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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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15577  10 
theory Cfun 
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imports Cont 

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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, rule CfunI) 
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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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less_cfun :: "[('a > 'b),('a > 'b)]=>bool" 
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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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text {* 
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Derive old type definition rules for @{term Abs_CFun} \& @{term Rep_CFun}. 
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@{term Rep_CFun} and @{term Abs_CFun} should be replaced by 
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@{term Rep_Cfun} and @{term Abs_Cfun} in future. 
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*} 
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lemma Rep_Cfun: "Rep_CFun fo : CFun" 
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by (rule Rep_CFun) 
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lemma Rep_Cfun_inverse: "Abs_CFun (Rep_CFun fo) = fo" 
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by (rule Rep_CFun_inverse) 
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lemma Abs_Cfun_inverse: "f:CFun==>Rep_CFun(Abs_CFun f)=f" 
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by (erule Abs_CFun_inverse) 
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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 ==> Rep_CFun (Abs_CFun f) = f" 
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apply (rule Abs_Cfun_inverse) 
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apply (unfold CFun_def) 
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apply (erule mem_Collect_eq [THEN ssubst]) 
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done 
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text {* Simplification of application *} 
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lemma Cfunapp2: "cont f ==> (Abs_CFun f)$x = f x" 
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by (erule Abs_Cfun_inverse2 [THEN fun_cong]) 
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text {* Beta  equality for continuous functions *} 
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lemma beta_cfun: "cont(c1) ==> (LAM x .c1 x)$u = c1 u" 
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by (rule Cfunapp2) 
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15641  71 
text {* Eta  equality for continuous functions *} 
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lemma eta_cfun: "(LAM x. f$x) = f" 

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by (rule Rep_CFun_inverse) 

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subsection {* Type @{typ "'a > 'b"} is a partial order *} 
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instance ">" :: (cpo, cpo) sq_ord .. 
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defs (overloaded) 
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less_cfun_def: "(op <<) == (% fo1 fo2. Rep_CFun fo1 << Rep_CFun fo2 )" 
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lemma refl_less_cfun: "(f::'a>'b) << f" 
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by (unfold less_cfun_def, rule refl_less) 
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lemma antisym_less_cfun: 
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"[(f1::'a>'b) << f2; f2 << f1] ==> f1 = f2" 
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by (unfold less_cfun_def, rule Rep_CFun_inject[THEN iffD1], rule antisym_less) 
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lemma trans_less_cfun: 
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"[(f1::'a>'b) << f2; f2 << f3] ==> f1 << f3" 
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by (unfold less_cfun_def, rule trans_less) 
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instance ">" :: (cpo, cpo) po 
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by intro_classes 
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(assumption  rule refl_less_cfun antisym_less_cfun trans_less_cfun)+ 
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text {* for compatibility with old HOLCFVersion *} 
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lemma inst_cfun_po: "(op <<)=(%f1 f2. Rep_CFun f1 << Rep_CFun f2)" 
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apply (fold less_cfun_def) 
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apply (rule refl) 
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done 
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text {* lemmas about application of continuous functions *} 
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lemma cfun_cong: "[ f=g; x=y ] ==> f$x = g$y" 
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by simp 
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lemma cfun_fun_cong: "f=g ==> f$x = g$x" 
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by simp 
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lemma cfun_arg_cong: "x=y ==> f$x = f$y" 
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by simp 
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text {* access to @{term less_cfun} in class po *} 
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lemma less_cfun: "( f1 << f2 ) = (Rep_CFun(f1) << Rep_CFun(f2))" 
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by (simp add: inst_cfun_po) 
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subsection {* Type @{typ "'a > 'b"} is pointed *} 
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lemma minimal_cfun: "Abs_CFun(% x. UU) << f" 
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apply (subst less_cfun) 
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apply (subst Abs_Cfun_inverse2) 
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apply (rule cont_const) 
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apply (rule minimal_fun) 
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done 
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lemmas UU_cfun_def = minimal_cfun [THEN minimal2UU, symmetric, standard] 
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lemma least_cfun: "? x::'a>'b::pcpo.!y. x<<y" 
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apply (rule_tac x = "Abs_CFun (% x. UU) " in exI) 
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apply (rule minimal_cfun [THEN allI]) 
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done 
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subsection {* Monotonicity of application *} 
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text {* 
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@{term Rep_CFun} yields continuous functions in @{typ "'a => 'b"}. 
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This is continuity of @{term Rep_CFun} in its 'second' argument: 
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@{prop "cont_Rep_CFun2 ==> monofun_Rep_CFun2 & contlub_Rep_CFun2"} 
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*} 
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lemma cont_Rep_CFun2: "cont (Rep_CFun fo)" 
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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_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard] 
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 {* @{thm monofun_Rep_CFun2} *} (* monofun(Rep_CFun(?fo)) *) 
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152 

