# 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 Operators_Properties.
Require Import Transitive_Closure.

Import ListNotations.

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).

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_tail :
forall (x : List) (a b : A),
Descl (b :: x ++ [a]) -> clos_refl_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 ++ [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.