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 :=
  | _dot__1 : symb
  | _if_2 : symb
  | a : symb
  | b : symb
  | b' : symb
  | c : symb
  | d : symb
  | d' : symb
  | e : symb
  | f : symb
  | g : symb
  | h : symb
  | i : 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._dot__1 => 2
  | M._if_2 => 3
  | M.a => 0
  | M.b => 0
  | M.b' => 0
  | M.c => 0
  | M.d => 0
  | M.d' => 0
  | M.e => 0
  | M.f => 2
  | M.g => 2
  | M.h => 2
  | M.i => 3
  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 _dot__1 x2 x1 := F0 M._dot__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition _if_2 x3 x2 x1 := F0 M._if_2 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a := F0 M.a Vnil.
  Definition b := F0 M.b Vnil.
  Definition b' := F0 M.b' Vnil.
  Definition c := F0 M.c Vnil.
  Definition d := F0 M.d Vnil.
  Definition d' := F0 M.d' Vnil.
  Definition e := F0 M.e Vnil.
  Definition f x2 x1 := F0 M.f (Vcons x2 (Vcons x1 Vnil)).
  Definition g x2 x1 := F0 M.g (Vcons x2 (Vcons x1 Vnil)).
  Definition h x2 x1 := F0 M.h (Vcons x2 (Vcons x1 Vnil)).
  Definition i x3 x2 x1 := F0 M.i (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
End S0.

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

Definition R :=
   R0 (S0.f (S0.g (S0.i S0.a S0.b S0.b') S0.c) S0.d)
      (S0._if_2 S0.e (S0.f (S0._dot__1 S0.b S0.c) S0.d') (S0.f (S0._dot__1 S0.b' S0.c) S0.d'))
:: R0 (S0.f (S0.g (S0.h S0.a S0.b) S0.c) S0.d)
      (S0._if_2 S0.e (S0.f (S0._dot__1 S0.b (S0.g (S0.h S0.a S0.b) S0.c)) S0.d) (S0.f S0.c S0.d'))
:: @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_dot__1 x2 x1 := F1 (hd_symb s1_p M._dot__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _dot__1 x2 x1 := F1 (int_symb s1_p M._dot__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 ha := F1 (hd_symb s1_p M.a) Vnil.
  Definition a := F1 (int_symb s1_p M.a) Vnil.
  Definition hb := F1 (hd_symb s1_p M.b) Vnil.
  Definition b := F1 (int_symb s1_p M.b) Vnil.
  Definition hb' := F1 (hd_symb s1_p M.b') Vnil.
  Definition b' := F1 (int_symb s1_p M.b') Vnil.
  Definition hc := F1 (hd_symb s1_p M.c) Vnil.
  Definition c := F1 (int_symb s1_p M.c) Vnil.
  Definition hd := F1 (hd_symb s1_p M.d) Vnil.
  Definition d := F1 (int_symb s1_p M.d) Vnil.
  Definition hd' := F1 (hd_symb s1_p M.d') Vnil.
  Definition d' := F1 (int_symb s1_p M.d') Vnil.
  Definition he := F1 (hd_symb s1_p M.e) Vnil.
  Definition e := F1 (int_symb s1_p M.e) 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 hg x2 x1 := F1 (hd_symb s1_p M.g) (Vcons x2 (Vcons x1 Vnil)).
  Definition g x2 x1 := F1 (int_symb s1_p M.g) (Vcons x2 (Vcons x1 Vnil)).
  Definition hh x2 x1 := F1 (hd_symb s1_p M.h) (Vcons x2 (Vcons x1 Vnil)).
  Definition h x2 x1 := F1 (int_symb s1_p M.h) (Vcons x2 (Vcons x1 Vnil)).
  Definition hi x3 x2 x1 := F1 (hd_symb s1_p M.i) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition i x3 x2 x1 := F1 (int_symb s1_p M.i) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hf (S1.g (S1.h (S1.a) (S1.b)) (S1.c)) (S1.d))
         (S1.hf (S1.c) (S1.d'))
   :: nil)

:: (  R1 (S1.hf (S1.g (S1.h (S1.a) (S1.b)) (S1.c)) (S1.d))
         (S1.hf (S1._dot__1 (S1.b) (S1.g (S1.h (S1.a) (S1.b)) (S1.c))) (S1.d))
   :: nil)

:: (  R1 (S1.hf (S1.g (S1.i (S1.a) (S1.b) (S1.b')) (S1.c)) (S1.d))
         (S1.hf (S1._dot__1 (S1.b') (S1.c)) (S1.d'))
   :: nil)

:: (  R1 (S1.hf (S1.g (S1.i (S1.a) (S1.b) (S1.b')) (S1.c)) (S1.d))
         (S1.hf (S1._dot__1 (S1.b) (S1.c)) (S1.d'))
   :: 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.
Qed.