src/HOL/Set.thy
author wenzelm
Thu Sep 28 14:42:21 2000 +0200 (2000-09-28)
changeset 10106 1b63e30437ee
parent 8148 5ef0b624aadb
child 10131 546686f0a6fb
permissions -rw-r--r--
fixed \<Union>, \<Inter> syntax;
     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 
    10 (** Core syntax **)
    11 
    12 global
    13 
    14 types
    15   'a set
    16 
    17 arities
    18   set :: (term) term
    19 
    20 instance
    21   set :: (term) {ord, minus}
    22 
    23 syntax
    24   "op :"        :: ['a, 'a set] => bool             ("op :")
    25 
    26 consts
    27   "{}"          :: 'a set                           ("{}")
    28   UNIV          :: 'a set
    29   insert        :: ['a, 'a set] => 'a set
    30   Collect       :: ('a => bool) => 'a set               (*comprehension*)
    31   Int           :: ['a set, 'a set] => 'a set       (infixl 70)
    32   Un            :: ['a set, 'a set] => 'a set       (infixl 65)
    33   UNION, INTER  :: ['a set, 'a => 'b set] => 'b set     (*general*)
    34   Union, Inter  :: (('a set) set) => 'a set             (*of a set*)
    35   Pow           :: 'a set => 'a set set                 (*powerset*)
    36   range         :: ('a => 'b) => 'b set                 (*of function*)
    37   Ball, Bex     :: ['a set, 'a => bool] => bool         (*bounded quantifiers*)
    38   "image"       :: ['a => 'b, 'a set] => ('b set)   (infixr "``" 90)
    39   (*membership*)
    40   "op :"        :: ['a, 'a set] => bool             ("(_/ : _)" [50, 51] 50)
    41 
    42 
    43 (** Additional concrete syntax **)
    44 
    45 syntax
    46 
    47   (* Infix syntax for non-membership *)
    48 
    49   "op ~:"       :: ['a, 'a set] => bool               ("op ~:")
    50   "op ~:"       :: ['a, 'a set] => bool               ("(_/ ~: _)" [50, 51] 50)
    51 
    52 
    53   "@Finset"     :: args => 'a set                     ("{(_)}")
    54   "@Coll"       :: [pttrn, bool] => 'a set            ("(1{_./ _})")
    55   "@SetCompr"   :: ['a, idts, bool] => 'a set         ("(1{_ |/_./ _})")
    56 
    57   (* Big Intersection / Union *)
    58 
    59   "@INTER1"     :: [pttrns, 'b set] => 'b set         ("(3INT _./ _)" 10)
    60   "@UNION1"     :: [pttrns, 'b set] => 'b set         ("(3UN _./ _)" 10)
    61   "@INTER"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3INT _:_./ _)" 10)
    62   "@UNION"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3UN _:_./ _)" 10)
    63 
    64   (* Bounded Quantifiers *)
    65   "_Ball"       :: [pttrn, 'a set, bool] => bool      ("(3ALL _:_./ _)" [0, 0, 10] 10)
    66   "_Bex"        :: [pttrn, 'a set, bool] => bool      ("(3EX _:_./ _)" [0, 0, 10] 10)
    67 
    68 syntax (HOL)
    69   "_Ball"       :: [pttrn, 'a set, bool] => bool      ("(3! _:_./ _)" [0, 0, 10] 10)
    70   "_Bex"        :: [pttrn, 'a set, bool] => bool      ("(3? _:_./ _)" [0, 0, 10] 10)
    71 
    72 translations
    73   "range f"     == "f``UNIV"
    74   "x ~: y"      == "~ (x : y)"
    75   "{x, xs}"     == "insert x {xs}"
    76   "{x}"         == "insert x {}"
    77   "{x. P}"      == "Collect (%x. P)"
    78   "UN x y. B"   == "UN x. UN y. B"
    79   "UN x. B"     == "UNION UNIV (%x. B)"
    80   "INT x y. B"  == "INT x. INT y. B"
    81   "INT x. B"    == "INTER UNIV (%x. B)"
    82   "UN x:A. B"   == "UNION A (%x. B)"
    83   "INT x:A. B"  == "INTER A (%x. B)"
    84   "ALL x:A. P"  == "Ball A (%x. P)"
    85   "EX x:A. P"   == "Bex A (%x. P)"
    86 
    87 syntax ("" output)
    88   "_setle"      :: ['a set, 'a set] => bool           ("op <=")
    89   "_setle"      :: ['a set, 'a set] => bool           ("(_/ <= _)" [50, 51] 50)
    90   "_setless"    :: ['a set, 'a set] => bool           ("op <")
    91   "_setless"    :: ['a set, 'a set] => bool           ("(_/ < _)" [50, 51] 50)
    92 
    93 syntax (symbols)
    94   "_setle"      :: ['a set, 'a set] => bool           ("op \\<subseteq>")
    95   "_setle"      :: ['a set, 'a set] => bool           ("(_/ \\<subseteq> _)" [50, 51] 50)
    96   "_setless"    :: ['a set, 'a set] => bool           ("op \\<subset>")
    97   "_setless"    :: ['a set, 'a set] => bool           ("(_/ \\<subset> _)" [50, 51] 50)
    98   "op Int"      :: ['a set, 'a set] => 'a set         (infixl "\\<inter>" 70)
    99   "op Un"       :: ['a set, 'a set] => 'a set         (infixl "\\<union>" 65)
