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(* Title: CCL/Trancl.thy
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Author: Martin Coen, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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*)
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section \<open>Transitive closure of a relation\<close>
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theory Trancl
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imports CCL
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begin
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definition trans :: "i set \<Rightarrow> o" (*transitivity predicate*)
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where "trans(r) == (ALL x y z. <x,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <x,z>:r)"
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definition id :: "i set" (*the identity relation*)
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where "id == {p. EX x. p = <x,x>}"
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definition relcomp :: "[i set,i set] \<Rightarrow> i set" (infixr "O" 60) (*composition of relations*)
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where "r O s == {xz. EX x y z. xz = <x,z> \<and> <x,y>:s \<and> <y,z>:r}"
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definition rtrancl :: "i set \<Rightarrow> i set" ("(_^*)" [100] 100)
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where "r^* == lfp(\<lambda>s. id Un (r O s))"
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definition trancl :: "i set \<Rightarrow> i set" ("(_^+)" [100] 100)
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where "r^+ == r O rtrancl(r)"
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subsection \<open>Natural deduction for \<open>trans(r)\<close>\<close>
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lemma transI: "(\<And>x y z. \<lbrakk><x,y>:r; <y,z>:r\<rbrakk> \<Longrightarrow> <x,z>:r) \<Longrightarrow> trans(r)"
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unfolding trans_def by blast
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lemma transD: "\<lbrakk>trans(r); <a,b>:r; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c>:r"
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unfolding trans_def by blast
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