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153 
lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard] 
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154 
 {* @{thm contlub_Rep_CFun2} *} (* contlub(Rep_CFun(?fo)) *) 
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155 

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156 
text {* expanded thms @{thm [source] cont_Rep_CFun2}, @{thm [source] contlub_Rep_CFun2} look nice with mixfix syntax *} 
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157 

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158 
lemmas cont_cfun_arg = cont_Rep_CFun2 [THEN contE, THEN spec, THEN mp] 
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159 
 {* @{thm cont_cfun_arg} *} (* chain(x1) ==> range (%i. fo3$(x1 i)) << fo3$(lub (range ?x1)) *) 
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160 

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161 
lemmas contlub_cfun_arg = contlub_Rep_CFun2 [THEN contlubE, THEN spec, THEN mp] 
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162 
 {* @{thm contlub_cfun_arg} *} (* chain(?x1) ==> ?fo4$(lub (range ?x1)) = lub (range (%i. ?fo4$(?x1 i))) *) 
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163 

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text {* @{term Rep_CFun} is monotone in its 'first' argument *} 
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165 

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lemma monofun_Rep_CFun1: "monofun(Rep_CFun)" 
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167 
apply (rule monofunI [rule_format]) 
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apply (erule less_cfun [THEN subst]) 
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169 
done 
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170 

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text {* monotonicity of application @{term Rep_CFun} in mixfix syntax @{text "[_]_"} *} 
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172 

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lemma monofun_cfun_fun: "f1 << f2 ==> f1$x << f2$x" 
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174 
apply (rule_tac x = "x" in spec) 
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175 
apply (rule less_fun [THEN subst]) 
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apply (erule monofun_Rep_CFun1 [THEN monofunE [rule_format]]) 
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177 
done 
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178 

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179 
lemmas monofun_cfun_arg = monofun_Rep_CFun2 [THEN monofunE [rule_format], standard] 
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 {* @{thm monofun_cfun_arg} *} (* ?x2 << ?x1 ==> ?fo5$?x2 << ?fo5$?x1 *) 
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181 

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lemma chain_monofun: "chain Y ==> chain (%i. f\<cdot>(Y i))" 
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183 
apply (rule chainI) 
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184 
apply (rule monofun_cfun_arg) 
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185 
apply (erule chainE) 
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186 
done 
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187 

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text {* monotonicity of @{term Rep_CFun} in both arguments in mixfix syntax @{text "[_]_"} *} 
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189 

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lemma monofun_cfun: "[f1<<f2;x1<<x2] ==> f1$x1 << f2$x2" 
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191 
apply (rule trans_less) 
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192 
apply (erule monofun_cfun_arg) 
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193 
apply (erule monofun_cfun_fun) 
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194 
done 
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195 

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lemma strictI: "f$x = UU ==> f$UU = UU" 
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197 
apply (rule eq_UU_iff [THEN iffD2]) 
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198 
apply (erule subst) 
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199 
apply (rule minimal [THEN monofun_cfun_arg]) 
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200 
done 
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201 

