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 cpair_strict)
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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 spair_Abs_Sprod:
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  "(:a, b:) = Abs_Sprod <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>"
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apply (unfold spair_def)
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apply (simp add: cont_Abs_Sprod spair_lemma)
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done
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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 (cases z rule: Abs_Sprod_cases)
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apply (simp add: Sprod_def)
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apply (erule disjE)
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apply (simp add: Abs_Sprod_strict)
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apply (rule disjI2)
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apply (rule_tac x="cfst\<cdot>y" in exI)
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apply (rule_tac x="csnd\<cdot>y" in exI)
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apply (simp add: spair_Abs_Sprod Abs_Sprod_inject spair_lemma)
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apply (simp add: surjective_pairing_Cprod2)
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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: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict)
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lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"
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by (simp add: spair_Abs_Sprod strictify_conv_if cpair_strict Abs_Sprod_strict)
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lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>"
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by auto
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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 (erule contrapos_np, auto)
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lemma spair_defined [simp]:
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  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>"
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by (simp add: spair_Abs_Sprod Abs_Sprod_defined Sprod_def)
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lemma spair_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>"
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by (erule contrapos_pp, 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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apply (simp add: spair_Abs_Sprod)
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apply (simp add: Abs_Sprod_inject [OF _ spair_lemma] Sprod_def)
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apply (simp add: strictify_conv_if)
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done
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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 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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apply (unfold spair_def)
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apply (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)
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done
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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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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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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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end