author  huffman 
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permissions  rwrr 
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(* Title: HOLCF/Cfun.thy 
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ID: $Id$ 
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Author: Franz Regensburger 
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Definition of the type > of continuous functions. 
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*) 
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header {* The type of continuous functions *} 
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theory Cfun 
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imports TypedefPcpo 
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begin 
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defaultsort cpo 
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subsection {* Definition of continuous function type *} 
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typedef (CFun) ('a, 'b) ">" (infixr 0) = "{f::'a => 'b. cont f}" 
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by (rule exI, fast intro: cont_const) 
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syntax 
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Rep_CFun :: "('a > 'b) => ('a => 'b)" ("_$_" [999,1000] 999) 
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(* application *) 
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Abs_CFun :: "('a => 'b) => ('a > 'b)" (binder "LAM " 10) 
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(* abstraction *) 
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syntax (xsymbols) 
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">" :: "[type, type] => type" ("(_ \<rightarrow>/ _)" [1,0]0) 
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"LAM " :: "[idts, 'a => 'b] => ('a > 'b)" 
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("(3\<Lambda>_./ _)" [0, 10] 10) 
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Rep_CFun :: "('a > 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999) 
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syntax (HTML output) 
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Rep_CFun :: "('a > 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999) 
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subsection {* Class instances *} 
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instance ">" :: (cpo, cpo) sq_ord .. 
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defs (overloaded) 
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less_cfun_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>f g. Rep_CFun f \<sqsubseteq> Rep_CFun g)" 
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lemma adm_CFun: "adm (\<lambda>f. f \<in> CFun)" 
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by (simp add: CFun_def, rule admI, rule cont_lub_fun) 

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lemma UU_CFun: "\<bottom> \<in> CFun" 

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by (simp add: CFun_def inst_fun_pcpo cont_const) 

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instance ">" :: (cpo, cpo) po 
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by (rule typedef_po [OF type_definition_CFun less_cfun_def]) 
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instance ">" :: (cpo, cpo) cpo 
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by (rule typedef_cpo [OF type_definition_CFun less_cfun_def adm_CFun]) 

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instance ">" :: (cpo, pcpo) pcpo 

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by (rule typedef_pcpo_UU [OF type_definition_CFun less_cfun_def UU_CFun]) 

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lemmas cont_Rep_CFun = 

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typedef_cont_Rep [OF type_definition_CFun less_cfun_def adm_CFun] 

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lemmas cont_Abs_CFun = 

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typedef_cont_Abs [OF type_definition_CFun less_cfun_def adm_CFun] 

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lemmas Rep_CFun_strict = 
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typedef_Rep_strict [OF type_definition_CFun less_cfun_def UU_CFun] 
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lemmas Abs_CFun_strict = 
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typedef_Abs_strict [OF type_definition_CFun less_cfun_def UU_CFun] 
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text {* Additional lemma about the isomorphism between 
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@{typ "'a > 'b"} and @{term CFun} *} 
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lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f" 
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by (simp add: Abs_CFun_inverse CFun_def) 
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text {* Betaequality for continuous functions *} 
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lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u" 
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by (simp add: Abs_CFun_inverse2) 
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text {* Etaequality for continuous functions *} 
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lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f" 
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by (rule Rep_CFun_inverse) 
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text {* Extensionality for continuous functions *} 
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lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g" 
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by (simp add: Rep_CFun_inject [symmetric] ext) 
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text {* lemmas about application of continuous functions *} 
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lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y" 
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by simp 
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lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x" 
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by simp 
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lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y" 
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by simp 
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subsection {* Continuity of application *} 
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lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)" 
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by (rule cont_Rep_CFun [THEN cont2cont_CF1L]) 
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lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)" 
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apply (rule_tac P = "cont" in CollectD) 
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apply (fold CFun_def) 
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apply (rule Rep_CFun) 
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done 
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lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono] 
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lemmas contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub] 
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lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard] 
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lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard] 
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lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard] 
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lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard] 
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text {* contlub, cont properties of @{term Rep_CFun} in each argument *} 
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lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(lub (range Y)) = (\<Squnion>i. f\<cdot>(Y i))" 
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by (rule contlub_Rep_CFun2 [THEN contlubE]) 
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lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) << f\<cdot>(lub (range Y))" 
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by (rule cont_Rep_CFun2 [THEN contE]) 
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lemma contlub_cfun_fun: "chain F \<Longrightarrow> lub (range F)\<cdot>x = (\<Squnion>i. F i\<cdot>x)" 
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by (rule contlub_Rep_CFun1 [THEN contlubE]) 
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lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) << lub (range F)\<cdot>x" 
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by (rule cont_Rep_CFun1 [THEN contE]) 
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text {* Extensionality wrt. @{term "op <<"} in @{typ "'a > 'b"} *} 
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lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g" 
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by (simp add: less_cfun_def less_fun_def) 
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text {* monotonicity of application *} 
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lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x" 
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by (simp add: less_cfun_def less_fun_def) 
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lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y" 
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by (rule monofun_Rep_CFun2 [THEN monofunE]) 
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lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y" 
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by (rule trans_less [OF monofun_cfun_fun monofun_cfun_arg]) 
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text {* ch2ch  rules for the type @{typ "'a > 'b"} *} 
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lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
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by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
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lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
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by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
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lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)" 
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by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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lemma ch2ch_Rep_CFun: "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" 
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apply (rule chainI) 
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apply (rule monofun_cfun) 
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apply (erule chainE) 
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apply (erule chainE) 
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done 
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text {* contlub, cont properties of @{term Rep_CFun} in both arguments *} 
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lemma contlub_cfun: 
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"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))" 
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apply (simp only: contlub_cfun_fun) 
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apply (simp only: contlub_cfun_arg) 
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apply (rule diag_lub) 
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apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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apply (erule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
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done 
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lemma cont_cfun: 
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"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) << (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)" 
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apply (rule thelubE) 
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apply (simp only: ch2ch_Rep_CFun) 
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apply (simp only: contlub_cfun) 
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done 
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186 

