261 lines
16 KiB
Plaintext
261 lines
16 KiB
Plaintext
(*****************************************************************************
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* Featherweight-OCL --- A Formal Semantics for UML-OCL Version OCL 2.5
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* for the OMG Standard.
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* http://www.brucker.ch/projects/hol-testgen/
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*
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* UML_Library.thy --- Library definitions.
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* This file is part of HOL-TestGen.
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*
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* Copyright (c) 2012-2015 Université Paris-Saclay, Univ. Paris-Sud, France
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* 2013-2015 IRT SystemX, France
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*
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions are
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* met:
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*
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* * Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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*
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* * Redistributions in binary form must reproduce the above
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* copyright notice, this list of conditions and the following
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* disclaimer in the documentation and/or other materials provided
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* with the distribution.
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*
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* * Neither the name of the copyright holders nor the names of its
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* contributors may be used to endorse or promote products derived
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* from this software without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
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* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
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* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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******************************************************************************)
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theory UML_Library
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imports (* Basic Type Operations *)
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"basic_types/UML_Boolean"
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"basic_types/UML_Void"
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"basic_types/UML_Integer"
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"basic_types/UML_Real"
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"basic_types/UML_String"
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(* Collection Type Operations *)
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"collection_types/UML_Pair"
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"collection_types/UML_Bag"
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"collection_types/UML_Set"
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"collection_types/UML_Sequence"
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begin
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section\<open>Miscellaneous Stuff\<close>
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subsection\<open>Definition: asBoolean\<close>
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definition OclAsBoolean\<^sub>I\<^sub>n\<^sub>t :: "('\<AA>) Integer \<Rightarrow> ('\<AA>) Boolean" ("(_)->oclAsType\<^sub>I\<^sub>n\<^sub>t'(Boolean')")
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where "OclAsBoolean\<^sub>I\<^sub>n\<^sub>t X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau>
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then \<lfloor>\<lfloor>\<lceil>\<lceil>X \<tau>\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor>
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else invalid \<tau>)"
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interpretation OclAsBoolean\<^sub>I\<^sub>n\<^sub>t : profile_mono\<^sub>d OclAsBoolean\<^sub>I\<^sub>n\<^sub>t "\<lambda>x. \<lfloor>\<lfloor>\<lceil>\<lceil>x\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor>"
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by unfold_locales (auto simp: OclAsBoolean\<^sub>I\<^sub>n\<^sub>t_def bot_option_def)
