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(* Title: HOLCF/Sprod.thy 
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ID: $Id$ 
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Author: Franz Regensburger and Brian Huffman 
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Strict product with typedef. 
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*) 
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header {* The type of strict products *} 
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theory Sprod 
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imports Cprod 
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begin 
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defaultsort pcpo 
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subsection {* Definition of strict product type *} 
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pcpodef (Sprod) ('a, 'b) "**" (infixr "**" 20) = 
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"{p::'a \<times> 'b. p = \<bottom> \<or> (cfst\<cdot>p \<noteq> \<bottom> \<and> csnd\<cdot>p \<noteq> \<bottom>)}" 
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by simp 
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instance "**" :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po 
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by (rule typedef_finite_po [OF type_definition_Sprod]) 
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instance "**" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin 
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by (rule typedef_chfin [OF type_definition_Sprod less_Sprod_def]) 
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syntax (xsymbols) 
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"**" :: "[type, type] => type" ("(_ \<otimes>/ _)" [21,20] 20) 
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syntax (HTML output) 
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"**" :: "[type, type] => type" ("(_ \<otimes>/ _)" [21,20] 20) 
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lemma spair_lemma: 
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"<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a> \<in> Sprod" 
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by (simp add: Sprod_def strictify_conv_if) 
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subsection {* Definitions of constants *} 
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definition 
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sfst :: "('a ** 'b) \<rightarrow> 'a" where 
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"sfst = (\<Lambda> p. cfst\<cdot>(Rep_Sprod p))" 
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definition 
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ssnd :: "('a ** 'b) \<rightarrow> 'b" where 
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"ssnd = (\<Lambda> p. csnd\<cdot>(Rep_Sprod p))" 
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definition 
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spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)" where 
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"spair = (\<Lambda> a b. Abs_Sprod 
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<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>)" 
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definition 
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ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c" where 
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"ssplit = (\<Lambda> f. strictify\<cdot>(\<Lambda> p. f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))" 
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syntax 
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"@stuple" :: "['a, args] => 'a ** 'b" ("(1'(:_,/ _:'))") 
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translations 
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"(:x, y, z:)" == "(:x, (:y, z:):)" 
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"(:x, y:)" == "CONST spair\<cdot>x\<cdot>y" 
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translations 
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"\<Lambda>(CONST spair\<cdot>x\<cdot>y). t" == "CONST ssplit\<cdot>(\<Lambda> x y. t)" 
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subsection {* Case analysis *} 
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lemma Rep_Sprod_spair: 
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"Rep_Sprod (:a, b:) = <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>" 

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unfolding spair_def 

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by (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma) 

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lemmas Rep_Sprod_simps = 

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Rep_Sprod_inject [symmetric] less_Sprod_def 

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Rep_Sprod_strict Rep_Sprod_spair 

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lemma Exh_Sprod2: 
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"z = \<bottom> \<or> (\<exists>a b. z = (:a, b:) \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)" 
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apply (insert Rep_Sprod [of z]) 
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apply (simp add: Rep_Sprod_simps eq_cprod) 

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apply (simp add: Sprod_def) 
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apply (erule disjE, simp) 
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apply (simp add: strictify_conv_if) 

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apply fast 

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done 
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lemma sprodE [cases type: **]: 
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"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x y. \<lbrakk>p = (:x, y:); x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 
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by (cut_tac z=p in Exh_Sprod2, auto) 
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lemma sprod_induct [induct type: **]: 
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"\<lbrakk>P \<bottom>; \<And>x y. \<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P (:x, y:)\<rbrakk> \<Longrightarrow> P x" 
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by (cases x, simp_all) 
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subsection {* Properties of @{term spair} *} 
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lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>" 
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by (simp add: Rep_Sprod_simps strictify_conv_if) 
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lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>" 
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by (simp add: Rep_Sprod_simps strictify_conv_if) 
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lemma spair_strict_iff [simp]: "((:x, y:) = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)" 

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by (simp add: Rep_Sprod_simps strictify_conv_if) 

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lemma spair_less_iff: 

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"((:a, b:) \<sqsubseteq> (:c, d:)) = (a = \<bottom> \<or> b = \<bottom> \<or> (a \<sqsubseteq> c \<and> b \<sqsubseteq> d))" 

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by (simp add: Rep_Sprod_simps strictify_conv_if) 

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lemma spair_eq_iff: 

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"((:a, b:) = (:c, d:)) = 

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(a = c \<and> b = d \<or> (a = \<bottom> \<or> b = \<bottom>) \<and> (c = \<bottom> \<or> d = \<bottom>))" 

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by (simp add: Rep_Sprod_simps strictify_conv_if) 

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lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>" 
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by simp 
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lemma spair_strict_rev: "(:x, y:) \<noteq> \<bottom> \<Longrightarrow> x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>" 
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by simp 
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lemma spair_defined: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>" 
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by simp 

