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(* Title: HOL/trancl.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1992 University of Cambridge
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Transitive closure of a relation
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rtrancl is refl/transitive closure; trancl is transitive closure
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*)
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120
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Trancl = Lfp + Prod +
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consts
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trans :: "('a * 'a)set => bool" (*transitivity predicate*)
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id :: "('a * 'a)set"
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rtrancl :: "('a * 'a)set => ('a * 'a)set" ("(_^*)" [100] 100)
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trancl :: "('a * 'a)set => ('a * 'a)set" ("(_^+)" [100] 100)
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O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
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defs
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trans_def "trans(r) == (!x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
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comp_def (*composition of relations*)
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"r O s == {xz. ? x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
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id_def (*the identity relation*)
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"id == {p. ? x. p = <x,x>}"
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rtrancl_def "r^* == lfp(%s. id Un (r O s))"
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trancl_def "r^+ == r O rtrancl(r)"
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end
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