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 :=
  | _eq__2 : symb
  | _plus__plus__1 : symb
  | f : symb
  | false : symb
  | g : symb
  | max : symb
  | max' : symb
  | mem : symb
  | nil : symb
  | not : symb
  | null : symb
  | or : symb
  | true : symb
  | u : 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__2 => 2
  | M._plus__plus__1 => 2
  | M.f => 2
  | M.false => 0
  | M.g => 2
  | M.max => 1
  | M.max' => 2
  | M.mem => 2
  | M.nil => 0
  | M.not => 1
  | M.null => 1
  | M.or => 2
  | M.true => 0
  | M.u => 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__2 x2 x1 := F0 M._eq__2 (Vcons x2 (Vcons x1 Vnil)).
  Definition _plus__plus__1 x2 x1 := F0 M._plus__plus__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition f x2 x1 := F0 M.f (Vcons x2 (Vcons x1 Vnil)).
  Definition false := F0 M.false Vnil.
  Definition g x2 x1 := F0 M.g (Vcons x2 (Vcons x1 Vnil)).
  Definition max x1 := F0 M.max (Vcons x1 Vnil).
  Definition max' x2 x1 := F0 M.max' (Vcons x2 (Vcons x1 Vnil)).
  Definition mem x2 x1 := F0 M.mem (Vcons x2 (Vcons x1 Vnil)).
  Definition nil := F0 M.nil Vnil.
  Definition not x1 := F0 M.not (Vcons x1 Vnil).
  Definition null x1 := F0 M.null (Vcons x1 Vnil).
  Definition or x2 x1 := F0 M.or (Vcons x2 (Vcons x1 Vnil)).
  Definition true := F0 M.true Vnil.
  Definition u := F0 M.u Vnil.
End S0.

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

Definition R :=
   R0 (S0.f (V0 0) S0.nil)
      (S0.g S0.nil (V0 0))
:: R0 (S0.f (V0 0) (S0.g (V0 1) (V0 2)))
      (S0.g (S0.f (V0 0) (V0 1)) (V0 2))
:: R0 (S0._plus__plus__1 (V0 0) S0.nil)
      (V0 0)
:: R0 (S0._plus__plus__1 (V0 0) (S0.g (V0 1) (V0 2)))
      (S0.g (S0._plus__plus__1 (V0 0) (V0 1)) (V0 2))
:: R0 (S0.null S0.nil)
      S0.true
:: R0 (S0.null (S0.g (V0 0) (V0 1)))
      S0.false
:: R0 (S0.mem S0.nil (V0 0))
      S0.false
:: R0 (S0.mem (S0.g (V0 0) (V0 1)) (V0 2))
      (S0.or (S0._eq__2 (V0 1) (V0 2)) (S0.mem (V0 0) (V0 2)))
:: R0 (S0.mem (V0 0) (S0.max (V0 0)))
      (S0.not (S0.null (V0 0)))
:: R0 (S0.max (S0.g (S0.g S0.nil (V0 0)) (V0 1)))
      (S0.max' (V0 0) (V0 1))
:: R0 (S0.max (S0.g (S0.g (S0.g (V0 0) (V0 1)) (V0 2)) S0.u))
      (S0.max' (S0.max (S0.g (S0.g (V0 0) (V0 1)) (V0 2))) S0.u)
:: @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__2 x2 x1 := F1 (hd_symb s1_p M._eq__2) (Vcons x2 (Vcons x1 Vnil)).
  Definition _eq__2 x2 x1 := F1 (int_symb s1_p M._eq__2) (Vcons x2 (Vcons x1 Vnil)).
  Definition h_plus__plus__1 x2 x1 := F1 (hd_symb s1_p M._plus__plus__1) (Vcons x2 (Vcons x1 Vnil)).
  Definition _plus__plus__1 x2 x1 := F1 (int_symb s1_p M._plus__plus__1) (Vcons x2 (Vcons x1 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 hfalse := F1 (hd_symb s1_p M.false) Vnil.
  Definition false := F1 (int_symb s1_p M.false) 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 hmax x1 := F1 (hd_symb s1_p M.max) (Vcons x1 Vnil).
  Definition max x1 := F1 (int_symb s1_p M.max) (Vcons x1 Vnil).
  Definition hmax' x2 x1 := F1 (hd_symb s1_p M.max') (Vcons x2 (Vcons x1 Vnil)).
  Definition max' x2 x1 := F1 (int_symb s1_p M.max') (Vcons x2 (Vcons x1 Vnil)).
  Definition hmem x2 x1 := F1 (hd_symb s1_p M.mem) (Vcons x2 (Vcons x1 Vnil)).
  Definition mem x2 x1 := F1 (int_symb s1_p M.mem) (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 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 hnull x1 := F1 (hd_symb s1_p M.null) (Vcons x1 Vnil).
  Definition null x1 := F1 (int_symb s1_p M.null) (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.
  Definition hu := F1 (hd_symb s1_p M.u) Vnil.
  Definition u := F1 (int_symb s1_p M.u) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmax (S1.g (S1.g (S1.g (V1 0) (V1 1)) (V1 2)) (S1.u)))
         (S1.hmax (S1.g (S1.g (V1 0) (V1 1)) (V1 2)))
   :: nil)

