author paulson
Tue, 09 Jul 2002 23:05:26 +0200
changeset 13328 703de709a64b
parent 13269 3ba9be497c33
child 14171 0cab06e3bbd0
permissions -rw-r--r--
better document preparation

(*  Title:      ZF/AC.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge


header{*The Axiom of Choice*}

theory AC = Main:

text{*This definition comes from Halmos (1960), page 59.*}
axioms AC: "[| a: A;  !!x. x:A ==> (EX y. y:B(x)) |] ==> EX z. z : Pi(A,B)"

(*The same as AC, but no premise a \<in> A*)
lemma AC_Pi: "[| !!x. x \<in> A ==> (\<exists>y. y \<in> B(x)) |] ==> \<exists>z. z \<in> Pi(A,B)"
apply (case_tac "A=0")
apply (simp add: Pi_empty1)
(*The non-trivial case*)
apply (blast intro: AC)

(*Using dtac, this has the advantage of DELETING the universal quantifier*)
lemma AC_ball_Pi: "\<forall>x \<in> A. \<exists>y. y \<in> B(x) ==> \<exists>y. y \<in> Pi(A,B)"
apply (rule AC_Pi)
apply (erule bspec, assumption)

lemma AC_Pi_Pow: "\<exists>f. f \<in> (\<Pi>X \<in> Pow(C)-{0}. X)"
apply (rule_tac B1 = "%x. x" in AC_Pi [THEN exE])
apply (erule_tac [2] exI, blast)

lemma AC_func:
     "[| !!x. x \<in> A ==> (\<exists>y. y \<in> x) |] ==> \<exists>f \<in> A->Union(A). \<forall>x \<in> A. f`x \<in> x"
apply (rule_tac B1 = "%x. x" in AC_Pi [THEN exE])
prefer 2 apply (blast dest: apply_type intro: Pi_type, blast) 

lemma non_empty_family: "[| 0 \<notin> A;  x \<in> A |] ==> \<exists>y. y \<in> x"
by (subgoal_tac "x \<noteq> 0", blast+)

lemma AC_func0: "0 \<notin> A ==> \<exists>f \<in> A->Union(A). \<forall>x \<in> A. f`x \<in> x"
apply (rule AC_func)
apply (simp_all add: non_empty_family) 

lemma AC_func_Pow: "\<exists>f \<in> (Pow(C)-{0}) -> C. \<forall>x \<in> Pow(C)-{0}. f`x \<in> x"
apply (rule AC_func0 [THEN bexE])
apply (rule_tac [2] bexI)
prefer 2 apply assumption
apply (erule_tac [2] fun_weaken_type, blast+)

lemma AC_Pi0: "0 \<notin> A ==> \<exists>f. f \<in> (\<Pi>x \<in> A. x)"
apply (rule AC_Pi)
apply (simp_all add: non_empty_family)