(* Title: HOL/Prod.thy
ID: Prod.thy,v 1.5 1994/08/19 09:04:27 lcp Exp
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Ordered Pairs and the Cartesian product type.
The unit type.
*)
Prod = Fun +
(** Products **)
(* type definition *)
consts
Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
defs
Pair_Rep_def "Pair_Rep == (%a b. %x y. x=a & y=b)"
subtype (Prod)
('a, 'b) "*" (infixr 20)
= "{f. ? a b. f = Pair_Rep(a::'a, b::'b)}"
(* abstract constants and syntax *)
consts
fst :: "'a * 'b => 'a"
snd :: "'a * 'b => 'b"
split :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
Pair :: "['a, 'b] => 'a * 'b"
Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set"
syntax
"@Tuple" :: "args => 'a * 'b" ("(1<_>)")
translations
"<x, y, z>" == "<x, <y, z>>"
"<x, y>" == "Pair(x, y)"
"<x>" => "x"
defs
Pair_def "Pair(a, b) == Abs_Prod(Pair_Rep(a, b))"
fst_def "fst(p) == @a. ? b. p = <a, b>"
snd_def "snd(p) == @b. ? a. p = <a, b>"
split_def "split(c, p) == c(fst(p), snd(p))"
prod_fun_def "prod_fun(f, g) == split(%x y.<f(x), g(y)>)"
Sigma_def "Sigma(A, B) == UN x:A. UN y:B(x). {<x, y>}"
(** Unit **)
subtype (Unit)
unit = "{p. p = True}"
consts
Unity :: "unit" ("<>")
defs
Unity_def "Unity == Abs_Unit(True)"
end