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(* Title: ZF/zf.thy


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ID: $Id$


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Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory


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Copyright 1993 University of Cambridge


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ZermeloFraenkel Set Theory


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*)


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ZF = FOL +


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types

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i, is 0

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arities


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i :: term


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consts


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"0" :: "i" ("0") (*the empty set*)


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Pow :: "i => i" (*power sets*)


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Inf :: "i" (*infinite set*)


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(* Bounded Quantifiers *)


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"@Ball" :: "[idt, i, o] => o" ("(3ALL _:_./ _)" 10)


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"@Bex" :: "[idt, i, o] => o" ("(3EX _:_./ _)" 10)


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Ball :: "[i, i => o] => o"


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Bex :: "[i, i => o] => o"


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(* General Union and Intersection *)


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"@INTER" :: "[idt, i, i] => i" ("(3INT _:_./ _)" 10)


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"@UNION" :: "[idt, i, i] => i" ("(3UN _:_./ _)" 10)


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Union, Inter :: "i => i"


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(* Variations on Replacement *)


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"@Replace" :: "[idt, idt, i, o] => i" ("(1{_ ./ _: _, _})")


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"@RepFun" :: "[i, idt, i] => i" ("(1{_ ./ _: _})")


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"@Collect" :: "[idt, i, o] => i" ("(1{_: _ ./ _})")


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PrimReplace :: "[i, [i, i] => o] => i"


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Replace :: "[i, [i, i] => o] => i"


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RepFun :: "[i, i => i] => i"


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Collect :: "[i, i => o] => i"


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(* Descriptions *)


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The :: "(i => o) => i" (binder "THE " 10)

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if :: "[o, i, i] => i"


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(* Enumerations of type i *)


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"" :: "i => is" ("_")


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"@Enum" :: "[i, is] => is" ("_,/ _")


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(* Finite Sets *)


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"@Finset" :: "is => i" ("{(_)}")


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Upair, cons :: "[i, i] => i"


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succ :: "i => i"


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(* Ordered Pairing and nTuples *)


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"@Tuple" :: "[i, is] => i" ("<(_,/ _)>")


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Pair :: "[i, i] => i"


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fst, snd :: "i => i"


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split :: "[[i,i] => i, i] => i"


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fsplit :: "[[i,i] => o, i] => o"


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(* Sigma and Pi Operators *)


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"@PROD" :: "[idt, i, i] => i" ("(3PROD _:_./ _)" 10)


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"@SUM" :: "[idt, i, i] => i" ("(3SUM _:_./ _)" 10)


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"@lam" :: "[idt, i, i] => i" ("(3lam _:_./ _)" 10)


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Pi, Sigma :: "[i, i => i] => i"


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(* Relations and Functions *)


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domain :: "i => i"


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range :: "i => i"


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field :: "i => i"


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converse :: "i => i"


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Lambda :: "[i, i => i] => i"


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restrict :: "[i, i] => i"


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(* Infixes in order of decreasing precedence *)


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"``" :: "[i, i] => i" (infixl 90) (*image*)


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"``" :: "[i, i] => i" (infixl 90) (*inverse image*)


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"`" :: "[i, i] => i" (infixl 90) (*function application*)


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(*Except for their translations, * and > are rightassociating infixes*)


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" *" :: "[i, i] => i" ("(_ */ _)" [81, 80] 80) (*Cartesian product*)


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"Int" :: "[i, i] => i" (infixl 70) (*binary intersection*)


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"Un" :: "[i, i] => i" (infixl 65) (*binary union*)


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"" :: "[i, i] => i" (infixl 65) (*set difference*)


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" >" :: "[i, i] => i" ("(_ >/ _)" [61, 60] 60) (*function space*)


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"<=" :: "[i, i] => o" (infixl 50) (*subset relation*)


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":" :: "[i, i] => o" (infixl 50) (*membership relation*)

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"~:" :: "[i, i] => o" (infixl 50) (*negated membership relation*)

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translations


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"{x, xs}" == "cons(x, {xs})"


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"{x}" == "cons(x, 0)"

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"<x, y, z>" == "<x, <y, z>>"


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"<x, y>" == "Pair(x, y)"

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"{x:A. P}" == "Collect(A, %x. P)"


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"{y. x:A, Q}" == "Replace(A, %x y. Q)"


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"{f. x:A}" == "RepFun(A, %x. f)"


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"INT x:A. B" == "Inter({B. x:A})"


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"UN x:A. B" == "Union({B. x:A})"


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"PROD x:A. B" => "Pi(A, %x. B)"


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"SUM x:A. B" => "Sigma(A, %x. B)"

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"A > B" => "Pi(A, _K(B))"