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202 
subsection {* Type @{typ "'a > 'b"} is a cpo *} 
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203 

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204 
text {* ch2ch  rules for the type @{typ "'a > 'b"} use MF2 lemmas from Cont.thy *} 
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205 

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206 
lemma ch2ch_Rep_CFunR: "chain(Y) ==> chain(%i. f$(Y i))" 
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207 
by (erule monofun_Rep_CFun2 [THEN ch2ch_MF2R]) 
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208 

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209 
lemmas ch2ch_Rep_CFunL = monofun_Rep_CFun1 [THEN ch2ch_MF2L, standard] 
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 {* @{thm ch2ch_Rep_CFunL} *} (* chain(?F) ==> chain (%i. ?F i$?x) *) 
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211 

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212 
text {* the lub of a chain of continous functions is monotone: uses MF2 lemmas from Cont.thy *} 
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213 

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214 
lemma lub_cfun_mono: "chain(F) ==> monofun(% x. lub(range(% j.(F j)$x)))" 
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215 
apply (rule lub_MF2_mono) 
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216 
apply (rule monofun_Rep_CFun1) 
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217 
apply (rule monofun_Rep_CFun2 [THEN allI]) 
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218 
apply assumption 
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219 
done 
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220 

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221 
text {* a lemma about the exchange of lubs for type @{typ "'a > 'b"}: uses MF2 lemmas from Cont.thy *} 
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222 

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223 
lemma ex_lubcfun: "[ chain(F); chain(Y) ] ==> 
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lub(range(%j. lub(range(%i. F(j)$(Y i))))) = 
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225 
lub(range(%i. lub(range(%j. F(j)$(Y i)))))" 
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226 
apply (rule ex_lubMF2) 
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227 
apply (rule monofun_Rep_CFun1) 
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228 
apply (rule monofun_Rep_CFun2 [THEN allI]) 
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229 
apply assumption 
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230 
apply assumption 
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231 
done 
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232 

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233 
text {* the lub of a chain of cont. functions is continuous *} 
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234 

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235 
lemma cont_lubcfun: "chain(F) ==> cont(% x. lub(range(% j. F(j)$x)))" 
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236 
apply (rule monocontlub2cont) 
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237 
apply (erule lub_cfun_mono) 
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238 
apply (rule contlubI [rule_format]) 
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239 
apply (subst contlub_cfun_arg [THEN ext]) 
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240 
apply assumption 
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241 
apply (erule ex_lubcfun) 
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242 
apply assumption 
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243 
done 
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244 

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text {* type @{typ "'a > 'b"} is chain complete *} 
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246 

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lemma lub_cfun: "chain(CCF) ==> range(CCF) << (LAM x. lub(range(% i. CCF(i)$x)))" 
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248 
apply (rule is_lubI) 
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249 
apply (rule ub_rangeI) 
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250 
apply (subst less_cfun) 
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251 
apply (subst Abs_Cfun_inverse2) 
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252 
apply (erule cont_lubcfun) 
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253 
apply (rule lub_fun [THEN is_lubD1, THEN ub_rangeD]) 
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254 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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255 
apply (subst less_cfun) 
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256 
apply (subst Abs_Cfun_inverse2) 
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257 
apply (erule cont_lubcfun) 
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258 
apply (rule lub_fun [THEN is_lub_lub]) 
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259 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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260 
apply (erule monofun_Rep_CFun1 [THEN ub2ub_monofun]) 
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261 
done 
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262 

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263 
lemmas thelub_cfun = lub_cfun [THEN thelubI, standard] 
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264 
 {* @{thm thelub_cfun} *} (* 
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chain(?CCF1) ==> lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i$x))) 
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266 
*) 
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267 

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268 
lemma cpo_cfun: "chain(CCF::nat=>('a>'b)) ==> ? x. range(CCF) << x" 
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269 
apply (rule exI) 
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270 
apply (erule lub_cfun) 
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271 
done 
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272 

