author  huffman 
Tue, 08 Mar 2005 00:11:49 +0100  
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parent 15577  e16da3068ad6 
child 15600  a59f07556a8d 
permissions  rwrr 
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(* Title: HOLCF/Cfun1.thy 
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ID: $Id$ 
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Author: Franz Regensburger 
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License: GPL (GNU GENERAL PUBLIC LICENSE) 
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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  12 
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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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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146 

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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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149 

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150 
lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard] 
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 {* @{thm contlub_Rep_CFun2} *} (* contlub(Rep_CFun(?fo)) *) 
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152 

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text {* expanded thms @{thm [source] cont_Rep_CFun2}, @{thm [source] contlub_Rep_CFun2} look nice with mixfix syntax *} 
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154 

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155 
lemmas cont_cfun_arg = cont_Rep_CFun2 [THEN contE, THEN spec, THEN mp] 
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156 
 {* @{thm cont_cfun_arg} *} (* chain(x1) ==> range (%i. fo3$(x1 i)) << fo3$(lub (range ?x1)) *) 
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157 

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158 
lemmas contlub_cfun_arg = contlub_Rep_CFun2 [THEN contlubE, THEN spec, THEN mp] 
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159 
 {* @{thm contlub_cfun_arg} *} (* chain(?x1) ==> ?fo4$(lub (range ?x1)) = lub (range (%i. ?fo4$(?x1 i))) *) 
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160 

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161 
text {* @{term Rep_CFun} is monotone in its 'first' argument *} 
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162 

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lemma monofun_Rep_CFun1: "monofun(Rep_CFun)" 
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164 
apply (rule monofunI [rule_format]) 
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165 
apply (erule less_cfun [THEN subst]) 
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166 
done 
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167 

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text {* monotonicity of application @{term Rep_CFun} in mixfix syntax @{text "[_]_"} *} 
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169 

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lemma monofun_cfun_fun: "f1 << f2 ==> f1$x << f2$x" 
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171 
apply (rule_tac x = "x" in spec) 
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172 
apply (rule less_fun [THEN subst]) 
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apply (erule monofun_Rep_CFun1 [THEN monofunE [rule_format]]) 
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174 
done 
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175 

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176 
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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178 

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lemma chain_monofun: "chain Y ==> chain (%i. f\<cdot>(Y i))" 
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180 
apply (rule chainI) 
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181 
apply (rule monofun_cfun_arg) 
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182 
apply (erule chainE) 
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183 
done 
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184 

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text {* monotonicity of @{term Rep_CFun} in both arguments in mixfix syntax @{text "[_]_"} *} 
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186 

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187 
lemma monofun_cfun: "[f1<<f2;x1<<x2] ==> f1$x1 << f2$x2" 
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188 
apply (rule trans_less) 
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189 
apply (erule monofun_cfun_arg) 
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190 
apply (erule monofun_cfun_fun) 
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191 
done 
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192 

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lemma strictI: "f$x = UU ==> f$UU = UU" 
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194 
apply (rule eq_UU_iff [THEN iffD2]) 
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195 
apply (erule subst) 
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196 
apply (rule minimal [THEN monofun_cfun_arg]) 
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197 
done 
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198 

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199 
subsection {* Type @{typ "'a > 'b"} is a cpo *} 
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200 

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201 
text {* ch2ch  rules for the type @{typ "'a > 'b"} use MF2 lemmas from Cont.thy *} 
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202 

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203 
lemma ch2ch_Rep_CFunR: "chain(Y) ==> chain(%i. f$(Y i))" 
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204 
by (erule monofun_Rep_CFun2 [THEN ch2ch_MF2R]) 
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205 

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206 
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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208 

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209 
text {* the lub of a chain of continous functions is monotone: uses MF2 lemmas from Cont.thy *} 
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210 

