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(*         *      The Rocq Prover / The Rocq Development Team           *)
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(** Author: Cristina Cornes  From : Constructing Recursion Operators in Type Theory L. Paulson  JSC (1986) 2, 325-355  *)
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).
(* Hint Resolve d_one d_nil t_step. *)

(* 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.
End Wf_Lexicographic_Exponentiation.