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 : symb
  | B : symb
  | C : symb
  | cons : symb
  | f : symb
  | f' : symb
  | f'' : symb
  | foldf : symb
  | g : symb
  | nil : symb
  | triple : 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 => 0
  | M.B => 0
  | M.C => 0
  | M.cons => 2
  | M.f => 2
  | M.f' => 2
  | M.f'' => 1
  | M.foldf => 2
  | M.g => 1
  | M.nil => 0
  | M.triple => 3
  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 := F0 M.A Vnil.
  Definition B := F0 M.B Vnil.
  Definition C := F0 M.C Vnil.
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition f x2 x1 := F0 M.f (Vcons x2 (Vcons x1 Vnil)).
  Definition f' x2 x1 := F0 M.f' (Vcons x2 (Vcons x1 Vnil)).
  Definition f'' x1 := F0 M.f'' (Vcons x1 Vnil).
  Definition foldf x2 x1 := F0 M.foldf (Vcons x2 (Vcons x1 Vnil)).
  Definition g x1 := F0 M.g (Vcons x1 Vnil).
  Definition nil := F0 M.nil Vnil.
  Definition triple x3 x2 x1 := F0 M.triple (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
End S0.

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

Definition R :=
   R0 (S0.g S0.A)
      S0.A
:: R0 (S0.g S0.B)
      S0.A
:: R0 (S0.g S0.B)
      S0.B
:: R0 (S0.g S0.C)
      S0.A
:: R0 (S0.g S0.C)
      S0.B
:: R0 (S0.g S0.C)
      S0.C
:: R0 (S0.foldf (V0 0) S0.nil)
      (V0 0)
:: R0 (S0.foldf (V0 0) (S0.cons (V0 1) (V0 2)))
      (S0.f (S0.foldf (V0 0) (V0 2)) (V0 1))
:: R0 (S0.f (V0 0) (V0 1))
      (S0.f' (V0 0) (S0.g (V0 1)))
:: R0 (S0.f' (S0.triple (V0 0) (V0 1) (V0 2)) S0.C)
      (S0.triple (V0 0) (V0 1) (S0.cons S0.C (V0 2)))
:: R0 (S0.f' (S0.triple (V0 0) (V0 1) (V0 2)) S0.B)
      (S0.f (S0.triple (V0 0) (V0 1) (V0 2)) S0.A)
:: R0 (S0.f' (S0.triple (V0 0) (V0 1) (V0 2)) S0.A)
      (S0.f'' (S0.foldf (S0.triple (S0.cons S0.A (V0 0)) S0.nil (V0 2)) (V0 1)))
:: R0 (S0.f'' (S0.triple (V0 0) (V0 1) (V0 2)))
      (S0.foldf (S0.triple (V0 0) (V0 1) S0.nil) (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 hA := F1 (hd_symb s1_p M.A) Vnil.
  Definition A := F1 (int_symb s1_p M.A) Vnil.
  Definition hB := F1 (hd_symb s1_p M.B) Vnil.
  Definition B := F1 (int_symb s1_p M.B) Vnil.
  Definition hC := F1 (hd_symb s1_p M.C) Vnil.
  Definition C := F1 (int_symb s1_p M.C) 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 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 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 hf'' x1 := F1 (hd_symb s1_p M.f'') (Vcons x1 Vnil).
  Definition f'' x1 := F1 (int_symb s1_p M.f'') (Vcons x1 Vnil).
  Definition hfoldf x2 x1 := F1 (hd_symb s1_p M.foldf) (Vcons x2 (Vcons x1 Vnil)).
  Definition foldf x2 x1 := F1 (int_symb s1_p M.foldf) (Vcons x2 (Vcons x1 Vnil)).
  Definition hg x1 := F1 (hd_symb s1_p M.g) (Vcons x1 Vnil).
  Definition g x1 := F1 (int_symb s1_p M.g) (Vcons x1 Vnil).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition htriple x3 x2 x1 := F1 (hd_symb s1_p M.triple) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition triple x3 x2 x1 := F1 (int_symb s1_p M.triple) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
End S1.