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211 
lemma lub_cfun_mono: "chain(F) ==> monofun(% x. lub(range(% j.(F j)$x)))" 
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212 
apply (rule lub_MF2_mono) 
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213 
apply (rule monofun_Rep_CFun1) 
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214 
apply (rule monofun_Rep_CFun2 [THEN allI]) 
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215 
apply assumption 
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216 
done 
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217 

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218 
text {* a lemma about the exchange of lubs for type @{typ "'a > 'b"}: uses MF2 lemmas from Cont.thy *} 
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219 

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220 
lemma ex_lubcfun: "[ chain(F); chain(Y) ] ==> 
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lub(range(%j. lub(range(%i. F(j)$(Y i))))) = 
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222 
lub(range(%i. lub(range(%j. F(j)$(Y i)))))" 
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223 
apply (rule ex_lubMF2) 
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224 
apply (rule monofun_Rep_CFun1) 
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225 
apply (rule monofun_Rep_CFun2 [THEN allI]) 
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226 
apply assumption 
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227 
apply assumption 
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228 
done 
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229 

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230 
text {* the lub of a chain of cont. functions is continuous *} 
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231 

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232 
lemma cont_lubcfun: "chain(F) ==> cont(% x. lub(range(% j. F(j)$x)))" 
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233 
apply (rule monocontlub2cont) 
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234 
apply (erule lub_cfun_mono) 
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235 
apply (rule contlubI [rule_format]) 
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236 
apply (subst contlub_cfun_arg [THEN ext]) 
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237 
apply assumption 
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238 
apply (erule ex_lubcfun) 
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239 
apply assumption 
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240 
done 
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241 

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242 
text {* type @{typ "'a > 'b"} is chain complete *} 
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243 

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lemma lub_cfun: "chain(CCF) ==> range(CCF) << (LAM x. lub(range(% i. CCF(i)$x)))" 
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245 
apply (rule is_lubI) 
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246 
apply (rule ub_rangeI) 
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247 
apply (subst less_cfun) 
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248 
apply (subst Abs_Cfun_inverse2) 
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249 
apply (erule cont_lubcfun) 
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250 
apply (rule lub_fun [THEN is_lubD1, THEN ub_rangeD]) 
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251 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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252 
apply (subst less_cfun) 
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253 
apply (subst Abs_Cfun_inverse2) 
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254 
apply (erule cont_lubcfun) 
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255 
apply (rule lub_fun [THEN is_lub_lub]) 
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256 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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257 
apply (erule monofun_Rep_CFun1 [THEN ub2ub_monofun]) 
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258 
done 
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259 

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260 
lemmas thelub_cfun = lub_cfun [THEN thelubI, standard] 
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261 
 {* @{thm thelub_cfun} *} (* 
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262 
chain(?CCF1) ==> lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i$x))) 
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263 
*) 
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264 

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lemma cpo_cfun: "chain(CCF::nat=>('a>'b)) ==> ? x. range(CCF) << x" 
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266 
apply (rule exI) 
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267 
apply (erule lub_cfun) 
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268 
done 
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269 

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270 
instance ">" :: (cpo, cpo) cpo 
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271 
by intro_classes (rule cpo_cfun) 
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272 

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273 
subsection {* Miscellaneous *} 
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274 

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text {* Extensionality in @{typ "'a > 'b"} *} 
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276 

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277 
lemma ext_cfun: "(!!x. f$x = g$x) ==> f = g" 
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278 
apply (rule Rep_CFun_inject [THEN iffD1]) 
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279 
apply (rule ext) 
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280 
apply simp 
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281 
done 
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282 

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283 
text {* Monotonicity of @{term Abs_CFun} *} 
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284 

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285 
lemma semi_monofun_Abs_CFun: "[ cont(f); cont(g); f<<g] ==> Abs_CFun(f)<<Abs_CFun(g)" 
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286 
by (simp add: less_cfun Abs_Cfun_inverse2) 
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287 

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288 
text {* Extensionality wrt. @{term "op <<"} in @{typ "'a > 'b"} *} 
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289 

