src/ZF/AC/AC0_AC1.ML

result() -> qed; Step_tac -> Safe_tac

(*  Title:      ZF/AC/AC0_AC1.ML
    ID:         $Id$
    Author:     Krzysztof Grabczewski

AC0 is equivalent to AC1
AC0 comes from Suppes, AC1 from Rubin & Rubin
*)

goal thy "!!A. 0~:A ==> A <= Pow(Union(A))-{0}";
by (Fast_tac 1);
qed "subset_Pow_Union";

goal thy "!!f. [| f:(PROD X:A. X); D<=A |] ==> EX g. g:(PROD X:D. X)";
by (fast_tac (!claset addSIs [restrict_type, apply_type]) 1);
val lemma1 = result();

goalw thy AC_defs "!!Z. AC0 ==> AC1"; 
by (fast_tac (FOL_cs addSEs [lemma1, subset_Pow_Union]) 1);
qed "AC0_AC1";

goalw thy AC_defs "!!Z. AC1 ==> AC0";
by (Deepen_tac 0 1);
(*Large search space.  Faster proof by
  by (fast_tac (!claset addSIs [notI, singletonI] addSEs [notE, DiffE]) 1);
*)
qed "AC1_AC0";