author  huffman 
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permissions  rwrr 
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(* Title: HOLCF/Ssum.thy 
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Author: Franz Regensburger and Brian Huffman 
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*) 
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header {* The type of strict sums *} 
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theory Ssum 
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imports Tr 
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begin 
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defaultsort pcpo 
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subsection {* Definition of strict sum type *} 
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pcpodef (Ssum) ('a, 'b) "++" (infixr "++" 10) = 
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"{p :: tr \<times> ('a \<times> 'b). 
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(fst p \<sqsubseteq> TT \<longleftrightarrow> snd (snd p) = \<bottom>) \<and> 
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(fst p \<sqsubseteq> FF \<longleftrightarrow> fst (snd p) = \<bottom>)}" 

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by simp_all 
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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_Ssum]) 
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instance "++" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin 
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by (rule typedef_chfin [OF type_definition_Ssum below_Ssum_def]) 
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syntax (xsymbols) 
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"++" :: "[type, type] => type" ("(_ \<oplus>/ _)" [21, 20] 20) 
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syntax (HTML output) 
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"++" :: "[type, type] => type" ("(_ \<oplus>/ _)" [21, 20] 20) 
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subsection {* Definitions of constructors *} 
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definition 
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sinl :: "'a \<rightarrow> ('a ++ 'b)" where 
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"sinl = (\<Lambda> a. Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>))" 
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definition 
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sinr :: "'b \<rightarrow> ('a ++ 'b)" where 
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"sinr = (\<Lambda> b. Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b))" 
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lemma sinl_Ssum: "(strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>) \<in> Ssum" 
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by (simp add: Ssum_def strictify_conv_if) 
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lemma sinr_Ssum: "(strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b) \<in> Ssum" 
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by (simp add: Ssum_def strictify_conv_if) 
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lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)" 
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by (unfold sinl_def, simp add: cont_Abs_Ssum sinl_Ssum) 
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lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)" 
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by (unfold sinr_def, simp add: cont_Abs_Ssum sinr_Ssum) 
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lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)" 
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by (simp add: sinl_Abs_Ssum Abs_Ssum_inverse sinl_Ssum) 
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lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)" 
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by (simp add: sinr_Abs_Ssum Abs_Ssum_inverse sinr_Ssum) 
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subsection {* Properties of @{term sinl} and @{term sinr} *} 
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text {* Ordering *} 
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lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)" 
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by (simp add: below_Ssum_def Rep_Ssum_sinl strictify_conv_if) 
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lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)" 
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by (simp add: below_Ssum_def Rep_Ssum_sinr strictify_conv_if) 
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lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)" 
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by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if) 
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lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)" 
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by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if) 
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text {* Equality *} 
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lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)" 
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by (simp add: po_eq_conv) 
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lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)" 
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by (simp add: po_eq_conv) 
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lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)" 
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by (subst po_eq_conv, simp) 
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lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)" 
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by (subst po_eq_conv, simp) 
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lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y" 
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by (rule sinl_eq [THEN iffD1]) 
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lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y" 
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by (rule sinr_eq [THEN iffD1]) 
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text {* Strictness *} 
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lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>" 
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lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>" 
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by (simp add: sinr_Abs_Ssum Abs_Ssum_strict) 
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lemma sinl_defined_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)" 
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by (cut_tac sinl_eq [of "x" "\<bottom>"], simp) 
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lemma sinr_defined_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)" 
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by (cut_tac sinr_eq [of "x" "\<bottom>"], simp) 
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lemma sinl_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>" 
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by simp 
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lemma sinr_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>" 
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by simp 
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text {* Compactness *} 
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lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)" 
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by (rule compact_Ssum, simp add: Rep_Ssum_sinl strictify_conv_if) 
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lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)" 
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lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x" 
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by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinl]], simp) 
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lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x" 
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by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinr]], simp) 
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lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x" 
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by (safe elim!: compact_sinl compact_sinlD) 
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lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x" 
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by (safe elim!: compact_sinr compact_sinrD) 
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subsection {* Case analysis *} 
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16921  140 
lemma Exh_Ssum: 
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"z = \<bottom> \<or> (\<exists>a. z = sinl\<cdot>a \<and> a \<noteq> \<bottom>) \<or> (\<exists>b. z = sinr\<cdot>b \<and> b \<noteq> \<bottom>)" 
31115  142 
apply (induct z rule: Abs_Ssum_induct) 
143 
apply (case_tac y, rename_tac t a b) 

