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 :=
  | _0_1 : symb
  | a__adx : symb
  | a__hd : symb
  | a__incr : symb
  | a__nats : symb
  | a__tl : symb
  | a__zeros : symb
  | adx : symb
  | cons : symb
  | hd : symb
  | incr : symb
  | mark : symb
  | nats : symb
  | s : symb
  | tl : symb
  | zeros : 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.a__adx => 1
  | M.a__hd => 1
  | M.a__incr => 1
  | M.a__nats => 0
  | M.a__tl => 1
  | M.a__zeros => 0
  | M.adx => 1
  | M.cons => 2
  | M.hd => 1
  | M.incr => 1
  | M.mark => 1
  | M.nats => 0
  | M.s => 1
  | M.tl => 1
  | M.zeros => 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 _0_1 := F0 M._0_1 Vnil.
  Definition a__adx x1 := F0 M.a__adx (Vcons x1 Vnil).
  Definition a__hd x1 := F0 M.a__hd (Vcons x1 Vnil).
  Definition a__incr x1 := F0 M.a__incr (Vcons x1 Vnil).
  Definition a__nats := F0 M.a__nats Vnil.
  Definition a__tl x1 := F0 M.a__tl (Vcons x1 Vnil).
  Definition a__zeros := F0 M.a__zeros Vnil.
  Definition adx x1 := F0 M.adx (Vcons x1 Vnil).
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition hd x1 := F0 M.hd (Vcons x1 Vnil).
  Definition incr x1 := F0 M.incr (Vcons x1 Vnil).
  Definition mark x1 := F0 M.mark (Vcons x1 Vnil).
  Definition nats := F0 M.nats Vnil.
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition tl x1 := F0 M.tl (Vcons x1 Vnil).
  Definition zeros := F0 M.zeros Vnil.
End S0.

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

Definition R :=
   R0 S0.a__nats
      (S0.a__adx S0.a__zeros)
:: R0 S0.a__zeros
      (S0.cons S0._0_1 S0.zeros)
:: R0 (S0.a__incr (S0.cons (V0 0) (V0 1)))
      (S0.cons (S0.s (V0 0)) (S0.incr (V0 1)))
:: R0 (S0.a__adx (S0.cons (V0 0) (V0 1)))
      (S0.a__incr (S0.cons (V0 0) (S0.adx (V0 1))))
:: R0 (S0.a__hd (S0.cons (V0 0) (V0 1)))
      (S0.mark (V0 0))
:: R0 (S0.a__tl (S0.cons (V0 0) (V0 1)))
      (S0.mark (V0 1))
:: R0 (S0.mark S0.nats)
      S0.a__nats
:: R0 (S0.mark (S0.adx (V0 0)))
      (S0.a__adx (S0.mark (V0 0)))
:: R0 (S0.mark S0.zeros)
      S0.a__zeros
:: R0 (S0.mark (S0.incr (V0 0)))
      (S0.a__incr (S0.mark (V0 0)))
:: R0 (S0.mark (S0.hd (V0 0)))
      (S0.a__hd (S0.mark (V0 0)))
:: R0 (S0.mark (S0.tl (V0 0)))
      (S0.a__tl (S0.mark (V0 0)))
:: R0 (S0.mark (S0.cons (V0 0) (V0 1)))
      (S0.cons (V0 0) (V0 1))
:: R0 (S0.mark S0._0_1)
      S0._0_1
:: R0 (S0.mark (S0.s (V0 0)))
      (S0.s (V0 0))
:: R0 S0.a__nats
      S0.nats
:: R0 (S0.a__adx (V0 0))
      (S0.adx (V0 0))
:: R0 S0.a__zeros
      S0.zeros
:: R0 (S0.a__incr (V0 0))
      (S0.incr (V0 0))
:: R0 (S0.a__hd (V0 0))
      (S0.hd (V0 0))
:: R0 (S0.a__tl (V0 0))
      (S0.tl (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 ha__adx x1 := F1 (hd_symb s1_p M.a__adx) (Vcons x1 Vnil).
  Definition a__adx x1 := F1 (int_symb s1_p M.a__adx) (Vcons x1 Vnil).
  Definition ha__hd x1 := F1 (hd_symb s1_p M.a__hd) (Vcons x1 Vnil).
  Definition a__hd x1 := F1 (int_symb s1_p M.a__hd) (Vcons x1 Vnil).
  Definition ha__incr x1 := F1 (hd_symb s1_p M.a__incr) (Vcons x1 Vnil).
  Definition a__incr x1 := F1 (int_symb s1_p M.a__incr) (Vcons x1 Vnil).
  Definition ha__nats := F1 (hd_symb s1_p M.a__nats) Vnil.
  Definition a__nats := F1 (int_symb s1_p M.a__nats) Vnil.
  Definition ha__tl x1 := F1 (hd_symb s1_p M.a__tl) (Vcons x1 Vnil).
  Definition a__tl x1 := F1 (int_symb s1_p M.a__tl) (Vcons x1 Vnil).
  Definition ha__zeros := F1 (hd_symb s1_p M.a__zeros) Vnil.
  Definition a__zeros := F1 (int_symb s1_p M.a__zeros) Vnil.
  Definition hadx x1 := F1 (hd_symb s1_p M.adx) (Vcons x1 Vnil).
  Definition adx x1 := F1 (int_symb s1_p M.adx) (Vcons x1 Vnil).
  Definition hcons x2 x1 := F1 (hd_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F1 (int_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)).
  Definition hhd x1 := F1 (hd_symb s1_p M.hd) (Vcons x1 Vnil).
  Definition hd x1 := F1 (int_symb s1_p M.hd) (Vcons x1 Vnil).
  Definition hincr x1 := F1 (hd_symb s1_p M.incr) (Vcons x1 Vnil).
  Definition incr x1 := F1 (int_symb s1_p M.incr) (Vcons x1 Vnil).
  Definition hmark x1 := F1 (hd_symb s1_p M.mark) (Vcons x1 Vnil).
  Definition mark x1 := F1 (int_symb s1_p M.mark) (Vcons x1 Vnil).
  Definition hnats := F1 (hd_symb s1_p M.nats) Vnil.
  Definition nats := F1 (int_symb s1_p M.nats) 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).
  Definition htl x1 := F1 (hd_symb s1_p M.tl) (Vcons x1 Vnil).
  Definition tl x1 := F1 (int_symb s1_p M.tl) (Vcons x1 Vnil).
  Definition hzeros := F1 (hd_symb s1_p M.zeros) Vnil.
  Definition zeros := F1 (int_symb s1_p M.zeros) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmark (S1.incr (V1 0)))
         (S1.ha__incr (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.hmark (S1.zeros))
         (S1.ha__zeros)
   :: nil)

