# Theory AC

Up to index of Isabelle/ZF

theory AC
imports Main
begin

    (*  Title:      ZF/AC.thy
        ID:         $Id: AC.thy,v 1.10 2005/06/17 14:15:09 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1994  University of Cambridge

    *)

    header{*The Axiom of Choice*}

    theory AC imports Main begin

    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 ∈ 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 ∈ (Π 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->Union(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->Union(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 ∈ (Π x ∈ A. x)"
    apply (rule AC_Pi)
    apply (simp_all add: non_empty_family)
    done

    end

lemma AC_Pi:

      (!!x. x ∈ A ==> ∃y. y ∈ B(x)) ==> ∃z. z ∈ Pi(A, B)

lemma AC_ball_Pi:

      ∀x∈A. ∃y. y ∈ B(x) ==> ∃y. y ∈ Pi(A, B)

lemma AC_Pi_Pow:

      ∃f. f ∈ (ΠX∈Pow(C) - {0}. X)

lemma AC_func:

      (!!x. x ∈ A ==> ∃y. y ∈ x) ==> ∃f∈A -> \<Union>A. ∀x∈A. f ` x ∈ x

lemma non_empty_family:

      [| 0 ∉ A; x ∈ A |] ==> ∃y. y ∈ x

lemma AC_func0:

      0 ∉ A ==> ∃f∈A -> \<Union>A. ∀x∈A. f ` x ∈ x

lemma AC_func_Pow:

      ∃f∈Pow(C) - {0} -> C. ∀x∈Pow(C) - {0}. f ` x ∈ x

lemma AC_Pi0:

      0 ∉ A ==> ∃f. f ∈ (Πx∈A. x)