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(* Title: ZF/Rel.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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Relations in Zermelo-Fraenkel Set Theory
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*)
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Rel = ZF +
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consts
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refl,irrefl,equiv :: "[i,i]=>o"
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sym,asym,antisym,trans :: "i=>o"
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trans_on :: "[i,i]=>o" ("trans[_]'(_')")
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defs
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refl_def "refl(A,r) == (ALL x: A. <x,x> : r)"
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irrefl_def "irrefl(A,r) == 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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asym_def "asym(r) == ALL x y. <x,y>:r --> ~ <y,x>:r"
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antisym_def "antisym(r) == ALL x y.<x,y>:r --> <y,x>:r --> x=y"
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trans_def "trans(r) == ALL x y z. <x,y>: r --> <y,z>: r --> <x,z>: r"
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trans_on_def "trans[A](r) == ALL x:A. ALL y:A. ALL z:A. \
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\ <x,y>: r --> <y,z>: r --> <x,z>: r"
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equiv_def "equiv(A,r) == r <= A*A & refl(A,r) & sym(r) & trans(r)"
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end
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