(* Title: HOL/Fun.thy
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Notions about functions.
*)
Fun = Vimage +
instance set :: (term) order
(subset_refl,subset_trans,subset_antisym,psubset_eq)
consts
Id :: 'a => 'a
o :: ['b => 'c, 'a => 'b, 'a] => 'c (infixl 55)
inj, surj :: ('a => 'b) => bool (*inj/surjective*)
inj_on :: ['a => 'b, 'a set] => bool
inv :: ('a => 'b) => ('b => 'a)
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
nonterminals
updbinds updbind
syntax
(* Let expressions *)
"_updbind" :: ['a, 'a] => updbind ("(2_ :=/ _)")
"" :: updbind => updbinds ("_")
"_updbinds" :: [updbind, updbinds] => updbinds ("_,/ _")
"_Update" :: ['a, updbinds] => 'a ("_/'((_)')" [900,0] 900)
translations
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
"f(x:=y)" == "fun_upd f x y"
defs
Id_def "Id == %x. x"
o_def "f o g == %x. f(g(x))"
inj_def "inj f == ! x y. f(x)=f(y) --> x=y"
inj_on_def "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
surj_def "surj f == ! y. ? x. y=f(x)"
inv_def "inv(f::'a=>'b) == % y. @x. f(x)=y"
fun_upd_def "f(a:=b) == % x. if x=a then b else f x"
end