(* Title: ZF/AC/DC.thy
ID: $Id$
Author: Krzysztof Grabczewski
Theory file for the proofs concernind the Axiom of Dependent Choice
*)
DC = AC_Equiv + Hartog + Cardinal_aux + DC_lemmas +
consts
DC :: i => o
DC0 :: o
ff :: [i, i, i, i] => i
rules
DC_def "DC(a) ==
ALL X R. R<=Pow(X)*X &
(ALL Y:Pow(X). Y lesspoll a --> (EX x:X. <Y,x> : R))
--> (EX f:a->X. ALL b<a. <f``b,f`b> : R)"
DC0_def "DC0 == ALL A B R. R <= A*B & R~=0 & range(R) <= domain(R)
--> (EX f:nat->domain(R). ALL n:nat. <f`n,f`succ(n)>:R)"
ff_def "ff(b, X, Q, R) ==
transrec(b, %c r. THE x. first(x, {x:X. <r``c, x> : R}, Q))"
locale DC0_imp =
fixes
XX :: i
RR :: i
X :: i
R :: i
assumes
all_ex "ALL Y:Pow(X). Y lesspoll nat --> (EX x:X. <Y, x> : R)"
defines
XX_def "XX == (UN n:nat. {f:n->X. ALL k:n. <f``k, f`k> : R})"
RR_def "RR == {<z1,z2>:XX*XX. domain(z2)=succ(domain(z1))
& restrict(z2, domain(z1)) = z1}"
locale imp_DC0 =
fixes
XX :: i
RR :: i
x :: i
R :: i
f :: i
allRR :: o
assumes
defines
XX_def "XX == (UN n:nat.
{f:succ(n)->domain(R). ALL k:n. <f`k, f`succ(k)> : R})"
RR_def
"RR == {<z1,z2>:Fin(XX)*XX.
(domain(z2)=succ(UN f:z1. domain(f))
& (ALL f:z1. restrict(z2, domain(f)) = f))
| (~ (EX g:XX. domain(g)=succ(UN f:z1. domain(f))
& (ALL f:z1. restrict(g, domain(f)) = f)) & z2={<0,x>})}"
allRR_def
"allRR == ALL b<nat.
<f``b, f`b> :
{<z1,z2>:Fin(XX)*XX. (domain(z2)=succ(UN f:z1. domain(f))
& (UN f:z1. domain(f)) = b
& (ALL f:z1. restrict(z2,domain(f)) = f))}"
end