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text {* strictness *} 
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188 

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lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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apply (rule UU_I) 
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apply (erule subst) 
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apply (rule minimal [THEN monofun_cfun_arg]) 
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done 
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text {* the lub of a chain of continous functions is monotone *} 
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lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)" 
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apply (drule ch2ch_monofun [OF monofun_Rep_CFun]) 
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apply (simp add: thelub_fun [symmetric]) 
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apply (erule monofun_lub_fun) 
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apply (simp add: monofun_Rep_CFun2) 
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done 
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203 

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text {* a lemma about the exchange of lubs for type @{typ "'a > 'b"}: uses MF2 lemmas from Cont.thy *} 
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lemma ex_lub_cfun: "[ chain(F); chain(Y) ] ==> 
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lub(range(%j. lub(range(%i. F(j)$(Y i))))) = 
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lub(range(%i. lub(range(%j. F(j)$(Y i)))))" 
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by (simp add: diag_lub ch2ch_Rep_CFunL ch2ch_Rep_CFunR) 
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text {* the lub of a chain of cont. functions is continuous *} 
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lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)" 
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apply (rule cont2cont_lub) 
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apply (erule monofun_Rep_CFun [THEN ch2ch_monofun]) 
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apply (rule cont_Rep_CFun2) 
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done 
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text {* type @{typ "'a > 'b"} is chain complete *} 
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lemma lub_cfun: "chain F \<Longrightarrow> range F << (LAM x. LUB i. F i$x)" 
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apply (subst thelub_fun [symmetric]) 
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apply (erule monofun_Rep_CFun [THEN ch2ch_monofun]) 
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apply (erule typedef_is_lub [OF type_definition_CFun less_cfun_def adm_CFun]) 
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done 
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lemmas thelub_cfun = lub_cfun [THEN thelubI, standard] 
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 {* @{thm thelub_cfun} *} (* chain F \<Longrightarrow> lub (range F) = (\<Lambda>x. \<Squnion>i. F i\<cdot>x) *) 
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subsection {* Miscellaneous *} 
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text {* Monotonicity of @{term Abs_CFun} *} 
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lemma semi_monofun_Abs_CFun: "[ cont(f); cont(g); f<<g] ==> Abs_CFun(f)<<Abs_CFun(g)" 
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by (simp add: less_cfun_def Abs_CFun_inverse2) 
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text {* for compatibility with old HOLCFVersion *} 
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lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)" 
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by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict) 
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240 

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subsection {* Continuity of application *} 
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242 

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text {* cont2cont lemma for @{term Rep_CFun} *} 
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lemma cont2cont_Rep_CFun: 
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"\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x)\<cdot>(t x))" 
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by (best intro: cont2cont_app2 cont_const cont_Rep_CFun cont_Rep_CFun2) 
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text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
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lemma cont2mono_LAM: 
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assumes p1: "!!x. cont(c1 x)" 
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assumes p2: "!!y. monofun(%x. c1 x y)" 
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shows "monofun(%x. LAM y. c1 x y)" 
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apply (rule monofunI) 
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apply (rule less_cfun_ext) 
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apply (simp add: p1) 
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apply (erule p2 [THEN monofunE]) 
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done 
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260 