:: (  R1 (S1.hmem (V1 0) (S1.max (V1 0)))
         (S1.hnull (V1 0))
   :: nil)

:: (  R1 (S1.hmem (S1.g (V1 0) (V1 1)) (V1 2))
         (S1.hmem (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.h_plus__plus__1 (V1 0) (S1.g (V1 1) (V1 2)))
         (S1.h_plus__plus__1 (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hf (V1 0) (S1.g (V1 1) (V1 2)))
         (S1.hf (V1 0) (V1 1))
   :: 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.f) =>
         nil
    | (int_symb M.f) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.g) =>
         nil
    | (int_symb M.g) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._plus__plus__1) =>
         nil
    | (int_symb M._plus__plus__1) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.null) =>
         nil
    | (int_symb M.null) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.mem) =>
         nil
    | (int_symb M.mem) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.or) =>
         nil
    | (int_symb M.or) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._eq__2) =>
         nil
    | (int_symb M._eq__2) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.max) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.max) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.max') =>
         nil
    | (int_symb M.max') =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         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.f) =>
         nil
    | (int_symb M.f) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.g) =>
         nil
    | (int_symb M.g) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._plus__plus__1) =>
         nil
    | (int_symb M._plus__plus__1) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.null) =>
         nil
    | (int_symb M.null) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.mem) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.mem) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M._eq__2) =>
         nil
    | (int_symb M._eq__2) =>
         nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.max') =>
         nil
    | (int_symb M.max') =>
         nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    end.

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

End PIS2.

Module PI2 := PolyInt PIS2.

(* polynomial interpretation 3 *)

Module PIS3 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.f) =>
         nil
    | (int_symb M.f) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.g) =>
         nil
    | (int_symb M.g) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._plus__plus__1) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M._plus__plus__1) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.null) =>
         nil
    | (int_symb M.null) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.mem) =>
         nil
    | (int_symb M.mem) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M._eq__2) =>
         nil
    | (int_symb M._eq__2) =>
         nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.max') =>
         nil
    | (int_symb M.max') =>
         nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

(* polynomial interpretation 4 *)

Module PIS4 (*<: TPolyInt*).

  Definition sig := s1.

  Definition trsInt f :=
    match f as f return poly (@ASignature.arity s1 f) with
    | (hd_symb M.f) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.f) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.g) =>
         nil
    | (int_symb M.g) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._plus__plus__1) =>
         nil
    | (int_symb M._plus__plus__1) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.null) =>
         nil
    | (int_symb M.null) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.mem) =>
         nil
    | (int_symb M.mem) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M._eq__2) =>
         nil
    | (int_symb M._eq__2) =>
         nil
    | (hd_symb M.max) =>
         nil
    | (int_symb M.max) =>
         nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M.max') =>
         nil
    | (int_symb M.max') =>
         nil
    | (hd_symb M.u) =>
         nil
    | (int_symb M.u) =>
         nil
    end.

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

End PIS4.

Module PI4 := PolyInt PIS4.

(* 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.
right. PI1.prove_termination.
termination_trivial.
left. co_scc.
right. PI2.prove_termination.
termination_trivial.
right. PI3.prove_termination.
termination_trivial.
right. PI4.prove_termination.
termination_trivial.
Qed.