:: (  R1 (S1.ha__adx (S1.cons (V1 0) (V1 1)))
         (S1.ha__incr (S1.cons (V1 0) (S1.adx (V1 1))))
   :: nil)

:: (  R1 (S1.hmark (S1.adx (V1 0)))
         (S1.ha__adx (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.ha__nats)
         (S1.ha__zeros)
   :: nil)

:: (  R1 (S1.ha__nats)
         (S1.ha__adx (S1.a__zeros))
   :: nil)

:: (  R1 (S1.hmark (S1.nats))
         (S1.ha__nats)
   :: nil)

:: (  R1 (S1.hmark (S1.adx (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.incr (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.hd (V1 0)))
         (S1.ha__hd (S1.mark (V1 0)))
   :: R1 (S1.ha__hd (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.hd (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.tl (V1 0)))
         (S1.ha__tl (S1.mark (V1 0)))
   :: R1 (S1.ha__tl (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 1))
   :: R1 (S1.hmark (S1.tl (V1 0)))
         (S1.hmark (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__nats) =>
         nil
    | (int_symb M.a__nats) =>
         nil
    | (hd_symb M.a__adx) =>
         nil
    | (int_symb M.a__adx) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__incr) =>
         nil
    | (int_symb M.a__incr) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.incr) =>
         nil
    | (int_symb M.incr) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.adx) =>
         nil
    | (int_symb M.adx) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__hd) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__hd) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__tl) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__tl) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nats) =>
         nil
    | (int_symb M.nats) =>
         nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         (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__nats) =>
         nil
    | (int_symb M.a__nats) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.a__adx) =>
         nil
    | (int_symb M.a__adx) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__incr) =>
         nil
    | (int_symb M.a__incr) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.incr) =>
         nil
    | (int_symb M.incr) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.adx) =>
         nil
    | (int_symb M.adx) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__hd) =>
         nil
    | (int_symb M.a__hd) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__tl) =>
         nil
    | (int_symb M.a__tl) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nats) =>
         nil
    | (int_symb M.nats) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         (3%Z, (Vcons 1 Vnil))
      :: 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.a__nats) =>
         nil
    | (int_symb M.a__nats) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.a__adx) =>
         nil
    | (int_symb M.a__adx) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__zeros) =>
         nil
    | (int_symb M.a__zeros) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.zeros) =>
         nil
    | (int_symb M.zeros) =>
         nil
    | (hd_symb M.a__incr) =>
         nil
    | (int_symb M.a__incr) =>
         (2%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.incr) =>
         nil
    | (int_symb M.incr) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.adx) =>
         nil
    | (int_symb M.adx) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__hd) =>
         nil
    | (int_symb M.a__hd) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__tl) =>
         nil
    | (int_symb M.a__tl) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.nats) =>
         nil
    | (int_symb M.nats) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: 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.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI1.prove_termination.
PI2.prove_termination.
PI3.prove_termination.
termination_trivial.
Qed.