(*****************************************************************************
 * Featherweight-OCL --- A Formal Semantics for UML-OCL Version OCL 2.5
 *                       for the OMG Standard.
 *                       http://www.brucker.ch/projects/hol-testgen/
 *
 * UML_Library.thy --- Library definitions.
 * This file is part of HOL-TestGen.
 *
 * Copyright (c) 2012-2015 Université Paris-Saclay, Univ. Paris-Sud, France
 *               2013-2015 IRT SystemX, France
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theory  UML_Library
imports (* Basic Type Operations *)
        "basic_types/UML_Boolean"
        "basic_types/UML_Void"
        "basic_types/UML_Integer"
        "basic_types/UML_Real"
        "basic_types/UML_String"
        
        (* Collection Type Operations *)
        "collection_types/UML_Pair"
        "collection_types/UML_Bag"
        "collection_types/UML_Set"
        "collection_types/UML_Sequence"
begin

section\<open>Miscellaneous Stuff\<close>

subsection\<open>Definition: asBoolean\<close>

definition OclAsBoolean\<^sub>I\<^sub>n\<^sub>t  :: "('\<AA>) Integer \<Rightarrow> ('\<AA>) Boolean" ("(_)->oclAsType\<^sub>I\<^sub>n\<^sub>t'(Boolean')")
where     "OclAsBoolean\<^sub>I\<^sub>n\<^sub>t X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> 
                              then \<lfloor>\<lfloor>\<lceil>\<lceil>X \<tau>\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor>
                              else invalid \<tau>)"

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>"
                                by unfold_locales (auto simp: OclAsBoolean\<^sub>I\<^sub>n\<^sub>t_def bot_option_def)

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')")
where     "OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> 
                              then \<lfloor>\<lfloor>\<lceil>\<lceil>X \<tau>\<rceil>\<rceil> \<noteq> 0\<rfloor>\<rfloor>
                              else invalid \<tau>)"

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>"
                                 by unfold_locales (auto simp: OclAsBoolean\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def bot_option_def)

subsection\<open>Definition: asInteger\<close>

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')")
where     "OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> 
                              then \<lfloor>\<lfloor>floor \<lceil>\<lceil>X \<tau>\<rceil>\<rceil>\<rfloor>\<rfloor>
                              else invalid \<tau>)"

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>"
                                 by unfold_locales (auto simp: OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def bot_option_def)

subsection\<open>Definition: asReal\<close>

definition OclAsReal\<^sub>I\<^sub>n\<^sub>t  :: "('\<AA>) Integer \<Rightarrow> ('\<AA>) Real" ("(_)->oclAsType\<^sub>I\<^sub>n\<^sub>t'(Real')")
where     "OclAsReal\<^sub>I\<^sub>n\<^sub>t X = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> 
                              then \<lfloor>\<lfloor>real_of_int \<lceil>\<lceil>X \<tau>\<rceil>\<rceil>\<rfloor>\<rfloor>
                              else invalid \<tau>)"

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>"
                             by unfold_locales (auto simp: OclAsReal\<^sub>I\<^sub>n\<^sub>t_def bot_option_def)

lemma Integer_subtype_of_Real: 
  assumes "\<tau> \<Turnstile> \<delta> X"
  shows   "\<tau> \<Turnstile> X ->oclAsType\<^sub>I\<^sub>n\<^sub>t(Real) ->oclAsType\<^sub>R\<^sub>e\<^sub>a\<^sub>l(Integer) \<triangleq> X"
  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)
  apply(subst (2 4) cp_defined, simp add: true_def)
  by (metis assms bot_option_def drop.elims foundation16 null_option_def)

subsection\<open>Definition: asPair\<close>

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'(')")
where     "OclAsPair\<^sub>S\<^sub>e\<^sub>q S = (if S->size\<^sub>S\<^sub>e\<^sub>q() \<doteq> \<two>
                            then Pair{S->at\<^sub>S\<^sub>e\<^sub>q(\<zero>),S->at\<^sub>S\<^sub>e\<^sub>q(\<one>)}
                            else invalid
                            endif)"

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'(')")
where     "OclAsPair\<^sub>S\<^sub>e\<^sub>t S = (if S->size\<^sub>S\<^sub>e\<^sub>t() \<doteq> \<two>
                            then let v = S->any\<^sub>S\<^sub>e\<^sub>t() in
                                 Pair{v,S->excluding\<^sub>S\<^sub>e\<^sub>t(v)->any\<^sub>S\<^sub>e\<^sub>t()}
                            else invalid
                            endif)"

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'(')")
where     "OclAsPair\<^sub>B\<^sub>a\<^sub>g S = (if S->size\<^sub>B\<^sub>a\<^sub>g() \<doteq> \<two>
                            then let v = S->any\<^sub>B\<^sub>a\<^sub>g() in
                                 Pair{v,S->excluding\<^sub>B\<^sub>a\<^sub>g(v)->any\<^sub>B\<^sub>a\<^sub>g()}
                            else invalid
                            endif)"

subsection\<open>Definition: asSet\<close>

definition OclAsSet\<^sub>S\<^sub>e\<^sub>q   :: "[('\<AA>,'\<alpha>::null)Sequence]\<Rightarrow>('\<AA>,'\<alpha>)Set" ("(_)->asSet\<^sub>S\<^sub>e\<^sub>q'(')")
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)))"