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290 
lemma less_cfun2: "(!!x. f$x << g$x) ==> f << g" 
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291 
apply (rule_tac t = "f" in Rep_Cfun_inverse [THEN subst]) 
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292 
apply (rule_tac t = "g" in Rep_Cfun_inverse [THEN subst]) 
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293 
apply (rule semi_monofun_Abs_CFun) 
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parents:
diff
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294 
apply (rule cont_Rep_CFun2) 
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parents:
diff
changeset

295 
apply (rule cont_Rep_CFun2) 
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parents:
diff
changeset

296 
apply (rule less_fun [THEN iffD2]) 
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parents:
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297 
apply simp 
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298 
done 
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299 

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300 
subsection {* Class instance of @{typ "'a > 'b"} for class pcpo *} 
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301 

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302 
instance ">" :: (cpo, pcpo) pcpo 
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303 
by (intro_classes, rule least_cfun) 
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304 

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305 
text {* for compatibility with old HOLCFVersion *} 
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306 
lemma inst_cfun_pcpo: "UU = Abs_CFun(%x. UU)" 
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307 
apply (simp add: UU_def UU_cfun_def) 
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308 
done 
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changeset

309 

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310 
defaultsort pcpo 
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311 

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312 
subsection {* Continuity of application *} 
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313 

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314 
text {* the contlub property for @{term Rep_CFun} its 'first' argument *} 
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315 

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316 
lemma contlub_Rep_CFun1: "contlub(Rep_CFun)" 
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317 
apply (rule contlubI [rule_format]) 
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318 
apply (rule ext) 
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319 
apply (subst thelub_cfun) 
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320 
apply assumption 
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parents:
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321 
apply (subst Cfunapp2) 
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parents:
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322 
apply (erule cont_lubcfun) 
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323 
apply (subst thelub_fun) 
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324 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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325 
apply (rule refl) 
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326 
done 
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parents:
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changeset

327 

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328 
text {* the cont property for @{term Rep_CFun} in its first argument *} 
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329 

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330 
lemma cont_Rep_CFun1: "cont(Rep_CFun)" 
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331 
apply (rule monocontlub2cont) 
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parents:
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changeset

332 
apply (rule monofun_Rep_CFun1) 
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parents:
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changeset

333 
apply (rule contlub_Rep_CFun1) 
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changeset

334 
done 
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parents:
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changeset

335 

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336 
text {* contlub, cont properties of @{term Rep_CFun} in its first argument in mixfix @{text "_[_]"} *} 
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337 

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338 
lemma contlub_cfun_fun: 
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339 
"chain(FY) ==> 
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340 
lub(range FY)$x = lub(range (%i. FY(i)$x))" 
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parents:
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changeset

341 
apply (rule trans) 
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parents:
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changeset

342 
apply (erule contlub_Rep_CFun1 [THEN contlubE, THEN spec, THEN mp, THEN fun_cong]) 
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parents:
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343 
apply (subst thelub_fun) 
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parents:
diff
changeset

344 
apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun]) 
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parents:
diff
changeset

345 
apply (rule refl) 
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diff
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346 
done 
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parents:
diff
changeset

347 

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348 
lemma cont_cfun_fun: 
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349 
"chain(FY) ==> 
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350 
range(%i. FY(i)$x) << lub(range FY)$x" 
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parents:
diff
changeset

351 
apply (rule thelubE) 
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parents:
diff
changeset

352 
apply (erule ch2ch_Rep_CFunL) 
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parents:
diff
changeset

353 
apply (erule contlub_cfun_fun [symmetric]) 
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diff
changeset

354 
done 
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parents:
diff
changeset

355 

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356 
text {* contlub, cont properties of @{term Rep_CFun} in both argument in mixfix @{text "_[_]"} *} 
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357 

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358 
lemma contlub_cfun: 
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diff
changeset

359 
"[chain(FY);chain(TY)] ==> 
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changeset

360 
(lub(range FY))$(lub(range TY)) = lub(range(%i. FY(i)$(TY i)))" 
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parents:
diff
changeset

