src/ZF/ZF.thy
 author clasohm Tue Feb 06 12:27:17 1996 +0100 (1996-02-06) changeset 1478 2b8c2a7547ab parent 1401 0c439768f45c child 2286 c2f76a5bad65 permissions -rw-r--r--
expanded tabs
1 (*  Title:      ZF/ZF.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
4     Copyright   1993  University of Cambridge
6 Zermelo-Fraenkel Set Theory
7 *)
9 ZF = FOL + Let +
11 types
12   i
14 arities
15   i :: term
17 consts
19   "0"         :: i                  ("0")   (*the empty set*)
20   Pow         :: i => i                     (*power sets*)
21   Inf         :: i                          (*infinite set*)
23   (* Bounded Quantifiers *)
25   Ball, Bex   :: [i, i => o] => o
27   (* General Union and Intersection *)
29   Union,Inter :: i => i
31   (* Variations on Replacement *)
33   PrimReplace :: [i, [i, i] => o] => i
34   Replace     :: [i, [i, i] => o] => i
35   RepFun      :: [i, i => i] => i
36   Collect     :: [i, i => o] => i
38   (* Descriptions *)
40   The         :: (i => o) => i      (binder "THE " 10)
41   if          :: [o, i, i] => i
43   (* Finite Sets *)
45   Upair, cons :: [i, i] => i
46   succ        :: i => i
48   (* Ordered Pairing *)
50   Pair        :: [i, i] => i
51   fst, snd    :: i => i
52   split       :: [[i, i] => 'a, i] => 'a::logic  (*for pattern-matching*)
54   (* Sigma and Pi Operators *)
56   Sigma, Pi   :: [i, i => i] => i
58   (* Relations and Functions *)
60   domain      :: i => i
61   range       :: i => i
62   field       :: i => i
63   converse    :: i => i
64   function    :: i => o         (*is a relation a function?*)
65   Lambda      :: [i, i => i] => i
66   restrict    :: [i, i] => i
68   (* Infixes in order of decreasing precedence *)
70   "``"        :: [i, i] => i    (infixl 90) (*image*)
71   "-``"       :: [i, i] => i    (infixl 90) (*inverse image*)
72   "`"         :: [i, i] => i    (infixl 90) (*function application*)
73 (*"*"         :: [i, i] => i    (infixr 80) (*Cartesian product*)*)
74   "Int"       :: [i, i] => i    (infixl 70) (*binary intersection*)
75   "Un"        :: [i, i] => i    (infixl 65) (*binary union*)
76   "-"         :: [i, i] => i    (infixl 65) (*set difference*)
77 (*"->"        :: [i, i] => i    (infixr 60) (*function space*)*)
78   "<="        :: [i, i] => o    (infixl 50) (*subset relation*)
79   ":"         :: [i, i] => o    (infixl 50) (*membership relation*)
80 (*"~:"        :: [i, i] => o    (infixl 50) (*negated membership relation*)*)
83 types
84   is
85   pttrns
87 syntax
88   ""          :: i => is                   ("_")
89   "@Enum"     :: [i, is] => is             ("_,/ _")
90   "~:"        :: [i, i] => o               (infixl 50)
91   "@Finset"   :: is => i                   ("{(_)}")
92   "@Tuple"    :: [i, is] => i              ("<(_,/ _)>")
93   "@Collect"  :: [pttrn, i, o] => i        ("(1{_: _ ./ _})")
94   "@Replace"  :: [pttrn, pttrn, i, o] => i ("(1{_ ./ _: _, _})")