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273 
instance ">" :: (cpo, cpo) cpo 
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274 
by intro_classes (rule cpo_cfun) 
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275 

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276 
subsection {* Miscellaneous *} 
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277 

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text {* Extensionality in @{typ "'a > 'b"} *} 
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279 

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280 
lemma ext_cfun: "(!!x. f$x = g$x) ==> f = g" 
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281 
apply (rule Rep_CFun_inject [THEN iffD1]) 
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282 
apply (rule ext) 
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283 
apply simp 
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284 
done 
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285 

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286 
text {* Monotonicity of @{term Abs_CFun} *} 
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287 

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288 
lemma semi_monofun_Abs_CFun: "[ cont(f); cont(g); f<<g] ==> Abs_CFun(f)<<Abs_CFun(g)" 
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289 
by (simp add: less_cfun Abs_Cfun_inverse2) 
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290 

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291 
text {* Extensionality wrt. @{term "op <<"} in @{typ "'a > 'b"} *} 
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292 

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293 
lemma less_cfun2: "(!!x. f$x << g$x) ==> f << g" 
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294 
apply (rule_tac t = "f" in Rep_Cfun_inverse [THEN subst]) 
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295 
apply (rule_tac t = "g" in Rep_Cfun_inverse [THEN subst]) 
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296 
apply (rule semi_monofun_Abs_CFun) 
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parents:
diff
changeset

297 
apply (rule cont_Rep_CFun2) 
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parents:
diff
changeset

298 
apply (rule cont_Rep_CFun2) 
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parents:
diff
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299 
apply (rule less_fun [THEN iffD2]) 
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300 
apply simp 
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301 
done 
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302 

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303 
subsection {* Class instance of @{typ "'a > 'b"} for class pcpo *} 
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304 

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305 
instance ">" :: (cpo, pcpo) pcpo 
15576
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306 
by (intro_classes, rule least_cfun) 
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307 

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308 
text {* for compatibility with old HOLCFVersion *} 
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309 
lemma inst_cfun_pcpo: "UU = Abs_CFun(%x. UU)" 
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310 
apply (simp add: UU_def UU_cfun_def) 
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311 
done 
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changeset

312 

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313 
subsection {* Continuity of application *} 
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314 

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315 
text {* the contlub property for @{term Rep_CFun} its 'first' argument *} 
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316 

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317 
lemma contlub_Rep_CFun1: "contlub(Rep_CFun)" 
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318 
apply (rule contlubI [rule_format]) 
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319 
apply (rule ext) 
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320 
apply (subst thelub_cfun) 
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321 
apply assumption 
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parents:
diff
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322 
apply (subst Cfunapp2) 
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parents:
diff
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323 
apply (erule cont_lubcfun) 
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parents:
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324 
apply (subst thelub_fun) 
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325 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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326 
apply (rule refl) 
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327 
done 
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parents:
diff
changeset

328 

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329 
text {* the cont property for @{term Rep_CFun} in its first argument *} 
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330 

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331 
lemma cont_Rep_CFun1: "cont(Rep_CFun)" 
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parents:
diff
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332 
apply (rule monocontlub2cont) 
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parents:
diff
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333 
apply (rule monofun_Rep_CFun1) 
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parents:
diff
changeset

334 
apply (rule contlub_Rep_CFun1) 
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parents:
diff
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335 
done 
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parents:
diff
changeset

336 

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337 
text {* contlub, cont properties of @{term Rep_CFun} in its first argument in mixfix @{text "_[_]"} *} 
15576
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338 

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339 
lemma contlub_cfun_fun: 
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340 
"chain(FY) ==> 
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341 
lub(range FY)$x = lub(range (%i. FY(i)$x))" 
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parents:
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342 
apply (rule trans) 
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parents:
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changeset

343 
apply (erule contlub_Rep_CFun1 [THEN contlubE, THEN spec, THEN mp, THEN fun_cong]) 
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parents:
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344 
apply (subst thelub_fun) 
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parents:
diff
changeset

