(* 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 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 :: "(i => o) => i" (binder "THE " 10)
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 :: "[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 and ~: left associative 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*)
"~:" :: "[i, i] => o" ("(_ ~:/ _)" [50, 51] 50) (*negated membership relation*)
translations
"{x, xs}" == "cons(x, {xs})"
"{x}" == "cons(x, 0)"
"<x, y, z>" == "<x, <y, z>>"
"<x, y>" == "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)"
"A -> B" => "Pi(A, _K(B))"
"A * B" => "Sigma(A, _K(B))"
"lam x:A. f" == "Lambda(A, %x. f)"
"ALL x:A. P" == "Ball(A, %x. P)"
"EX x:A. P" == "Bex(A, %x. P)"
"x ~: y" == "~ (x : y)"
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,b> == {{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=<a,b> & y=c(a,b)"
fsplit_def "fsplit(R,z) == EX x y. z=<x,y> & R(x,y)"
Sigma_def "Sigma(A,B) == UN x:A. UN y:B(x). {<x,y>}"
(* Operations on relations *)
(*converse of relation r, inverse of function*)
converse_def "converse(r) == {z. w:r, EX x y. w=<x,y> & z=<y,x>}"
domain_def "domain(r) == {x. w:r, EX y. w=<x,y>}"
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. <x,y> : r}"
vimage_def "r -`` A == converse(r)``A"
(* Abstraction, application and Cartesian product of a family of sets *)
lam_def "Lambda(A,b) == {<x,b(x)> . x:A}"
apply_def "f`a == THE y. <a,y> : f"
Pi_def "Pi(A,B) == {f: Pow(Sigma(A,B)). ALL x:A. EX! y. <x,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 print_translation =
[("Pi", dependent_tr' ("@PROD", " ->")),
("Sigma", dependent_tr' ("@SUM", " *"))];