src/HOL/HOLCF/Sprod.thy
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(*  Title:      HOL/HOLCF/Sprod.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 products *}
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theory Sprod
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imports Cfun
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begin
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default_sort pcpo
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subsection {* Definition of strict product type *}
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definition "sprod = {p::'a \<times> 'b. p = \<bottom> \<or> (fst p \<noteq> \<bottom> \<and> snd p \<noteq> \<bottom>)}"
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pcpodef (open) ('a, 'b) sprod (infixr "**" 20) = "sprod :: ('a \<times> 'b) set"
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  unfolding sprod_def by simp_all
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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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type_notation (xsymbols)
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  sprod  ("(_ \<otimes>/ _)" [21,20] 20)
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type_notation (HTML output)
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  sprod  ("(_ \<otimes>/ _)" [21,20] 20)
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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 (seq\<cdot>b\<cdot>a, seq\<cdot>a\<cdot>b))"
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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 p. seq\<cdot>p\<cdot>(f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))"
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syntax
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  "_stuple" :: "[logic, args] \<Rightarrow> logic"  ("(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 spair_sprod: "(seq\<cdot>b\<cdot>a, seq\<cdot>a\<cdot>b) \<in> sprod"
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by (simp add: sprod_def seq_conv_if)
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lemma Rep_sprod_spair: "Rep_sprod (:a, b:) = (seq\<cdot>b\<cdot>a, seq\<cdot>a\<cdot>b)"
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by (simp add: spair_def cont_Abs_sprod Abs_sprod_inverse spair_sprod)
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lemmas Rep_sprod_simps =
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  Rep_sprod_inject [symmetric] below_sprod_def
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  prod_eq_iff below_prod_def
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  Rep_sprod_strict Rep_sprod_spair
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lemma sprodE [case_names bottom spair, cases type: sprod]:
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  obtains "p = \<bottom>" | x y where "p = (:x, y:)" and "x \<noteq> \<bottom>" and "y \<noteq> \<bottom>"
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using Rep_sprod [of p] by (auto simp add: sprod_def Rep_sprod_simps)
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lemma sprod_induct [case_names bottom spair, 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 \emph{spair} *}
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lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>"
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by (simp add: Rep_sprod_simps)
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lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"
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by (simp add: Rep_sprod_simps)
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lemma spair_bottom_iff [simp]: "((:x, y:) = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
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by (simp add: Rep_sprod_simps seq_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 seq_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 seq_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_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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by (simp add: spair_below_iff)
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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: "\<bottom> = (:\<bottom>, \<bottom>:)"
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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 \emph{sfst} and \emph{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_bottom_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)"
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by (cases p, simp_all)
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lemma ssnd_bottom_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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555c8951f05c added lemmas sfst_defined_iff, ssnd_defined_iff, sfst_defined, ssnd_defined
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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 spair_sfst_ssnd: "(: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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by (simp add: Rep_sprod_simps sfst_def ssnd_def cont_Rep_sprod)
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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 sfst_below_iff: "sfst\<cdot>x \<sqsubseteq> y \<longleftrightarrow> 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 \<longleftrightarrow> 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 seq_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 \emph{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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end