src/HOLCF/Sprod.thy
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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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422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
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   139
lemma sfst_strict [simp]: "sfst\<cdot>\<bottom> = \<bottom>"
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
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   140
by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_strict)
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efb95d0d01f7 converted to new-style theories, and combined numbered files
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   141
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
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   142
lemma ssnd_strict [simp]: "ssnd\<cdot>\<bottom> = \<bottom>"
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
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parents: 16082
diff changeset
   143
by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_strict)
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   144
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
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   145
lemma sfst_spair [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>(:x, y:) = x"
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diff changeset
   146
by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_spair)
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   147
16212
422f836f6b39 renamed strict, defined, and inject lemmas; renamed sfst2, ssnd2 to sfst_spair, ssnd_spair
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   148
lemma ssnd_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>(:x, y:) = y"
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   149
by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_spair)
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   150
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555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
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   151
lemma sfst_defined_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)"
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   152
by (cases p, simp_all)
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diff changeset
   153
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
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   154
lemma ssnd_defined_iff [simp]: "(ssnd\<cdot>p = \<bottom>) = (p = \<bottom>)"
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5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
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diff changeset
   155
by (cases p, simp_all)
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868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
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parents: 16212
diff changeset
   156
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555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
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   157
lemma sfst_defined: "p \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>p \<noteq> \<bottom>"
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   158
by simp
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   159
555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
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   160
lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>"
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   161
by simp
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   162
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   163
lemma surjective_pairing_Sprod2: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p"
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   164
by (cases p, simp_all)
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   165
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   166
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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   167
apply (simp add: less_Sprod_def sfst_def ssnd_def cont_Rep_Sprod)
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   168
apply (rule less_cprod)
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   169
done
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   170
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   171
lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)"
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   172
by (auto simp add: po_eq_conv less_sprod)
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   173
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868eddbcaf6e added theorems less_sprod, spair_less, spair_eq, spair_inject
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   174
lemma spair_less:
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   175
  "\<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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   176
apply (cases "a = \<bottom>", simp)
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   177
apply (cases "b = \<bottom>", simp)
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   178
apply (simp add: less_sprod)
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   179
done
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   180
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   181
lemma sfst_less_iff: "sfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd\<cdot>x:)"
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   182
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)
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   183
apply (simp add: less_sprod)
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   184
done
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   185
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   186
lemma ssnd_less_iff: "ssnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:sfst\<cdot>x, y:)"
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   187
apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)
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   188
apply (simp add: less_sprod)
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   189
done
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   190
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   191
subsection {* Compactness *}
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   192
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   193
lemma compact_sfst: "compact x \<Longrightarrow> compact (sfst\<cdot>x)"
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   194
by (rule compactI, simp add: sfst_less_iff)
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   195
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   196
lemma compact_ssnd: "compact x \<Longrightarrow> compact (ssnd\<cdot>x)"
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   197
by (rule compactI, simp add: ssnd_less_iff)
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   198
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   199
lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)"
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   200
by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)
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   201
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   202
lemma compact_spair_iff:
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   203
  "compact (:x, y:) = (x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y))"
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   204
apply (safe elim!: compact_spair)
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   205
apply (drule compact_sfst, simp)
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   206
apply (drule compact_ssnd, simp)
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   207
apply simp
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   208
apply simp
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   209
done
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   210
16059
dab0d004732f Simplified version of strict product theory, using TypedefPcpo
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   211
subsection {* Properties of @{term ssplit} *}
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   212
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dab0d004732f Simplified version of strict product theory, using TypedefPcpo
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   213
lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>"
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50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
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   214
by (simp add: ssplit_def)
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   215
16920
ded12c9e88c2 cleaned up
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   216
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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50c3384ca6c4 reordered and arranged for document generation, cleaned up some proofs
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   217
by (simp add: ssplit_def)
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diff changeset
   218
16553
aa36d41e4263 add csplit3, ssplit3, fup3 as simp rules
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   219
lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z"
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5957e3d72fec declare sprodE as cases rule; new induction rule sprod_induct
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   220
by (cases z, simp_all)
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efb95d0d01f7 converted to new-style theories, and combined numbered files
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   221
25827
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
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   222
subsection {* Strict product preserves flatness *}
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
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   223
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
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   224
instance "**" :: (flat, flat) flat
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   225
apply (intro_classes, clarify)
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   226
apply (rule_tac p=x in sprodE, simp)
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   227
apply (rule_tac p=y in sprodE, simp)
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   228
apply (simp add: flat_less_iff spair_less)
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   229
done
c2adeb1bae5c new instance proofs for classes finite_po, chfin, flat
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   230
25914
ff835e25ae87 clean up some proofs;
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   231
subsection {* Strict product is a bifinite domain *}
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   232
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   233
instance "**" :: (bifinite, bifinite) approx ..
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   234
ff835e25ae87 clean up some proofs;
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   235
defs (overloaded)
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   236
  approx_sprod_def:
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   237
    "approx \<equiv> \<lambda>n. \<Lambda>(:x, y:). (:approx n\<cdot>x, approx n\<cdot>y:)"
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   238
ff835e25ae87 clean up some proofs;
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   239
instance "**" :: (bifinite, bifinite) bifinite
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   240
proof
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   241
  fix i :: nat and x :: "'a \<otimes> 'b"
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   242
  show "chain (\<lambda>i. approx i\<cdot>x)"
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   243
    unfolding approx_sprod_def by simp
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   244
  show "(\<Squnion>i. approx i\<cdot>x) = x"
ff835e25ae87 clean up some proofs;
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diff changeset
   245
    unfolding approx_sprod_def
ff835e25ae87 clean up some proofs;
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diff changeset
   246
    by (simp add: lub_distribs eta_cfun)
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   247
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
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diff changeset
   248
    unfolding approx_sprod_def
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   249
    by (simp add: ssplit_def strictify_conv_if)
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   250
  have "Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x} \<subseteq> {x. approx i\<cdot>x = x}"
ff835e25ae87 clean up some proofs;
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parents: 25881
diff changeset
   251
    unfolding approx_sprod_def
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   252
    apply (clarify, rule_tac p=x in sprodE)
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   253
     apply (simp add: Rep_Sprod_strict)
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   254
    apply (simp add: Rep_Sprod_spair spair_eq_iff)
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   255
    done
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diff changeset
   256
  hence "finite (Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x})"
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diff changeset
   257
    using finite_fixes_approx by (rule finite_subset)
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   258
  thus "finite {x::'a \<otimes> 'b. approx i\<cdot>x = x}"
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   259
    by (rule finite_imageD, simp add: inj_on_def Rep_Sprod_inject)
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   260
qed
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   261
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   262
lemma approx_spair [simp]:
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   263
  "approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)"
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diff changeset
   264
unfolding approx_sprod_def
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   265
by (simp add: ssplit_def strictify_conv_if)
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diff changeset
   266
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efb95d0d01f7 converted to new-style theories, and combined numbered files
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diff changeset
   267
end