10294
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theory Relations = Main:
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ML "Pretty.setmargin 64"
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(*Id is only used in UNITY*)
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(*refl, antisym,trans,univalent,\<dots> ho hum*)
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text{*
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@{thm[display]"Id_def"}
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\rulename{Id_def}
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*}
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text{*
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@{thm[display]"comp_def"}
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\rulename{comp_def}
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*}
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text{*
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@{thm[display]"R_O_Id"}
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\rulename{R_O_Id}
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*}
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text{*
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@{thm[display]"comp_mono"}
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\rulename{comp_mono}
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*}
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text{*
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@{thm[display]"converse_iff"}
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\rulename{converse_iff}
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*}
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text{*
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@{thm[display]"converse_comp"}
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\rulename{converse_comp}
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*}
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text{*
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@{thm[display]"Image_iff"}
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\rulename{Image_iff}
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*}
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text{*
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@{thm[display]"Image_UN"}
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\rulename{Image_UN}
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*}
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text{*
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@{thm[display]"Domain_iff"}
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\rulename{Domain_iff}
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*}
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text{*
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@{thm[display]"Range_iff"}
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\rulename{Range_iff}
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*}
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text{*
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@{thm[display]"relpow.simps"}
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\rulename{relpow.simps}
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@{thm[display]"rtrancl_unfold"}
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\rulename{rtrancl_unfold}
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@{thm[display]"rtrancl_refl"}
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\rulename{rtrancl_refl}
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@{thm[display]"r_into_rtrancl"}
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\rulename{r_into_rtrancl}
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@{thm[display]"rtrancl_trans"}
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\rulename{rtrancl_trans}
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@{thm[display]"rtrancl_induct"}
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\rulename{rtrancl_induct}
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@{thm[display]"rtrancl_idemp"}
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\rulename{rtrancl_idemp}
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@{thm[display]"r_into_trancl"}
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\rulename{r_into_trancl}
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@{thm[display]"trancl_trans"}
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\rulename{trancl_trans}
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@{thm[display]"trancl_into_rtrancl"}
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\rulename{trancl_into_rtrancl}
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@{thm[display]"trancl_converse"}
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\rulename{trancl_converse}
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*}
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text{*Relations. transitive closure*}
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lemma rtrancl_converseD: "(x,y) \<in> (r^-1)^* \<Longrightarrow> (y,x) \<in> r^*"
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apply (erule rtrancl_induct)
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apply (rule rtrancl_refl)
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apply (blast intro: r_into_rtrancl rtrancl_trans)
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done
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text{*
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proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
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\isanewline
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goal\ {\isacharparenleft}lemma\ rtrancl{\isacharunderscore}converseD{\isacharparenright}{\isacharcolon}\isanewline
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{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ {\isacharparenleft}r{\isacharcircum}{\isacharminus}\isadigit{1}{\isacharparenright}{\isacharcircum}{\isacharasterisk}\ {\isasymLongrightarrow}\ {\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}\isanewline
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\ \isadigit{1}{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}\isanewline
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\ \isadigit{2}{\isachardot}\ {\isasymAnd}y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymin}\ {\isacharparenleft}r{\isacharcircum}{\isacharminus}\isadigit{1}{\isacharparenright}{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharminus}\isadigit{1}{\isacharsemicolon}\ {\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}{\isasymrbrakk}\isanewline
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\ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isacharparenleft}z{\isacharcomma}\ x{\isacharparenright}\ {\isasymin}\ r{\isacharcircum}{\isacharasterisk}
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*}
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lemma rtrancl_converseI: "(y,x) \<in> r^* \<Longrightarrow> (x,y) \<in> (r^-1)^*"
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apply (erule rtrancl_induct)
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apply (rule rtrancl_refl)
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apply (blast intro: r_into_rtrancl rtrancl_trans)
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done
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lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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apply (auto intro: rtrancl_converseI dest: rtrancl_converseD)
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done
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lemma "A \<subseteq> Id"
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apply (rule subsetI)
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apply (auto)
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oops
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text{*
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proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
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\isanewline
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goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
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A\ {\isasymsubseteq}\ Id\isanewline
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\ \isadigit{1}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ x\ {\isasymin}\ Id
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proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{2}\isanewline
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\isanewline
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goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
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A\ {\isasymsubseteq}\ Id\isanewline
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\ \isadigit{1}{\isachardot}\ {\isasymAnd}a\ b{\isachardot}\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ A\ {\isasymLongrightarrow}\ a\ {\isacharequal}\ b
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*}
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text{*questions: do we cover force? (Why not?)
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Do we include tables of operators in ASCII and X-symbol notation like in the Logics manuals?*}
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text{*rejects*}
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lemma "(a \<in> {z. P z} \<union> {y. Q y}) = P a \<or> Q a"
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apply (blast)
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done
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text{*Pow, Inter too little used*}
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lemma "(A \<subset> B) = (A \<subseteq> B \<and> A \<noteq> B)"
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apply (simp add: psubset_def)
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done
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(*
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text{*
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@{thm[display]"DD"}
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\rulename{DD}
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*}
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*)
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end
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