(* Title: ZF/OrderArith.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Towards ordinal arithmetic
*)
OrderArith = Order + Sum +
consts
radd, rmult :: "[i,i,i,i]=>i"
rvimage :: "[i,i,i]=>i"
defs
(*disjoint sum of two relations; underlies ordinal addition*)
radd_def "radd(A,r,B,s) ==
{z: (A+B) * (A+B).
(EX x y. z = <Inl(x), Inr(y)>) |
(EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r) |
(EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
(*lexicographic product of two relations; underlies ordinal multiplication*)
rmult_def "rmult(A,r,B,s) ==
{z: (A*B) * (A*B).
EX x' y' x y. z = <<x',y'>, <x,y>> &
(<x',x>: r | (x'=x & <y',y>: s))}"
(*inverse image of a relation*)
rvimage_def "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
end