author  huffman 
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permissions  rwrr 
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(* Title: HOLCF/Sprod.thy 
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Author: Franz Regensburger and Brian Huffman 
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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 Bifinite 
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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) sprod (infixr "**" 20) = 
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"{p::'a \<times> 'b. p = \<bottom> \<or> (fst p \<noteq> \<bottom> \<and> snd p \<noteq> \<bottom>)}" 
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by simp_all 
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instance sprod :: ("{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 sprod :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin 
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by (rule typedef_chfin [OF type_definition_Sprod below_Sprod_def]) 
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syntax (xsymbols) 
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sprod :: "[type, type] => type" ("(_ \<otimes>/ _)" [21,20] 20) 
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syntax (HTML output) 
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sprod :: "[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. fst (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. snd (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] below_Sprod_def 
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Rep_Sprod_strict Rep_Sprod_spair 
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lemma Exh_Sprod: 
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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 Pair_fst_snd_eq) 
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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: sprod]: 
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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_Sprod, auto) 
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lemma sprod_induct [induct type: sprod]: 
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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_below_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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lemma sprodE2: "(\<And>x y. p = (:x, y:) \<Longrightarrow> Q) \<Longrightarrow> Q" 
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by (cases p, simp only: inst_sprod_pcpo2, 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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159 

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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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162 

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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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165 

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lemma below_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: below_Sprod_def sfst_def ssnd_def cont_Rep_Sprod) 
31114  168 
apply (simp only: below_prod_def) 
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done 
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170 

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lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)" 
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by (auto simp add: po_eq_conv below_sprod) 
16751  173 

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lemma spair_below: 
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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: below_sprod) 
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done 
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180 

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lemma sfst_below_iff: "sfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd\<cdot>x:)" 
25881  182 
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp) 
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apply (simp add: below_sprod) 
25881  184 
done 
185 

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lemma ssnd_below_iff: "ssnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:sfst\<cdot>x, y:)" 
25881  187 
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp) 
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188 
apply (simp add: below_sprod) 
25881  189 
done 
190 

191 
subsection {* Compactness *} 

192 

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

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by (rule compactI, simp add: sfst_below_iff) 
25881  195 

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

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197 
by (rule compactI, simp add: ssnd_below_iff) 
25881  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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212 

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lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>" 
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214 
by (simp add: ssplit_def) 
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215 

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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218 

16553  219 
lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z" 
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by (cases z, simp_all) 
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221 

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222 
subsection {* Strict product preserves flatness *} 
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223 

35525  224 
instance sprod :: (flat, flat) flat 
27310  225 
proof 
226 
fix x y :: "'a \<otimes> 'b" 

227 
assume "x \<sqsubseteq> y" thus "x = \<bottom> \<or> x = y" 

228 
apply (induct x, simp) 

229 
apply (induct y, simp) 

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apply (simp add: spair_below_iff flat_below_iff) 
27310  231 
done 
232 
qed 

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233 

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subsection {* Map function for strict products *} 
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235 

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236 
definition 
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sprod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<otimes> 'c \<rightarrow> 'b \<otimes> 'd" 
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238 
where 
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"sprod_map = (\<Lambda> f g. ssplit\<cdot>(\<Lambda> x y. (:f\<cdot>x, g\<cdot>y:)))" 
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240 

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241 
lemma sprod_map_strict [simp]: "sprod_map\<cdot>a\<cdot>b\<cdot>\<bottom> = \<bottom>" 
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unfolding sprod_map_def by simp 
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243 

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244 
lemma sprod_map_spair [simp]: 
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"x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)" 
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246 
by (simp add: sprod_map_def) 
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247 

35491  248 
lemma sprod_map_spair': 
249 
"f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)" 

250 
by (cases "x = \<bottom> \<or> y = \<bottom>") auto 

251 

33808  252 
lemma sprod_map_ID: "sprod_map\<cdot>ID\<cdot>ID = ID" 
253 
unfolding sprod_map_def by (simp add: expand_cfun_eq eta_cfun) 

254 

33587  255 
lemma sprod_map_map: 
256 
"\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow> 

257 
sprod_map\<cdot>f1\<cdot>g1\<cdot>(sprod_map\<cdot>f2\<cdot>g2\<cdot>p) = 

258 
sprod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p" 

259 
apply (induct p, simp) 

260 
apply (case_tac "f2\<cdot>x = \<bottom>", simp) 

