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).

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.