src/HOL/Set.thy
author nipkow
Fri May 16 17:40:41 1997 +0200 (1997-05-16)
changeset 3222 726a9b069947
parent 2965 afbda7e26f15
child 3370 5c5fdce3a4e4
permissions -rw-r--r--
Distributed Psubset stuff to basic set theory files, incl Finite.
Added stuff by bu.
     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 types
    13   'a set
    14 
    15 arities
    16   set :: (term) term
    17 
    18 instance
    19   set :: (term) {ord, minus}
    20 
    21 consts
    22   "{}"          :: 'a set                           ("{}")
    23   insert        :: ['a, 'a set] => 'a set
    24   Collect       :: ('a => bool) => 'a set               (*comprehension*)
    25   Compl         :: ('a set) => 'a set                   (*complement*)
    26   Int           :: ['a set, 'a set] => 'a set       (infixl 70)
    27   Un            :: ['a set, 'a set] => 'a set       (infixl 65)
    28   UNION, INTER  :: ['a set, 'a => 'b set] => 'b set     (*general*)
    29   UNION1        :: ['a => 'b set] => 'b set         (binder "UN " 10)
    30   INTER1        :: ['a => 'b set] => 'b set         (binder "INT " 10)
    31   Union, Inter  :: (('a set) set) => 'a set             (*of a set*)
    32   Pow           :: 'a set => 'a set set                 (*powerset*)
    33   range         :: ('a => 'b) => 'b set                 (*of function*)
    34   Ball, Bex     :: ['a set, 'a => bool] => bool         (*bounded quantifiers*)
    35   "``"          :: ['a => 'b, 'a set] => ('b set)   (infixr 90)
    36   (*membership*)
    37   "op :"        :: ['a, 'a set] => bool             ("(_/ : _)" [50, 51] 50)
    38 
    39 
    40 
    41 (** Additional concrete syntax **)
    42 
    43 syntax
    44 
    45   "op :"        :: ['a, 'a set] => bool               ("op :")
    46 
    47   UNIV          :: 'a set
    48 
    49   (* Infix syntax for non-membership *)
    50 
    51   "op ~:"       :: ['a, 'a set] => bool               ("(_/ ~: _)" [50, 51] 50)
    52   "op ~:"       :: ['a, 'a set] => bool               ("op ~:")
    53 
    54   "@Finset"     :: args => 'a set                     ("{(_)}")
    55 
    56   "@Coll"       :: [pttrn, bool] => 'a set            ("(1{_./ _})")
    57   "@SetCompr"   :: ['a, idts, bool] => 'a set         ("(1{_ |/_./ _})")
    58 
    59   (* Big Intersection / Union *)
    60 
    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 
    66   "@Ball"       :: [pttrn, 'a set, bool] => bool      ("(3! _:_./ _)" [0, 0, 10] 10)
    67   "@Bex"        :: [pttrn, 'a set, bool] => bool      ("(3? _:_./ _)" [0, 0, 10] 10)
    68   "*Ball"       :: [pttrn, 'a set, bool] => bool      ("(3ALL _:_./ _)" [0, 0, 10] 10)
    69   "*Bex"        :: [pttrn, 'a set, bool] => bool      ("(3EX _:_./ _)" [0, 0, 10] 10)
    70 
    71 translations
    72   "UNIV"        == "Compl {}"
    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   "INT x:A. B"  == "INTER A (%x. B)"
    79   "UN x:A. B"   == "UNION A (%x. B)"
    80   "! x:A. P"    == "Ball A (%x. P)"
    81   "? x:A. P"    == "Bex A (%x. P)"
    82   "ALL x:A. P"  => "Ball A (%x. P)"
    83   "EX x:A. P"   => "Bex A (%x. P)"
    84 
    85 syntax ("" output)
    86   "_setle"      :: ['a set, 'a set] => bool           ("(_/ <= _)" [50, 51] 50)
    87   "_setle"      :: ['a set, 'a set] => bool           ("op <=")
    88   "_setless"    :: ['a set, 'a set] => bool           ("(_/ < _)" [50, 51] 50)
    89   "_setless"    :: ['a set, 'a set] => bool           ("op <")
    90 
    91 syntax (symbols)
    92   "_setle"      :: ['a set, 'a set] => bool           ("(_/ \\<subseteq> _)" [50, 51] 50)
    93   "_setle"      :: ['a set, 'a set] => bool           ("op \\<subseteq>")
    94   "_setless"    :: ['a set, 'a set] => bool           ("(_/ \\<subset> _)" [50, 51] 50)
    95   "_setless"    :: ['a set, 'a set] => bool           ("op \\<subset>")
    96   "op Int"      :: ['a set, 'a set] => 'a set         (infixl "\\<inter>" 70)
    97   "op Un"       :: ['a set, 'a set] => 'a set         (infixl "\\<union>" 65)
    98   "op :"        :: ['a, 'a set] => bool               ("(_/ \\<in> _)" [50, 51] 50)
    99   "op :"        :: ['a, 'a set] => bool               ("op \\<in>")
   100   "op ~:"       :: ['a, 'a set] => bool               ("(_/ \\<notin> _)" [50, 51] 50)
   101   "op ~:"       :: ['a, 'a set] => bool               ("op \\<notin>")