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text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *} 
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262 

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lemma cont2cont_LAM: 
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assumes p1: "!!x. cont(c1 x)" 
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assumes p2: "!!y. cont(%x. c1 x y)" 
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shows "cont(%x. LAM y. c1 x y)" 
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apply (rule cont_Abs_CFun) 
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apply (simp add: p1 CFun_def) 

269 
apply (simp add: p2 cont2cont_CF1L_rev) 

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done 
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text {* cont2cont tactic *} 
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lemmas cont_lemmas1 = 
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cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM 

276 

277 
text {* 

278 
Continuity simproc by Brian Huffman. 

279 
Given the term @{term "cont f"}, the procedure tries to 

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construct the theorem @{prop "cont f == True"}. If this 

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theorem cannot be completely solved by the introduction 

282 
rules, then the procedure returns a conditional rewrite 

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rule with the unsolved subgoals as premises. 

284 
*} 

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ML_setup {* 
287 
local 

288 
val rules = thms "cont_lemmas1"; 

289 
fun solve_cont sg _ t = 

290 
let val tr = instantiate' [] [SOME (cterm_of sg t)] Eq_TrueI; 

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val tac = REPEAT_ALL_NEW (resolve_tac rules) 1; 

292 
in Option.map fst (Seq.pull (tac tr)) end; 

293 
in 

294 
val cont_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 

295 
"continuity" ["cont f"] solve_cont; 

296 
end; 

297 
Addsimprocs [cont_proc]; 

298 
*} 

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(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*) 
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(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*) 
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text {* function application is strict in its first argument *} 
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lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>" 
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by (simp add: Rep_CFun_strict) 
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307 

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text {* some lemmata for functions with flat/chfin domain/range types *} 
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309 

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lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
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==> !s. ? n. lub(range(Y))$s = Y n$s" 
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apply (rule allI) 
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apply (subst contlub_cfun_fun) 
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apply assumption 
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apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) 
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done 
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317 

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subsection {* Continuous injectionretraction pairs *} 
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319 

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text {* Continuous retractions are strict. *} 
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lemma retraction_strict: 
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"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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apply (rule UU_I) 
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apply (drule_tac x="\<bottom>" in spec) 
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apply (erule subst) 
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apply (rule monofun_cfun_arg) 
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apply (rule minimal) 
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done 
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330 

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lemma injection_eq: 
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"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" 
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apply (rule iffI) 
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apply (drule_tac f=f in cfun_arg_cong) 
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apply simp 
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336 
apply simp 
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done 
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338 

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lemma injection_defined_rev: 
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"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" 
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apply (drule_tac f=f in cfun_arg_cong) 
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342 
apply (simp add: retraction_strict) 
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done 
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344 

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lemma injection_defined: 
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"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" 
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347 
by (erule contrapos_nn, rule injection_defined_rev) 
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348 

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text {* propagation of flatness and chainfiniteness by retractions *} 
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350 

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351 
lemma chfin2chfin: 
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"\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y 
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\<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" 
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354 
apply clarify 
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355 
apply (drule_tac f=g in chain_monofun) 
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356 
apply (drule chfin [rule_format]) 
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357 
apply (unfold max_in_chain_def) 
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358 
apply (simp add: injection_eq) 
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359 
done 
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360 

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361 
lemma flat2flat: 
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362 
"\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y 
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\<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y" 
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364 
apply clarify 
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365 
apply (drule_tac f=g in monofun_cfun_arg) 
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366 
apply (drule ax_flat [rule_format]) 
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367 
apply (erule disjE) 
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368 
apply (simp add: injection_defined_rev) 
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369 
apply (simp add: injection_eq) 
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370 
done 
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371 

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text {* a result about functions with flat codomain *} 
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373 

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374 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
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375 
by (drule ax_flat [rule_format], simp) 
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376 

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377 
lemma flat_codom: 
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378 
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" 
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379 
apply (case_tac "f\<cdot>x = \<bottom>") 
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380 
apply (rule disjI1) 
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381 
apply (rule UU_I) 
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382 
apply (erule_tac t="\<bottom>" in subst) 
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383 
apply (rule minimal [THEN monofun_cfun_arg]) 
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384 
apply clarify 
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385 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
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386 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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387 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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388 
done 
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389 