345 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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parents:
diff
changeset

346 
apply (rule refl) 
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diff
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347 
done 
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converted to newstyle theories, and combined numbered files
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parents:
diff
changeset

348 

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parents:
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349 
lemma cont_cfun_fun: 
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350 
"chain(FY) ==> 
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parents:
diff
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351 
range(%i. FY(i)$x) << lub(range FY)$x" 
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parents:
diff
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352 
apply (rule thelubE) 
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parents:
diff
changeset

353 
apply (erule ch2ch_Rep_CFunL) 
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parents:
diff
changeset

354 
apply (erule contlub_cfun_fun [symmetric]) 
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diff
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355 
done 
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parents:
diff
changeset

356 

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357 
text {* contlub, cont properties of @{term Rep_CFun} in both argument in mixfix @{text "_[_]"} *} 
15576
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358 

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diff
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359 
lemma contlub_cfun: 
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diff
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360 
"[chain(FY);chain(TY)] ==> 
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diff
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361 
(lub(range FY))$(lub(range TY)) = lub(range(%i. FY(i)$(TY i)))" 
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parents:
diff
changeset

362 
apply (rule contlub_CF2) 
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huffman
parents:
diff
changeset

363 
apply (rule cont_Rep_CFun1) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

364 
apply (rule allI) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

365 
apply (rule cont_Rep_CFun2) 
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huffman
parents:
diff
changeset

366 
apply assumption 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

367 
apply assumption 
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huffman
parents:
diff
changeset

368 
done 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

369 

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370 
lemma cont_cfun: 
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parents:
diff
changeset

371 
"[chain(FY);chain(TY)] ==> 
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huffman
parents:
diff
changeset

372 
range(%i.(FY i)$(TY i)) << (lub (range FY))$(lub(range TY))" 
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parents:
diff
changeset

373 
apply (rule thelubE) 
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huffman
parents:
diff
changeset

374 
apply (rule monofun_Rep_CFun1 [THEN ch2ch_MF2LR]) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

375 
apply (rule allI) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

376 
apply (rule monofun_Rep_CFun2) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

377 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

378 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

379 
apply (erule contlub_cfun [symmetric]) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

380 
apply assumption 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

381 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

382 

15589
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383 
text {* cont2cont lemma for @{term Rep_CFun} *} 
15576
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huffman
parents:
diff
changeset

384 

efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

385 
lemma cont2cont_Rep_CFun: "[cont(%x. ft x);cont(%x. tt x)] ==> cont(%x. (ft x)$(tt x))" 
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huffman
parents:
diff
changeset

386 
apply (best intro: cont2cont_app2 cont_const cont_Rep_CFun1 cont_Rep_CFun2) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

387 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

388 

15589
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changeset

389 
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
15576
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huffman
parents:
diff
changeset

390 

efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

391 
lemma cont2mono_LAM: 
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huffman
parents:
diff
changeset

392 
assumes p1: "!!x. cont(c1 x)" 
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huffman
parents:
diff
changeset

393 
assumes p2: "!!y. monofun(%x. c1 x y)" 
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huffman
parents:
diff
changeset

394 
shows "monofun(%x. LAM y. c1 x y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

395 
apply (rule monofunI) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

396 
apply (intro strip) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

397 
apply (subst less_cfun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

398 
apply (subst less_fun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

399 
apply (rule allI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

400 
apply (subst beta_cfun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

401 
apply (rule p1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

402 
apply (subst beta_cfun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

403 
apply (rule p1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

404 
apply (erule p2 [THEN monofunE, THEN spec, THEN spec, THEN mp]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

405 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

406 

15589
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huffman
parents:
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diff
changeset

407 
text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

408 

efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

409 
lemma cont2cont_LAM: 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

410 
assumes p1: "!!x. cont(c1 x)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

