Theory AC

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

section‹The Axiom of Choice›

theory AC imports ZF begin

text‹This definition comes from Halmos (1960), page 59.›
axiomatization where
  AC: "[| a ∈ A;  !!x. x ∈ A ==> (∃y. y ∈ B(x)) |] ==> ∃z. z ∈ Pi(A,B)"

(*The same as AC, but no premise @{term"a ∈ A"}*)
lemma AC_Pi: "[| !!x. x ∈ A ==> (∃y. y ∈ B(x)) |] ==> ∃z. z ∈ Pi(A,B)"
apply (case_tac "A=0")
apply (simp add: Pi_empty1)
(*The non-trivial case*)
apply (blast intro: AC)
done

(*Using dtac, this has the advantage of DELETING the universal quantifier*)
lemma AC_ball_Pi: "∀x ∈ A. ∃y. y ∈ B(x) ==> ∃y. y ∈ Pi(A,B)"
apply (rule AC_Pi)
apply (erule bspec, assumption)
done

lemma AC_Pi_Pow: "∃f. f ∈ (∏X ∈ Pow(C)-{0}. X)"
apply (rule_tac B1 = "%x. x" in AC_Pi [THEN exE])
apply (erule_tac [2] exI, blast)
done

lemma AC_func:
     "[| !!x. x ∈ A ==> (∃y. y ∈ x) |] ==> ∃f ∈ A->⋃(A). ∀x ∈ A. f`x ∈ x"
apply (rule_tac B1 = "%x. x" in AC_Pi [THEN exE])
prefer 2 apply (blast dest: apply_type intro: Pi_type, blast)
done

lemma non_empty_family: "[| 0 ∉ A;  x ∈ A |] ==> ∃y. y ∈ x"
by (subgoal_tac "x ≠ 0", blast+)

lemma AC_func0: "0 ∉ A ==> ∃f ∈ A->⋃(A). ∀x ∈ A. f`x ∈ x"
apply (rule AC_func)
apply (simp_all add: non_empty_family)
done

lemma AC_func_Pow: "∃f ∈ (Pow(C)-{0}) -> C. ∀x ∈ Pow(C)-{0}. f`x ∈ x"
apply (rule AC_func0 [THEN bexE])
apply (rule_tac [2] bexI)
prefer 2 apply assumption
apply (erule_tac [2] fun_weaken_type, blast+)
done

lemma AC_Pi0: "0 ∉ A ==> ∃f. f ∈ (∏x ∈ A. x)"
apply (rule AC_Pi)
apply (simp_all add: non_empty_family)
done

end