src/HOL/Set.thy
author clasohm
Fri Mar 03 12:02:25 1995 +0100 (1995-03-03)
changeset 923 ff1574a81019
child 1068 e0f2dffab506
permissions -rw-r--r--
new version of HOL with curried function application
     1 (*  Title:      HOL/Set.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1993  University of Cambridge
     5 *)
     6 
     7 Set = Ord +
     8 
     9 types
    10   'a set
    11 
    12 arities
    13   set :: (term) term
    14 
    15 instance
    16   set :: (term) {ord, minus}
    17 
    18 consts
    19   "{}"          :: "'a set"                           ("{}")
    20   insert        :: "['a, 'a set] => 'a set"
    21   Collect       :: "('a => bool) => 'a set"               (*comprehension*)
    22   Compl         :: "('a set) => 'a set"                   (*complement*)
    23   Int           :: "['a set, 'a set] => 'a set"       (infixl 70)
    24   Un            :: "['a set, 'a set] => 'a set"       (infixl 65)
    25   UNION, INTER  :: "['a set, 'a => 'b set] => 'b set"     (*general*)
    26   UNION1        :: "['a => 'b set] => 'b set"         (binder "UN " 10)
    27   INTER1        :: "['a => 'b set] => 'b set"         (binder "INT " 10)
    28   Union, Inter  :: "(('a set)set) => 'a set"              (*of a set*)
    29   Pow           :: "'a set => 'a set set"                 (*powerset*)
    30   range         :: "('a => 'b) => 'b set"                 (*of function*)
    31   Ball, Bex     :: "['a set, 'a => bool] => bool"         (*bounded quantifiers*)
    32   inj, surj     :: "('a => 'b) => bool"                   (*inj/surjective*)
    33   inj_onto      :: "['a => 'b, 'a set] => bool"
    34   "``"          :: "['a => 'b, 'a set] => ('b set)"   (infixl 90)
    35   ":"           :: "['a, 'a set] => bool"             (infixl 50) (*membership*)
    36 
    37 
    38 syntax
    39 
    40   "~:"          :: "['a, 'a set] => bool"             (infixl 50)
    41 
    42   "@Finset"     :: "args => 'a set"                   ("{(_)}")
    43 
    44   "@Coll"       :: "[idt, bool] => 'a set"            ("(1{_./ _})")
    45   "@SetCompr"   :: "['a, idts, bool] => 'a set"       ("(1{_ |/_./ _})")
    46 
    47   (* Big Intersection / Union *)
    48 
    49   "@INTER"      :: "[idt, 'a set, 'b set] => 'b set"  ("(3INT _:_./ _)" 10)
    50   "@UNION"      :: "[idt, 'a set, 'b set] => 'b set"  ("(3UN _:_./ _)" 10)
    51 
    52   (* Bounded Quantifiers *)
    53 
    54   "@Ball"       :: "[idt, 'a set, bool] => bool"      ("(3! _:_./ _)" 10)
    55   "@Bex"        :: "[idt, 'a set, bool] => bool"      ("(3? _:_./ _)" 10)
    56   "*Ball"       :: "[idt, 'a set, bool] => bool"      ("(3ALL _:_./ _)" 10)
    57   "*Bex"        :: "[idt, 'a set, bool] => bool"      ("(3EX _:_./ _)" 10)
    58 
    59 translations
    60   "x ~: y"      == "~ (x : y)"
    61   "{x, xs}"     == "insert x {xs}"
    62   "{x}"         == "insert x {}"
    63   "{x. P}"      == "Collect (%x. P)"
    64   "INT x:A. B"  == "INTER A (%x. B)"
    65   "UN x:A. B"   == "UNION A (%x. B)"
    66   "! x:A. P"    == "Ball A (%x. P)"
    67   "? x:A. P"    == "Bex A (%x. P)"
    68   "ALL x:A. P"  => "Ball A (%x. P)"
    69   "EX x:A. P"   => "Bex A (%x. P)"
    70 
    71 
    72 rules
    73 
    74   (* Isomorphisms between Predicates and Sets *)
    75 
    76   mem_Collect_eq    "(a : {x.P(x)}) = P(a)"
    77   Collect_mem_eq    "{x.x:A} = A"
    78 
    79 
    80 defs
    81   Ball_def      "Ball A P       == ! x. x:A --> P(x)"
    82   Bex_def       "Bex A P        == ? x. x:A & P(x)"
    83   subset_def    "A <= B         == ! x:A. x:B"
    84   Compl_def     "Compl(A)       == {x. ~x:A}"
    85   Un_def        "A Un B         == {x.x:A | x:B}"
    86   Int_def       "A Int B        == {x.x:A & x:B}"
    87   set_diff_def  "A - B          == {x. x:A & ~x:B}"
    88   INTER_def     "INTER A B      == {y. ! x:A. y: B(x)}"
    89   UNION_def     "UNION A B      == {y. ? x:A. y: B(x)}"
    90   INTER1_def    "INTER1(B)      == INTER {x.True} B"
    91   UNION1_def    "UNION1(B)      == UNION {x.True} B"
    92   Inter_def     "Inter(S)       == (INT x:S. x)"
    93   Union_def     "Union(S)       == (UN x:S. x)"
    94   Pow_def       "Pow(A)         == {B. B <= A}"
    95   empty_def     "{}             == {x. False}"
    96   insert_def    "insert a B     == {x.x=a} Un B"
    97   range_def     "range(f)       == {y. ? x. y=f(x)}"
    98   image_def     "f``A           == {y. ? x:A. y=f(x)}"
    99   inj_def       "inj(f)         == ! x y. f(x)=f(y) --> x=y"
   100   inj_onto_def  "inj_onto f A   == ! x:A. ! y:A. f(x)=f(y) --> x=y"
   101   surj_def      "surj(f)        == ! y. ? x. y=f(x)"
   102 
   103 end
   104 
   105 ML
   106 
   107 local
   108 
   109 (* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P}      *)
   110 (* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)
   111 
   112 val ex_tr = snd(mk_binder_tr("? ","Ex"));
   113 
   114 fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1
   115   | nvars(_) = 1;
   116 
   117 fun setcompr_tr[e,idts,b] =
   118   let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e
   119       val P = Syntax.const("op &") $ eq $ b
   120       val exP = ex_tr [idts,P]
   121   in Syntax.const("Collect") $ Abs("",dummyT,exP) end;
   122 
   123 val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));
   124 
   125 fun setcompr_tr'[Abs(_,_,P)] =
   126   let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)
   127         | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ _, n) =
   128             if n>0 andalso m=n andalso
   129               ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))
   130             then () else raise Match
   131 
   132       fun tr'(_ $ abs) =
   133         let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]
   134         in Syntax.const("@SetCompr") $ e $ idts $ Q end
   135   in ok(P,0); tr'(P) end;
   136 
   137 in
   138 
   139 val parse_translation = [("@SetCompr", setcompr_tr)];
   140 val print_translation = [("Collect", setcompr_tr')];
   141 val print_ast_translation =
   142   map HOL.alt_ast_tr' [("@Ball", "*Ball"), ("@Bex", "*Bex")];
   143 
   144 end;
   145