src/ZF/ZF.thy
 author clasohm Thu, 16 Sep 1993 12:20:38 +0200 changeset 0 a5a9c433f639 child 37 cebe01deba80 permissions -rw-r--r--
Initial revision
```
(*  Title:      ZF/zf.thy
ID:         \$Id\$
Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory

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))"
"<x, y, z>"           <= "PAIR(x, <y, z>)"
"<x, y, z>"           == "Pair(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)"
"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,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 parse_translation =
[(" ->", ndependent_tr "Pi"),
(" *", ndependent_tr "Sigma")];

val print_translation =
[("Pi", dependent_tr' ("@PROD", " ->")),
("Sigma", dependent_tr' ("@SUM", " *"))];
```