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 :=
  | app : symb
  | compose : symb
  | cons : symb
  | hd : symb
  | init : symb
  | last : symb
  | nil : symb
  | reverse : symb
  | reverse2 : symb
  | tl : 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.app => 2
  | M.compose => 0
  | M.cons => 0
  | M.hd => 0
  | M.init => 0
  | M.last => 0
  | M.nil => 0
  | M.reverse => 0
  | M.reverse2 => 0
  | M.tl => 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 app x2 x1 := F0 M.app (Vcons x2 (Vcons x1 Vnil)).
  Definition compose := F0 M.compose Vnil.
  Definition cons := F0 M.cons Vnil.
  Definition hd := F0 M.hd Vnil.
  Definition init := F0 M.init Vnil.
  Definition last := F0 M.last Vnil.
  Definition nil := F0 M.nil Vnil.
  Definition reverse := F0 M.reverse Vnil.
  Definition reverse2 := F0 M.reverse2 Vnil.
  Definition tl := F0 M.tl Vnil.
End S0.

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

Definition R :=
   R0 (S0.app (S0.app (S0.app S0.compose (V0 0)) (V0 1)) (V0 2))
      (S0.app (V0 1) (S0.app (V0 0) (V0 2)))
:: R0 (S0.app S0.reverse (V0 0))
      (S0.app (S0.app S0.reverse2 (V0 0)) S0.nil)
:: R0 (S0.app (S0.app S0.reverse2 S0.nil) (V0 0))
      (V0 0)
:: R0 (S0.app (S0.app S0.reverse2 (S0.app (S0.app S0.cons (V0 0)) (V0 1))) (V0 2))
      (S0.app (S0.app S0.reverse2 (V0 1)) (S0.app (S0.app S0.cons (V0 0)) (V0 2)))
:: R0 (S0.app S0.hd (S0.app (S0.app S0.cons (V0 0)) (V0 1)))
      (V0 0)
:: R0 (S0.app S0.tl (S0.app (S0.app S0.cons (V0 0)) (V0 1)))
      (V0 1)
:: R0 S0.last
      (S0.app (S0.app S0.compose S0.hd) S0.reverse)
:: R0 S0.init
      (S0.app (S0.app S0.compose S0.reverse) (S0.app (S0.app S0.compose S0.tl) S0.reverse))
:: @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 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 hcompose := F1 (hd_symb s1_p M.compose) Vnil.
  Definition compose := F1 (int_symb s1_p M.compose) Vnil.
  Definition hcons := F1 (hd_symb s1_p M.cons) Vnil.
  Definition cons := F1 (int_symb s1_p M.cons) Vnil.
  Definition hhd := F1 (hd_symb s1_p M.hd) Vnil.
  Definition hd := F1 (int_symb s1_p M.hd) Vnil.
  Definition hinit := F1 (hd_symb s1_p M.init) Vnil.
  Definition init := F1 (int_symb s1_p M.init) Vnil.
  Definition hlast := F1 (hd_symb s1_p M.last) Vnil.
  Definition last := F1 (int_symb s1_p M.last) Vnil.
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hreverse := F1 (hd_symb s1_p M.reverse) Vnil.
  Definition reverse := F1 (int_symb s1_p M.reverse) Vnil.
  Definition hreverse2 := F1 (hd_symb s1_p M.reverse2) Vnil.
  Definition reverse2 := F1 (int_symb s1_p M.reverse2) Vnil.
  Definition htl := F1 (hd_symb s1_p M.tl) Vnil.
  Definition tl := F1 (int_symb s1_p M.tl) Vnil.
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.happ (S1.app (S1.reverse2) (S1.app (S1.app (S1.cons) (V1 0)) (V1 1))) (V1 2))
         (S1.happ (S1.cons) (V1 0))
   :: nil)

:: (  R1 (S1.happ (S1.app (S1.reverse2) (S1.app (S1.app (S1.cons) (V1 0)) (V1 1))) (V1 2))
         (S1.happ (S1.reverse2) (V1 1))
   :: nil)