361 
apply (rule contlub_CF2) 
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huffman
parents:
diff
changeset

362 
apply (rule cont_Rep_CFun1) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

364 
apply (rule cont_Rep_CFun2) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

365 
apply assumption 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

366 
apply assumption 
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parents:
diff
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367 
done 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

368 

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parents:
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changeset

369 
lemma cont_cfun: 
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parents:
diff
changeset

370 
"[chain(FY);chain(TY)] ==> 
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huffman
parents:
diff
changeset

371 
range(%i.(FY i)$(TY i)) << (lub (range FY))$(lub(range TY))" 
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parents:
diff
changeset

372 
apply (rule thelubE) 
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parents:
diff
changeset

373 
apply (rule monofun_Rep_CFun1 [THEN ch2ch_MF2LR]) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

374 
apply (rule allI) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

375 
apply (rule monofun_Rep_CFun2) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

376 
apply assumption 
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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 (erule contlub_cfun [symmetric]) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

379 
apply assumption 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

381 

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382 
text {* cont2cont lemma for @{term Rep_CFun} *} 
15576
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diff
changeset

383 

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

384 
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

385 
apply (best intro: cont2cont_app2 cont_const cont_Rep_CFun1 cont_Rep_CFun2) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

386 
done 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

387 

15589
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changeset

388 
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
15576
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huffman
parents:
diff
changeset

389 

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

390 
lemma cont2mono_LAM: 
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huffman
parents:
diff
changeset

391 
assumes p1: "!!x. cont(c1 x)" 
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huffman
parents:
diff
changeset

392 
assumes p2: "!!y. monofun(%x. c1 x y)" 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

393 
shows "monofun(%x. LAM y. c1 x y)" 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

394 
apply (rule monofunI) 
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

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

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

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

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

400 
apply (rule p1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

402 
apply (rule p1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

403 
apply (erule p2 [THEN monofunE, THEN spec, THEN spec, THEN mp]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

404 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

405 

15589
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parents:
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diff
changeset

406 
text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *} 
15576
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converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

407 

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

408 
lemma cont2cont_LAM: 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

409 
assumes p1: "!!x. cont(c1 x)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

410 
assumes p2: "!!y. cont(%x. c1 x y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

412 
apply (rule monocontlub2cont) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

413 
apply (rule p1 [THEN cont2mono_LAM]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

414 
apply (rule p2 [THEN cont2mono]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

415 
apply (rule contlubI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

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

418 
apply (rule p1 [THEN cont2mono_LAM, THEN ch2ch_monofun]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

419 
apply (rule p2 [THEN cont2mono]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

420 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

421 
apply (rule_tac f = "Abs_CFun" in arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

422 
apply (rule ext) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

423 
apply (subst p1 [THEN beta_cfun, THEN ext]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

424 
apply (erule p2 [THEN cont2contlub, THEN contlubE, THEN spec, THEN mp]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

425 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

426 

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

427 
text {* cont2cont tactic *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

428 

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

429 
lemmas cont_lemmas1 = cont_const cont_id cont_Rep_CFun2 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

430 
cont2cont_Rep_CFun cont2cont_LAM 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

431 

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

432 
declare cont_lemmas1 [simp] 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

433 

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

434 
text {* HINT: @{text cont_tac} is now installed in simplifier in Lift.ML ! *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

435 

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

436 
(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

437 
(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

438 

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

439 
text {* function application @{text "_[_]"} is strict in its first arguments *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

440 

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

441 
lemma strict_Rep_CFun1 [simp]: "(UU::'a::cpo>'b)$x = (UU::'b)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

442 
by (simp add: inst_cfun_pcpo beta_cfun) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

443 

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

444 
text {* Instantiate the simplifier *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

445 

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

446 
declare beta_cfun [simp] 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

447 

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

448 
text {* use @{text cont_tac} as autotac. *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

449 

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

450 
text {* HINT: @{text cont_tac} is now installed in simplifier in Lift.ML ! *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