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"A * B" => "Sigma(A, _K(B))"

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"lam x:A. f" == "Lambda(A, %x. f)"


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"ALL x:A. P" == "Ball(A, %x. P)"


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"EX x:A. P" == "Bex(A, %x. P)"

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"x ~: y" == "~ (x:y)"


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rules


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(* Bounded Quantifiers *)


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Ball_def "Ball(A,P) == ALL x. x:A > P(x)"


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Bex_def "Bex(A,P) == EX x. x:A & P(x)"


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subset_def "A <= B == ALL x:A. x:B"


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(* ZF axioms  see Suppes p.238


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Axioms for Union, Pow and Replace state existence only,


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uniqueness is derivable using extensionality. *)


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extension "A = B <> A <= B & B <= A"


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union_iff "A : Union(C) <> (EX B:C. A:B)"


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power_set "A : Pow(B) <> A <= B"


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succ_def "succ(i) == cons(i,i)"


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(*We may name this set, though it is not uniquely defined. *)


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infinity "0:Inf & (ALL y:Inf. succ(y): Inf)"


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(*This formulation facilitates case analysis on A. *)

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foundation "A=0  (EX x:A. ALL y:x. y~:A)"

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(* Schema axiom since predicate P is a higherorder variable *)


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replacement "(ALL x:A. ALL y z. P(x,y) & P(x,z) > y=z) ==> \


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\ b : PrimReplace(A,P) <> (EX x:A. P(x,b))"


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(* Derived form of replacement, restricting P to its functional part.


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The resulting set (for functional P) is the same as with


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PrimReplace, but the rules are simpler. *)


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Replace_def "Replace(A,P) == PrimReplace(A, %x y. (EX!z.P(x,z)) & P(x,y))"


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(* Functional form of replacement  analgous to ML's map functional *)


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RepFun_def "RepFun(A,f) == {y . x:A, y=f(x)}"


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(* Separation and Pairing can be derived from the Replacement


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and Powerset Axioms using the following definitions. *)


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Collect_def "Collect(A,P) == {y . x:A, x=y & P(x)}"


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(*Unordered pairs (Upair) express binary union/intersection and cons;


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set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...) *)


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Upair_def "Upair(a,b) == {y. x:Pow(Pow(0)), (x=0 & y=a)  (x=Pow(0) & y=b)}"


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cons_def "cons(a,A) == Upair(a,a) Un A"


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(* Difference, general intersection, binary union and small intersection *)


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Diff_def "A  B == { x:A . ~(x:B) }"


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Inter_def "Inter(A) == { x:Union(A) . ALL y:A. x:y}"


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Un_def "A Un B == Union(Upair(A,B))"


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Int_def "A Int B == Inter(Upair(A,B))"


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(* Definite descriptions  via Replace over the set "1" *)


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the_def "The(P) == Union({y . x:{0}, P(y)})"


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if_def "if(P,a,b) == THE z. P & z=a  ~P & z=b"


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(* Ordered pairs and disjoint union of a family of sets *)


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(* this "symmetric" definition works better than {{a}, {a,b}} *)


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Pair_def "<a,b> == {{a,a}, {a,b}}"


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fst_def "fst == split(%x y.x)"


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snd_def "snd == split(%x y.y)"


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split_def "split(c,p) == THE y. EX a b. p=<a,b> & y=c(a,b)"


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fsplit_def "fsplit(R,z) == EX x y. z=<x,y> & R(x,y)"


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Sigma_def "Sigma(A,B) == UN x:A. UN y:B(x). {<x,y>}"


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(* Operations on relations *)


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(*converse of relation r, inverse of function*)


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converse_def "converse(r) == {z. w:r, EX x y. w=<x,y> & z=<y,x>}"


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domain_def "domain(r) == {x. w:r, EX y. w=<x,y>}"


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range_def "range(r) == domain(converse(r))"


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field_def "field(r) == domain(r) Un range(r)"


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image_def "r `` A == {y : range(r) . EX x:A. <x,y> : r}"


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vimage_def "r `` A == converse(r)``A"


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(* Abstraction, application and Cartesian product of a family of sets *)


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lam_def "Lambda(A,b) == {<x,b(x)> . x:A}"


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apply_def "f`a == THE y. <a,y> : f"


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Pi_def "Pi(A,B) == {f: Pow(Sigma(A,B)). ALL x:A. EX! y. <x,y>: f}"


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(* Restrict the function f to the domain A *)


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restrict_def "restrict(f,A) == lam x:A.f`x"


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end


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ML


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(* 'Dependent' type operators *)


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val print_translation =


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[("Pi", dependent_tr' ("@PROD", " >")),


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("Sigma", dependent_tr' ("@SUM", " *"))];

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