src/HOL/HOLCF/Ssum.thy
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(*  Title:      HOL/HOLCF/Ssum.thy
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    Author:     Franz Regensburger
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    Author:     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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default_sort pcpo
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subsection {* Definition of strict sum type *}
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definition
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  "ssum =
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    {p :: tr \<times> ('a \<times> 'b). p = \<bottom> \<or>
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      (fst p = TT \<and> fst (snd p) \<noteq> \<bottom> \<and> snd (snd p) = \<bottom>) \<or>
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      (fst p = FF \<and> fst (snd p) = \<bottom> \<and> snd (snd p) \<noteq> \<bottom>)}"
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pcpodef (open) ('a, 'b) ssum (infixr "++" 10) = "ssum :: (tr \<times> 'a \<times> 'b) set"
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  unfolding ssum_def by simp_all
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instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
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by (rule typedef_chfin [OF type_definition_ssum below_ssum_def])
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type_notation (xsymbols)
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  ssum  ("(_ \<oplus>/ _)" [21, 20] 20)
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type_notation (HTML output)
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  ssum  ("(_ \<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 (seq\<cdot>a\<cdot>TT, 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 (seq\<cdot>b\<cdot>FF, \<bottom>, b))"
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lemma sinl_ssum: "(seq\<cdot>a\<cdot>TT, a, \<bottom>) \<in> ssum"
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by (simp add: ssum_def seq_conv_if)
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lemma sinr_ssum: "(seq\<cdot>b\<cdot>FF, \<bottom>, b) \<in> ssum"
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by (simp add: ssum_def seq_conv_if)
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lemma Rep_ssum_sinl: "Rep_ssum (sinl\<cdot>a) = (seq\<cdot>a\<cdot>TT, a, \<bottom>)"
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by (simp add: sinl_def cont_Abs_ssum Abs_ssum_inverse sinl_ssum)
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lemma Rep_ssum_sinr: "Rep_ssum (sinr\<cdot>b) = (seq\<cdot>b\<cdot>FF, \<bottom>, b)"
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by (simp add: sinr_def cont_Abs_ssum Abs_ssum_inverse sinr_ssum)
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lemmas Rep_ssum_simps =
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  Rep_ssum_inject [symmetric] below_ssum_def
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  prod_eq_iff below_prod_def
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  Rep_ssum_strict Rep_ssum_sinl Rep_ssum_sinr
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subsection {* Properties of \emph{sinl} and \emph{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: Rep_ssum_simps seq_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: Rep_ssum_simps seq_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: Rep_ssum_simps seq_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: Rep_ssum_simps seq_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: Rep_ssum_simps)
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lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"
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by (simp add: Rep_ssum_simps)
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lemma sinl_bottom_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"
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using sinl_eq [of "x" "\<bottom>"] by simp
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lemma sinr_bottom_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"
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using sinr_eq [of "x" "\<bottom>"] by simp
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lemma sinl_defined: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"
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by simp
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lemma sinr_defined: "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)
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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)
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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 ssumE [case_names bottom sinl sinr, cases type: ssum]:
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  obtains "p = \<bottom>"
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  | x where "p = sinl\<cdot>x" and "x \<noteq> \<bottom>"
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  | y where "p = sinr\<cdot>y" and "y \<noteq> \<bottom>"
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using Rep_ssum [of p] by (auto simp add: ssum_def Rep_ssum_simps)
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lemma ssum_induct [case_names bottom sinl sinr, induct type: ssum]:
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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 [case_names sinl sinr]:
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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) (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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  "case s of (XCONST sinl :: 'a)\<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) (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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subsection {* Strict sum preserves flatness *}
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instance ssum :: (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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end