Require Import ADPUnif.
Require Import ADecomp.
Require Import ADuplicateSymb.
Require Import AGraph.
Require Import ATrs.
Require Import List.
Require Import LogicUtil.
Require Import SN.
Require Import VecUtil.

Open Scope nat_scope.
(* termination problem *)

Module M.
  Inductive symb : Type :=
  | _eq__1 : symb
  | _if_2 : symb
  | and : symb
  | false : symb
  | implies : symb
  | not : symb
  | or : 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._eq__1 => 2
  | M._if_2 => 3
  | M.and => 2
  | M.false => 0
  | M.implies => 2
  | M.not => 1
  | M.or => 2
  | 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 _eq__1 x2 x1 := F0 M._eq__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition _if_2 x3 x2 x1 := F0 M._if_2 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition and x2 x1 := F0 M.and (Vcons x2 (Vcons x1 Vnil)).
  Definition false := F0 M.false Vnil.
  Definition implies x2 x1 := F0 M.implies (Vcons x2 (Vcons x1 Vnil)).
  Definition not x1 := F0 M.not (Vcons x1 Vnil).
  Definition or x2 x1 := F0 M.or (Vcons x2 (Vcons x1 Vnil)).
  Definition true := F0 M.true Vnil.
End S0.

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

Definition R :=
   R0 (S0.not (V0 0))
      (S0._if_2 (V0 0) S0.false S0.true)
:: R0 (S0.and (V0 0) (V0 1))
      (S0._if_2 (V0 0) (V0 1) S0.false)
:: R0 (S0.or (V0 0) (V0 1))
      (S0._if_2 (V0 0) S0.true (V0 1))
:: R0 (S0.implies (V0 0) (V0 1))
      (S0._if_2 (V0 0) (V0 1) S0.true)
:: R0 (S0._eq__1 (V0 0) (V0 0))
      S0.true
:: R0 (S0._eq__1 (V0 0) (V0 1))
      (S0._if_2 (V0 0) (V0 1) (S0.not (V0 1)))
:: R0 (S0._if_2 S0.true (V0 0) (V0 1))
      (V0 0)
:: R0 (S0._if_2 S0.false (V0 0) (V0 1))
      (V0 1)
:: R0 (S0._if_2 (V0 0) (V0 0) (S0._if_2 (V0 0) S0.false S0.true))
      S0.true
:: R0 (S0._eq__1 (V0 0) (V0 1))
      (S0._if_2 (V0 0) (V0 1) (S0._if_2 (V0 1) S0.false S0.true))
:: @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_eq__1 x2 x1 := F1 (hd_symb s1_p M._eq__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _eq__1 x2 x1 := F1 (int_symb s1_p M._eq__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition h_if_2 x3 x2 x1 := F1 (hd_symb s1_p M._if_2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _if_2 x3 x2 x1 := F1 (int_symb s1_p M._if_2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hand x2 x1 := F1 (hd_symb s1_p M.and) (Vcons x2 (Vcons x1 Vnil)).
  Definition and x2 x1 := F1 (int_symb s1_p M.and) (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 himplies x2 x1 := F1 (hd_symb s1_p M.implies) (Vcons x2 (Vcons x1 Vnil)).
  Definition implies x2 x1 := F1 (int_symb s1_p M.implies) (Vcons x2 (Vcons x1 Vnil)).
  Definition hnot x1 := F1 (hd_symb s1_p M.not) (Vcons x1 Vnil).
  Definition not x1 := F1 (int_symb s1_p M.not) (Vcons x1 Vnil).
  Definition hor x2 x1 := F1 (hd_symb s1_p M.or) (Vcons x2 (Vcons x1 Vnil)).
  Definition or x2 x1 := F1 (int_symb s1_p M.or) (Vcons x2 (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.

(* graph decomposition 1 *)

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

   (  R1 (S1.h_eq__1 (V1 0) (V1 1))
         (S1.h_if_2 (V1 1) (S1.false) (S1.true))
   :: nil)

:: (  R1 (S1.h_eq__1 (V1 0) (V1 1))
         (S1.h_if_2 (V1 0) (V1 1) (S1._if_2 (V1 1) (S1.false) (S1.true)))
   :: nil)

:: (  R1 (S1.h_eq__1 (V1 0) (V1 1))
         (S1.h_if_2 (V1 0) (V1 1) (S1.not (V1 1)))
   :: nil)

:: (  R1 (S1.himplies (V1 0) (V1 1))
         (S1.h_if_2 (V1 0) (V1 1) (S1.true))
   :: nil)

:: (  R1 (S1.hor (V1 0) (V1 1))
         (S1.h_if_2 (V1 0) (S1.true) (V1 1))
   :: nil)

:: (  R1 (S1.hand (V1 0) (V1 1))
         (S1.h_if_2 (V1 0) (V1 1) (S1.false))
   :: nil)

:: (  R1 (S1.hnot (V1 0))
         (S1.h_if_2 (V1 0) (S1.false) (S1.true))
   :: nil)

:: (  R1 (S1.h_eq__1 (V1 0) (V1 1))
         (S1.hnot (V1 1))
   :: nil)

:: nil.

(* 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.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.