Inductive Ltl : List -> List -> Prop :=
    | Lt_nil (a:A) (x:List) : Ltl Nil (a :: x)
    | Lt_hd (a b:A) : leA a b -> forall x y:list A, Ltl (a :: x) (b :: y)
    | Lt_tl (a:A) (x y:List) : Ltl x y -> Ltl (a :: x) (a :: y).Inductive Ltl : List -> List -> Prop :=
    | Lt_nil (a:A) (x:List) : Ltl Nil (a :: x)
    | Lt_hd (a b:A) : leA a b -> forall x y:list A, Ltl (a :: x) (b :: y)
    | Lt_tl (a:A) (x y:List) : Ltl x y -> Ltl (a :: x) (a :: y).Inductive Ltl : List -> List -> Prop :=
    | Lt_nil (a:A) (x:List) : Ltl Nil (a :: x)
    | Lt_hd (a b:A) : leA a b -> forall x y:list A, Ltl (a :: x) (b :: y)
    | Lt_tl (a:A) (x y:List) : Ltl x y -> Ltl (a :: x) (a :: y).Inductive Ltl : List -> List -> Prop :=
    | Lt_nil (a:A) (x:List) : Ltl Nil (a :: x)
    | Lt_hd (a b:A) : leA a b -> forall x y:list A, Ltl (a :: x) (b :: y)
    | Lt_tl (a:A) (x y:List) : Ltl x y -> Ltl (a :: x) (a :: y).Inductive Desc : List -> Prop :=
    | d_nil : Desc Nil
    | d_one (x:A) : Desc (x :: Nil)
    | d_conc (x y:A) (l:List) :
        leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil).Inductive Desc : List -> Prop :=
    | d_nil : Desc Nil
    | d_one (x:A) : Desc (x :: Nil)
    | d_conc (x y:A) (l:List) :
        leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil).Inductive Desc : List -> Prop :=
    | d_nil : Desc Nil
    | d_one (x:A) : Desc (x :: Nil)
    | d_conc (x y:A) (l:List) :
        leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil).Inductive Desc : List -> Prop :=
    | d_nil : Desc Nil
    | d_one (x:A) : Desc (x :: Nil)
    | d_conc (x y:A) (l:List) :
        leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil).Definition Pow : Set := sig Desc.Definition lex_exp (a b:Pow) : Prop := Ltl (proj1_sig a) (proj1_sig b).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 Nil := (nil (A:=A)).(* useless but symmetric *)
  Notation Cons := (cons (A:=A)).Notation "<< x , y >>" := (exist Descl x y) (at level 0, x, y at level 100).

(* Specifications: *)
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 ++ Cons a Nil) -> Descl x.
Lemma desc_tail : forall (x:List) (a b:A), Descl (Cons b (x ++ Cons a Nil)) -> clos_trans A leA a b.
Lemma dist_aux : forall z:List, Descl z -> forall x y:List, z = x ++ y -> Descl x /\ Descl y.
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 ++ Cons a Nil) /\ ltl (x ++ Cons a Nil) (Cons b Nil) -> clos_trans A leA a b.
Lemma ltl_unit : forall (x:List) (a b:A), Descl (x ++ Cons a Nil) -> ltl (x ++ Cons a Nil) (Cons b Nil) -> ltl x (Cons b Nil).
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.