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 :=
  | _0_2 : symb
  | _if_4 : symb
  | _lt__3 : symb
  | _minus__1 : symb
  | gcd : symb
  | s : 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._0_2 => 0
  | M._if_4 => 3
  | M._lt__3 => 2
  | M._minus__1 => 2
  | M.gcd => 2
  | M.s => 1
  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 _0_2 := F0 M._0_2 Vnil.
  Definition _if_4 x3 x2 x1 := F0 M._if_4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _lt__3 x2 x1 := F0 M._lt__3 (Vcons x2 (Vcons x1 Vnil)).
  Definition _minus__1 x2 x1 := F0 M._minus__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition gcd x2 x1 := F0 M.gcd (Vcons x2 (Vcons x1 Vnil)).
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.gcd (V0 0) S0._0_2)
      (V0 0)
:: R0 (S0.gcd S0._0_2 (V0 0))
      (V0 0)
:: R0 (S0.gcd (S0.s (V0 0)) (S0.s (V0 1)))
      (S0._if_4 (S0._lt__3 (V0 0) (V0 1)) (S0.gcd (S0.s (V0 0)) (S0._minus__1 (V0 1) (V0 0))) (S0.gcd (S0._minus__1 (V0 0) (V0 1)) (S0.s (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 h_0_2 := F1 (hd_symb s1_p M._0_2) Vnil.
  Definition _0_2 := F1 (int_symb s1_p M._0_2) Vnil.
  Definition h_if_4 x3 x2 x1 := F1 (hd_symb s1_p M._if_4) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _if_4 x3 x2 x1 := F1 (int_symb s1_p M._if_4) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition h_lt__3 x2 x1 := F1 (hd_symb s1_p M._lt__3) (Vcons x2 (Vcons x1 Vnil)).
  Definition _lt__3 x2 x1 := F1 (int_symb s1_p M._lt__3) (Vcons x2 (Vcons x1 Vnil)).
  Definition h_minus__1 x2 x1 := F1 (hd_symb s1_p M._minus__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _minus__1 x2 x1 := F1 (int_symb s1_p M._minus__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition hgcd x2 x1 := F1 (hd_symb s1_p M.gcd) (Vcons x2 (Vcons x1 Vnil)).
  Definition gcd x2 x1 := F1 (int_symb s1_p M.gcd) (Vcons x2 (Vcons x1 Vnil)).
  Definition hs x1 := F1 (hd_symb s1_p M.s) (Vcons x1 Vnil).
  Definition s x1 := F1 (int_symb s1_p M.s) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hgcd (S1.s (V1 0)) (S1.s (V1 1)))
         (S1.hgcd (S1._minus__1 (V1 0) (V1 1)) (S1.s (V1 1)))
   :: nil)

:: (  R1 (S1.hgcd (S1.s (V1 0)) (S1.s (V1 1)))
         (S1.hgcd (S1.s (V1 0)) (S1._minus__1 (V1 1) (V1 0)))
   :: 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.
Qed.