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_1 : symb
  | add : symb
  | cons : symb
  | dbl : symb
  | first : symb
  | nil : symb
  | recip : symb
  | s : symb
  | sqr : symb
  | terms : 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_1 => 0
  | M.add => 2
  | M.cons => 1
  | M.dbl => 1
  | M.first => 2
  | M.nil => 0
  | M.recip => 1
  | M.s => 0
  | M.sqr => 1
  | M.terms => 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_1 := F0 M._0_1 Vnil.
  Definition add x2 x1 := F0 M.add (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x1 := F0 M.cons (Vcons x1 Vnil).
  Definition dbl x1 := F0 M.dbl (Vcons x1 Vnil).
  Definition first x2 x1 := F0 M.first (Vcons x2 (Vcons x1 Vnil)).
  Definition nil := F0 M.nil Vnil.
  Definition recip x1 := F0 M.recip (Vcons x1 Vnil).
  Definition s := F0 M.s Vnil.
  Definition sqr x1 := F0 M.sqr (Vcons x1 Vnil).
  Definition terms x1 := F0 M.terms (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.terms (V0 0))
      (S0.cons (S0.recip (S0.sqr (V0 0))))
:: R0 (S0.sqr S0._0_1)
      S0._0_1
:: R0 (S0.sqr S0.s)
      S0.s
:: R0 (S0.dbl S0._0_1)
      S0._0_1
:: R0 (S0.dbl S0.s)
      S0.s
:: R0 (S0.add S0._0_1 (V0 0))
      (V0 0)
:: R0 (S0.add S0.s (V0 0))
      S0.s
:: R0 (S0.first S0._0_1 (V0 0))
      S0.nil
:: R0 (S0.first S0.s (S0.cons (V0 0)))
      (S0.cons (V0 0))
:: @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_1 := F1 (hd_symb s1_p M._0_1) Vnil.
  Definition _0_1 := F1 (int_symb s1_p M._0_1) Vnil.
  Definition hadd x2 x1 := F1 (hd_symb s1_p M.add) (Vcons x2 (Vcons x1 Vnil)).
  Definition add x2 x1 := F1 (int_symb s1_p M.add) (Vcons x2 (Vcons x1 Vnil)).
  Definition hcons x1 := F1 (hd_symb s1_p M.cons) (Vcons x1 Vnil).
  Definition cons x1 := F1 (int_symb s1_p M.cons) (Vcons x1 Vnil).
  Definition hdbl x1 := F1 (hd_symb s1_p M.dbl) (Vcons x1 Vnil).
  Definition dbl x1 := F1 (int_symb s1_p M.dbl) (Vcons x1 Vnil).
  Definition hfirst x2 x1 := F1 (hd_symb s1_p M.first) (Vcons x2 (Vcons x1 Vnil)).
  Definition first x2 x1 := F1 (int_symb s1_p M.first) (Vcons x2 (Vcons x1 Vnil)).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hrecip x1 := F1 (hd_symb s1_p M.recip) (Vcons x1 Vnil).
  Definition recip x1 := F1 (int_symb s1_p M.recip) (Vcons x1 Vnil).
  Definition hs := F1 (hd_symb s1_p M.s) Vnil.
  Definition s := F1 (int_symb s1_p M.s) Vnil.
  Definition hsqr x1 := F1 (hd_symb s1_p M.sqr) (Vcons x1 Vnil).
  Definition sqr x1 := F1 (int_symb s1_p M.sqr) (Vcons x1 Vnil).
  Definition hterms x1 := F1 (hd_symb s1_p M.terms) (Vcons x1 Vnil).
  Definition terms x1 := F1 (int_symb s1_p M.terms) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hterms (V1 0))
         (S1.hsqr (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.
Qed.