411 
assumes p2: "!!y. cont(%x. c1 x y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

412 
shows "cont(%x. LAM y. c1 x y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

413 
apply (rule monocontlub2cont) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

414 
apply (rule p1 [THEN cont2mono_LAM]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

415 
apply (rule p2 [THEN cont2mono]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

416 
apply (rule contlubI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

417 
apply (intro strip) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

418 
apply (subst thelub_cfun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

419 
apply (rule p1 [THEN cont2mono_LAM, THEN ch2ch_monofun]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

420 
apply (rule p2 [THEN cont2mono]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

421 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

422 
apply (rule_tac f = "Abs_CFun" in arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

423 
apply (rule ext) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

424 
apply (subst p1 [THEN beta_cfun, THEN ext]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

425 
apply (erule p2 [THEN cont2contlub, THEN contlubE, THEN spec, THEN mp]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

426 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

427 

15641  428 
text {* cont2cont Lemma for @{term "%x. LAM y. c1 x$y"} *} 
429 

430 
lemma cont2cont_eta: "cont c1 ==> cont (%x. LAM y. c1 x$y)" 

431 
by (simp only: eta_cfun) 

432 

15589
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huffman
parents:
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diff
changeset

433 
text {* cont2cont tactic *} 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

434 

16055  435 
lemmas cont_lemmas1 = 
436 
cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM 

437 

438 
text {* 

439 
Continuity simproc by Brian Huffman. 

440 
Given the term @{term "cont f"}, the procedure tries to 

441 
construct the theorem @{prop "cont f == True"}. If this 

442 
theorem cannot be completely solved by the introduction 

443 
rules, then the procedure returns a conditional rewrite 

444 
rule with the unsolved subgoals as premises. 

445 
*} 

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446 

16055  447 
ML_setup {* 
448 
local 

449 
val rules = thms "cont_lemmas1"; 

450 
fun solve_cont sg _ t = 

451 
let val tr = instantiate' [] [SOME (cterm_of sg t)] Eq_TrueI; 

452 
val tac = REPEAT_ALL_NEW (resolve_tac rules) 1; 

453 
in Option.map fst (Seq.pull (tac tr)) end; 

454 
in 

455 
val cont_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 

456 
"continuity" ["cont f"] solve_cont; 

457 
end; 

458 
Addsimprocs [cont_proc]; 

459 
*} 

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changeset

460 

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461 
text {* HINT: @{text cont_tac} is now installed in simplifier in Lift.ML ! *} 
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changeset

462 

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changeset

463 
(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*) 
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464 
(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*) 
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changeset

465 

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466 
text {* function application @{text "_[_]"} is strict in its first arguments *} 
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changeset

467 

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468 
lemma strict_Rep_CFun1 [simp]: "\<bottom>\<cdot>x = \<bottom>" 
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469 
by (simp add: inst_cfun_pcpo beta_cfun) 
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changeset

470 

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471 
text {* Instantiate the simplifier *} 
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diff
changeset

472 

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473 
declare beta_cfun [simp] 
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changeset

474 

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475 
text {* some lemmata for functions with flat/chfin domain/range types *} 
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changeset

476 

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diff
changeset

477 
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
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changeset

478 
==> !s. ? n. lub(range(Y))$s = Y n$s" 
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parents:
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changeset

479 
apply (rule allI) 
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parents:
diff
changeset

480 
apply (subst contlub_cfun_fun) 
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parents:
diff
changeset

481 
apply assumption 
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parents:
diff
changeset

482 
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) 
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parents:
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changeset

483 
done 
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parents:
diff
changeset

484 

16085
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485 
subsection {* Continuous injectionretraction pairs *} 
15589
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diff
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486 

16085
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487 
text {* Continuous retractions are strict. *} 
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changeset

488 

16085
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parents:
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diff
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489 
lemma retraction_strict: 
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parents:
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changeset

490 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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changeset

491 
apply (rule UU_I) 
16085
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parents:
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diff
changeset

