author  lcp 
Fri, 28 Apr 1995 11:24:32 +0200  
changeset 1074  d60f203eeddf 
parent 760  f0200e91b272 
child 1461  6bcb44e4d6e5 
permissions  rwrr 
484  1 
(* Title: ZF/AC.ML 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1994 University of Cambridge 

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For AC.thy. The Axiom of Choice 

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*) 

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open AC; 

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(*The same as AC, but no premise a:A*) 

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val [nonempty] = goal AC.thy 

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"[ !!x. x:A ==> (EX y. y:B(x)) ] ==> EX z. z : Pi(A,B)"; 

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by (excluded_middle_tac "A=0" 1); 

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by (asm_simp_tac (ZF_ss addsimps [Pi_empty1]) 2 THEN fast_tac ZF_cs 2); 

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(*The nontrivial case*) 

1074
d60f203eeddf
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
760
diff
changeset

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by (fast_tac (eq_cs addIs [AC, nonempty]) 1); 
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qed "AC_Pi"; 
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(*Using dtac, this has the advantage of DELETING the universal quantifier*) 

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goal AC.thy "!!A B. ALL x:A. EX y. y:B(x) ==> EX y. y : Pi(A,B)"; 

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by (resolve_tac [AC_Pi] 1); 

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by (eresolve_tac [bspec] 1); 

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by (assume_tac 1); 

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qed "AC_ball_Pi"; 
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goal AC.thy "EX f. f: (PROD X: Pow(C){0}. X)"; 

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by (res_inst_tac [("B1", "%x.x")] (AC_Pi RS exE) 1); 

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by (etac exI 2); 

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by (fast_tac eq_cs 1); 

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qed "AC_Pi_Pow"; 
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val [nonempty] = goal AC.thy 

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"[ !!x. x:A ==> (EX y. y:x) \ 

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\ ] ==> EX f: A>Union(A). ALL x:A. f`x : x"; 

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by (res_inst_tac [("B1", "%x.x")] (AC_Pi RS exE) 1); 

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by (etac nonempty 1); 

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by (fast_tac (ZF_cs addDs [apply_type] addIs [Pi_type]) 1); 

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qed "AC_func"; 
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goal ZF.thy "!!x A. [ 0 ~: A; x: A ] ==> EX y. y:x"; 
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by (subgoal_tac "x ~= 0" 1); 

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by (ALLGOALS (fast_tac eq_cs)); 

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qed "non_empty_family"; 
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goal AC.thy "!!A. 0 ~: A ==> EX f: A>Union(A). ALL x:A. f`x : x"; 

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by (rtac AC_func 1); 

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by (REPEAT (ares_tac [non_empty_family] 1)); 

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qed "AC_func0"; 
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goal AC.thy "EX f: (Pow(C){0}) > C. ALL x:(Pow(C){0}). f`x : x"; 

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by (resolve_tac [AC_func0 RS bexE] 1); 

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by (rtac bexI 2); 

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by (assume_tac 2); 

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by (eresolve_tac [fun_weaken_type] 2); 

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by (ALLGOALS (fast_tac ZF_cs)); 

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qed "AC_func_Pow"; 
484  58 

1074
d60f203eeddf
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
760
diff
changeset

59 
goal AC.thy "!!A. 0 ~: A ==> EX f. f: (PROD x:A. x)"; 
d60f203eeddf
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
760
diff
changeset

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by (rtac AC_Pi 1); 
d60f203eeddf
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
760
diff
changeset

61 
by (REPEAT (ares_tac [non_empty_family] 1)); 
d60f203eeddf
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
760
diff
changeset

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qed "AC_Pi0"; 
d60f203eeddf
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
760
diff
changeset

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