(* Title: ZF/AC/AC_Equiv.thy
ID: $Id$
Author: Krzysztof Gr`abczewski
Axioms AC1 -- AC19 come from "Equivalents of the Axiom of Choice, II"
by H. Rubin and J.E. Rubin, 1985.
Axiom AC0 comes from "Axiomatic Set Theory" by P. Suppes, 1972.
Some Isabelle proofs of equivalences of these axioms are formalizations of
proofs presented by Rubin. The others are based on Rubin's proofs, but
slightly changed.
*)
AC_Equiv = CardinalArith + Univ + Transrec2 +
consts
(* Well Ordering Theorems *)
WO1, WO2, WO3, WO5, WO6, WO7, WO8 :: "o"
WO4 :: "i => o"
(* Axioms of Choice *)
AC0, AC1, AC2, AC3, AC4, AC5, AC6, AC7, AC8, AC9,
AC11, AC12, AC14, AC15, AC17, AC18, AC19 :: "o"
AC10, AC13 :: "i => o"
AC16 :: "[i, i] => o"
(* Auxiliary definitions used in theorems *)
first :: "[i, i, i] => o"
exists_first :: "[i, i] => o"
pairwise_disjoint :: "i => o"
sets_of_size_between :: "[i, i, i] => o"
(* Auxiliary definitions used in proofs *)
NN :: "i => i"
uu :: "[i, i, i, i] => i"
(* Other useful definitions *)
vv1 :: "[i, i, i] => i"
ww1 :: "[i, i, i] => i"
vv2 :: "[i, i, i, i] => i"
ww2 :: "[i, i, i, i] => i"
GG :: "[i, i, i] => i"
GG2 :: "[i, i, i] => i"
HH :: "[i, i, i] => i"
recfunAC16 :: "[i, i, i, i] => i"
defs
(* Well Ordering Theorems *)
WO1_def "WO1 == ALL A. EX R. well_ord(A,R)"
WO2_def "WO2 == ALL A. EX a. Ord(a) & A eqpoll a"
WO3_def "WO3 == ALL A. EX a. Ord(a) & (EX b. b <= a & A eqpoll b)"
WO4_def "WO4(m) == ALL A. EX a f. Ord(a) & domain(f)=a & \
\ (UN b<a. f`b) = A & (ALL b<a. f`b lepoll m)"
WO5_def "WO5 == EX m:nat. 1 le m & WO4(m)"
WO6_def "WO6 == ALL A. EX m:nat. 1 le m & (EX a f. Ord(a) & domain(f)=a \
\ & (UN b<a. f`b) = A & (ALL b<a. f`b lepoll m))"
WO7_def "WO7 == ALL A. Finite(A) <-> (ALL R. well_ord(A,R) --> \
\ well_ord(A,converse(R)))"
WO8_def "WO8 == ALL A. (0~:A --> (EX f. f : (PROD X:A. X))) --> \
\ (EX R. well_ord(A,R))"
(* Axioms of Choice *)
AC0_def "AC0 == ALL A. EX f. f:(PROD X:Pow(A)-{0}. X)"
AC1_def "AC1 == ALL A. 0~:A --> (EX f. f:(PROD X:A. X))"
AC2_def "AC2 == ALL A. 0~:A & pairwise_disjoint(A) \
\ --> (EX C. ALL B:A. EX y. B Int C = {y})"
AC3_def "AC3 == ALL A B. ALL f:A->B. EX g. g:(PROD x:{a:A. f`a~=0}. f`x)"
AC4_def "AC4 == ALL R A B. (R<=A*B --> (EX f. f:(PROD x:domain(R). R``{x})))"
AC5_def "AC5 == ALL A B. ALL f:A->B. EX g:range(f)->A. \
\ ALL x:domain(g). f`(g`x) = x"
AC6_def "AC6 == ALL A. 0~:A --> (PROD B:A. B)~=0"
AC7_def "AC7 == ALL A. 0~:A & (ALL B1:A. ALL B2:A. B1 eqpoll B2) \
\ --> (PROD B:A. B)~=0"
AC8_def "AC8 == ALL A. (ALL B:A. EX B1 B2. B=<B1,B2> & B1 eqpoll B2) \
\ --> (EX f. ALL B:A. f`B : bij(fst(B),snd(B)))"
AC9_def "AC9 == ALL A. (ALL B1:A. ALL B2:A. B1 eqpoll B2) --> \
\ (EX f. ALL B1:A. ALL B2:A. f`<B1,B2> : bij(B1,B2))"
AC10_def "AC10(n) == ALL A. (ALL B:A. ~Finite(B)) --> \
\ (EX f. ALL B:A. (pairwise_disjoint(f`B) & \
\ sets_of_size_between(f`B, 2, succ(n)) & Union(f`B)=B))"
