src/ZF/ex/equiv.thy
author lcp
Tue Aug 16 18:58:42 1994 +0200 (1994-08-16)
changeset 532 851df239ac8b
parent 0 a5a9c433f639
permissions -rw-r--r--
ZF/Makefile,ROOT.ML, ZF/ex/Integ.thy: updated for EquivClass
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(*  Title: 	ZF/ex/equiv.thy
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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Equivalence relations in Zermelo-Fraenkel Set Theory 
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*)
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Equiv = Trancl +
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consts
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    refl,equiv 	::      "[i,i]=>o"
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    sym         ::      "i=>o"
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    "'/"        ::      "[i,i]=>i"  (infixl 90)  (*set of equiv classes*)
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    congruent	::	"[i,i=>i]=>o"
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    congruent2  ::      "[i,[i,i]=>i]=>o"
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rules
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    refl_def      "refl(A,r) == r <= (A*A) & (ALL x: A. <x,x> : r)"
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    sym_def       "sym(r) == ALL x y. <x,y>: r --> <y,x>: r"
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    equiv_def     "equiv(A,r) == refl(A,r) & sym(r) & trans(r)"
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    quotient_def  "A/r == {r``{x} . x:A}"
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    congruent_def "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"
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    congruent2_def
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       "congruent2(r,b) == ALL y1 z1 y2 z2. \
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\           <y1,z1>:r --> <y2,z2>:r --> b(y1,y2) = b(z1,z2)"
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end