(* Title: HOL/Sum.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
The disjoint sum of two types.
*)
Sum = mono + Prod +
(* type definition *)
constdefs
Inl_Rep :: ['a, 'a, 'b, bool] => bool
"Inl_Rep == (%a. %x y p. x=a & p)"
Inr_Rep :: ['b, 'a, 'b, bool] => bool
"Inr_Rep == (%b. %x y p. y=b & ~p)"
global
typedef (Sum)
('a, 'b) "+" (infixr 10)
= "{f. (? a. f = Inl_Rep(a::'a)) | (? b. f = Inr_Rep(b::'b))}"
(* abstract constants and syntax *)
consts
Inl :: "'a => 'a + 'b"
Inr :: "'b => 'a + 'b"
sum_case :: "['a => 'c, 'b => 'c, 'a + 'b] => 'c"
(*disjoint sum for sets; the operator + is overloaded with wrong type!*)
Plus :: "['a set, 'b set] => ('a + 'b) set" (infixr 65)
Part :: ['a set, 'b => 'a] => 'a set
translations
"case p of Inl x => a | Inr y => b" == "sum_case (%x. a) (%y. b) p"
local
defs
Inl_def "Inl == (%a. Abs_Sum(Inl_Rep(a)))"
Inr_def "Inr == (%b. Abs_Sum(Inr_Rep(b)))"
sum_case_def "sum_case f g p == @z. (!x. p=Inl(x) --> z=f(x))
& (!y. p=Inr(y) --> z=g(y))"
sum_def "A Plus B == (Inl``A) Un (Inr``B)"
(*for selecting out the components of a mutually recursive definition*)
Part_def "Part A h == A Int {x. ? z. x = h(z)}"
end