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definition OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l :: "('\<AA>) Real \<Rightarrow> ('\<AA>) Boolean" ("(_)->oclAsType\<^sub>R\<^sub>e\<^sub>a\<^sub>l'(Boolean')")
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where "OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau>
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then \<lfloor>\<lfloor>\<lceil>\<lceil>X \<tau>\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor>
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else invalid \<tau>)"
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interpretation OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l : profile_mono\<^sub>d OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l "\<lambda>x. \<lfloor>\<lfloor>\<lceil>\<lceil>x\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor>"
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by unfold_locales (auto simp: OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def bot_option_def)
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subsection\<open>Definition: asInteger\<close>
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definition OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l :: "('\<AA>) Real \<Rightarrow> ('\<AA>) Integer" ("(_)->oclAsType\<^sub>R\<^sub>e\<^sub>a\<^sub>l'(Integer')")
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where "OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau>
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then \<lfloor>\<lfloor>floor \<lceil>\<lceil>X \<tau>\<rceil>\<rceil>\<rfloor>\<rfloor>
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else invalid \<tau>)"
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interpretation OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l : profile_mono\<^sub>d OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l "\<lambda>x. \<lfloor>\<lfloor>floor \<lceil>\<lceil>x\<rceil>\<rceil>\<rfloor>\<rfloor>"
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by unfold_locales (auto simp: OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def bot_option_def)
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subsection\<open>Definition: asReal\<close>
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definition OclAsReal\<^sub>I\<^sub>n\<^sub>t :: "('\<AA>) Integer \<Rightarrow> ('\<AA>) Real" ("(_)->oclAsType\<^sub>I\<^sub>n\<^sub>t'(Real')")
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where "OclAsReal\<^sub>I\<^sub>n\<^sub>t X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau>
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then \<lfloor>\<lfloor>real_of_int \<lceil>\<lceil>X \<tau>\<rceil>\<rceil>\<rfloor>\<rfloor>
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else invalid \<tau>)"
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interpretation OclAsReal\<^sub>I\<^sub>n\<^sub>t : profile_mono\<^sub>d OclAsReal\<^sub>I\<^sub>n\<^sub>t "\<lambda>x. \<lfloor>\<lfloor>real_of_int \<lceil>\<lceil>x\<rceil>\<rceil>\<rfloor>\<rfloor>"
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by unfold_locales (auto simp: OclAsReal\<^sub>I\<^sub>n\<^sub>t_def bot_option_def)
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lemma Integer_subtype_of_Real:
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assumes "\<tau> \<Turnstile> \<delta> X"
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shows "\<tau> \<Turnstile> X ->oclAsType\<^sub>I\<^sub>n\<^sub>t(Real) ->oclAsType\<^sub>R\<^sub>e\<^sub>a\<^sub>l(Integer) \<triangleq> X"
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apply(insert assms, simp add: OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def OclAsReal\<^sub>I\<^sub>n\<^sub>t_def OclValid_def StrongEq_def)
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apply(subst (2 4) cp_defined, simp add: true_def)
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by (metis assms bot_option_def drop.elims foundation16 null_option_def)
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subsection\<open>Definition: asPair\<close>
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definition OclAsPair\<^sub>S\<^sub>e\<^sub>q :: "[('\<AA>,'\<alpha>::null)Sequence]\<Rightarrow>('\<AA>,'\<alpha>::null,'\<alpha>::null) Pair" ("(_)->asPair\<^sub>S\<^sub>e\<^sub>q'(')")
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where "OclAsPair\<^sub>S\<^sub>e\<^sub>q S = (if S->size\<^sub>S\<^sub>e\<^sub>q() \<doteq> \<two>
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then Pair{S->at\<^sub>S\<^sub>e\<^sub>q(\<zero>),S->at\<^sub>S\<^sub>e\<^sub>q(\<one>)}
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else invalid
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endif)"
