diff -r 000000000000 -r a5a9c433f639 src/ZF/ZF.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/ZF/ZF.thy Thu Sep 16 12:20:38 1993 +0200 @@ -0,0 +1,227 @@ +(* Title: ZF/zf.thy + ID: $Id$ + Author: Lawrence C Paulson and Martin D Coen, CU Computer Laboratory + Copyright 1993 University of Cambridge + +Zermelo-Fraenkel Set Theory +*) + +ZF = FOL + + +types + i, is, syntax 0 + +arities + i :: term + + +consts + + "0" :: "i" ("0") (*the empty set*) + Pow :: "i => i" (*power sets*) + Inf :: "i" (*infinite set*) + + (* Bounded Quantifiers *) + + "@Ball" :: "[idt, i, o] => o" ("(3ALL _:_./ _)" 10) + "@Bex" :: "[idt, i, o] => o" ("(3EX _:_./ _)" 10) + Ball :: "[i, i => o] => o" + Bex :: "[i, i => o] => o" + + (* General Union and Intersection *) + + "@INTER" :: "[idt, i, i] => i" ("(3INT _:_./ _)" 10) + "@UNION" :: "[idt, i, i] => i" ("(3UN _:_./ _)" 10) + Union, Inter :: "i => i" + + (* Variations on Replacement *) + + "@Replace" :: "[idt, idt, i, o] => i" ("(1{_ ./ _: _, _})") + "@RepFun" :: "[i, idt, i] => i" ("(1{_ ./ _: _})") + "@Collect" :: "[idt, i, o] => i" ("(1{_: _ ./ _})") + PrimReplace :: "[i, [i, i] => o] => i" + Replace :: "[i, [i, i] => o] => i" + RepFun :: "[i, i => i] => i" + Collect :: "[i, i => o] => i" + + (* Descriptions *) + + "@THE" :: "[idt, o] => i" ("(3THE _./ _)" 10) + The :: "[i => o] => i" + if :: "[o, i, i] => i" + + (* Enumerations of type i *) + + "" :: "i => is" ("_") + "@Enum" :: "[i, is] => is" ("_,/ _") + + (* Finite Sets *) + + "@Finset" :: "is => i" ("{(_)}") + Upair, cons :: "[i, i] => i" + succ :: "i => i" + + (* Ordered Pairing and n-Tuples *) + + "@Tuple" :: "[i, is] => i" ("<(_,/ _)>") + PAIR :: "syntax" + Pair :: "[i, i] => i" + fst, snd :: "i => i" + split :: "[[i,i] => i, i] => i" + fsplit :: "[[i,i] => o, i] => o" + + (* Sigma and Pi Operators *) + + "@PROD" :: "[idt, i, i] => i" ("(3PROD _:_./ _)" 10) + "@SUM" :: "[idt, i, i] => i" ("(3SUM _:_./ _)" 10) + "@lam" :: "[idt, i, i] => i" ("(3lam _:_./ _)" 10) + Pi, Sigma :: "[i, i => i] => i" + + (* Relations and Functions *) + + domain :: "i => i" + range :: "i => i" + field :: "i => i" + converse :: "i => i" + Lambda :: "[i, i => i] => i" + restrict :: "[i, i] => i" + + (* Infixes in order of decreasing precedence *) + + "``" :: "[i, i] => i" (infixl 90) (*image*) + "-``" :: "[i, i] => i" (infixl 90) (*inverse image*) + "`" :: "[i, i] => i" (infixl 90) (*function application*) + + (*Except for their translations, * and -> are right-associating infixes*) + " *" :: "[i, i] => i" ("(_ */ _)" [81, 80] 80) (*Cartesian product*) + "Int" :: "[i, i] => i" (infixl 70) (*binary intersection*) + "Un" :: "[i, i] => i" (infixl 65) (*binary union*) + "-" :: "[i, i] => i" (infixl 65) (*set difference*) + " ->" :: "[i, i] => i" ("(_ ->/ _)" [61, 60] 60) (*function space*) + "<=" :: "[i, i] => o" (infixl 50) (*subset relation*) + ":" :: "[i, i] => o" (infixl 50) (*membership relation*) + + +translations + "{x, xs}" == "cons(x, {xs})" + "{x}" == "cons(x, 0)" + + "PAIR(x, Pair(y, z))" <= "Pair(x, Pair(y, z))" + "PAIR(x, PAIR(y, z))" <= "Pair(x, PAIR(y, z))" + "" <= "PAIR(x, )" + "" == "Pair(x, )" + "" == "Pair(x, y)" + + "{x:A. P}" == "Collect(A, %x. P)" + "{y. x:A, Q}" == "Replace(A, %x y. Q)" + "{f. x:A}" == "RepFun(A, %x. f)" + "INT x:A. B" == "Inter({B. x:A})" + "UN x:A. B" == "Union({B. x:A})" + "PROD x:A. B" => "Pi(A, %x. B)" + "SUM x:A. B" => "Sigma(A, %x. B)" + "THE x. P" == "The(%x. P)" + "lam x:A. f" == "Lambda(A, %x. f)" + "ALL