src/HOL/Set.thy
author paulson
Mon Oct 07 10:28:44 1996 +0200 (1996-10-07)
changeset 2056 93c093620c28
parent 2006 72754e060aa2
child 2261 d926157c0a6a
permissions -rw-r--r--
Removed commands made redundant by new one-point rules
     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)   (infixr 90)
    35      (*membership*)
    36   "op :"        :: ['a, 'a set] => bool             ("(_/ : _)" [50,51] 50)
    37 
    38 
    39 syntax
    40 
    41   UNIV         :: 'a set
    42 
    43      (*infix synatx for non-membership*)
    44   "op ~:"        :: ['a, 'a set] => bool             ("(_/ ~: _)" [50,51] 50)
    45 
    46   "@Finset"     :: args => 'a set                   ("{(_)}")
    47 
    48   "@Coll"       :: [pttrn, bool] => 'a set          ("(1{_./ _})")
    49   "@SetCompr"   :: ['a, idts, bool] => 'a set       ("(1{_ |/_./ _})")
    50 
    51   (* Big Intersection / Union *)
    52 
    53   "@INTER"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3INT _:_./ _)" 10)
    54   "@UNION"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3UN _:_./ _)" 10)
    55 
    56   (* Bounded Quantifiers *)
    57 
    58   "@Ball"       :: [pttrn, 'a set, bool] => bool      ("(3! _:_./ _)" 10)
    59   "@Bex"        :: [pttrn, 'a set, bool] => bool      ("(3? _:_./ _)" 10)
    60   "*Ball"       :: [pttrn, 'a set, bool] => bool      ("(3ALL _:_./ _)" 10)
    61   "*Bex"        :: [pttrn, 'a set, bool] => bool      ("(3EX _:_./ _)" 10)
    62 
    63 translations
    64   "UNIV"        == "Compl {}"
    65   "range(f)"    == "f``UNIV"
    66   "x ~: y"      == "~ (x : y)"
    67   "{x, xs}"     == "insert x {xs}"
    68   "{x}"         == "insert x {}"
    69   "{x. P}"      == "Collect (%x. P)"
    70   "INT x:A. B"  == "INTER A (%x. B)"
    71   "UN x:A. B"   == "UNION A (%x. B)"
    72   "! x:A. P"    == "Ball A (%x. P)"
    73   "? x:A. P"    == "Bex A (%x. P)"
    74   "ALL x:A. P"  => "Ball A (%x. P)"
    75   "EX x:A. P"   => "Bex A (%x. P)"
    76 
    77 
    78 rules
    79 
    80   (* Isomorphisms between Predicates and Sets *)
    81 
    82   mem_Collect_eq    "(a : {x.P(x)}) = P(a)"
    83   Collect_mem_eq    "{x.x:A} = A"
    84 
    85 
    86 defs
    87   Ball_def      "Ball A P       == ! x. x:A --> P(x)"
    88   Bex_def       "Bex A P        == ? x. x:A & P(x)"
    89   subset_def    "A <= B         == ! x:A. x:B"
    90   Compl_def     "Compl(A)       == {x. ~x:A}"
    91   Un_def        "A Un B         == {x.x:A | x:B}"
    92   Int_def       "A Int B        == {x.x:A & x:B}"
    93   set_diff_def  "A - B          == {x. x:A & ~x:B}"
    94   INTER_def     "INTER A B      == {y. ! x:A. y: B(x)}"
    95   UNION_def     "UNION A B      == {y. ? x:A. y: B(x)}"
    96   INTER1_def    "INTER1(B)      == INTER {x.True} B"
    97   UNION1_def    "UNION1(B)      == UNION {x.True} B"
    98   Inter_def     "Inter(S)       == (INT x:S. x)"
    99   Union_def     "Union(S)       == (UN x:S. x)"
   100   Pow_def       "Pow(A)         == {B. B <= A}"
   101   empty_def     "{}             == {x. False}"
   102   insert_def    "insert a B     == {x.x=a} Un B"
   103   image_def     "f``A           == {y. ? x:A. y=f(x)}"
   104   inj_def       "inj(f)         == ! x y. f(x)=f(y) --> x=y"
   105   inj_onto_def  "inj_onto f A   == ! x:A. ! y:A. f(x)=f(y) --> x=y"
   106   surj_def      "surj(f)        == ! y. ? x. y=f(x)"
   107 
   108 (* start 8bit 1 *)
   109 (* end 8bit 1 *)
   110 
   111 end
   112 
   113 ML
   114 
   115 local
   116 
   117 (* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P}      *)
   118 (* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)
   119 
   120 val ex_tr = snd(mk_binder_tr("? ","Ex"));
   121 
   122 fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1
   123   | nvars(_) = 1;
   124 
   125 fun setcompr_tr[e,idts,b] =
   126   let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e
   127       val P = Syntax.const("op &") $ eq $ b
   128       val exP = ex_tr [idts,P]
   129   in Syntax.const("Collect") $ Abs("",dummyT,exP) end;
   130 
   131 val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));
   132 
   133 fun setcompr_tr'[Abs(_,_,P)] =
   134   let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)
   135         | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ _, n) =
   136             if n>0 andalso m=n andalso
   137               ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))
   138             then () else raise Match
   139 
   140       fun tr'(_ $ abs) =
   141         let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]
   142         in Syntax.const("@SetCompr") $ e $ idts $ Q end
   143   in ok(P,0); tr'(P) end;
   144 
   145 in
   146 
   147 val parse_translation = [("@SetCompr", setcompr_tr)];
   148 val print_translation = [("Collect", setcompr_tr')];
   149 val print_ast_translation =
   150   map HOL.alt_ast_tr' [("@Ball", "*Ball"), ("@Bex", "*Bex")];
   151 
   152 end;
   153