451 
(*simpset_ref() := simpset() addsolver (K (DEPTH_SOLVE_1 o cont_tac));*) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

452 

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

453 
text {* some lemmata for functions with flat/chfin domain/range types *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

454 

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

455 
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

456 
==> !s. ? n. lub(range(Y))$s = Y n$s" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

457 
apply (rule allI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

458 
apply (subst contlub_cfun_fun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

459 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

460 
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

461 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

462 

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

463 
subsection {* Continuous isomorphisms *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

464 

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

465 
text {* Continuous isomorphisms are strict. A proof for embedding projection pairs is similar. *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

466 

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

467 
lemma iso_strict: 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

468 
"!!f g.[!y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a) ] 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

469 
==> f$UU=UU & g$UU=UU" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

470 
apply (rule conjI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

471 
apply (rule UU_I) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

472 
apply (rule_tac s = "f$ (g$ (UU::'b))" and t = "UU::'b" in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

473 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

474 
apply (rule minimal [THEN monofun_cfun_arg]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

475 
apply (rule UU_I) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

476 
apply (rule_tac s = "g$ (f$ (UU::'a))" and t = "UU::'a" in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

477 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

478 
apply (rule minimal [THEN monofun_cfun_arg]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

479 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

480 

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

481 
lemma isorep_defined: "[!x. rep$(ab$x)=x;!y. ab$(rep$y)=y; z~=UU] ==> rep$z ~= UU" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

482 
apply (erule contrapos_nn) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

483 
apply (drule_tac f = "ab" in cfun_arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

484 
apply (erule box_equals) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

485 
apply fast 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

486 
apply (erule iso_strict [THEN conjunct1]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

487 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

488 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

489 

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

490 
lemma isoabs_defined: "[!x. rep$(ab$x) = x;!y. ab$(rep$y)=y ; z~=UU] ==> ab$z ~= UU" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

491 
apply (erule contrapos_nn) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

492 
apply (drule_tac f = "rep" in cfun_arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

493 
apply (erule box_equals) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

494 
apply fast 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

495 
apply (erule iso_strict [THEN conjunct2]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

496 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

497 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

498 

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

499 
text {* propagation of flatness and chainfiniteness by continuous isomorphisms *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

500 

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

501 
lemma chfin2chfin: "!!f g.[! Y::nat=>'a. chain Y > (? n. max_in_chain n Y); 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

502 
!y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a::chfin) ] 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

503 
==> ! Y::nat=>'b. chain Y > (? n. max_in_chain n Y)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

504 
apply (unfold max_in_chain_def) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

505 
apply (clarify) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

506 
apply (rule exE) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

507 
apply (rule_tac P = "chain (%i. g$ (Y i))" in mp) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

508 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

509 
apply (erule ch2ch_Rep_CFunR) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

510 
apply (rule exI) 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

511 
apply (clarify) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

512 
apply (rule_tac s = "f$ (g$ (Y x))" and t = "Y (x) " in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

513 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

514 
apply (rule_tac s = "f$ (g$ (Y j))" and t = "Y (j) " in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

515 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

516 
apply (rule cfun_arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

517 
apply (rule mp) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

518 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

519 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

520 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

521 

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

522 
lemma flat2flat: "!!f g.[!x y::'a. x<<y > x=UU  x=y; 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

523 
!y. f$(g$y)=(y::'b); !x. g$(f$x)=(x::'a)] ==> !x y::'b. x<<y > x=UU  x=y" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

524 
apply (intro strip) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

525 
apply (rule disjE) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

526 
apply (rule_tac P = "g$x<<g$y" in mp) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

527 
apply (erule_tac [2] monofun_cfun_arg) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

528 
apply (drule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

529 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

530 
apply (rule disjI1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

531 
apply (rule trans) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

532 
apply (rule_tac s = "f$ (g$x) " and t = "x" in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

533 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

534 
apply (erule cfun_arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

535 
apply (rule iso_strict [THEN conjunct1]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