144 
apply (case_tac t rule: trE) 

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apply (rule disjI1) 
31115  146 
apply (simp add: Ssum_def Abs_Ssum_strict) 
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apply (rule disjI2, rule disjI1, rule_tac x=a in exI) 
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apply (simp add: sinl_Abs_Ssum Ssum_def) 
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apply (rule disjI2, rule disjI2, rule_tac x=b in exI) 
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apply (simp add: sinr_Abs_Ssum Ssum_def) 
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done 
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lemma ssumE [cases type: ++]: 
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"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; 
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\<And>x. \<lbrakk>p = sinl\<cdot>x; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q; 
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\<And>y. \<lbrakk>p = sinr\<cdot>y; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 
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by (cut_tac z=p in Exh_Ssum, auto) 
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25756  159 
lemma ssum_induct [induct type: ++]: 
160 
"\<lbrakk>P \<bottom>; 

161 
\<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x); 

162 
\<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x" 

163 
by (cases x, simp_all) 

164 

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lemma ssumE2: 
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"\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 
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by (cases p, simp only: sinl_strict [symmetric], simp, simp) 
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lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x" 
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by (cases p, rule_tac x="\<bottom>" in exI, simp_all) 
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171 

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lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x" 
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by (cases p, rule_tac x="\<bottom>" in exI, simp_all) 
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174 

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subsection {* Case analysis combinator *} 
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176 

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definition 
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sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where 
31115  179 
"sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y fi) (Rep_Ssum s))" 
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180 

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translations 
26046  182 
"case s of XCONST sinl\<cdot>x \<Rightarrow> t1  XCONST sinr\<cdot>y \<Rightarrow> t2" == "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s" 
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183 

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translations 
26046  185 
"\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>" 
186 
"\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)" 

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lemma beta_sscase: 
31115  189 
"sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y fi) (Rep_Ssum s)" 
190 
unfolding sscase_def by (simp add: cont_Rep_Ssum [THEN cont_compose]) 

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191 

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lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>" 
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unfolding beta_sscase by (simp add: Rep_Ssum_strict) 
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lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x" 
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unfolding beta_sscase by (simp add: Rep_Ssum_sinl) 
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lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y" 
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unfolding beta_sscase by (simp add: Rep_Ssum_sinr) 
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200 

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lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z" 
25756  202 
by (cases z, simp_all) 
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203 

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subsection {* Strict sum preserves flatness *} 
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205 

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instance "++" :: (flat, flat) flat 
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apply (intro_classes, clarify) 
31115  208 
apply (case_tac x, simp) 
209 
apply (case_tac y, simp_all add: flat_below_iff) 

210 
apply (case_tac y, simp_all add: flat_below_iff) 

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done 
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212 

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subsection {* Map function for strict sums *} 
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214 

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definition 
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ssum_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<oplus> 'c \<rightarrow> 'b \<oplus> 'd" 
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where 
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"ssum_map = (\<Lambda> f g. sscase\<cdot>(sinl oo f)\<cdot>(sinr oo g))" 
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219 

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lemma ssum_map_strict [simp]: "ssum_map\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>" 
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unfolding ssum_map_def by simp 
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222 

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lemma ssum_map_sinl [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)" 
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unfolding ssum_map_def by simp 
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225 

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lemma ssum_map_sinr [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)" 
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unfolding ssum_map_def by simp 
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228 

33587  229 
lemma ssum_map_map: 
230 
"\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow> 

231 
ssum_map\<cdot>f1\<cdot>g1\<cdot>(ssum_map\<cdot>f2\<cdot>g2\<cdot>p) = 

232 
ssum_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p" 

233 
apply (induct p, simp) 

234 
apply (case_tac "f2\<cdot>x = \<bottom>", simp, simp) 