(* graph decomposition 1 *)

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

   (  R1 (S1.hf (V1 0) (V1 1))
         (S1.hg (V1 1))
   :: nil)

:: (  R1 (S1.hf (V1 0) (V1 1))
         (S1.hf' (V1 0) (S1.g (V1 1)))
   :: R1 (S1.hf' (S1.triple (V1 0) (V1 1) (V1 2)) (S1.B))
         (S1.hf (S1.triple (V1 0) (V1 1) (V1 2)) (S1.A))
   :: R1 (S1.hf' (S1.triple (V1 0) (V1 1) (V1 2)) (S1.A))
         (S1.hf'' (S1.foldf (S1.triple (S1.cons (S1.A) (V1 0)) (S1.nil) (V1 2)) (V1 1)))
   :: R1 (S1.hf'' (S1.triple (V1 0) (V1 1) (V1 2)))
         (S1.hfoldf (S1.triple (V1 0) (V1 1) (S1.nil)) (V1 2))
   :: R1 (S1.hfoldf (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hf (S1.foldf (V1 0) (V1 2)) (V1 1))
   :: R1 (S1.hfoldf (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hfoldf (V1 0) (V1 2))
   :: R1 (S1.hf' (S1.triple (V1 0) (V1 1) (V1 2)) (S1.A))
         (S1.hfoldf (S1.triple (S1.cons (S1.A) (V1 0)) (S1.nil) (V1 2)) (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.g) =>
         nil
    | (int_symb M.g) =>
         nil
    | (hd_symb M.A) =>
         nil
    | (int_symb M.A) =>
         nil
    | (hd_symb M.B) =>
         nil
    | (int_symb M.B) =>
         nil
    | (hd_symb M.C) =>
         nil
    | (int_symb M.C) =>
         nil
    | (hd_symb M.foldf) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.foldf) =>
         (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.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.f) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.f) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.f') =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.f') =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.triple) =>
         nil
    | (int_symb M.triple) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.f'') =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (int_symb M.f'') =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%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.hf' (S1.triple (V1 0) (V1 1) (V1 2)) (S1.A))
         (S1.hfoldf (S1.triple (S1.cons (S1.A) (V1 0)) (S1.nil) (V1 2)) (V1 1))
   :: nil)

:: (  R1 (S1.hf'' (S1.triple (V1 0) (V1 1) (V1 2)))
         (S1.hfoldf (S1.triple (V1 0) (V1 1) (S1.nil)) (V1 2))
   :: nil)

:: (  R1 (S1.hf' (S1.triple (V1 0) (V1 1) (V1 2)) (S1.A))
         (S1.hf'' (S1.foldf (S1.triple (S1.cons (S1.A) (V1 0)) (S1.nil) (V1 2)) (V1 1)))
   :: nil)

:: (  R1 (S1.hf' (S1.triple (V1 0) (V1 1) (V1 2)) (S1.B))
         (S1.hf (S1.triple (V1 0) (V1 1) (V1 2)) (S1.A))
   :: R1 (S1.hf (V1 0) (V1 1))
         (S1.hf' (V1 0) (S1.g (V1 1)))
   :: 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.g) =>
         nil
    | (int_symb M.g) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.A) =>
         nil
    | (int_symb M.A) =>
         nil
    | (hd_symb M.B) =>
         nil
    | (int_symb M.B) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.C) =>
         nil
    | (int_symb M.C) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.foldf) =>
         nil
    | (int_symb M.foldf) =>
         (2%Z, (Vcons 1 (Vcons 0 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.f) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.f) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.f') =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.f') =>
         nil
    | (hd_symb M.triple) =>
         nil
    | (int_symb M.triple) =>
         nil
    | (hd_symb M.f'') =>
         nil
    | (int_symb M.f'') =>
         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.hf (V1 0) (V1 1))
         (S1.hf' (V1 0) (S1.g (V1 1)))
   :: nil)

:: nil.

(* 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.
left. co_scc.
left. co_scc.
left. co_scc.
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.
left. co_scc.
Qed.