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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
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