src/ZF/OrderArith.thy
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Thu, 22 Jun 1995 17:13:05 +0200
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(*  Title: 	ZF/OrderArith.thy
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Towards ordinal arithmetic 
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*)
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OrderArith = Order + Sum + 
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consts
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  radd, rmult      :: "[i,i,i,i]=>i"
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  rvimage          :: "[i,i,i]=>i"
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defs
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  (*disjoint sum of two relations; underlies ordinal addition*)
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  radd_def "radd(A,r,B,s) == 
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                {z: (A+B) * (A+B).  
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                    (EX x y. z = <Inl(x), Inr(y)>)   | 	 
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                    (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   | 	 
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                    (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
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  (*lexicographic product of two relations; underlies ordinal multiplication*)
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  rmult_def "rmult(A,r,B,s) == 
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                {z: (A*B) * (A*B).  
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                    EX x' y' x y. z = <<x',y'>, <x,y>> &	 
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                       (<x',x>: r | (x'=x & <y',y>: s))}"
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  (*inverse image of a relation*)
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  rvimage_def "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
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end