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(* Title: ZF/OrderType.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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850
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Order types and ordinal arithmetic.
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435
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The order type of a well-ordering is the least ordinal isomorphic to it.
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*)
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467
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OrderType = OrderArith + Ordinal +
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435
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consts
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ordermap :: "[i,i]=>i"
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ordertype :: "[i,i]=>i"
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Ord_alt :: "i => o"
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"**" :: "[i,i]=>i" (infixl 70)
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"++" :: "[i,i]=>i" (infixl 65)
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753
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defs
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ordermap_def
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"ordermap(A,r) == lam x:A. wfrec[A](r, x, %x f. f `` pred(A,x,r))"
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ordertype_def "ordertype(A,r) == ordermap(A,r)``A"
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850
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Ord_alt_def (*alternative definition of ordinal numbers*)
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"Ord_alt(X) == well_ord(X, Memrel(X)) & (ALL u:X. u=pred(X, u, Memrel(X)))"
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(*ordinal multiplication*)
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omult_def "i ** j == ordertype(j*i, rmult(j,Memrel(j),i,Memrel(i)))"
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(*ordinal addition*)
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oadd_def "i ++ j == ordertype(i+j, radd(i,Memrel(i),j,Memrel(j)))"
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end
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