261 
apply (case_tac "g2\<cdot>y = \<bottom>", simp) 

262 
apply simp 

263 
done 

264 

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lemma ep_pair_sprod_map: 
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266 
assumes "ep_pair e1 p1" and "ep_pair e2 p2" 
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267 
shows "ep_pair (sprod_map\<cdot>e1\<cdot>e2) (sprod_map\<cdot>p1\<cdot>p2)" 
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268 
proof 
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269 
interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact 
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270 
interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact 
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271 
fix x show "sprod_map\<cdot>p1\<cdot>p2\<cdot>(sprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x" 
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272 
by (induct x) simp_all 
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273 
fix y show "sprod_map\<cdot>e1\<cdot>e2\<cdot>(sprod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" 
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274 
apply (induct y, simp) 
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275 
apply (case_tac "p1\<cdot>x = \<bottom>", simp, case_tac "p2\<cdot>y = \<bottom>", simp) 
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276 
apply (simp add: monofun_cfun e1p1.e_p_below e2p2.e_p_below) 
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277 
done 
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278 
qed 
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279 

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280 
lemma deflation_sprod_map: 
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281 
assumes "deflation d1" and "deflation d2" 
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282 
shows "deflation (sprod_map\<cdot>d1\<cdot>d2)" 
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283 
proof 
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284 
interpret d1: deflation d1 by fact 
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285 
interpret d2: deflation d2 by fact 
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286 
fix x 
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287 
show "sprod_map\<cdot>d1\<cdot>d2\<cdot>(sprod_map\<cdot>d1\<cdot>d2\<cdot>x) = sprod_map\<cdot>d1\<cdot>d2\<cdot>x" 
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288 
apply (induct x, simp) 
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289 
apply (case_tac "d1\<cdot>x = \<bottom>", simp, case_tac "d2\<cdot>y = \<bottom>", simp) 
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290 
apply (simp add: d1.idem d2.idem) 
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291 
done 
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292 
show "sprod_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x" 
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293 
apply (induct x, simp) 
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294 
apply (simp add: monofun_cfun d1.below d2.below) 
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295 
done 
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296 
qed 
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297 

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298 
lemma finite_deflation_sprod_map: 
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299 
assumes "finite_deflation d1" and "finite_deflation d2" 
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300 
shows "finite_deflation (sprod_map\<cdot>d1\<cdot>d2)" 
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301 
proof (intro finite_deflation.intro finite_deflation_axioms.intro) 
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302 
interpret d1: finite_deflation d1 by fact 
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303 
interpret d2: finite_deflation d2 by fact 
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304 
have "deflation d1" and "deflation d2" by fact+ 
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305 
thus "deflation (sprod_map\<cdot>d1\<cdot>d2)" by (rule deflation_sprod_map) 
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306 
have "{x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq> insert \<bottom> 
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((\<lambda>(x, y). (:x, y:)) ` ({x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}))" 
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by (rule subsetI, case_tac x, auto simp add: spair_eq_iff) 
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thus "finite {x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x}" 
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by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes) 
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qed 
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312 

25914  313 
subsection {* Strict product is a bifinite domain *} 
314 

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instantiation sprod :: (bifinite, bifinite) bifinite 
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begin 
25914  317 

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definition 
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approx_sprod_def: 
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"approx = (\<lambda>n. sprod_map\<cdot>(approx n)\<cdot>(approx n))" 
25914  321 

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instance proof 
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fix i :: nat and x :: "'a \<otimes> 'b" 
27310  324 
show "chain (approx :: nat \<Rightarrow> 'a \<otimes> 'b \<rightarrow> 'a \<otimes> 'b)" 
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show "(\<Squnion>i. approx i\<cdot>x) = x" 

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unfolding approx_sprod_def sprod_map_def 
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by (simp add: lub_distribs eta_cfun) 
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show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" 

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unfolding approx_sprod_def sprod_map_def 
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by (simp add: ssplit_def strictify_conv_if) 
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show "finite {x::'a \<otimes> 'b. approx i\<cdot>x = x}" 
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by (intro finite_deflation.finite_fixes 
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finite_deflation_sprod_map 
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finite_deflation_approx) 
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qed 
338 

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end 
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25914  341 
lemma approx_spair [simp]: 
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"approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)" 

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unfolding approx_sprod_def sprod_map_def 
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by (simp add: ssplit_def strictify_conv_if) 
345 

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end 