492 
apply (drule_tac x="\<bottom>" in spec) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

493 
apply (erule subst) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

494 
apply (rule monofun_cfun_arg) 
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rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

495 
apply (rule minimal) 
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496 
done 
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huffman
parents:
diff
changeset

497 

16085
c004b9bc970e
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parents:
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diff
changeset

498 
lemma injection_eq: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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parents:
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diff
changeset

499 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

500 
apply (rule iffI) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

501 
apply (drule_tac f=f in cfun_arg_cong) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

502 
apply simp 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

503 
apply simp 
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huffman
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changeset

504 
done 
efb95d0d01f7
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huffman
parents:
diff
changeset

505 

16085
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rewrote continuous isomorphism section, cleaned up
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parents:
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diff
changeset

506 
lemma injection_defined_rev: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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parents:
16070
diff
changeset

507 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

508 
apply (drule_tac f=f in cfun_arg_cong) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

509 
apply (simp add: retraction_strict) 
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huffman
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diff
changeset

510 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

511 

16085
c004b9bc970e
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huffman
parents:
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diff
changeset

512 
lemma injection_defined: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

513 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

514 
by (erule contrapos_nn, rule injection_defined_rev) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

515 

c004b9bc970e
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huffman
parents:
16070
diff
changeset

516 
text {* propagation of flatness and chainfiniteness by retractions *} 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

517 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

518 
lemma chfin2chfin: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

519 
"\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

520 
\<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

521 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

522 
apply (drule_tac f=g in chain_monofun) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

523 
apply (drule chfin [rule_format]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

524 
apply (unfold max_in_chain_def) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

525 
apply (simp add: injection_eq) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

526 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

527 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

528 
lemma flat2flat: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

529 
"\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

530 
\<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

531 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

532 
apply (drule_tac fo=g in monofun_cfun_arg) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

533 
apply (drule ax_flat [rule_format]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

534 
apply (erule disjE) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

535 
apply (simp add: injection_defined_rev) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

536 
apply (simp add: injection_eq) 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

537 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

538 

15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

539 
text {* a result about functions with flat codomain *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

540 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

541 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

542 
by (drule ax_flat [rule_format], simp) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

543 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

544 
lemma flat_codom: 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

545 
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

546 
apply (case_tac "f\<cdot>x = \<bottom>") 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

547 
apply (rule disjI1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

548 
apply (rule UU_I) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

549 
apply (erule_tac t="\<bottom>" in subst) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

550 
apply (rule minimal [THEN monofun_cfun_arg]) 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

551 
apply clarify 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

552 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

553 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

554 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

555 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

556 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

557 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

558 
subsection {* Identity and composition *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

559 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

560 
consts 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

561 
ID :: "'a \<rightarrow> 'a" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

562 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

563 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

564 
syntax "@oo" :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

565 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

566 
translations "f1 oo f2" == "cfcomp$f1$f2" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

567 

69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

568 
defs 
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

569 
ID_def: "ID \<equiv> (\<Lambda> x. x)" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

570 
oo_def: "cfcomp \<equiv> (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

571 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

572 
lemma ID1 [simp]: "ID\<cdot>x = x" 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

573 
by (simp add: ID_def) 
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huffman
parents:
diff
changeset

574 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

575 
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

576 
by (simp add: oo_def) 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

577 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

578 
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

579 
by (simp add: cfcomp1) 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

580 

15589
69bea57212ef
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huffman
parents:
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changeset

581 
text {* 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

582 
Show that interpretation of (pcpo,@{text "_>_"}) is a category. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

583 
The class of objects is interpretation of syntactical class pcpo. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

584 
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a > 'b"}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

585 
The identity arrow is interpretation of @{term ID}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

586 
The composition of f and g is interpretation of @{text "oo"}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

587 
*} 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

588 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

589 
lemma ID2 [simp]: "f oo ID = f" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

590 
by (rule ext_cfun, simp) 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