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lemma spair_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>" 
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by simp 
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lemma spair_eq: 
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"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ((:x, y:) = (:a, b:)) = (x = a \<and> y = b)" 
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by (simp add: spair_eq_iff) 
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lemma spair_inject: 
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"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>; (:x, y:) = (:a, b:)\<rbrakk> \<Longrightarrow> x = a \<and> y = b" 
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by (rule spair_eq [THEN iffD1]) 
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lemma inst_sprod_pcpo2: "UU = (:UU,UU:)" 
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by simp 
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subsection {* Properties of @{term sfst} and @{term ssnd} *} 
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lemma sfst_strict [simp]: "sfst\<cdot>\<bottom> = \<bottom>" 
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by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_strict) 
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lemma ssnd_strict [simp]: "ssnd\<cdot>\<bottom> = \<bottom>" 
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by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_strict) 
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lemma sfst_spair [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>(:x, y:) = x" 
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by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_spair) 
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lemma ssnd_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>(:x, y:) = y" 
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by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_spair) 
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lemma sfst_defined_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)" 
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by (cases p, simp_all) 
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lemma ssnd_defined_iff [simp]: "(ssnd\<cdot>p = \<bottom>) = (p = \<bottom>)" 
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by (cases p, simp_all) 
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lemma sfst_defined: "p \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>p \<noteq> \<bottom>" 
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by simp 
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lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>" 
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by simp 
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lemma surjective_pairing_Sprod2: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p" 
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by (cases p, simp_all) 
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16751  166 
lemma less_sprod: "x \<sqsubseteq> y = (sfst\<cdot>x \<sqsubseteq> sfst\<cdot>y \<and> ssnd\<cdot>x \<sqsubseteq> ssnd\<cdot>y)" 
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apply (simp add: less_Sprod_def sfst_def ssnd_def cont_Rep_Sprod) 
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apply (rule less_cprod) 
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done 
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16751  171 
lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)" 
172 
by (auto simp add: po_eq_conv less_sprod) 

173 

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lemma spair_less: 
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"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<sqsubseteq> (:a, b:) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> b)" 
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apply (cases "a = \<bottom>", simp) 
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apply (cases "b = \<bottom>", simp) 
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apply (simp add: less_sprod) 
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done 
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25881  181 
lemma sfst_less_iff: "sfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd\<cdot>x:)" 
182 
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp) 

183 
apply (simp add: less_sprod) 

184 
done 

185 

186 
lemma ssnd_less_iff: "ssnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:sfst\<cdot>x, y:)" 

187 
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp) 

188 
apply (simp add: less_sprod) 

189 
done 

190 

191 
subsection {* Compactness *} 

192 

193 
lemma compact_sfst: "compact x \<Longrightarrow> compact (sfst\<cdot>x)" 

194 
by (rule compactI, simp add: sfst_less_iff) 

195 

196 
lemma compact_ssnd: "compact x \<Longrightarrow> compact (ssnd\<cdot>x)" 

197 
by (rule compactI, simp add: ssnd_less_iff) 

198 

199 
lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)" 

200 
by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if) 

201 

202 
lemma compact_spair_iff: 

203 
"compact (:x, y:) = (x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y))" 

204 
apply (safe elim!: compact_spair) 

205 
apply (drule compact_sfst, simp) 

206 
apply (drule compact_ssnd, simp) 

207 
apply simp 

208 
apply simp 

209 
done 

210 

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subsection {* Properties of @{term ssplit} *} 
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lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>" 
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by (simp add: ssplit_def) 
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16920  216 
lemma ssplit2 [simp]: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ssplit\<cdot>f\<cdot>(:x, y:) = f\<cdot>x\<cdot>y" 
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by (simp add: ssplit_def) 
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16553  219 
lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z" 
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by (cases z, simp_all) 
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subsection {* Strict product preserves flatness *} 
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instance "**" :: (flat, flat) flat 
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apply (intro_classes, clarify) 
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apply (rule_tac p=x in sprodE, simp) 
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apply (rule_tac p=y in sprodE, simp) 
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apply (simp add: flat_less_iff spair_less) 
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done 
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25914  231 
subsection {* Strict product is a bifinite domain *} 
232 

233 
instance "**" :: (bifinite, bifinite) approx .. 

234 

235 
defs (overloaded) 

236 
approx_sprod_def: 

237 
"approx \<equiv> \<lambda>n. \<Lambda>(:x, y:). (:approx n\<cdot>x, approx n\<cdot>y:)" 

238 

239 
instance "**" :: (bifinite, bifinite) bifinite 

240 
proof 

241 
fix i :: nat and x :: "'a \<otimes> 'b" 

242 
show "chain (\<lambda>i. approx i\<cdot>x)" 

243 
unfolding approx_sprod_def by simp 

244 
show "(\<Squnion>i. approx i\<cdot>x) = x" 

245 
unfolding approx_sprod_def 

246 
by (simp add: lub_distribs eta_cfun) 

247 
show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" 

248 
unfolding approx_sprod_def 

249 
by (simp add: ssplit_def strictify_conv_if) 

250 
have "Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x} \<subseteq> {x. approx i\<cdot>x = x}" 

251 
unfolding approx_sprod_def 

252 
apply (clarify, rule_tac p=x in sprodE) 

253 
apply (simp add: Rep_Sprod_strict) 

254 
apply (simp add: Rep_Sprod_spair spair_eq_iff) 

255 
done 

256 
hence "finite (Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x})" 

257 
using finite_fixes_approx by (rule finite_subset) 

258 
thus "finite {x::'a \<otimes> 'b. approx i\<cdot>x = x}" 

259 
by (rule finite_imageD, simp add: inj_on_def Rep_Sprod_inject) 

260 
qed 

261 

262 
lemma approx_spair [simp]: 

263 
"approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)" 

264 
unfolding approx_sprod_def 

265 
by (simp add: ssplit_def strictify_conv_if) 

266 

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end 