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definition OclAsPair\<^sub>S\<^sub>e\<^sub>t :: "[('\<AA>,'\<alpha>::null)Set]\<Rightarrow>('\<AA>,'\<alpha>::null,'\<alpha>::null) Pair" ("(_)->asPair\<^sub>S\<^sub>e\<^sub>t'(')")
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where "OclAsPair\<^sub>S\<^sub>e\<^sub>t S = (if S->size\<^sub>S\<^sub>e\<^sub>t() \<doteq> \<two>
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then let v = S->any\<^sub>S\<^sub>e\<^sub>t() in
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Pair{v,S->excluding\<^sub>S\<^sub>e\<^sub>t(v)->any\<^sub>S\<^sub>e\<^sub>t()}
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else invalid
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endif)"
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definition OclAsPair\<^sub>B\<^sub>a\<^sub>g :: "[('\<AA>,'\<alpha>::null)Bag]\<Rightarrow>('\<AA>,'\<alpha>::null,'\<alpha>::null) Pair" ("(_)->asPair\<^sub>B\<^sub>a\<^sub>g'(')")
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where "OclAsPair\<^sub>B\<^sub>a\<^sub>g S = (if S->size\<^sub>B\<^sub>a\<^sub>g() \<doteq> \<two>
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then let v = S->any\<^sub>B\<^sub>a\<^sub>g() in
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Pair{v,S->excluding\<^sub>B\<^sub>a\<^sub>g(v)->any\<^sub>B\<^sub>a\<^sub>g()}
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else invalid
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endif)"
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subsection\<open>Definition: asSet\<close>
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definition OclAsSet\<^sub>S\<^sub>e\<^sub>q :: "[('\<AA>,'\<alpha>::null)Sequence]\<Rightarrow>('\<AA>,'\<alpha>)Set" ("(_)->asSet\<^sub>S\<^sub>e\<^sub>q'(')")
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where "OclAsSet\<^sub>S\<^sub>e\<^sub>q S = (S->iterate\<^sub>S\<^sub>e\<^sub>q(b; x = Set{} | x ->including\<^sub>S\<^sub>e\<^sub>t(b)))"
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definition OclAsSet\<^sub>P\<^sub>a\<^sub>i\<^sub>r :: "[('\<AA>,'\<alpha>::null,'\<alpha>::null) Pair]\<Rightarrow>('\<AA>,'\<alpha>)Set" ("(_)->asSet\<^sub>P\<^sub>a\<^sub>i\<^sub>r'(')")
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where "OclAsSet\<^sub>P\<^sub>a\<^sub>i\<^sub>r S = Set{S .First(), S .Second()}"
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definition OclAsSet\<^sub>B\<^sub>a\<^sub>g :: "('\<AA>,'\<alpha>::null) Bag\<Rightarrow>('\<AA>,'\<alpha>)Set" ("(_)->asSet\<^sub>B\<^sub>a\<^sub>g'(')")
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where "OclAsSet\<^sub>B\<^sub>a\<^sub>g S = (\<lambda> \<tau>. if (\<delta> S) \<tau> = true \<tau>
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then Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e\<lfloor>\<lfloor> Rep_Set_base S \<tau> \<rfloor>\<rfloor>
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else if (\<upsilon> S) \<tau> = true \<tau> then null \<tau>
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else invalid \<tau>)"
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subsection\<open>Definition: asSequence\<close>
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definition OclAsSeq\<^sub>S\<^sub>e\<^sub>t :: "[('\<AA>,'\<alpha>::null)Set]\<Rightarrow>('\<AA>,'\<alpha>)Sequence" ("(_)->asSequence\<^sub>S\<^sub>e\<^sub>t'(')")
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where "OclAsSeq\<^sub>S\<^sub>e\<^sub>t S = (S->iterate\<^sub>S\<^sub>e\<^sub>t(b; x = Sequence{} | x ->including\<^sub>S\<^sub>e\<^sub>q(b)))"
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definition OclAsSeq\<^sub>B\<^sub>a\<^sub>g :: "[('\<AA>,'\<alpha>::null)Bag]\<Rightarrow>('\<AA>,'\<alpha>)Sequence" ("(_)->asSequence\<^sub>B\<^sub>a\<^sub>g'(')")
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where "OclAsSeq\<^sub>B\<^sub>a\<^sub>g S = (S->iterate\<^sub>B\<^sub>a\<^sub>g(b; x = Sequence{} | x ->including\<^sub>S\<^sub>e\<^sub>q(b)))"
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definition OclAsSeq\<^sub>P\<^sub>a\<^sub>i\<^sub>r :: "[('\<AA>,'\<alpha>::null,'\<alpha>::null) Pair]\<Rightarrow>('\<AA>,'\<alpha>)Sequence" ("(_)->asSequence\<^sub>P\<^sub>a\<^sub>i\<^sub>r'(')")
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where "OclAsSeq\<^sub>P\<^sub>a\<^sub>i\<^sub>r S = Sequence{S .First(), S .Second()}"
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subsection\<open>Definition: asBag\<close>
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definition OclAsBag\<^sub>S\<^sub>e\<^sub>q :: "[('\<AA>,'\<alpha>::null)Sequence]\<Rightarrow>('\<AA>,'\<alpha>)Bag" ("(_)->asBag\<^sub>S\<^sub>e\<^sub>q'(')")
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where "OclAsBag\<^sub>S\<^sub>e\<^sub>q S = (\<lambda>\<tau>. Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>\<lambda>s. if list_ex ((=) s) \<lceil>\<lceil>Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S \<tau>)\<rceil>\<rceil> then 1 else 0\<rfloor>\<rfloor>)"
