--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Set.thy Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,145 @@
+(* 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" :: "[idt, bool] => 'a set" ("(1{_./ _})")
+ "@SetCompr" :: "['a, idts, bool] => 'a set" ("(1{_ |/_./ _})")
+
+ (* Big Intersection / Union *)
+
+ "@INTER" :: "[idt, 'a set, 'b set] => 'b set" ("(3INT _:_./ _)" 10)
+ "@UNION" :: "[idt, 'a set, 'b set] => 'b set" ("(3UN _:_./ _)" 10)
+
+ (* Bounded Quantifiers *)
+
+ "@Ball" :: "[idt, 'a set, bool] => bool" ("(3! _:_./ _)" 10)
+ "@Bex" :: "[idt, 'a set, bool] => bool" ("(3? _:_./ _)" 10)
+ "*Ball" :: "[idt, 'a set, bool] => bool" ("(3ALL _:_./ _)" 10)
+ "*Bex" :: "[idt, '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)"
+
+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;
+