95   "@RepFun"   :: [i, pttrn, i] => i        ("(1{_ ./ _: _})" [51,0,51])
96   "@INTER"    :: [pttrn, i, i] => i        ("(3INT _:_./ _)" 10)
97   "@UNION"    :: [pttrn, i, i] => i        ("(3UN _:_./ _)" 10)
98   "@PROD"     :: [pttrn, i, i] => i        ("(3PROD _:_./ _)" 10)
99   "@SUM"      :: [pttrn, i, i] => i        ("(3SUM _:_./ _)" 10)
100   "->"        :: [i, i] => i               (infixr 60)
101   "*"         :: [i, i] => i               (infixr 80)
102   "@lam"      :: [pttrn, i, i] => i        ("(3lam _:_./ _)" 10)
103   "@Ball"     :: [pttrn, i, o] => o        ("(3ALL _:_./ _)" 10)
104   "@Bex"      :: [pttrn, i, o] => o        ("(3EX _:_./ _)" 10)
106   (** Patterns -- extends pre-defined type "pttrn" used in abstractions **)
108   "@pttrn"  :: pttrns => pttrn            ("<_>")
109   ""        ::  pttrn           => pttrns ("_")
110   "@pttrns" :: [pttrn,pttrns]   => pttrns ("_,/_")
112 translations
113   "x ~: y"      == "~ (x : y)"
114   "{x, xs}"     == "cons(x, {xs})"
115   "{x}"         == "cons(x, 0)"
116   "{x:A. P}"    == "Collect(A, %x. P)"
117   "{y. x:A, Q}" == "Replace(A, %x y. Q)"
118   "{b. x:A}"    == "RepFun(A, %x. b)"
119   "INT x:A. B"  == "Inter({B. x:A})"
120   "UN x:A. B"   == "Union({B. x:A})"
121   "PROD x:A. B" => "Pi(A, %x. B)"
122   "SUM x:A. B"  => "Sigma(A, %x. B)"
123   "A -> B"      => "Pi(A, _K(B))"
124   "A * B"       => "Sigma(A, _K(B))"
125   "lam x:A. f"  == "Lambda(A, %x. f)"
126   "ALL x:A. P"  == "Ball(A, %x. P)"
127   "EX x:A. P"   == "Bex(A, %x. P)"
129   "<x, y, z>"   == "<x, <y, z>>"
130   "<x, y>"      == "Pair(x, y)"
131   "%<x,y,zs>.b"   == "split(%x <y,zs>.b)"
132   "%<x,y>.b"      == "split(%x y.b)"
133 (* The <= direction fails if split has more than one argument because
134    ast-matching fails.  Otherwise it would work fine *)
136 defs
138   (* Bounded Quantifiers *)
139   Ball_def      "Ball(A, P) == ALL x. x:A --> P(x)"
140   Bex_def       "Bex(A, P) == EX x. x:A & P(x)"
142   subset_def    "A <= B == ALL x:A. x:B"
143   succ_def      "succ(i) == cons(i, i)"
145 rules
147   (* ZF axioms -- see Suppes p.238
148      Axioms for Union, Pow and Replace state existence only,
149      uniqueness is derivable using extensionality. *)
151   extension     "A = B <-> A <= B & B <= A"
152   Union_iff     "A : Union(C) <-> (EX B:C. A:B)"
153   Pow_iff       "A : Pow(B) <-> A <= B"
155   (*We may name this set, though it is not uniquely defined.*)
156   infinity      "0:Inf & (ALL y:Inf. succ(y): Inf)"
158   (*This formulation facilitates case analysis on A.*)
159   foundation    "A=0 | (EX x:A. ALL y:x. y~:A)"
161   (*Schema axiom since predicate P is a higher-order variable*)
162   replacement   "(ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==>
163                          b : PrimReplace(A,P) <-> (EX x:A. P(x,b))"
165 defs
167   (* Derived form of replacement, restricting P to its functional part.