591 

16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset

592 
lemma ID3 [simp]: "ID oo f = f" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

593 
by (rule ext_cfun, simp) 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

594 

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595 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
15589
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596 
by (rule ext_cfun, simp) 
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597 

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598 

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599 
subsection {* Strictified functions *} 
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600 

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601 
defaultsort pcpo 
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602 

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603 
consts 
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604 
Istrictify :: "('a \<rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" 
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605 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" 
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606 

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607 
defs 
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608 
Istrictify_def: "Istrictify f x \<equiv> if x = \<bottom> then \<bottom> else f\<cdot>x" 
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609 
strictify_def: "strictify \<equiv> (\<Lambda> f x. Istrictify f x)" 
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610 

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611 
text {* results about strictify *} 
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612 

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613 
lemma Istrictify1: "Istrictify f \<bottom> = \<bottom>" 
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614 
by (simp add: Istrictify_def) 
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615 

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616 
lemma Istrictify2: "x \<noteq> \<bottom> \<Longrightarrow> Istrictify f x = f\<cdot>x" 
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617 
by (simp add: Istrictify_def) 
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618 

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619 
lemma monofun_Istrictify1: "monofun (\<lambda>f. Istrictify f x)" 
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620 
apply (rule monofunI [rule_format]) 
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621 
apply (simp add: Istrictify_def monofun_cfun_fun) 
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622 
done 
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623 

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624 
lemma monofun_Istrictify2: "monofun (\<lambda>x. Istrictify f x)" 
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625 
apply (rule monofunI [rule_format]) 
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626 
apply (simp add: Istrictify_def monofun_cfun_arg) 
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627 
apply clarify 
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628 
apply (simp add: eq_UU_iff) 
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629 
done 
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630 

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631 
lemma contlub_Istrictify1: "contlub (\<lambda>f. Istrictify f x)" 
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632 
apply (rule contlubI [rule_format]) 
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633 
apply (case_tac "x = \<bottom>") 
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634 
apply (simp add: Istrictify1) 
16093  635 
apply (simp add: thelub_const) 
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636 
apply (simp add: Istrictify2) 
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637 
apply (erule contlub_cfun_fun) 
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638 
done 
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639 

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640 
lemma contlub_Istrictify2: "contlub (\<lambda>x. Istrictify f x)" 
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641 
apply (rule contlubI [rule_format]) 
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642 
apply (case_tac "lub (range Y) = \<bottom>") 
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643 
apply (simp add: Istrictify1 chain_UU_I) 
16093  644 
apply (simp add: thelub_const) 
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645 
apply (simp add: Istrictify2) 
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646 
apply (simp add: contlub_cfun_arg) 
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647 
apply (rule lub_equal2) 
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648 
apply (rule chain_mono2 [THEN exE]) 
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649 
apply (erule chain_UU_I_inverse2) 
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650 
apply (assumption) 
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651 
apply (blast intro: Istrictify2 [symmetric]) 
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652 
apply (erule chain_monofun) 
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653 
apply (erule monofun_Istrictify2 [THEN ch2ch_monofun]) 
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654 
done 
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655 

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656 
lemmas cont_Istrictify1 = 
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657 
monocontlub2cont [OF monofun_Istrictify1 contlub_Istrictify1, standard] 
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658 

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659 
lemmas cont_Istrictify2 = 
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660 
monocontlub2cont [OF monofun_Istrictify2 contlub_Istrictify2, standard] 
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661 

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662 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
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663 
apply (unfold strictify_def) 
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664 
apply (simp add: cont_Istrictify1 cont_Istrictify2) 
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665 
apply (rule Istrictify1) 
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666 
done 
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667 

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668 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
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669 
apply (unfold strictify_def) 
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670 
apply (simp add: cont_Istrictify1 cont_Istrictify2) 
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671 
apply (erule Istrictify2) 
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672 
done 
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673 

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674 
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" 
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675 
by simp 
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676 

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677 
end 