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__app : symb
  | a__from : symb
  | a__prefix : symb
  | a__zWadr : symb
  | app : symb
  | cons : symb
  | from : symb
  | mark : symb
  | nil : symb
  | prefix : symb
  | s : symb
  | zWadr : 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__app => 2
  | M.a__from => 1
  | M.a__prefix => 1
  | M.a__zWadr => 2
  | M.app => 2
  | M.cons => 2
  | M.from => 1
  | M.mark => 1
  | M.nil => 0
  | M.prefix => 1
  | M.s => 1
  | M.zWadr => 2
  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__app x2 x1 := F0 M.a__app (Vcons x2 (Vcons x1 Vnil)).
  Definition a__from x1 := F0 M.a__from (Vcons x1 Vnil).
  Definition a__prefix x1 := F0 M.a__prefix (Vcons x1 Vnil).
  Definition a__zWadr x2 x1 := F0 M.a__zWadr (Vcons x2 (Vcons x1 Vnil)).
  Definition app x2 x1 := F0 M.app (Vcons x2 (Vcons x1 Vnil)).
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition from x1 := F0 M.from (Vcons x1 Vnil).
  Definition mark x1 := F0 M.mark (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition prefix x1 := F0 M.prefix (Vcons x1 Vnil).
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition zWadr x2 x1 := F0 M.zWadr (Vcons x2 (Vcons x1 Vnil)).
End S0.

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

Definition R :=
   R0 (S0.a__app S0.nil (V0 0))
      (S0.mark (V0 0))
:: R0 (S0.a__app (S0.cons (V0 0) (V0 1)) (V0 2))
      (S0.cons (S0.mark (V0 0)) (S0.app (V0 1) (V0 2)))
:: R0 (S0.a__from (V0 0))
      (S0.cons (S0.mark (V0 0)) (S0.from (S0.s (V0 0))))
:: R0 (S0.a__zWadr S0.nil (V0 0))
      S0.nil
:: R0 (S0.a__zWadr (V0 0) S0.nil)
      S0.nil
:: R0 (S0.a__zWadr (S0.cons (V0 0) (V0 1)) (S0.cons (V0 2) (V0 3)))
      (S0.cons (S0.a__app (S0.mark (V0 2)) (S0.cons (S0.mark (V0 0)) S0.nil)) (S0.zWadr (V0 1) (V0 3)))
:: R0 (S0.a__prefix (V0 0))
      (S0.cons S0.nil (S0.zWadr (V0 0) (S0.prefix (V0 0))))
:: R0 (S0.mark (S0.app (V0 0) (V0 1)))
      (S0.a__app (S0.mark (V0 0)) (S0.mark (V0 1)))
:: R0 (S0.mark (S0.from (V0 0)))
      (S0.a__from (S0.mark (V0 0)))
:: R0 (S0.mark (S0.zWadr (V0 0) (V0 1)))
      (S0.a__zWadr (S0.mark (V0 0)) (S0.mark (V0 1)))
:: R0 (S0.mark (S0.prefix (V0 0)))
      (S0.a__prefix (S0.mark (V0 0)))
:: R0 (S0.mark S0.nil)
      S0.nil
:: R0 (S0.mark (S0.cons (V0 0) (V0 1)))
      (S0.cons (S0.mark (V0 0)) (V0 1))
:: R0 (S0.mark (S0.s (V0 0)))
      (S0.s (S0.mark (V0 0)))
:: R0 (S0.a__app (V0 0) (V0 1))
      (S0.app (V0 0) (V0 1))
:: R0 (S0.a__from (V0 0))
      (S0.from (V0 0))
:: R0 (S0.a__zWadr (V0 0) (V0 1))
      (S0.zWadr (V0 0) (V0 1))
:: R0 (S0.a__prefix (V0 0))
      (S0.prefix (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__app x2 x1 := F1 (hd_symb s1_p M.a__app) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__app x2 x1 := F1 (int_symb s1_p M.a__app) (Vcons x2 (Vcons x1 Vnil)).
  Definition ha__from x1 := F1 (hd_symb s1_p M.a__from) (Vcons x1 Vnil).
  Definition a__from x1 := F1 (int_symb s1_p M.a__from) (Vcons x1 Vnil).
  Definition ha__prefix x1 := F1 (hd_symb s1_p M.a__prefix) (Vcons x1 Vnil).
  Definition a__prefix x1 := F1 (int_symb s1_p M.a__prefix) (Vcons x1 Vnil).
  Definition ha__zWadr x2 x1 := F1 (hd_symb s1_p M.a__zWadr) (Vcons x2 (Vcons x1 Vnil)).
  Definition a__zWadr x2 x1 := F1 (int_symb s1_p M.a__zWadr) (Vcons x2 (Vcons x1 Vnil)).
  Definition happ x2 x1 := F1 (hd_symb s1_p M.app) (Vcons x2 (Vcons x1 Vnil)).
  Definition app x2 x1 := F1 (int_symb s1_p M.app) (Vcons x2 (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 hfrom x1 := F1 (hd_symb s1_p M.from) (Vcons x1 Vnil).
  Definition from x1 := F1 (int_symb s1_p M.from) (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 hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hprefix x1 := F1 (hd_symb s1_p M.prefix) (Vcons x1 Vnil).
  Definition prefix x1 := F1 (int_symb s1_p M.prefix) (Vcons x1 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 hzWadr x2 x1 := F1 (hd_symb s1_p M.zWadr) (Vcons x2 (Vcons x1 Vnil)).
  Definition zWadr x2 x1 := F1 (int_symb s1_p M.zWadr) (Vcons x2 (Vcons x1 Vnil)).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hmark (S1.prefix (V1 0)))
         (S1.ha__prefix (S1.mark (V1 0)))
   :: nil)