168      The resulting set (for functional P) is the same as with
169      PrimReplace, but the rules are simpler. *)
171   Replace_def   "Replace(A,P) == PrimReplace(A, %x y. (EX!z.P(x,z)) & P(x,y))"
173   (* Functional form of replacement -- analgous to ML's map functional *)
175   RepFun_def    "RepFun(A,f) == {y . x:A, y=f(x)}"
177   (* Separation and Pairing can be derived from the Replacement
178      and Powerset Axioms using the following definitions. *)
180   Collect_def   "Collect(A,P) == {y . x:A, x=y & P(x)}"
182   (*Unordered pairs (Upair) express binary union/intersection and cons;
183     set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*)
185   Upair_def   "Upair(a,b) == {y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
186   cons_def    "cons(a,A) == Upair(a,a) Un A"
188   (* Difference, general intersection, binary union and small intersection *)
190   Diff_def      "A - B    == { x:A . ~(x:B) }"
191   Inter_def     "Inter(A) == { x:Union(A) . ALL y:A. x:y}"
192   Un_def        "A Un  B  == Union(Upair(A,B))"
193   Int_def       "A Int B  == Inter(Upair(A,B))"
195   (* Definite descriptions -- via Replace over the set "1" *)
197   the_def       "The(P)    == Union({y . x:{0}, P(y)})"
198   if_def        "if(P,a,b) == THE z. P & z=a | ~P & z=b"
200   (* Ordered pairs and disjoint union of a family of sets *)
202   (* this "symmetric" definition works better than {{a}, {a,b}} *)
203   Pair_def      "<a,b>  == {{a,a}, {a,b}}"
204   fst_def       "fst(p) == THE a. EX b. p=<a,b>"
205   snd_def       "snd(p) == THE b. EX a. p=<a,b>"
206   split_def     "split(c,p) == c(fst(p), snd(p))"
207   Sigma_def     "Sigma(A,B) == UN x:A. UN y:B(x). {<x,y>}"
209   (* Operations on relations *)
211   (*converse of relation r, inverse of function*)
212   converse_def  "converse(r) == {z. w:r, EX x y. w=<x,y> & z=<y,x>}"
214   domain_def    "domain(r) == {x. w:r, EX y. w=<x,y>}"
215   range_def     "range(r) == domain(converse(r))"
216   field_def     "field(r) == domain(r) Un range(r)"
217   function_def  "function(r) == ALL x y. <x,y>:r -->
218                                 (ALL y'. <x,y'>:r --> y=y')"
219   image_def     "r `` A  == {y : range(r) . EX x:A. <x,y> : r}"
220   vimage_def    "r -`` A == converse(r)``A"
222   (* Abstraction, application and Cartesian product of a family of sets *)
224   lam_def       "Lambda(A,b) == {<x,b(x)> . x:A}"
225   apply_def     "f`a == THE y. <a,y> : f"
226   Pi_def        "Pi(A,B)  == {f: Pow(Sigma(A,B)). A<=domain(f) & function(f)}"
228   (* Restrict the function f to the domain A *)
229   restrict_def  "restrict(f,A) == lam x:A.f`x"
231 end
234 ML
236 (* Pattern-matching and 'Dependent' type operators *)
237 (*
238 local open Syntax
240 fun pttrn s = const"@pttrn" \$ s;
241 fun pttrns s t = const"@pttrns" \$ s \$ t;
243 fun split2(Abs(x,T,t)) =
244       let val (pats,u) = split1 t
245       in (pttrns (Free(x,T)) pats, subst_bounds([free x],u)) end
246   | split2(Const("split",_) \$ r) =
247       let val (pats,s) = split2(r)
248           val (pats2,t) = split1(s)
249       in (pttrns (pttrn pats) pats2, t) end
250 and split1(Abs(x,T,t)) =  (Free(x,T), subst_bounds([free x],t))
251   | split1(Const("split",_)\$t) = split2(t);
253 fun split_tr'(t::args) =
254   let val (pats,ft) = split2(t)
255   in list_comb(const"_lambda" \$ pttrn pats \$ ft, args) end;
257 in
258 *)
259 val print_translation =
260   [(*("split", split_tr'),*)
261    ("Pi",    dependent_tr' ("@PROD", "op ->")),
262    ("Sigma", dependent_tr' ("@SUM", "op *"))];
263 (*
264 end;
265 *)