x:A. P" == "Ball(A, %x. P)" + "EX x:A. P" == "Bex(A, %x. P)" + + +rules + + (* Bounded Quantifiers *) +Ball_def "Ball(A,P) == ALL x. x:A --> P(x)" +Bex_def "Bex(A,P) == EX x. x:A & P(x)" +subset_def "A <= B == ALL x:A. x:B" + + (* ZF axioms -- see Suppes p.238 + Axioms for Union, Pow and Replace state existence only, + uniqueness is derivable using extensionality. *) + +extension "A = B <-> A <= B & B <= A" +union_iff "A : Union(C) <-> (EX B:C. A:B)" +power_set "A : Pow(B) <-> A <= B" +succ_def "succ(i) == cons(i,i)" + + (*We may name this set, though it is not uniquely defined. *) +infinity "0:Inf & (ALL y:Inf. succ(y): Inf)" + + (*This formulation facilitates case analysis on A. *) +foundation "A=0 | (EX x:A. ALL y:x. ~ y:A)" + + (* Schema axiom since predicate P is a higher-order variable *) +replacement "(ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==> \ +\ b : PrimReplace(A,P) <-> (EX x:A. P(x,b))" + + (* Derived form of replacement, restricting P to its functional part. + The resulting set (for functional P) is the same as with + PrimReplace, but the rules are simpler. *) +Replace_def "Replace(A,P) == PrimReplace(A, %x y. (EX!z.P(x,z)) & P(x,y))" + + (* Functional form of replacement -- analgous to ML's map functional *) +RepFun_def "RepFun(A,f) == {y . x:A, y=f(x)}" + + (* Separation and Pairing can be derived from the Replacement + and Powerset Axioms using the following definitions. *) + +Collect_def "Collect(A,P) == {y . x:A, x=y & P(x)}" + + (*Unordered pairs (Upair) express binary union/intersection and cons; + set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...) *) +Upair_def "Upair(a,b) == {y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}" +cons_def "cons(a,A) == Upair(a,a) Un A" + + (* Difference, general intersection, binary union and small intersection *) + +Diff_def "A - B == { x:A . ~(x:B) }" +Inter_def "Inter(A) == { x:Union(A) . ALL y:A. x:y}" +Un_def "A Un B == Union(Upair(A,B))" +Int_def "A Int B == Inter(Upair(A,B))" + + (* Definite descriptions -- via Replace over the set "1" *) + +the_def "The(P) == Union({y . x:{0}, P(y)})" +if_def "if(P,a,b) == THE z. P & z=a | ~P & z=b" + + (* Ordered pairs and disjoint union of a family of sets *) + + (* this "symmetric" definition works better than {{a}, {a,b}} *) +Pair_def " == {{a,a}, {a,b}}" +fst_def "fst == split(%x y.x)" +snd_def "snd == split(%x y.y)" +split_def "split(c,p) == THE y. EX a b. p= & y=c(a,b)" +fsplit_def "fsplit(R,z) == EX x y. z= & R(x,y)" +Sigma_def "Sigma(A,B) == UN x:A. UN y:B(x). {}" + + (* Operations on relations *) + +(*converse of relation r, inverse of function*) +converse_def "converse(r) == {z. w:r, EX x y. w= & z=}" + +domain_def "domain(r) == {x. w:r, EX y. w=}" +range_def "range(r) == domain(converse(r))" +field_def "field(r) == domain(r) Un range(r)" +image_def "r `` A == {y : range(r) . EX x:A. : r}" +vimage_def "r -`` A == converse(r)``A" + + (* Abstraction, application and Cartesian product of a family of sets *) + +lam_def "Lambda(A,b) == { . x:A}" +apply_def "f`a == THE y. : f" +Pi_def "Pi(A,B) == {f: Pow(Sigma(A,B)). ALL x:A. EX! y. : f}" + + (* Restrict the function f to the domain A *) +restrict_def "restrict(f,A) == lam x:A.f`x" + +end + + +ML + +(* 'Dependent' type operators *) + +val parse_translation = + [(" ->", ndependent_tr "Pi"), + (" *", ndependent_tr "Sigma")]; + +val print_translation = + [("Pi", dependent_tr' ("@PROD", " ->")), + ("Sigma", dependent_tr' ("@SUM", " *"))];