536 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

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

538 
apply (rule disjI2) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

539 
apply (rule_tac s = "f$ (g$x) " and t = "x" in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

540 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

541 
apply (rule_tac s = "f$ (g$y) " and t = "y" in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

542 
apply (erule spec) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

543 
apply (erule cfun_arg_cong) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

544 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

545 

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

546 
text {* a result about functions with flat codomain *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

547 

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

548 
lemma flat_codom: "f$(x::'a)=(c::'b::flat) ==> f$(UU::'a)=(UU::'b)  (!z. f$(z::'a)=c)" 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

549 
apply (case_tac "f$ (x::'a) = (UU::'b) ") 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

550 
apply (rule disjI1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

551 
apply (rule UU_I) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

552 
apply (rule_tac s = "f$ (x) " and t = "UU::'b" in subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

553 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

554 
apply (rule minimal [THEN monofun_cfun_arg]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

555 
apply (case_tac "f$ (UU::'a) = (UU::'b) ") 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

556 
apply (erule disjI1) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

557 
apply (rule disjI2) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

558 
apply (rule allI) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

559 
apply (erule subst) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

560 
apply (rule_tac a = "f$ (UU::'a) " in refl [THEN box_equals]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

561 
apply (rule_tac fo5 = "f" in minimal [THEN monofun_cfun_arg, THEN ax_flat [THEN spec, THEN spec, THEN mp], THEN disjE]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

562 
apply simp 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

563 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

564 
apply (rule_tac fo5 = "f" in minimal [THEN monofun_cfun_arg, THEN ax_flat [THEN spec, THEN spec, THEN mp], THEN disjE]) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

565 
apply simp 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

566 
apply assumption 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

567 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

568 

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

569 
subsection {* Strictified functions *} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

570 

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

571 
consts 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

572 
Istrictify :: "('a>'b)=>'a=>'b" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

573 
strictify :: "('a>'b)>'a>'b" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

574 
defs 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

575 

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

576 
Istrictify_def: "Istrictify f x == if x=UU then UU else f$x" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

577 
strictify_def: "strictify == (LAM f x. Istrictify f x)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

578 

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

579 
text {* results about strictify *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

580 

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

581 
lemma Istrictify1: 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

582 
"Istrictify(f)(UU)= (UU)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

583 
apply (unfold Istrictify_def) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

584 
apply (simp (no_asm)) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

585 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

586 

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

587 
lemma Istrictify2: 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

588 
"~x=UU ==> Istrictify(f)(x)=f$x" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

589 
by (simp add: Istrictify_def) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

590 

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

591 
lemma monofun_Istrictify1: "monofun(Istrictify)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

592 
apply (rule monofunI [rule_format]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

593 
apply (rule less_fun [THEN iffD2, rule_format]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

594 
apply (case_tac "xa=UU") 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

595 
apply (simp add: Istrictify1) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

596 
apply (simp add: Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

597 
apply (erule monofun_cfun_fun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

598 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

599 

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

600 
lemma monofun_Istrictify2: "monofun(Istrictify(f))" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

601 
apply (rule monofunI [rule_format]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

602 
apply (case_tac "x=UU") 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

603 
apply (simp add: Istrictify1) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

604 
apply (frule notUU_I) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

605 
apply assumption 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

606 
apply (simp add: Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

607 
apply (erule monofun_cfun_arg) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

608 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

609 

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

610 
lemma contlub_Istrictify1: "contlub(Istrictify)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

611 
apply (rule contlubI [rule_format]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

612 
apply (rule ext) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

613 
apply (subst thelub_fun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

614 
apply (erule monofun_Istrictify1 [THEN ch2ch_monofun]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

615 
apply (case_tac "x=UU") 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

616 
apply (simp add: Istrictify1) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

617 
apply (simp add: lub_const [THEN thelubI]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

618 
apply (simp add: Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

619 
apply (erule contlub_cfun_fun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

