(*  Title:      HOL/Set.thy

    ID:         $Id$

    Author:     Tobias Nipkow

    Copyright   1993  University of Cambridge

*)

Set = Ord +

types

  'a set

arities

  set :: (term) term

instance

  set :: (term) {ord, minus}

consts

  "{}"          :: "'a set"                           ("{}")

  insert        :: "['a, 'a set] => 'a set"

  Collect       :: "('a => bool) => 'a set"               (*comprehension*)

  Compl         :: "('a set) => 'a set"                   (*complement*)

  Int           :: "['a set, 'a set] => 'a set"       (infixl 70)

  Un            :: "['a set, 'a set] => 'a set"       (infixl 65)

  UNION, INTER  :: "['a set, 'a => 'b set] => 'b set"     (*general*)

  UNION1        :: "['a => 'b set] => 'b set"         (binder "UN " 10)

  INTER1        :: "['a => 'b set] => 'b set"         (binder "INT " 10)

  Union, Inter  :: "(('a set)set) => 'a set"              (*of a set*)

  Pow           :: "'a set => 'a set set"                 (*powerset*)

  range         :: "('a => 'b) => 'b set"                 (*of function*)

  Ball, Bex     :: "['a set, 'a => bool] => bool"         (*bounded quantifiers*)

  inj, surj     :: "('a => 'b) => bool"                   (*inj/surjective*)

  inj_onto      :: "['a => 'b, 'a set] => bool"

  "``"          :: "['a => 'b, 'a set] => ('b set)"   (infixl 90)

  ":"           :: "['a, 'a set] => bool"             (infixl 50) (*membership*)

syntax

  "~:"          :: "['a, 'a set] => bool"             (infixl 50)

  "@Finset"     :: "args => 'a set"                   ("{(_)}")

  "@Coll"       :: "[pttrn, bool] => 'a set"          ("(1{_./ _})")

  "@SetCompr"   :: "['a, idts, bool] => 'a set"       ("(1{_ |/_./ _})")

  (* Big Intersection / Union *)

  "@INTER"      :: "[pttrn, 'a set, 'b set] => 'b set"  ("(3INT _:_./ _)" 10)

  "@UNION"      :: "[pttrn, 'a set, 'b set] => 'b set"  ("(3UN _:_./ _)" 10)

  (* Bounded Quantifiers *)

  "@Ball"       :: "[pttrn, 'a set, bool] => bool"      ("(3! _:_./ _)" 10)

  "@Bex"        :: "[pttrn, 'a set, bool] => bool"      ("(3? _:_./ _)" 10)

  "*Ball"       :: "[pttrn, 'a set, bool] => bool"      ("(3ALL _:_./ _)" 10)

  "*Bex"        :: "[pttrn, 'a set, bool] => bool"      ("(3EX _:_./ _)" 10)

translations

  "x ~: y"      == "~ (x : y)"

  "{x, xs}"     == "insert x {xs}"

  "{x}"         == "insert x {}"

  "{x. P}"      == "Collect (%x. P)"

  "INT x:A. B"  == "INTER A (%x. B)"

  "UN x:A. B"   == "UNION A (%x. B)"

  "! x:A. P"    == "Ball A (%x. P)"

  "? x:A. P"    == "Bex A (%x. P)"

  "ALL x:A. P"  => "Ball A (%x. P)"

  "EX x:A. P"   => "Bex A (%x. P)"

rules

  (* Isomorphisms between Predicates and Sets *)

  mem_Collect_eq    "(a : {x.P(x)}) = P(a)"

  Collect_mem_eq    "{x.x:A} = A"

defs

  Ball_def      "Ball A P       == ! x. x:A --> P(x)"

  Bex_def       "Bex A P        == ? x. x:A & P(x)"

  subset_def    "A <= B         == ! x:A. x:B"

  Compl_def     "Compl(A)       == {x. ~x:A}"

  Un_def        "A Un B         == {x.x:A | x:B}"

  Int_def       "A Int B        == {x.x:A & x:B}"

  set_diff_def  "A - B          == {x. x:A & ~x:B}"

  INTER_def     "INTER A B      == {y. ! x:A. y: B(x)}"

  UNION_def     "UNION A B      == {y. ? x:A. y: B(x)}"

  INTER1_def    "INTER1(B)      == INTER {x.True} B"

  UNION1_def    "UNION1(B)      == UNION {x.True} B"

  Inter_def     "Inter(S)       == (INT x:S. x)"

  Union_def     "Union(S)       == (UN x:S. x)"

  Pow_def       "Pow(A)         == {B. B <= A}"

  empty_def     "{}             == {x. False}"

  insert_def    "insert a B     == {x.x=a} Un B"

  range_def     "range(f)       == {y. ? x. y=f(x)}"

  image_def     "f``A           == {y. ? x:A. y=f(x)}"

  inj_def       "inj(f)         == ! x y. f(x)=f(y) --> x=y"

  inj_onto_def  "inj_onto f A   == ! x:A. ! y:A. f(x)=f(y) --> x=y"

  surj_def      "surj(f)        == ! y. ? x. y=f(x)"

(* start 8bit 1 *)

(* end 8bit 1 *)

end

ML

local

(* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P}      *)

(* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)

val ex_tr = snd(mk_binder_tr("? ","Ex"));

fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1

  | nvars(_) = 1;

fun setcompr_tr[e,idts,b] =

  let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e

      val P = Syntax.const("op &") $ eq $ b

      val exP = ex_tr [idts,P]

  in Syntax.const("Collect") $ Abs("",dummyT,exP) end;

val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));

fun setcompr_tr'[Abs(_,_,P)] =

  let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)

        | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ _, n) =

            if n>0 andalso m=n andalso

              ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))

            then () else raise Match

      fun tr'(_ $ abs) =

        let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]

        in Syntax.const("@SetCompr") $ e $ idts $ Q end

  in ok(P,0); tr'(P) end;

in

val parse_translation = [("@SetCompr", setcompr_tr)];

val print_translation = [("Collect", setcompr_tr')];

val print_ast_translation =

  map HOL.alt_ast_tr' [("@Ball", "*Ball"), ("@Bex", "*Bex")];

end;