src/ZF/AC.ML

-(*  Title: 	ZF/AC.ML
+(*  Title:      ZF/AC.ML
     ID:         $Id$
-    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1994  University of Cambridge
 
 For AC.thy.  The Axiom of Choice
 *)
 

 by (fast_tac (eq_cs addIs [AC, nonempty]) 1);
 qed "AC_Pi";
 
 (*Using dtac, this has the advantage of DELETING the universal quantifier*)
 goal AC.thy "!!A B. ALL x:A. EX y. y:B(x) ==> EX y. y : Pi(A,B)";
-by (resolve_tac [AC_Pi] 1);
-by (eresolve_tac [bspec] 1);
+by (rtac AC_Pi 1);
+by (etac bspec 1);
 by (assume_tac 1);
 qed "AC_ball_Pi";
 
 goal AC.thy "EX f. f: (PROD X: Pow(C)-{0}. X)";
 by (res_inst_tac [("B1", "%x.x")] (AC_Pi RS exE) 1);
 by (etac exI 2);
 by (fast_tac eq_cs 1);
 qed "AC_Pi_Pow";
 
 val [nonempty] = goal AC.thy
-     "[| !!x. x:A ==> (EX y. y:x)	\
+     "[| !!x. x:A ==> (EX y. y:x)       \
 \     |] ==> EX f: A->Union(A). ALL x:A. f`x : x";
 by (res_inst_tac [("B1", "%x.x")] (AC_Pi RS exE) 1);
 by (etac nonempty 1);
 by (fast_tac (ZF_cs addDs [apply_type] addIs [Pi_type]) 1);
 qed "AC_func";

 
 goal AC.thy "EX f: (Pow(C)-{0}) -> C. ALL x:(Pow(C)-{0}). f`x : x";
 by (resolve_tac [AC_func0 RS bexE] 1);
 by (rtac bexI 2);
 by (assume_tac 2);
-by (eresolve_tac [fun_weaken_type] 2);
+by (etac fun_weaken_type 2);
 by (ALLGOALS (fast_tac ZF_cs));
 qed "AC_func_Pow";
 
 goal AC.thy "!!A. 0 ~: A ==> EX f. f: (PROD x:A. x)";
 by (rtac AC_Pi 1);