# Library Coq.Wellfounded.Lexicographic_Exponentiation

Author: Cristina Cornes
From : Constructing Recursion Operators in Type Theory L. Paulson JSC (1986) 2, 325-355
Require Import List.
Require Import Relation_Operators.
Require Import Transitive_Closure.
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 Nil := (nil (A:=A)).
Notation Cons := (cons (A:=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 ++ 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.
End Wf_Lexicographic_Exponentiation.