Theory WO1_WO7

theory WO1_WO7
imports AC_Equiv
(*  Title:      ZF/AC/WO1_WO7.thy
Author: Lawrence C Paulson, CU Computer Laboratory
Copyright 1998 University of Cambridge

WO7 <-> LEMMA <-> WO1 (Rubin & Rubin p. 5)
LEMMA is the sentence denoted by (**)

Also, WO1 <-> WO8
*)


theory WO1_WO7
imports AC_Equiv
begin

definition
"LEMMA ==
∀X. ~Finite(X) --> (∃R. well_ord(X,R) & ~well_ord(X,converse(R)))"


(* ********************************************************************** *)
(* It is easy to see that WO7 is equivalent to (**) *)
(* ********************************************************************** *)

lemma WO7_iff_LEMMA: "WO7 <-> LEMMA"
apply (unfold WO7_def LEMMA_def)
apply (blast intro: Finite_well_ord_converse)
done

(* ********************************************************************** *)
(* It is also easy to show that LEMMA implies WO1. *)
(* ********************************************************************** *)

lemma LEMMA_imp_WO1: "LEMMA ==> WO1"
apply (unfold WO1_def LEMMA_def Finite_def eqpoll_def)
apply (blast intro!: well_ord_rvimage [OF bij_is_inj nat_implies_well_ord])
done

(* ********************************************************************** *)
(* The Rubins' proof of the other implication is contained within the *)
(* following sentence ∈ *)
(* "... each infinite ordinal is well ordered by < but not by >." *)
(* This statement can be proved by the following two theorems. *)
(* But moreover we need to show similar property for any well ordered *)
(* infinite set. It is not very difficult thanks to Isabelle order types *)
(* We show that if a set is well ordered by some relation and by its *)
(* converse, then apropriate order type is well ordered by the converse *)
(* of it's membership relation, which in connection with the previous *)
(* gives the conclusion. *)
(* ********************************************************************** *)

lemma converse_Memrel_not_wf_on:
"[| Ord(a); ~Finite(a) |] ==> ~wf[a](converse(Memrel(a)))"
apply (unfold wf_on_def wf_def)
apply (drule nat_le_infinite_Ord [THEN le_imp_subset], assumption)
apply (rule notI)
apply (erule_tac x = nat in allE, blast)
done

lemma converse_Memrel_not_well_ord:
"[| Ord(a); ~Finite(a) |] ==> ~well_ord(a,converse(Memrel(a)))"
apply (unfold well_ord_def)
apply (blast dest: converse_Memrel_not_wf_on)
done

lemma well_ord_rvimage_ordertype:
"well_ord(A,r) ==>
rvimage (ordertype(A,r), converse(ordermap(A,r)),r) =
Memrel(ordertype(A,r))"

by (blast intro: ordertype_ord_iso [THEN ord_iso_sym] ord_iso_rvimage_eq
Memrel_type [THEN subset_Int_iff [THEN iffD1]] trans)

lemma well_ord_converse_Memrel:
"[| well_ord(A,r); well_ord(A,converse(r)) |]
==> well_ord(ordertype(A,r), converse(Memrel(ordertype(A,r))))"

apply (subst well_ord_rvimage_ordertype [symmetric], assumption)
apply (rule rvimage_converse [THEN subst])
apply (blast intro: ordertype_ord_iso ord_iso_sym ord_iso_is_bij
bij_is_inj well_ord_rvimage)
done

lemma WO1_imp_LEMMA: "WO1 ==> LEMMA"
apply (unfold WO1_def LEMMA_def, clarify)
apply (blast dest: well_ord_converse_Memrel
Ord_ordertype [THEN converse_Memrel_not_well_ord]
intro: ordertype_ord_iso ord_iso_is_bij bij_is_inj lepoll_Finite
lepoll_def [THEN def_imp_iff, THEN iffD2] )
done

lemma WO1_iff_WO7: "WO1 <-> WO7"
apply (simp add: WO7_iff_LEMMA)
apply (blast intro: LEMMA_imp_WO1 WO1_imp_LEMMA)
done



(* ********************************************************************** *)
(* The proof of WO8 <-> WO1 (Rubin & Rubin p. 6) *)
(* ********************************************************************** *)

lemma WO1_WO8: "WO1 ==> WO8"
by (unfold WO1_def WO8_def, fast)


(* The implication "WO8 ==> WO1": a faithful image of Rubin & Rubin's proof*)
lemma WO8_WO1: "WO8 ==> WO1"
apply (unfold WO1_def WO8_def)
apply (rule allI)
apply (erule_tac x = "{{x}. x ∈ A}" in allE)
apply (erule impE)
apply (rule_tac x = "λa ∈ {{x}. x ∈ A}. THE x. a={x}" in exI)
apply (force intro!: lam_type simp add: singleton_eq_iff the_equality)
apply (blast intro: lam_sing_bij bij_is_inj well_ord_rvimage)
done

end