   102   "UN "         :: [idts, bool] => bool               ("(3\\<Union> _./ _)" 10)
   103   "INT "        :: [idts, bool] => bool               ("(3\\<Inter> _./ _)" 10)
   104   "@UNION"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3\\<Union> _\\<in>_./ _)" 10)
   105   "@INTER"      :: [pttrn, 'a set, 'b set] => 'b set  ("(3\\<Inter> _\\<in>_./ _)" 10)
   106   Union         :: (('a set) set) => 'a set           ("\\<Union> _" [90] 90)
   107   Inter         :: (('a set) set) => 'a set           ("\\<Inter> _" [90] 90)
   108   "@Ball"       :: [pttrn, 'a set, bool] => bool      ("(3\\<forall> _\\<in>_./ _)" [0, 0, 10] 10)
   109   "@Bex"        :: [pttrn, 'a set, bool] => bool      ("(3\\<exists> _\\<in>_./ _)" [0, 0, 10] 10)
   110 
   111 syntax (symbols output)
   112   "*Ball"       :: [pttrn, 'a set, bool] => bool      ("(3\\<forall> _\\<in>_./ _)" [0, 0, 10] 10)
   113   "*Bex"        :: [pttrn, 'a set, bool] => bool      ("(3\\<exists> _\\<in>_./ _)" [0, 0, 10] 10)
   114 
   115 translations
   116   "op \\<subseteq>" => "op <= :: [_ set, _ set] => bool"
   117   "op \\<subset>" => "op <  :: [_ set, _ set] => bool"
   118 
   119 
   120 
   121 (** Rules and definitions **)
   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 
   133   Ball_def      "Ball A P       == ! x. x:A --> P(x)"
   134   Bex_def       "Bex A P        == ? x. x:A & P(x)"
   135   subset_def    "A <= B         == ! x:A. x:B"
   136   psubset_def   "A < B          == (A::'a set) <= B & ~ A=B"
   137   Compl_def     "Compl A        == {x. ~x:A}"
   138   Un_def        "A Un B         == {x.x:A | x:B}"
   139   Int_def       "A Int B        == {x.x:A & x:B}"
   140   set_diff_def  "A - B          == {x. x:A & ~x:B}"
   141   INTER_def     "INTER A B      == {y. ! x:A. y: B(x)}"
   142   UNION_def     "UNION A B      == {y. ? x:A. y: B(x)}"
   143   INTER1_def    "INTER1 B       == INTER {x.True} B"
   144   UNION1_def    "UNION1 B       == UNION {x.True} B"
   145   Inter_def     "Inter S        == (INT x:S. x)"
   146   Union_def     "Union S        == (UN x:S. x)"
   147   Pow_def       "Pow A          == {B. B <= A}"
   148   empty_def     "{}             == {x. False}"
   149   insert_def    "insert a B     == {x.x=a} Un B"
   150   image_def     "f``A           == {y. ? x:A. y=f(x)}"
   151 
   152 end
   153 
   154 
   155 ML
   156 
   157 local
   158 
   159 (* Set inclusion *)
   160 
   161 fun le_tr' (*op <=*) (Type ("fun", (Type ("set", _) :: _))) ts =
   162       list_comb (Syntax.const "_setle", ts)
   163   | le_tr' (*op <=*) _ _ = raise Match;
   164 
   165 fun less_tr' (*op <*) (Type ("fun", (Type ("set", _) :: _))) ts =
   166       list_comb (Syntax.const "_setless", ts)
   167   | less_tr' (*op <*) _ _ = raise Match;
   168 
   169 
   170 (* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P}      *)
   171 (* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)
   172 
   173 val ex_tr = snd(mk_binder_tr("? ","Ex"));
   174 
   175 fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1
   176   | nvars(_) = 1;
   177 
   178 fun setcompr_tr[e,idts,b] =
   179   let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e
   180       val P = Syntax.const("op &") $ eq $ b
   181       val exP = ex_tr [idts,P]
   182   in Syntax.const("Collect") $ Abs("",dummyT,exP) end;
   183 
   184 val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));
   185 
   186 fun setcompr_tr'[Abs(_,_,P)] =
   187   let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)
   188         | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ _, n) =
   189             if n>0 andalso m=n andalso
   190               ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))
   191             then () else raise Match
   192 
   193       fun tr'(_ $ abs) =
   194         let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]
   195         in Syntax.const("@SetCompr") $ e $ idts $ Q end
   196   in ok(P,0); tr'(P) end;
   197 
   198 in
   199 
   200 val parse_translation = [("@SetCompr", setcompr_tr)];
   201 val print_translation = [("Collect", setcompr_tr')];
   202 val typed_print_translation = [("op <=", le_tr'), ("op <", less_tr')];
   203 val print_ast_translation =
   204   map HOL.alt_ast_tr' [("@Ball", "*Ball"), ("@Bex", "*Bex")];
   205 
   206 end;