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 :=
  | a : symb
  | b : symb
  | c : symb
  | d : symb
  | u : symb
  | v : symb
  | w : 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.a => 1
  | M.b => 1
  | M.c => 1
  | M.d => 1
  | M.u => 1
  | M.v => 1
  | M.w => 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 a x1 := F0 M.a (Vcons x1 Vnil).
  Definition b x1 := F0 M.b (Vcons x1 Vnil).
  Definition c x1 := F0 M.c (Vcons x1 Vnil).
  Definition d x1 := F0 M.d (Vcons x1 Vnil).
  Definition u x1 := F0 M.u (Vcons x1 Vnil).
  Definition v x1 := F0 M.v (Vcons x1 Vnil).
  Definition w x1 := F0 M.w (Vcons x1 Vnil).
End S0.

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

Definition R :=
   R0 (S0.a (S0.c (S0.d (V0 0))))
      (S0.c (V0 0))
:: R0 (S0.u (S0.b (S0.d (S0.d (V0 0)))))
      (S0.b (V0 0))
:: R0 (S0.v (S0.a (S0.a (V0 0))))
      (S0.u (S0.v (V0 0)))
:: R0 (S0.v (S0.a (S0.c (V0 0))))
      (S0.u (S0.b (S0.d (V0 0))))
:: R0 (S0.v (S0.c (V0 0)))
      (S0.b (V0 0))
:: R0 (S0.w (S0.a (S0.a (V0 0))))
      (S0.u (S0.w (V0 0)))
:: R0 (S0.w (S0.a (S0.c (V0 0))))
      (S0.u (S0.b (S0.d (V0 0))))
:: R0 (S0.w (S0.c (V0 0)))
      (S0.b (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 ha x1 := F1 (hd_symb s1_p M.a) (Vcons x1 Vnil).
  Definition a x1 := F1 (int_symb s1_p M.a) (Vcons x1 Vnil).
  Definition hb x1 := F1 (hd_symb s1_p M.b) (Vcons x1 Vnil).
  Definition b x1 := F1 (int_symb s1_p M.b) (Vcons x1 Vnil).
  Definition hc x1 := F1 (hd_symb s1_p M.c) (Vcons x1 Vnil).
  Definition c x1 := F1 (int_symb s1_p M.c) (Vcons x1 Vnil).
  Definition hd x1 := F1 (hd_symb s1_p M.d) (Vcons x1 Vnil).
  Definition d x1 := F1 (int_symb s1_p M.d) (Vcons x1 Vnil).
  Definition hu x1 := F1 (hd_symb s1_p M.u) (Vcons x1 Vnil).
  Definition u x1 := F1 (int_symb s1_p M.u) (Vcons x1 Vnil).
  Definition hv x1 := F1 (hd_symb s1_p M.v) (Vcons x1 Vnil).
  Definition v x1 := F1 (int_symb s1_p M.v) (Vcons x1 Vnil).
  Definition hw x1 := F1 (hd_symb s1_p M.w) (Vcons x1 Vnil).
  Definition w x1 := F1 (int_symb s1_p M.w) (Vcons x1 Vnil).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hw (S1.a (S1.c (V1 0))))
         (S1.hu (S1.b (S1.d (V1 0))))
   :: nil)

:: (  R1 (S1.hw (S1.a (S1.a (V1 0))))
         (S1.hu (S1.w (V1 0)))
   :: nil)

:: (  R1 (S1.hw (S1.a (S1.a (V1 0))))
         (S1.hw (V1 0))
   :: nil)

:: (  R1 (S1.hv (S1.a (S1.c (V1 0))))
         (S1.hu (S1.b (S1.d (V1 0))))
   :: nil)

:: (  R1 (S1.hv (S1.a (S1.a (V1 0))))
         (S1.hu (S1.v (V1 0)))
   :: nil)

:: (  R1 (S1.hv (S1.a (S1.a (V1 0))))
         (S1.hv (V1 0))
   :: nil)

:: nil.

(* 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.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.d) =>
         nil
    | (int_symb M.d) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.b) =>
         nil
    | (int_symb M.b) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.v) =>
         nil
    | (int_symb M.v) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.w) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.w) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS1.

Module PI1 := PolyInt PIS1.

(* polynomial interpretation 2 *)

Module PIS2 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.c) =>
         nil
    | (int_symb M.c) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.d) =>
         nil
    | (int_symb M.d) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.b) =>
         nil
    | (int_symb M.b) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.v) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.v) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.w) =>
         nil
    | (int_symb M.w) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS2.

Module PI2 := PolyInt PIS2.

(* 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.
right. PI1.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI2.prove_termination.
termination_trivial.
Qed.