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).

(* 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 ++ [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.