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(* Title: HOLCF/Ssum.thy 
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ID: $Id$ 
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Author: Franz Regensburger and Brian Huffman 
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Strict sum with typedef. 
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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 Cprod 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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(cfst\<cdot>p \<sqsubseteq> TT \<longleftrightarrow> csnd\<cdot>(csnd\<cdot>p) = \<bottom>) \<and> 
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(cfst\<cdot>p \<sqsubseteq> FF \<longleftrightarrow> cfst\<cdot>(csnd\<cdot>p) = \<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_Ssum]) 
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instance "++" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin 
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by (rule typedef_chfin [OF type_definition_Ssum less_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_less [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)" 
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by (simp add: less_Ssum_def Rep_Ssum_sinl strictify_conv_if) 
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lemma sinr_less [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)" 
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by (simp add: less_Ssum_def Rep_Ssum_sinr strictify_conv_if) 
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lemma sinl_less_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)" 
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by (simp add: less_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if) 
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lemma sinr_less_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)" 
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by (simp add: less_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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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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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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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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unfolding compact_def 
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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  143 
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>)" 
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apply (rule_tac x=z in Abs_Ssum_induct) 
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apply (rule_tac p=y in cprodE, rename_tac t x) 
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apply (rule_tac p=x in cprodE, rename_tac a b) 
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apply (rule_tac p=t in trE) 
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apply (rule disjI1) 
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apply (simp add: Ssum_def cpair_strict 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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156 

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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  163 
lemma ssum_induct [induct type: ++]: 
164 
"\<lbrakk>P \<bottom>; 

165 
\<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x); 

166 
\<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x" 

167 
by (cases x, simp_all) 

168 

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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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172 

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lemma less_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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lemma less_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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subsection {* Case analysis combinator *} 
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180 

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definition 
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sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where 
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"sscase = (\<Lambda> f g s. (\<Lambda><t, x, y>. If t then f\<cdot>x else g\<cdot>y fi)\<cdot>(Rep_Ssum s))" 
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translations 
26046  186 
"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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187 

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translations 
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"\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>" 
190 
"\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)" 

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lemma beta_sscase: 
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"sscase\<cdot>f\<cdot>g\<cdot>s = (\<Lambda><t, x, y>. If t then f\<cdot>x else g\<cdot>y fi)\<cdot>(Rep_Ssum s)" 
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unfolding sscase_def by (simp add: cont_Rep_Ssum) 
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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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lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z" 
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by (cases z, simp_all) 
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207 

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subsection {* Strict sum preserves flatness *} 
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209 

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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 ssumE, simp) 
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apply (rule_tac p=y in ssumE, simp_all add: flat_less_iff) 
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apply (rule_tac p=y in ssumE, simp_all add: flat_less_iff) 
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done 
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25915  217 
subsection {* Strict sum is a bifinite domain *} 
218 

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instantiation "++" :: (bifinite, bifinite) bifinite 
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begin 
25915  221 

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definition 
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approx_ssum_def: 
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"approx = (\<lambda>n. sscase\<cdot>(\<Lambda> x. sinl\<cdot>(approx n\<cdot>x))\<cdot>(\<Lambda> y. sinr\<cdot>(approx n\<cdot>y)))" 
25915  225 

226 
lemma approx_sinl [simp]: "approx i\<cdot>(sinl\<cdot>x) = sinl\<cdot>(approx i\<cdot>x)" 

227 
unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all 

228 

229 
lemma approx_sinr [simp]: "approx i\<cdot>(sinr\<cdot>x) = sinr\<cdot>(approx i\<cdot>x)" 

230 
unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all 

231 

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instance proof 
25915  233 
fix i :: nat and x :: "'a \<oplus> 'b" 
27310  234 
show "chain (approx :: nat \<Rightarrow> 'a \<oplus> 'b \<rightarrow> 'a \<oplus> 'b)" 
25915  235 
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236 
show "(\<Squnion>i. approx i\<cdot>x) = x" 

237 
unfolding approx_ssum_def 

238 
by (simp add: lub_distribs eta_cfun) 

239 
show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" 

240 
by (cases x, simp add: approx_ssum_def, simp, simp) 

241 
have "{x::'a \<oplus> 'b. approx i\<cdot>x = x} \<subseteq> 

242 
(\<lambda>x. sinl\<cdot>x) ` {x. approx i\<cdot>x = x} \<union> 

243 
(\<lambda>x. sinr\<cdot>x) ` {x. approx i\<cdot>x = x}" 

27310  244 
by (rule subsetI, case_tac x rule: ssumE2, simp, simp) 
25915  245 
thus "finite {x::'a \<oplus> 'b. approx i\<cdot>x = x}" 
246 
by (rule finite_subset, 

247 
intro finite_UnI finite_imageI finite_fixes_approx) 

248 
qed 

249 

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end 
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251 

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252 
end 