(* Title: HOL/Fun.thy
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Notions about functions.
*)
Fun = Typedef +
instance set :: (type) order
(subset_refl,subset_trans,subset_antisym,psubset_eq)
consts
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
nonterminals
updbinds updbind
syntax
"_updbind" :: ['a, 'a] => updbind ("(2_ :=/ _)")
"" :: updbind => updbinds ("_")
"_updbinds" :: [updbind, updbinds] => updbinds ("_,/ _")
"_Update" :: ['a, updbinds] => 'a ("_/'((_)')" [1000,0] 900)
translations
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
"f(x:=y)" == "fun_upd f x y"
defs
fun_upd_def "f(a:=b) == % x. if x=a then b else f x"
(* Hint: to define the sum of two functions (or maps), use sum_case.
A nice infix syntax could be defined (in Datatype.thy or below) by
consts
fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
translations
"fun_sum" == "sum_case"
*)
constdefs
id :: 'a => 'a
"id == %x. x"
o :: ['b => 'c, 'a => 'b, 'a] => 'c (infixl 55)
"f o g == %x. f(g(x))"
inj_on :: ['a => 'b, 'a set] => bool
"inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
syntax (xsymbols)
"op o" :: "['b => 'c, 'a => 'b, 'a] => 'c" (infixl "\\<circ>" 55)
syntax
inj :: ('a => 'b) => bool (*injective*)
translations
"inj f" == "inj_on f UNIV"
constdefs
surj :: ('a => 'b) => bool (*surjective*)
"surj f == ! y. ? x. y=f(x)"
bij :: ('a => 'b) => bool (*bijective*)
"bij f == inj f & surj f"
(*The Pi-operator, by Florian Kammueller*)
constdefs
Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"
"Pi A B == {f. ! x. if x:A then f(x) : B(x) else f(x) = arbitrary}"
restrict :: "['a => 'b, 'a set] => ('a => 'b)"
"restrict f A == (%x. if x : A then f x else arbitrary)"
syntax
"@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3PI _:_./ _)" 10)
funcset :: "['a set, 'b set] => ('a => 'b) set" (infixr 60)
"@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a => 'b)" ("(3%_:_./ _)" [0, 0, 3] 3)
syntax (xsymbols)
"@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a => 'b)" ("(3\\<lambda>_\\<in>_./ _)" [0, 0, 3] 3)
(*Giving funcset an arrow syntax (-> or =>) clashes with many existing theories*)
syntax (xsymbols)
"@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set" ("(3\\<Pi> _\\<in>_./ _)" 10)
translations
"PI x:A. B" => "Pi A (%x. B)"
"A funcset B" => "Pi A (_K B)"
"%x:A. f" == "restrict (%x. f) A"
constdefs
compose :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
"compose A g f == %x:A. g (f x)"
end
ML
val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];