src/HOLCF/Ssum.thy
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map functions for various types, with ep_pair/deflation/finite_deflation lemmas
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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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by (simp add: sinl_Abs_Ssum Abs_Ssum_strict)
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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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by (rule compact_Ssum, simp add: Rep_Ssum_sinr strictify_conv_if)
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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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unfolding compact_def
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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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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 (induct z rule: Abs_Ssum_induct)
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apply (case_tac y, rename_tac t a b)
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apply (case_tac t rule: trE)
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apply (rule disjI1)
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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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lemma ssum_induct [induct type: ++]:
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  "\<lbrakk>P \<bottom>;
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   \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x);
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   \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x"
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by (cases x, simp_all)
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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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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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subsection {* Case analysis combinator *}
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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) (Rep_Ssum s))"
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translations
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  "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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translations
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  "\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"
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  "\<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) (Rep_Ssum s)"
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unfolding sscase_def by (simp add: cont_Rep_Ssum [THEN cont_compose])
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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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subsection {* Strict sum preserves flatness *}
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instance "++" :: (flat, flat) flat
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apply (intro_classes, clarify)
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apply (case_tac x, simp)
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apply (case_tac y, simp_all add: flat_below_iff)
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apply (case_tac y, simp_all add: flat_below_iff)
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done
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subsection {* Map function for strict sums *}
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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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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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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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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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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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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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qed
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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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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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  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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    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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subsection {* Strict sum is a bifinite domain *}
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instantiation "++" :: (bifinite, bifinite) bifinite
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begin
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definition
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  approx_ssum_def:
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    "approx = (\<lambda>n. ssum_map\<cdot>(approx n)\<cdot>(approx n))"
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f1bce5261dec add instance for class bifinite
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lemma approx_sinl [simp]: "approx i\<cdot>(sinl\<cdot>x) = sinl\<cdot>(approx i\<cdot>x)"
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unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all
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lemma approx_sinr [simp]: "approx i\<cdot>(sinr\<cdot>x) = sinr\<cdot>(approx i\<cdot>x)"
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unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all
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instance proof
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  fix i :: nat and x :: "'a \<oplus> 'b"
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  show "chain (approx :: nat \<Rightarrow> 'a \<oplus> 'b \<rightarrow> 'a \<oplus> 'b)"
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    unfolding approx_ssum_def by simp
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  show "(\<Squnion>i. approx i\<cdot>x) = x"
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    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"
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    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)
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qed
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end
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end