AC11_def "AC11 == EX n:nat. 1 le n & AC10(n)"
AC12_def "AC12 == ALL A. (ALL B:A. ~Finite(B)) --> \
\ (EX n:nat. 1 le n & (EX f. ALL B:A. (pairwise_disjoint(f`B) & \
\ sets_of_size_between(f`B, 2, succ(n)) & Union(f`B)=B)))"
AC13_def "AC13(m) == ALL A. 0~:A --> (EX f. ALL B:A. f`B~=0 & \
\ f`B <= B & f`B lepoll m)"
AC14_def "AC14 == EX m:nat. 1 le m & AC13(m)"
AC15_def "AC15 == ALL A. 0~:A --> (EX m:nat. 1 le m & (EX f. ALL B:A. \
\ f`B~=0 & f`B <= B & f`B lepoll m))"
AC16_def "AC16(n, k) == ALL A. ~Finite(A) --> \
\ (EX T. T <= {X:Pow(A). X eqpoll succ(n)} & \
\ (ALL X:{X:Pow(A). X eqpoll succ(k)}. EX! Y. Y:T & X <= Y))"
AC17_def "AC17 == ALL A. ALL g: (Pow(A)-{0} -> A) -> Pow(A)-{0}. \
\ EX f: Pow(A)-{0} -> A. f`(g`f) : g`f"
(***problems! X is free, and is higher-order!
AC18_def "AC18 == ALL A. A~=0 --> (ALL F. (domain(F) = A & \
\ (ALL a:A. F`a ~= 0)) --> \
\ ((INT a:A. UN b:F`a. X(a,b)) = \
\ (UN f: PROD a:A. F`a. INT a:A. X(a, f`a))))"
***)
AC19_def "AC19 == ALL A. A~=0 & 0~:A --> ((INT a:A. UN b:a. b) = \
\ (UN f:{f: A->Union(A). ALL B:A. f`B:B}. INT a:A. f`a))"
(* Auxiliary definitions used in theorems *)
first_def "first(u, X, R) \
\ == u:X & (ALL v:X. v~=u --> <u,v> : R)"
exists_first_def "exists_first(X,R) \
\ == EX u:X. first(u, X, R)"
pairwise_disjoint_def "pairwise_disjoint(A) \
\ == ALL A1:A. ALL A2:A. A1 Int A2 ~= 0 --> A1=A2"
sets_of_size_between_def "sets_of_size_between(A,m,n) \
\ == ALL B:A. m lepoll B & B lepoll n"
(* Auxiliary definitions used in proofs *)
NN_def "NN(y) == {m:nat. EX a. EX f. Ord(a) & domain(f)=a \
\ & (UN b<a. f`b) = y & (ALL b<a. f`b lepoll m)}"
uu_def "uu(f, beta, gamma, delta) \
\ == (f`beta * f`gamma) Int f`delta"
(* Other useful definitions *)
vv1_def "vv1(f,b,m) == if(f`b ~= 0, \
\ domain(uu(f,b, \
\ LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & \
\ domain(uu(f,b,g,d)) lesspoll m)), \
\ LEAST d. domain(uu(f,b, \
\ LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & \
\ domain(uu(f,b,g,d)) lesspoll m)), d)) ~= 0 & \
\ domain(uu(f,b, \
\ LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & \
\ domain(uu(f,b,g,d)) lesspoll m)), d)) lesspoll m)), 0)"
ww1_def "ww1(f,b,m) == f`b - vv1(f,b,m)"
vv2_def "vv2(f,b,g,s) == \
\ if(f`g ~= 0, {uu(f,b,g,LEAST d. uu(f,b,g,d) ~= 0)`s}, 0)"
ww2_def "ww2(f,b,g,s) == f`g - vv2(f,b,g,s)"
GG_def "GG(f,x,a) == transrec(a, %b r. (lam z:Pow(x). \
\ if(z=0, x, f`z))`(x - {r`c. c:b}))"
GG2_def "GG2(f,x,a) == transrec(a, %b r. (lam z:Pow(x). \
\ if(z=0, {x}, f`z))`(x - Union({r`c. c:b})))"
HH_def "HH(f,x,a) == transrec(a, %b r. (lam z:Pow(x). \
\ if(z=0|f`z~:z, x, f`z))`(x - {r`c. c:b}))"
recfunAC16_def
"recfunAC16(f,fa,i,a) == \
\ transrec2(i, 0, \
\ %g r. if(EX y:r. fa`g <= y, r, \
\ r Un {f`(LEAST i. fa`g <= f`i & \
\ (ALL b<a. (fa`b <= f`i --> (ALL t:r. ~ fa`b <= t))))}))"
end