:: (  R1 (S1.happ (S1.reverse) (V1 0))
         (S1.happ (S1.reverse2) (V1 0))
   :: nil)

:: (  R1 (S1.happ (S1.app (S1.app (S1.compose) (V1 0)) (V1 1)) (V1 2))
         (S1.happ (V1 0) (V1 2))
   :: R1 (S1.happ (S1.app (S1.app (S1.compose) (V1 0)) (V1 1)) (V1 2))
         (S1.happ (V1 1) (S1.app (V1 0) (V1 2)))
   :: R1 (S1.happ (S1.reverse) (V1 0))
         (S1.happ (S1.app (S1.reverse2) (V1 0)) (S1.nil))
   :: R1 (S1.happ (S1.app (S1.reverse2) (S1.app (S1.app (S1.cons) (V1 0)) (V1 1))) (V1 2))
         (S1.happ (S1.app (S1.reverse2) (V1 1)) (S1.app (S1.app (S1.cons) (V1 0)) (V1 2)))
   :: R1 (S1.happ (S1.app (S1.reverse2) (S1.app (S1.app (S1.cons) (V1 0)) (V1 1))) (V1 2))
         (S1.happ (S1.app (S1.cons) (V1 0)) (V1 2))
   :: nil)

:: (  R1 (S1.hinit)
         (S1.happ (S1.compose) (S1.tl))
   :: nil)

:: (  R1 (S1.hinit)
         (S1.happ (S1.app (S1.compose) (S1.tl)) (S1.reverse))
   :: nil)

:: (  R1 (S1.hinit)
         (S1.happ (S1.compose) (S1.reverse))
   :: nil)

:: (  R1 (S1.hinit)
         (S1.happ (S1.app (S1.compose) (S1.reverse)) (S1.app (S1.app (S1.compose) (S1.tl)) (S1.reverse)))
   :: nil)

:: (  R1 (S1.hlast)
         (S1.happ (S1.compose) (S1.hd))
   :: nil)

:: (  R1 (S1.hlast)
         (S1.happ (S1.app (S1.compose) (S1.hd)) (S1.reverse))
   :: 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.app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.compose) =>
         nil
    | (int_symb M.compose) =>
         nil
    | (hd_symb M.reverse) =>
         nil
    | (int_symb M.reverse) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.reverse2) =>
         nil
    | (int_symb M.reverse2) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         nil
    | (hd_symb M.last) =>
         nil
    | (int_symb M.last) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.init) =>
         nil
    | (int_symb M.init) =>
         (3%Z, 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.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.compose) =>
         nil
    | (int_symb M.compose) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.reverse) =>
         nil
    | (int_symb M.reverse) =>
         nil
    | (hd_symb M.reverse2) =>
         nil
    | (int_symb M.reverse2) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         nil
    | (hd_symb M.last) =>
         nil
    | (int_symb M.last) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.init) =>
         nil
    | (int_symb M.init) =>
         (2%Z, 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.app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.compose) =>
         nil
    | (int_symb M.compose) =>
         nil
    | (hd_symb M.reverse) =>
         nil
    | (int_symb M.reverse) =>
         nil
    | (hd_symb M.reverse2) =>
         nil
    | (int_symb M.reverse2) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.last) =>
         nil
    | (int_symb M.last) =>
         (3%Z, Vnil)
      :: nil
    | (hd_symb M.init) =>
         nil
    | (int_symb M.init) =>
         (2%Z, Vnil)
      :: 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.app) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.app) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.compose) =>
         nil
    | (int_symb M.compose) =>
         nil
    | (hd_symb M.reverse) =>
         nil
    | (int_symb M.reverse) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.reverse2) =>
         nil
    | (int_symb M.reverse2) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         nil
    | (hd_symb M.hd) =>
         nil
    | (int_symb M.hd) =>
         nil
    | (hd_symb M.tl) =>
         nil
    | (int_symb M.tl) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.last) =>
         nil
    | (int_symb M.last) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.init) =>
         nil
    | (int_symb M.init) =>
         (3%Z, Vnil)
      :: 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.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI1.prove_termination.
PI2.prove_termination.
PI3.prove_termination.
PI4.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
Qed.