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(* Title: HOLCF/Domain_Aux.thy


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Author: Brian Huffman


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*)


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header {* Domain package support *}


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theory Domain_Aux


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imports Ssum Sprod Fixrec


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uses


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("Tools/Domain/domain_take_proofs.ML")


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begin


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subsection {* Continuous isomorphisms *}


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text {* A locale for continuous isomorphisms *}


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locale iso =


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fixes abs :: "'a \<rightarrow> 'b"


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fixes rep :: "'b \<rightarrow> 'a"


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assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"


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assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"


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begin


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lemma swap: "iso rep abs"


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by (rule iso.intro [OF rep_iso abs_iso])


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lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"


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proof


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assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"


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then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)


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then show "x \<sqsubseteq> y" by simp


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next


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assume "x \<sqsubseteq> y"


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then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)


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qed


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lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"


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by (rule iso.abs_below [OF swap])


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lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"


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by (simp add: po_eq_conv abs_below)


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lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"


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by (rule iso.abs_eq [OF swap])


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lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"


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proof 


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have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..


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then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)


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then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp


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then show ?thesis by (rule UU_I)


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qed


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lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"


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by (rule iso.abs_strict [OF swap])


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lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"


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proof 


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have "x = rep\<cdot>(abs\<cdot>x)" by simp


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also assume "abs\<cdot>x = \<bottom>"


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also note rep_strict


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finally show "x = \<bottom>" .


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qed


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lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"


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by (rule iso.abs_defin' [OF swap])


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lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"


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by (erule contrapos_nn, erule abs_defin')


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lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"


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by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)


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lemma abs_defined_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"


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by (auto elim: abs_defin' intro: abs_strict)


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lemma rep_defined_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"


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by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)


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lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"


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by (simp add: rep_defined_iff)


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lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"


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proof (unfold compact_def)


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assume "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> y)"


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with cont_Rep_CFun2


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have "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> abs\<cdot>y)" by (rule adm_subst)


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then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_below by simp


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qed


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lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"


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by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)


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lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"


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by (rule compact_rep_rev) simp


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lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"


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by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)


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lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"


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proof


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assume "x = abs\<cdot>y"


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then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp


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then show "rep\<cdot>x = y" by simp


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next


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assume "rep\<cdot>x = y"


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then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp


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then show "x = abs\<cdot>y" by simp


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qed


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end


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subsection {* Proofs about take functions *}


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text {*


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This section contains lemmas that are used in a module that supports


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the domain isomorphism package; the module contains proofs related


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to take functions and the finiteness predicate.


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*}


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lemma deflation_abs_rep:


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fixes abs and rep and d


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assumes abs_iso: "\<And>x. rep\<cdot>(abs\<cdot>x) = x"


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assumes rep_iso: "\<And>y. abs\<cdot>(rep\<cdot>y) = y"


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shows "deflation d \<Longrightarrow> deflation (abs oo d oo rep)"


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by (rule ep_pair.deflation_e_d_p) (simp add: ep_pair.intro assms)


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lemma deflation_chain_min:


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assumes chain: "chain d"


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assumes defl: "\<And>n. deflation (d n)"


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shows "d m\<cdot>(d n\<cdot>x) = d (min m n)\<cdot>x"


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proof (rule linorder_le_cases)


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assume "m \<le> n"


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with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)


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then have "d m\<cdot>(d n\<cdot>x) = d m\<cdot>x"


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by (rule deflation_below_comp1 [OF defl defl])


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moreover from `m \<le> n` have "min m n = m" by simp


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ultimately show ?thesis by simp


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next


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assume "n \<le> m"


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with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)


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then have "d m\<cdot>(d n\<cdot>x) = d n\<cdot>x"


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by (rule deflation_below_comp2 [OF defl defl])


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moreover from `n \<le> m` have "min m n = n" by simp


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ultimately show ?thesis by simp


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qed


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lemma lub_ID_take_lemma:


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assumes "chain t" and "(\<Squnion>n. t n) = ID"


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assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"


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proof 


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have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"


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using assms(3) by simp


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then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y"


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using assms(1) by (simp add: lub_distribs)


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then show "x = y"


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using assms(2) by simp


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qed


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lemma lub_ID_reach:


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assumes "chain t" and "(\<Squnion>n. t n) = ID"


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shows "(\<Squnion>n. t n\<cdot>x) = x"


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using assms by (simp add: lub_distribs)


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subsection {* Finiteness *}


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text {*


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Let a ``decisive'' function be a deflation that maps every input to


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either itself or bottom. Then if a domain's take functions are all


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decisive, then all values in the domain are finite.


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*}


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definition


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decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"


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where


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"decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"


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lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"


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unfolding decisive_def by simp


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lemma decisive_cases:


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assumes "decisive d" obtains "d\<cdot>x = x"  "d\<cdot>x = \<bottom>"


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using assms unfolding decisive_def by auto


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lemma decisive_bottom: "decisive \<bottom>"


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unfolding decisive_def by simp


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lemma decisive_ID: "decisive ID"


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unfolding decisive_def by simp


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lemma decisive_ssum_map:


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assumes f: "decisive f"


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assumes g: "decisive g"


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shows "decisive (ssum_map\<cdot>f\<cdot>g)"


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apply (rule decisiveI, rename_tac s)


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apply (case_tac s, simp_all)


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apply (rule_tac x=x in decisive_cases [OF f], simp_all)


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apply (rule_tac x=y in decisive_cases [OF g], simp_all)


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done


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lemma decisive_sprod_map:


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assumes f: "decisive f"


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assumes g: "decisive g"


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shows "decisive (sprod_map\<cdot>f\<cdot>g)"


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apply (rule decisiveI, rename_tac s)


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apply (case_tac s, simp_all)


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apply (rule_tac x=x in decisive_cases [OF f], simp_all)


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apply (rule_tac x=y in decisive_cases [OF g], simp_all)


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done


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lemma decisive_abs_rep:


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fixes abs rep


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assumes iso: "iso abs rep"


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assumes d: "decisive d"


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shows "decisive (abs oo d oo rep)"


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apply (rule decisiveI)


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apply (rule_tac x="rep\<cdot>x" in decisive_cases [OF d])


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apply (simp add: iso.rep_iso [OF iso])


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apply (simp add: iso.abs_strict [OF iso])


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done


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lemma lub_ID_finite:


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assumes chain: "chain d"


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assumes lub: "(\<Squnion>n. d n) = ID"


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assumes decisive: "\<And>n. decisive (d n)"


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shows "\<exists>n. d n\<cdot>x = x"


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proof 


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have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp


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have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)


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have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"


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using decisive unfolding decisive_def by simp


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hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"


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by auto


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hence "finite (range (\<lambda>n. d n\<cdot>x))"


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by (rule finite_subset, simp)


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with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"


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by (rule finite_range_imp_finch)


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then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"


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unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)


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with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)


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qed


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subsection {* ML setup *}


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use "Tools/Domain/domain_take_proofs.ML"


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end
