(* Title: HOL/set.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1993 University of Cambridge
*)
Set = Ord +
types
'a set
arities
set :: (term) term
set :: (term) ord
set :: (term) minus
consts
(* Constants *)
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*)
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*)
"~:" :: "['a, 'a set] => bool" ("(_ ~:/ _)" [50, 51] 50)
(* Finite Sets *)
insert :: "['a, 'a set] => 'a set"
"{}" :: "'a set" ("{}")
"@Finset" :: "args => 'a set" ("{(_)}")
(** Binding Constants **)
"@Coll" :: "[idt, bool] => 'a set" ("(1{_./ _})") (*collection*)
(* 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"
(* Definitions *)
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)"
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
val print_ast_translation =
map HOL.alt_ast_tr' [("@Ball", "*Ball"), ("@Bex", "*Bex")];