:: (  R1 (S1.hmark (S1.app (V1 0) (V1 1)))
         (S1.ha__app (S1.mark (V1 0)) (S1.mark (V1 1)))
   :: R1 (S1.ha__app (S1.nil) (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.app (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.app (V1 0) (V1 1)))
         (S1.hmark (V1 1))
   :: R1 (S1.hmark (S1.from (V1 0)))
         (S1.ha__from (S1.mark (V1 0)))
   :: R1 (S1.ha__from (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.from (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.zWadr (V1 0) (V1 1)))
         (S1.ha__zWadr (S1.mark (V1 0)) (S1.mark (V1 1)))
   :: R1 (S1.ha__zWadr (S1.cons (V1 0) (V1 1)) (S1.cons (V1 2) (V1 3)))
         (S1.ha__app (S1.mark (V1 2)) (S1.cons (S1.mark (V1 0)) (S1.nil)))
   :: R1 (S1.ha__app (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.zWadr (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.zWadr (V1 0) (V1 1)))
         (S1.hmark (V1 1))
   :: R1 (S1.hmark (S1.prefix (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.s (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.ha__zWadr (S1.cons (V1 0) (V1 1)) (S1.cons (V1 2) (V1 3)))
         (S1.hmark (V1 2))
   :: R1 (S1.ha__zWadr (S1.cons (V1 0) (V1 1)) (S1.cons (V1 2) (V1 3)))
         (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__app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__app) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__from) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__from) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.from) =>
         nil
    | (int_symb M.from) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a__zWadr) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.a__zWadr) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.zWadr) =>
         nil
    | (int_symb M.zWadr) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__prefix) =>
         nil
    | (int_symb M.a__prefix) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.prefix) =>
         nil
    | (int_symb M.prefix) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    end.

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

End PIS1.

Module PI1 := PolyInt PIS1.

(* graph decomposition 2 *)

Definition cs2 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.ha__from (V1 0))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.from (V1 0)))
         (S1.ha__from (S1.mark (V1 0)))
   :: R1 (S1.hmark (S1.from (V1 0)))
         (S1.hmark (V1 0))
   :: R1 (S1.hmark (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__app (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__app (S1.nil) (V1 0))
         (S1.hmark (V1 0))
   :: nil)

:: nil.

(* 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__app) =>
         nil
    | (int_symb M.a__app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__from) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.a__from) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.from) =>
         nil
    | (int_symb M.from) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__zWadr) =>
         nil
    | (int_symb M.a__zWadr) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.zWadr) =>
         nil
    | (int_symb M.zWadr) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__prefix) =>
         nil
    | (int_symb M.a__prefix) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.prefix) =>
         nil
    | (int_symb M.prefix) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    end.

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

End PIS2.

Module PI2 := PolyInt PIS2.

(* graph decomposition 3 *)

Definition cs3 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hmark (S1.cons (V1 0) (V1 1)))
         (S1.hmark (V1 0))
   :: nil)

:: (  R1 (S1.ha__from (V1 0))
         (S1.hmark (V1 0))
   :: nil)

:: nil.

(* 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__app) =>
         nil
    | (int_symb M.a__app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.mark) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.mark) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.app) =>
         nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__from) =>
         nil
    | (int_symb M.a__from) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.from) =>
         nil
    | (int_symb M.from) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M.a__zWadr) =>
         nil
    | (int_symb M.a__zWadr) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.zWadr) =>
         nil
    | (int_symb M.zWadr) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.a__prefix) =>
         nil
    | (int_symb M.a__prefix) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.prefix) =>
         nil
    | (int_symb M.prefix) =>
         (2%Z, (Vcons 0 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.
right. PI1.prove_termination.
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) cs2; subst D; subst R.
dpg_unif_N_correct.
right. PI2.prove_termination.
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) cs3; subst D; subst R.
dpg_unif_N_correct.
right. PI3.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.