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definition OclAsBag\<^sub>S\<^sub>e\<^sub>t :: "[('\<AA>,'\<alpha>::null)Set]\<Rightarrow>('\<AA>,'\<alpha>)Bag" ("(_)->asBag\<^sub>S\<^sub>e\<^sub>t'(')")
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where "OclAsBag\<^sub>S\<^sub>e\<^sub>t S = (\<lambda>\<tau>. Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>\<lambda>s. if s \<in> \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (S \<tau>)\<rceil>\<rceil> then 1 else 0\<rfloor>\<rfloor>)"
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lemma assumes "\<tau> \<Turnstile> \<delta> (S ->size\<^sub>S\<^sub>e\<^sub>t())" (* S is finite *)
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shows "OclAsBag\<^sub>S\<^sub>e\<^sub>t S = (S->iterate\<^sub>S\<^sub>e\<^sub>t(b; x = Bag{} | x ->including\<^sub>B\<^sub>a\<^sub>g(b)))"
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oops
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definition OclAsBag\<^sub>P\<^sub>a\<^sub>i\<^sub>r :: "[('\<AA>,'\<alpha>::null,'\<alpha>::null) Pair]\<Rightarrow>('\<AA>,'\<alpha>)Bag" ("(_)->asBag\<^sub>P\<^sub>a\<^sub>i\<^sub>r'(')")
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where "OclAsBag\<^sub>P\<^sub>a\<^sub>i\<^sub>r S = Bag{S .First(), S .Second()}"
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text_raw\<open>\isatagafp\<close>
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subsection\<open>Collection Types\<close>
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lemmas cp_intro'' [intro!,simp,code_unfold] =
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cp_intro'
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(* cp_intro''\<^sub>P\<^sub>a\<^sub>i\<^sub>r *)
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cp_intro''\<^sub>S\<^sub>e\<^sub>t
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cp_intro''\<^sub>S\<^sub>e\<^sub>q
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text_raw\<open>\endisatagafp\<close>
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subsection\<open>Test Statements\<close>
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lemma syntax_test: "Set{\<two>,\<one>} = (Set{}->including\<^sub>S\<^sub>e\<^sub>t(\<one>)->including\<^sub>S\<^sub>e\<^sub>t(\<two>))"
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by (rule refl)
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text\<open>Here is an example of a nested collection.\<close>
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lemma semantic_test2:
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assumes H:"(Set{\<two>} \<doteq> null) = (false::('\<AA>)Boolean)"
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shows "(\<tau>::('\<AA>)st) \<Turnstile> (Set{Set{\<two>},null}->includes\<^sub>S\<^sub>e\<^sub>t(null))"
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by(simp add: OclIncludes_execute\<^sub>S\<^sub>e\<^sub>t H)
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lemma short_cut'[simp,code_unfold]: "(\<eight> \<doteq> \<six>) = false"
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apply(rule ext)
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apply(simp add: StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r StrongEq_def OclInt8_def OclInt6_def
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true_def false_def invalid_def bot_option_def)
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done
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lemma short_cut''[simp,code_unfold]: "(\<two> \<doteq> \<one>) = false"
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apply(rule ext)
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apply(simp add: StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r StrongEq_def OclInt2_def OclInt1_def
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true_def false_def invalid_def bot_option_def)
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done
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lemma short_cut'''[simp,code_unfold]: "(\<one> \<doteq> \<two>) = false"
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apply(rule ext)
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apply(simp add: StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r StrongEq_def OclInt2_def OclInt1_def
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true_def false_def invalid_def bot_option_def)
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done
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Assert "\<tau> \<Turnstile> (\<zero> <\<^sub>i\<^sub>n\<^sub>t \<two>) and (\<zero> <\<^sub>i\<^sub>n\<^sub>t \<one>) "