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390 

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391 
subsection {* Identity and composition *} 
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392 

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393 
consts 
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394 
ID :: "'a \<rightarrow> 'a" 
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395 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" 
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396 

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397 
syntax "@oo" :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) 
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398 

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399 
translations "f1 oo f2" == "cfcomp$f1$f2" 
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400 

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401 
defs 
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402 
ID_def: "ID \<equiv> (\<Lambda> x. x)" 
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403 
oo_def: "cfcomp \<equiv> (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
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404 

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405 
lemma ID1 [simp]: "ID\<cdot>x = x" 
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406 
by (simp add: ID_def) 
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407 

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408 
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" 
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409 
by (simp add: oo_def) 
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410 

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411 
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" 
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412 
by (simp add: cfcomp1) 
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413 

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414 
text {* 
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415 
Show that interpretation of (pcpo,@{text "_>_"}) is a category. 
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416 
The class of objects is interpretation of syntactical class pcpo. 
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417 
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a > 'b"}. 
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418 
The identity arrow is interpretation of @{term ID}. 
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419 
The composition of f and g is interpretation of @{text "oo"}. 
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420 
*} 
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421 

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422 
lemma ID2 [simp]: "f oo ID = f" 
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423 
by (rule ext_cfun, simp) 
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424 

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425 
lemma ID3 [simp]: "ID oo f = f" 
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426 
by (rule ext_cfun, simp) 
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427 

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428 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
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429 
by (rule ext_cfun, simp) 
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430 

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431 

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432 
subsection {* Strictified functions *} 
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433 

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434 
defaultsort pcpo 
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435 

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436 
consts 
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437 
Istrictify :: "('a \<rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" 
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438 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" 
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439 

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440 
defs 
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441 
Istrictify_def: "Istrictify f x \<equiv> if x = \<bottom> then \<bottom> else f\<cdot>x" 
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442 
strictify_def: "strictify \<equiv> (\<Lambda> f x. Istrictify f x)" 
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443 

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444 
text {* results about strictify *} 
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445 

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446 
lemma Istrictify1: "Istrictify f \<bottom> = \<bottom>" 
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447 
by (simp add: Istrictify_def) 
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448 

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449 
lemma Istrictify2: "x \<noteq> \<bottom> \<Longrightarrow> Istrictify f x = f\<cdot>x" 
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450 
by (simp add: Istrictify_def) 
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451 

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452 
lemma cont_Istrictify1: "cont (\<lambda>f. Istrictify f x)" 
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453 
apply (case_tac "x = \<bottom>") 
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454 
apply (simp add: Istrictify1) 
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455 
apply (simp add: Istrictify2) 
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456 
done 
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457 

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458 
lemma monofun_Istrictify2: "monofun (\<lambda>x. Istrictify f x)" 
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459 
apply (rule monofunI) 
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460 
apply (simp add: Istrictify_def monofun_cfun_arg) 
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461 
apply clarify 
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462 
apply (simp add: eq_UU_iff) 
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463 
done 
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464 

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465 
lemma contlub_Istrictify2: "contlub (\<lambda>x. Istrictify f x)" 
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466 
apply (rule contlubI) 
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467 
apply (case_tac "lub (range Y) = \<bottom>") 
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468 
apply (simp add: Istrictify1 chain_UU_I thelub_const) 
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469 
apply (simp add: Istrictify2) 
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470 
apply (simp add: contlub_cfun_arg) 
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471 
apply (rule lub_equal2) 
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472 
apply (rule chain_mono2 [THEN exE]) 
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473 
apply (erule chain_UU_I_inverse2) 
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474 
apply (assumption) 
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475 
apply (blast intro: Istrictify2 [symmetric]) 
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476 
apply (erule chain_monofun) 
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477 
apply (erule monofun_Istrictify2 [THEN ch2ch_monofun]) 
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478 
done 
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479 

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480 
lemmas cont_Istrictify2 = 
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481 
monocontlub2cont [OF monofun_Istrictify2 contlub_Istrictify2, standard] 
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482 

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483 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
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484 
apply (unfold strictify_def) 
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485 
apply (simp add: cont_Istrictify1 cont_Istrictify2) 
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486 
apply (rule Istrictify1) 
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487 
done 
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488 

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489 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
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490 
apply (unfold strictify_def) 
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491 
apply (simp add: cont_Istrictify1 cont_Istrictify2) 
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492 
apply (erule Istrictify2) 
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493 
done 
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494 

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495 
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" 
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496 
by simp 
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497 

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498 
end 