   100   "op :"        :: ['a, 'a set] => bool               ("op \\<in>")
   101   "op :"        :: ['a, 'a set] => bool               ("(_/ \\<in> _)" [50, 51] 50)
   102   "op ~:"       :: ['a, 'a set] => bool               ("op \\<notin>")
   103   "op ~:"       :: ['a, 'a set] => bool               ("(_/ \\<notin> _)" [50, 51] 50)
   104   "@UNION1"     :: [pttrns, 'b set] => 'b set         ("(3\\<Union>_./ _)" 10)
   105   "@INTER1"     :: [pttrns, 'b set] => 'b set         ("(3\\<Inter>_./ _)" 10)
   106   "@UNION"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3\\<Union>_\\<in>_./ _)" 10)
   107   "@INTER"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3\\<Inter>_\\<in>_./ _)" 10)
   108   Union         :: (('a set) set) => 'a set           ("\\<Union>_" [90] 90)
   109   Inter         :: (('a set) set) => 'a set           ("\\<Inter>_" [90] 90)
   110   "_Ball"       :: [pttrn, 'a set, bool] => bool      ("(3\\<forall>_\\<in>_./ _)" [0, 0, 10] 10)
   111   "_Bex"        :: [pttrn, 'a set, bool] => bool      ("(3\\<exists>_\\<in>_./ _)" [0, 0, 10] 10)
   112 
   113 translations
   114   "op \\<subseteq>" => "op <= :: [_ set, _ set] => bool"
   115   "op \\<subset>" => "op <  :: [_ set, _ set] => bool"
   116 
   117 
   118 
   119 (** Rules and definitions **)
   120 
   121 local
   122 
   123 rules
   124 
   125   (* Isomorphisms between Predicates and Sets *)
   126 
   127   mem_Collect_eq    "(a : {x. P(x)}) = P(a)"
   128   Collect_mem_eq    "{x. x:A} = A"
   129 
   130 
   131 defs
   132   Ball_def      "Ball A P       == ! x. x:A --> P(x)"
   133   Bex_def       "Bex A P        == ? x. x:A & P(x)"
   134   subset_def    "A <= B         == ! x:A. x:B"
   135   psubset_def   "A < B          == (A::'a set) <= B & ~ A=B"
   136   Compl_def     "- A            == {x. ~x:A}"
   137   Un_def        "A Un B         == {x. x:A | x:B}"
   138   Int_def       "A Int B        == {x. x:A & x:B}"
   139   set_diff_def  "A - B          == {x. x:A & ~x:B}"
   140   INTER_def     "INTER A B      == {y. ! x:A. y: B(x)}"
   141   UNION_def     "UNION A B      == {y. ? x:A. y: B(x)}"
   142   Inter_def     "Inter S        == (INT x:S. x)"
   143   Union_def     "Union S        == (UN x:S. x)"
   144   Pow_def       "Pow A          == {B. B <= A}"
   145   empty_def     "{}             == {x. False}"
   146   UNIV_def      "UNIV           == {x. True}"
   147   insert_def    "insert a B     == {x. x=a} Un B"
   148   image_def     "f``A           == {y. ? x:A. y=f(x)}"
   149 
   150 
   151 end
   152 
   153 
   154 ML
   155 
   156 local
   157 
   158 (* Set inclusion *)
   159 
   160 fun le_tr' _ (*op <=*) (Type ("fun", (Type ("set", _) :: _))) ts =
   161       list_comb (Syntax.const "_setle", ts)
   162   | le_tr' _ (*op <=*) _ _ = raise Match;
   163 
   164 fun less_tr' _ (*op <*) (Type ("fun", (Type ("set", _) :: _))) ts =
   165       list_comb (Syntax.const "_setless", ts)
   166   | less_tr' _ (*op <*) _ _ = raise Match;
   167 
   168 
   169 (* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P}      *)
   170 (* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)
   171 
   172 val ex_tr = snd(mk_binder_tr("EX ","Ex"));
   173 
   174 fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1
   175   | nvars(_) = 1;
   176 
   177 fun setcompr_tr[e,idts,b] =
   178   let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e
   179       val P = Syntax.const("op &") $ eq $ b
   180       val exP = ex_tr [idts,P]
   181   in Syntax.const("Collect") $ Abs("",dummyT,exP) end;
   182 
   183 val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));
   184 
   185 fun setcompr_tr'[Abs(_,_,P)] =
   186   let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)
   187         | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ _, n) =
   188             if n>0 andalso m=n andalso
   189               ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))
   190             then () else raise Match
   191 
   192       fun tr'(_ $ abs) =
   193         let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]
   194         in Syntax.const("@SetCompr") $ e $ idts $ Q end
   195   in ok(P,0); tr'(P) end;
   196 
   197 in
   198 
   199 val parse_translation = [("@SetCompr", setcompr_tr)];
   200 val print_translation = [("Collect", setcompr_tr')];
   201 val typed_print_translation = [("op <=", le_tr'), ("op <", less_tr')];
   202 
   203 
   204 end;