620 
done 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

621 

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

622 
lemma contlub_Istrictify2: "contlub(Istrictify(f::'a > 'b))" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

623 
apply (rule contlubI [rule_format]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

624 
apply (case_tac "lub (range (Y))=UU") 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

625 
apply (simp add: Istrictify1 chain_UU_I) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

626 
apply (simp add: lub_const [THEN thelubI]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

627 
apply (simp add: Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

628 
apply (rule_tac s = "lub (range (%i. f$ (Y i)))" in trans) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

629 
apply (erule contlub_cfun_arg) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

630 
apply (rule lub_equal2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

631 
apply (rule chain_mono2 [THEN exE]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

632 
apply (erule chain_UU_I_inverse2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

633 
apply (assumption) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

634 
apply (blast intro: Istrictify2 [symmetric]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

635 
apply (erule chain_monofun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

636 
apply (erule monofun_Istrictify2 [THEN ch2ch_monofun]) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

637 
done 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

638 

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

639 
lemmas cont_Istrictify1 = contlub_Istrictify1 [THEN monofun_Istrictify1 [THEN monocontlub2cont], standard] 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

640 

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

641 
lemmas cont_Istrictify2 = contlub_Istrictify2 [THEN monofun_Istrictify2 [THEN monocontlub2cont], standard] 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

642 

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

643 
lemma strictify1 [simp]: "strictify$f$UU=UU" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

644 
apply (unfold strictify_def) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

645 
apply (subst beta_cfun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

646 
apply (simp add: cont_Istrictify2 cont_Istrictify1 cont2cont_CF1L) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

647 
apply (subst beta_cfun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

648 
apply (rule cont_Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

649 
apply (rule Istrictify1) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

650 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

651 

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

652 
lemma strictify2 [simp]: "~x=UU ==> strictify$f$x=f$x" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

653 
apply (unfold strictify_def) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

654 
apply (subst beta_cfun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

655 
apply (simp add: cont_Istrictify2 cont_Istrictify1 cont2cont_CF1L) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

656 
apply (subst beta_cfun) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

657 
apply (rule cont_Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

658 
apply (erule Istrictify2) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

659 
done 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

660 

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

661 
subsection {* Identity and composition *} 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

662 

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

663 
consts 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

664 
ID :: "('a::cpo) > 'a" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

665 
cfcomp :: "('b>'c)>(('a::cpo)>('b::cpo))>'a>('c::cpo)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

666 

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

667 
syntax "@oo" :: "('b>'c)=>('a>'b)=>'a>'c" ("_ oo _" [101,100] 100) 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

668 

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

669 
translations "f1 oo f2" == "cfcomp$f1$f2" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

670 

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

671 
defs 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

672 
ID_def: "ID ==(LAM x. x)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

673 
oo_def: "cfcomp == (LAM f g x. f$(g$x))" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

674 

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

675 
lemma ID1 [simp]: "ID$x=x" 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

676 
apply (unfold ID_def) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

677 
apply (subst beta_cfun) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

678 
apply (rule cont_id) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

679 
apply (rule refl) 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

680 
done 
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

681 

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

682 
lemma cfcomp1: "(f oo g)=(LAM x. f$(g$x))" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

683 
by (simp add: oo_def) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

684 

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

685 
lemma cfcomp2 [simp]: "(f oo g)$x=f$(g$x)" 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

686 
by (simp add: cfcomp1) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

687 

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

688 
text {* 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

689 
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

690 
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

691 
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

692 
The identity arrow is interpretation of @{term ID}. 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

693 
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

694 
*} 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

695 

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

696 
lemma ID2 [simp]: "f oo ID = f " 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

697 
by (rule ext_cfun, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

698 

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

699 
lemma ID3 [simp]: "ID oo f = f " 
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

700 
by (rule ext_cfun, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

701 

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

702 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset

703 
by (rule ext_cfun, simp) 
15576
efb95d0d01f7
converted to newstyle theories, and combined numbered files
huffman
parents:
diff
changeset

704 

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

705 
end 