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 :=
  | _dot__1 : symb
  | _eq__3 : symb
  | _if_4 : symb
  | _lt__eq__2 : symb
  | del : symb
  | min : symb
  | msort : symb
  | nil : 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._eq__3 => 2
  | M._if_4 => 3
  | M._lt__eq__2 => 2
  | M.del => 2
  | M.min => 2
  | M.msort => 1
  | M.nil => 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 _dot__1 x2 x1 := F0 M._dot__1 (Vcons x2 (Vcons x1 Vnil)).
  Definition _eq__3 x2 x1 := F0 M._eq__3 (Vcons x2 (Vcons x1 Vnil)).
  Definition _if_4 x3 x2 x1 := F0 M._if_4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _lt__eq__2 x2 x1 := F0 M._lt__eq__2 (Vcons x2 (Vcons x1 Vnil)).
  Definition del x2 x1 := F0 M.del (Vcons x2 (Vcons x1 Vnil)).
  Definition min x2 x1 := F0 M.min (Vcons x2 (Vcons x1 Vnil)).
  Definition msort x1 := F0 M.msort (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
End S0.

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

Definition R :=
   R0 (S0.msort S0.nil)
      S0.nil
:: R0 (S0.msort (S0._dot__1 (V0 0) (V0 1)))
      (S0._dot__1 (S0.min (V0 0) (V0 1)) (S0.msort (S0.del (S0.min (V0 0) (V0 1)) (S0._dot__1 (V0 0) (V0 1)))))
:: R0 (S0.min (V0 0) S0.nil)
      (V0 0)
:: R0 (S0.min (V0 0) (S0._dot__1 (V0 1) (V0 2)))
      (S0._if_4 (S0._lt__eq__2 (V0 0) (V0 1)) (S0.min (V0 0) (V0 2)) (S0.min (V0 1) (V0 2)))
:: R0 (S0.del (V0 0) S0.nil)
      S0.nil
:: R0 (S0.del (V0 0) (S0._dot__1 (V0 1) (V0 2)))
      (S0._if_4 (S0._eq__3 (V0 0) (V0 1)) (V0 2) (S0._dot__1 (V0 1) (S0.del (V0 0) (V0 2))))
:: @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_eq__3 x2 x1 := F1 (hd_symb s1_p M._eq__3) (Vcons x2 (Vcons x1 Vnil)).
  Definition _eq__3 x2 x1 := F1 (int_symb s1_p M._eq__3) (Vcons x2 (Vcons x1 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__eq__2 x2 x1 := F1 (hd_symb s1_p M._lt__eq__2) (Vcons x2 (Vcons x1 Vnil)).
  Definition _lt__eq__2 x2 x1 := F1 (int_symb s1_p M._lt__eq__2) (Vcons x2 (Vcons x1 Vnil)).
  Definition hdel x2 x1 := F1 (hd_symb s1_p M.del) (Vcons x2 (Vcons x1 Vnil)).
  Definition del x2 x1 := F1 (int_symb s1_p M.del) (Vcons x2 (Vcons x1 Vnil)).
  Definition hmin x2 x1 := F1 (hd_symb s1_p M.min) (Vcons x2 (Vcons x1 Vnil)).
  Definition min x2 x1 := F1 (int_symb s1_p M.min) (Vcons x2 (Vcons x1 Vnil)).
  Definition hmsort x1 := F1 (hd_symb s1_p M.msort) (Vcons x1 Vnil).
  Definition msort x1 := F1 (int_symb s1_p M.msort) (Vcons x1 Vnil).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hdel (V1 0) (S1._dot__1 (V1 1) (V1 2)))
         (S1.hdel (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hmin (V1 0) (S1._dot__1 (V1 1) (V1 2)))
         (S1.hmin (V1 1) (V1 2))
   :: R1 (S1.hmin (V1 0) (S1._dot__1 (V1 1) (V1 2)))
         (S1.hmin (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hmsort (S1._dot__1 (V1 0) (V1 1)))
         (S1.hdel (S1.min (V1 0) (V1 1)) (S1._dot__1 (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.hmsort (S1._dot__1 (V1 0) (V1 1)))
         (S1.hmin (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hmsort (S1._dot__1 (V1 0) (V1 1)))
         (S1.hmsort (S1.del (S1.min (V1 0) (V1 1)) (S1._dot__1 (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.msort) =>
         nil
    | (int_symb M.msort) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M._dot__1) =>
         nil
    | (int_symb M._dot__1) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.del) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.del) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_4) =>
         nil
    | (int_symb M._if_4) =>
         nil
    | (hd_symb M._lt__eq__2) =>
         nil
    | (int_symb M._lt__eq__2) =>
         nil
    | (hd_symb M._eq__3) =>
         nil
    | (int_symb M._eq__3) =>
         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.msort) =>
         nil
    | (int_symb M.msort) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M._dot__1) =>
         nil
    | (int_symb M._dot__1) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.min) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.del) =>
         nil
    | (int_symb M.del) =>
         nil
    | (hd_symb M._if_4) =>
         nil
    | (int_symb M._if_4) =>
         nil
    | (hd_symb M._lt__eq__2) =>
         nil
    | (int_symb M._lt__eq__2) =>
         nil
    | (hd_symb M._eq__3) =>
         nil
    | (int_symb M._eq__3) =>
         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.msort) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.msort) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M._dot__1) =>
         nil
    | (int_symb M._dot__1) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.min) =>
         nil
    | (int_symb M.min) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.del) =>
         nil
    | (int_symb M.del) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._if_4) =>
         nil
    | (int_symb M._if_4) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M._lt__eq__2) =>
         nil
    | (int_symb M._lt__eq__2) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M._eq__3) =>
         nil
    | (int_symb M._eq__3) =>
         nil
    end.

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

End PIS3.

Module PI3 := PolyInt PIS3.

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