235 
apply (case_tac "g2\<cdot>y = \<bottom>", simp, simp) 

236 
done 

237 

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lemma ep_pair_ssum_map: 
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assumes "ep_pair e1 p1" and "ep_pair e2 p2" 
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shows "ep_pair (ssum_map\<cdot>e1\<cdot>e2) (ssum_map\<cdot>p1\<cdot>p2)" 
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241 
proof 
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interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact 
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interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact 
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fix x show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x" 
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by (induct x) simp_all 
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fix y show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" 
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apply (induct y, simp) 
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apply (case_tac "p1\<cdot>x = \<bottom>", simp, simp add: e1p1.e_p_below) 
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apply (case_tac "p2\<cdot>y = \<bottom>", simp, simp add: e2p2.e_p_below) 
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done 
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251 
qed 
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252 

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lemma deflation_ssum_map: 
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assumes "deflation d1" and "deflation d2" 
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shows "deflation (ssum_map\<cdot>d1\<cdot>d2)" 
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256 
proof 
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interpret d1: deflation d1 by fact 
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interpret d2: deflation d2 by fact 
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259 
fix x 
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show "ssum_map\<cdot>d1\<cdot>d2\<cdot>(ssum_map\<cdot>d1\<cdot>d2\<cdot>x) = ssum_map\<cdot>d1\<cdot>d2\<cdot>x" 
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apply (induct x, simp) 
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apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.idem) 
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apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.idem) 
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264 
done 
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show "ssum_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x" 
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apply (induct x, simp) 
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apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.below) 
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apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.below) 
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done 
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qed 
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lemma finite_deflation_ssum_map: 
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assumes "finite_deflation d1" and "finite_deflation d2" 
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shows "finite_deflation (ssum_map\<cdot>d1\<cdot>d2)" 
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proof (intro finite_deflation.intro finite_deflation_axioms.intro) 
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interpret d1: finite_deflation d1 by fact 
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interpret d2: finite_deflation d2 by fact 
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have "deflation d1" and "deflation d2" by fact+ 
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thus "deflation (ssum_map\<cdot>d1\<cdot>d2)" by (rule deflation_ssum_map) 
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have "{x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq> 
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(\<lambda>x. sinl\<cdot>x) ` {x. d1\<cdot>x = x} \<union> 
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(\<lambda>x. sinr\<cdot>x) ` {x. d2\<cdot>x = x} \<union> {\<bottom>}" 
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by (rule subsetI, case_tac x, simp_all) 
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thus "finite {x. ssum_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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25915  288 
subsection {* Strict sum is a bifinite domain *} 
289 

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instantiation "++" :: (bifinite, bifinite) bifinite 
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begin 
25915  292 

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definition 
25915  294 
approx_ssum_def: 
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"approx = (\<lambda>n. ssum_map\<cdot>(approx n)\<cdot>(approx n))" 
25915  296 

297 
lemma approx_sinl [simp]: "approx i\<cdot>(sinl\<cdot>x) = sinl\<cdot>(approx i\<cdot>x)" 

298 
unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all 

299 

300 
lemma approx_sinr [simp]: "approx i\<cdot>(sinr\<cdot>x) = sinr\<cdot>(approx i\<cdot>x)" 

301 
unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all 

302 

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instance proof 
25915  304 
fix i :: nat and x :: "'a \<oplus> 'b" 
27310  305 
show "chain (approx :: nat \<Rightarrow> 'a \<oplus> 'b \<rightarrow> 'a \<oplus> 'b)" 
25915  306 
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307 
show "(\<Squnion>i. approx i\<cdot>x) = x" 

308 
unfolding approx_ssum_def 

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by (cases x, simp_all add: lub_distribs) 
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show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" 
311 
by (cases x, simp add: approx_ssum_def, simp, simp) 

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show "finite {x::'a \<oplus> 'b. approx i\<cdot>x = x}" 
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unfolding approx_ssum_def 
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by (intro finite_deflation.finite_fixes 
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finite_deflation_ssum_map 
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finite_deflation_approx) 
25915  317 
qed 
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end 
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end 