Library Stdlib.Wellfounded.Lexicographic_Exponentiation

Require Import List.
Require Import Relation_Operators.
Require Import Operators_Properties.
Require Import Inverse_Image.
Require Import Transitive_Closure.
Require Import List_Extension.
Import ListNotations.

Section Wf_Lexicographic_Exponentiation.
  Variable A : Set.
  Variable leA : A -> A -> Prop.

  Notation Power := (Pow A leA).
  Notation Lex_Exp := (lex_exp A leA).
  Notation ltl := (Ltl A leA).
  Notation Descl := (Desc A leA).

  Notation List := (list A).
  Notation "<< x , y >>" := (exist Descl x y) (at level 0, x, y at level 100).


  Lemma left_prefix : forall x y z : List, ltl (x ++ y) z -> ltl x z.

  Lemma right_prefix :
    forall x y z : List,
      ltl x (y ++ z) ->
      ltl x y \/ (exists y' : List, x = y ++ y' /\ ltl y' z).

  Lemma desc_prefix : forall (x : List) (a : A), Descl (x ++ [a]) -> Descl x.

  Lemma desc_hd l : Descl l -> Forall (fun b => clos_refl_trans A leA b (hd b l)) l.

  Lemma desc_tail :
    forall (x : List) (a b : A),
      Descl (b :: x ++ [a]) -> clos_refl_trans A leA a b.

  Lemma dist_Desc_concat :
    forall x y : List, Descl (x ++ y) -> Descl x /\ Descl y.

  Lemma desc_end :
    forall (a b : A) (x : List),
      Descl (x ++ [a]) /\ ltl (x ++ [a]) [b] -> clos_trans A leA a b.

  Lemma ltl_unit :
    forall (x : List) (a b : A),
      Descl (x ++ [a]) -> ltl (x ++ [a]) [b] -> ltl x [b].

  Lemma acc_app :
    forall (x1 x2 : List) (y1 : Descl (x1 ++ x2)),
      Acc Lex_Exp << x1 ++ x2, y1 >> ->
      forall (x : List) (y : Descl x),
        ltl x (x1 ++ x2) -> Acc Lex_Exp << x, y >>.

  Theorem wf_lex_exp : well_founded leA -> well_founded Lex_Exp.

End Wf_Lexicographic_Exponentiation.