src/HOLCF/Sprod.thy
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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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lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>"
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by simp
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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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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)
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apply (simp only: below_prod_def)
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done
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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)
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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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lemma sfst_below_iff: "sfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd\<cdot>x:)"
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apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)
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apply (simp add: below_sprod)
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done
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lemma ssnd_below_iff: "ssnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:sfst\<cdot>x, y:)"
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apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)
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apply (simp add: below_sprod)
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done
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subsection {* Compactness *}
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lemma compact_sfst: "compact x \<Longrightarrow> compact (sfst\<cdot>x)"
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by (rule compactI, simp add: sfst_below_iff)
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lemma compact_ssnd: "compact x \<Longrightarrow> compact (ssnd\<cdot>x)"
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by (rule compactI, simp add: ssnd_below_iff)
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lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)"
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by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)
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lemma compact_spair_iff:
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  "compact (:x, y:) = (x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y))"
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apply (safe elim!: compact_spair)
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apply (drule compact_sfst, simp)
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apply (drule compact_ssnd, simp)
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apply simp
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apply simp
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done
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subsection {* Properties of @{term ssplit} *}
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lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>"
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by (simp add: ssplit_def)
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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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lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z"
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by (cases z, simp_all)
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subsection {* Strict product preserves flatness *}
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instance sprod :: (flat, flat) flat
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proof
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  fix x y :: "'a \<otimes> 'b"
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  assume "x \<sqsubseteq> y" thus "x = \<bottom> \<or> x = y"
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    apply (induct x, simp)
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    apply (induct y, simp)
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    apply (simp add: spair_below_iff flat_below_iff)
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    done
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qed
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subsection {* Map function for strict products *}
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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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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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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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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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by (simp add: sprod_map_def)
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lemma sprod_map_spair':
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  "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:)"
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by (cases "x = \<bottom> \<or> y = \<bottom>") auto
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lemma sprod_map_ID: "sprod_map\<cdot>ID\<cdot>ID = ID"
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unfolding sprod_map_def by (simp add: expand_cfun_eq eta_cfun)
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lemma sprod_map_map:
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  "\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow>
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    sprod_map\<cdot>f1\<cdot>g1\<cdot>(sprod_map\<cdot>f2\<cdot>g2\<cdot>p) =
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     sprod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
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apply (induct p, simp)
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apply (case_tac "f2\<cdot>x = \<bottom>", simp)
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apply (case_tac "g2\<cdot>y = \<bottom>", simp)
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apply simp
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done
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lemma ep_pair_sprod_map:
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  assumes "ep_pair e1 p1" and "ep_pair e2 p2"
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  shows "ep_pair (sprod_map\<cdot>e1\<cdot>e2) (sprod_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 "sprod_map\<cdot>p1\<cdot>p2\<cdot>(sprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
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    by (induct x) simp_all
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  fix y show "sprod_map\<cdot>e1\<cdot>e2\<cdot>(sprod_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, case_tac "p2\<cdot>y = \<bottom>", simp)
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    apply (simp add: monofun_cfun e1p1.e_p_below e2p2.e_p_below)
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    done
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qed
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   279
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lemma deflation_sprod_map:
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  assumes "deflation d1" and "deflation d2"
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  shows "deflation (sprod_map\<cdot>d1\<cdot>d2)"
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   283
proof
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  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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  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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diff changeset
   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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diff changeset
   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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   307
        ((\<lambda>(x, y). (:x, y:)) ` ({x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}))"
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diff changeset
   308
    by (rule subsetI, case_tac x, auto simp add: spair_eq_iff)
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   309
  thus "finite {x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
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diff changeset
   310
    by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
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   311
qed
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   312
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subsection {* Strict product is a bifinite domain *}
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   314
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   315
instantiation sprod :: (bifinite, bifinite) bifinite
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begin
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   317
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   318
definition
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   319
  approx_sprod_def:
33504
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   320
    "approx = (\<lambda>n. sprod_map\<cdot>(approx n)\<cdot>(approx n))"
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   321
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instance proof
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   323
  fix i :: nat and x :: "'a \<otimes> 'b"
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   324
  show "chain (approx :: nat \<Rightarrow> 'a \<otimes> 'b \<rightarrow> 'a \<otimes> 'b)"
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diff changeset
   325
    unfolding approx_sprod_def by simp
ff835e25ae87 clean up some proofs;
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diff changeset
   326
  show "(\<Squnion>i. approx i\<cdot>x) = x"
33504
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diff changeset
   327
    unfolding approx_sprod_def sprod_map_def
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diff changeset
   328
    by (simp add: lub_distribs eta_cfun)
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diff changeset
   329
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
33504
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diff changeset
   330
    unfolding approx_sprod_def sprod_map_def
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diff changeset
   331
    by (simp add: ssplit_def strictify_conv_if)
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diff changeset
   332
  show "finite {x::'a \<otimes> 'b. approx i\<cdot>x = x}"
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diff changeset
   333
    unfolding approx_sprod_def
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diff changeset
   334
    by (intro finite_deflation.finite_fixes
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   335
              finite_deflation_sprod_map
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   336
              finite_deflation_approx)
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   337
qed
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   338
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   339
end
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   340
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   341
lemma approx_spair [simp]:
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   342
  "approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)"
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diff changeset
   343
unfolding approx_sprod_def sprod_map_def
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   344
by (simp add: ssplit_def strictify_conv_if)
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   345
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parents:
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   346
end