nil)
      S0.true
:: R0 (S0.is_empty (S0.cons (V0 0) (V0 1)))
      S0.false
:: R0 (S0.hd (S0.cons (V0 0) (V0 1)))
      (V0 0)
:: R0 (S0.tl (S0.cons (V0 0) (V0 1)))
      (V0 1)
:: R0 (S0.append (V0 0) (V0 1))
      (S0.ifappend (V0 0) (V0 1) (V0 0))
:: R0 (S0.ifappend (V0 0) (V0 1) S0.nil)
      (V0 1)
:: R0 (S0.ifappend (V0 0) (V0 1) (S0.cons (V0 2) (V0 3)))
      (S0.cons (V0 2) (S0.append (V0 3) (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 happend x2 x1 := F1 (hd_symb s1_p M.append) (Vcons x2 (Vcons x1 Vnil)).
  Definition append x2 x1 := F1 (int_symb s1_p M.append) (Vcons x2 (Vcons x1 Vnil)).
  Definition hcons x2 x1 := F1 (hd_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F1 (int_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition hfalse := F1 (hd_symb s1_p M.false) Vnil.
  Definition false := F1 (int_symb s1_p M.false) Vnil.
  Definition hhd x1 := F1 (hd_symb s1_p M.hd) (Vcons x1 Vnil).
  Definition hd x1 := F1 (int_symb s1_p M.hd) (Vcons x1 Vnil).
  Definition hifappend x3 x2 x1 := F1 (hd_symb s1_p M.ifappend) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition ifappend x3 x2 x1 := F1 (int_symb s1_p M.ifappend) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition his_empty x1 := F1 (hd_symb s1_p M.is_empty) (Vcons x1 Vnil).
  Definition is_empty x1 := F1 (int_symb s1_p M.is_empty) (Vcons x1 Vnil).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition htl x1 := F1 (hd_symb s1_p M.tl) (Vcons x1 Vnil).
  Definition tl x1 := F1 (int_symb s1_p M.tl) (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.is_empty) =>
         nil
    | (int_symb M.is_empty) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.false) =>
         nil
    | (int_symb M.false) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.ifappend) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.ifappend) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (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.

(* graph decomposition 1 *)

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

   (  R1 (S1.happend (V1 0) (V1 1))
         (S1.hifappend (V1 0) (V1 1) (V1 0))
   :: 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.