src/HOLCF/Sprod.thy
author huffman
Tue Jan 15 02:20:47 2008 +0100 (2008-01-15)
changeset 25914 ff835e25ae87
parent 25881 d80bd899ea95
child 26962 c8b20f615d6c
permissions -rw-r--r--
clean up some proofs;
add lemmas spair_strict_iff, spair_less_iff, spair_eq_iff;
add instance for class bifinite
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(*  Title:      HOLCF/Sprod.thy
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    ID:         $Id$
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    Author:     Franz Regensburger and Brian Huffman
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Strict product with typedef.
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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 Cprod
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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) "**" (infixr "**" 20) =
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        "{p::'a \<times> 'b. p = \<bottom> \<or> (cfst\<cdot>p \<noteq> \<bottom> \<and> csnd\<cdot>p \<noteq> \<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_Sprod])
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instance "**" :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
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by (rule typedef_chfin [OF type_definition_Sprod less_Sprod_def])
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syntax (xsymbols)
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  "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)
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syntax (HTML output)
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  "**"		:: "[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. cfst\<cdot>(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. csnd\<cdot>(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] less_Sprod_def
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  Rep_Sprod_strict Rep_Sprod_spair
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lemma Exh_Sprod2:
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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 eq_cprod)
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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: **]:
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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_Sprod2, auto)
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lemma sprod_induct [induct type: **]:
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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_less_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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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 less_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: less_Sprod_def sfst_def ssnd_def cont_Rep_Sprod)
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apply (rule less_cprod)
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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 less_sprod)
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lemma spair_less:
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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: less_sprod)
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done
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lemma sfst_less_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: less_sprod)
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done
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lemma ssnd_less_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: less_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_less_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_less_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 "**" :: (flat, flat) flat
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apply (intro_classes, clarify)
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apply (rule_tac p=x in sprodE, simp)
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apply (rule_tac p=y in sprodE, simp)
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apply (simp add: flat_less_iff spair_less)
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done
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subsection {* Strict product is a bifinite domain *}
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instance "**" :: (bifinite, bifinite) approx ..
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defs (overloaded)
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  approx_sprod_def:
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    "approx \<equiv> \<lambda>n. \<Lambda>(:x, y:). (:approx n\<cdot>x, approx n\<cdot>y:)"
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instance "**" :: (bifinite, bifinite) bifinite
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proof
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  fix i :: nat and x :: "'a \<otimes> 'b"
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  show "chain (\<lambda>i. approx i\<cdot>x)"
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    unfolding approx_sprod_def by simp
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  show "(\<Squnion>i. approx i\<cdot>x) = x"
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    unfolding approx_sprod_def
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    by (simp add: lub_distribs eta_cfun)
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  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
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    unfolding approx_sprod_def
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    by (simp add: ssplit_def strictify_conv_if)
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  have "Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x} \<subseteq> {x. approx i\<cdot>x = x}"
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    unfolding approx_sprod_def
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    apply (clarify, rule_tac p=x in sprodE)
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     apply (simp add: Rep_Sprod_strict)
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    apply (simp add: Rep_Sprod_spair spair_eq_iff)
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    done
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  hence "finite (Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x})"
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    using finite_fixes_approx by (rule finite_subset)
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  thus "finite {x::'a \<otimes> 'b. approx i\<cdot>x = x}"
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    by (rule finite_imageD, simp add: inj_on_def Rep_Sprod_inject)
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qed
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lemma approx_spair [simp]:
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  "approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)"
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unfolding approx_sprod_def
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by (simp add: ssplit_def strictify_conv_if)
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end