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text\<open>Elementary computations on Sets.\<close>
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declare OclSelect_body_def [simp]
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Assert "\<not> (\<tau> \<Turnstile> \<upsilon>(invalid::('\<AA>,'\<alpha>::null) Set))"
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Assert "\<tau> \<Turnstile> \<upsilon>(null::('\<AA>,'\<alpha>::null) Set)"
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Assert "\<not> (\<tau> \<Turnstile> \<delta>(null::('\<AA>,'\<alpha>::null) Set))"
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Assert "\<tau> \<Turnstile> \<upsilon>(Set{})"
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Assert "\<tau> \<Turnstile> \<upsilon>(Set{Set{\<two>},null})"
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Assert "\<tau> \<Turnstile> \<delta>(Set{Set{\<two>},null})"
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Assert "\<tau> \<Turnstile> (Set{\<two>,\<one>}->includes\<^sub>S\<^sub>e\<^sub>t(\<one>))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{\<two>}->includes\<^sub>S\<^sub>e\<^sub>t(\<one>)))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{\<two>,\<one>}->includes\<^sub>S\<^sub>e\<^sub>t(null)))"
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Assert "\<tau> \<Turnstile> (Set{\<two>,null}->includes\<^sub>S\<^sub>e\<^sub>t(null))"
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Assert "\<tau> \<Turnstile> (Set{null,\<two>}->includes\<^sub>S\<^sub>e\<^sub>t(null))"
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Assert "\<tau> \<Turnstile> ((Set{})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z))"
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Assert "\<tau> \<Turnstile> ((Set{\<two>,\<one>})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z))"
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Assert "\<not> (\<tau> \<Turnstile> ((Set{\<two>,\<one>})->exists\<^sub>S\<^sub>e\<^sub>t(z | z <\<^sub>i\<^sub>n\<^sub>t \<zero> )))"
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Assert "\<not> (\<tau> \<Turnstile> (\<delta>(Set{\<two>,null})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z)))"
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Assert "\<not> (\<tau> \<Turnstile> ((Set{\<two>,null})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z)))"
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Assert "\<tau> \<Turnstile> ((Set{\<two>,null})->exists\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{null::'a Boolean} \<doteq> Set{}))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{null::'a Integer} \<doteq> Set{}))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{true} \<doteq> Set{false}))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{true,true} \<doteq> Set{false}))"
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Assert "\<not> (\<tau> \<Turnstile> (Set{\<two>} \<doteq> Set{\<one>}))"
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Assert "\<tau> \<Turnstile> (Set{\<two>,null,\<two>} \<doteq> Set{null,\<two>})"
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Assert "\<tau> \<Turnstile> (Set{\<one>,null,\<two>} <> Set{null,\<two>})"
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Assert "\<tau> \<Turnstile> (Set{Set{\<two>,null}} \<doteq> Set{Set{null,\<two>}})"
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Assert "\<tau> \<Turnstile> (Set{Set{\<two>,null}} <> Set{Set{null,\<two>},null})"
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Assert "\<tau> \<Turnstile> (Set{null}->select\<^sub>S\<^sub>e\<^sub>t(x | not x) \<doteq> Set{null})"
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Assert "\<tau> \<Turnstile> (Set{null}->reject\<^sub>S\<^sub>e\<^sub>t(x | not x) \<doteq> Set{null})"
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lemma "const (Set{Set{\<two>,null}, invalid})" by(simp add: const_ss)
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text\<open>Elementary computations on Sequences.\<close>
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Assert "\<not> (\<tau> \<Turnstile> \<upsilon>(invalid::('\<AA>,'\<alpha>::null) Sequence))"
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Assert "\<tau> \<Turnstile> \<upsilon>(null::('\<AA>,'\<alpha>::null) Sequence)"
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Assert "\<not> (\<tau> \<Turnstile> \<delta>(null::('\<AA>,'\<alpha>::null) Sequence))"
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Assert "\<tau> \<Turnstile> \<upsilon>(Sequence{})"
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lemma "const (Sequence{Sequence{\<two>,null}, invalid})" by(simp add: const_ss)
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end
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