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'(')")
where     "OclAsSet\<^sub>P\<^sub>a\<^sub>i\<^sub>r S = Set{S .First(), S .Second()}"

definition OclAsSet\<^sub>B\<^sub>a\<^sub>g   :: "('\<AA>,'\<alpha>::null) Bag\<Rightarrow>('\<AA>,'\<alpha>)Set" ("(_)->asSet\<^sub>B\<^sub>a\<^sub>g'(')")
where     "OclAsSet\<^sub>B\<^sub>a\<^sub>g S =  (\<lambda> \<tau>. if (\<delta> S) \<tau> = true \<tau> 
                                 then Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e\<lfloor>\<lfloor> Rep_Set_base S \<tau> \<rfloor>\<rfloor> 
                                 else if (\<upsilon> S) \<tau> = true \<tau> then null \<tau> 
                                                          else invalid \<tau>)"

subsection\<open>Definition: asSequence\<close>

definition OclAsSeq\<^sub>S\<^sub>e\<^sub>t   :: "[('\<AA>,'\<alpha>::null)Set]\<Rightarrow>('\<AA>,'\<alpha>)Sequence" ("(_)->asSequence\<^sub>S\<^sub>e\<^sub>t'(')")
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)))"

definition OclAsSeq\<^sub>B\<^sub>a\<^sub>g   :: "[('\<AA>,'\<alpha>::null)Bag]\<Rightarrow>('\<AA>,'\<alpha>)Sequence" ("(_)->asSequence\<^sub>B\<^sub>a\<^sub>g'(')")
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)))"

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'(')")
where     "OclAsSeq\<^sub>P\<^sub>a\<^sub>i\<^sub>r S = Sequence{S .First(), S .Second()}"

subsection\<open>Definition: asBag\<close>

definition OclAsBag\<^sub>S\<^sub>e\<^sub>q   :: "[('\<AA>,'\<alpha>::null)Sequence]\<Rightarrow>('\<AA>,'\<alpha>)Bag" ("(_)->asBag\<^sub>S\<^sub>e\<^sub>q'(')")
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>)"

definition OclAsBag\<^sub>S\<^sub>e\<^sub>t   :: "[('\<AA>,'\<alpha>::null)Set]\<Rightarrow>('\<AA>,'\<alpha>)Bag" ("(_)->asBag\<^sub>S\<^sub>e\<^sub>t'(')")
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>)"

lemma assumes "\<tau> \<Turnstile> \<delta> (S ->size\<^sub>S\<^sub>e\<^sub>t())" (* S is finite *)
      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)))"
oops

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'(')")
where     "OclAsBag\<^sub>P\<^sub>a\<^sub>i\<^sub>r S = Bag{S .First(), S .Second()}"

text_raw\<open>\isatagafp\<close>
subsection\<open>Collection Types\<close>
lemmas cp_intro'' [intro!,simp,code_unfold] =
       cp_intro'
     (*  cp_intro''\<^sub>P\<^sub>a\<^sub>i\<^sub>r *)
       cp_intro''\<^sub>S\<^sub>e\<^sub>t
       cp_intro''\<^sub>S\<^sub>e\<^sub>q
text_raw\<open>\endisatagafp\<close>

subsection\<open>Test Statements\<close>

lemma syntax_test: "Set{\<two>,\<one>} = (Set{}->including\<^sub>S\<^sub>e\<^sub>t(\<one>)->including\<^sub>S\<^sub>e\<^sub>t(\<two>))"
by (rule refl)

text\<open>Here is an example of a nested collection.\<close>
lemma semantic_test2:
assumes H:"(Set{\<two>} \<doteq> null) = (false::('\<AA>)Boolean)"
shows   "(\<tau>::('\<AA>)st) \<Turnstile> (Set{Set{\<two>},null}->includes\<^sub>S\<^sub>e\<^sub>t(null))"
by(simp add: OclIncludes_execute\<^sub>S\<^sub>e\<^sub>t H)



lemma short_cut'[simp,code_unfold]: "(\<eight> \<doteq> \<six>) = false"
 apply(rule ext)
 apply(simp add: StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r StrongEq_def OclInt8_def OclInt6_def
                 true_def false_def invalid_def bot_option_def)
done

lemma short_cut''[simp,code_unfold]: "(\<two> \<doteq> \<one>) = false"
 apply(rule ext)
 apply(simp add: StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r StrongEq_def OclInt2_def OclInt1_def
                 true_def false_def invalid_def bot_option_def)
done
lemma short_cut'''[simp,code_unfold]: "(\<one> \<doteq> \<two>) = false"
 apply(rule ext)
 apply(simp add: StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r StrongEq_def OclInt2_def OclInt1_def
                 true_def false_def invalid_def bot_option_def)
done

Assert   "\<tau> \<Turnstile> (\<zero> <\<^sub>i\<^sub>n\<^sub>t \<two>) and (\<zero> <\<^sub>i\<^sub>n\<^sub>t \<one>) "


text\<open>Elementary computations on Sets.\<close>

declare OclSelect_body_def [simp]

Assert "\<not> (\<tau> \<Turnstile> \<upsilon>(invalid::('\<AA>,'\<alpha>::null) Set))"
Assert    "\<tau> \<Turnstile> \<upsilon>(null::('\<AA>,'\<alpha>::null) Set)"
Assert "\<not> (\<tau> \<Turnstile> \<delta>(null::('\<AA>,'\<alpha>::null) Set))"
Assert    "\<tau> \<Turnstile> \<upsilon>(Set{})"
Assert    "\<tau> \<Turnstile> \<upsilon>(Set{Set{\<two>},null})"
Assert    "\<tau> \<Turnstile> \<delta>(Set{Set{\<two>},null})"
Assert    "\<tau> \<Turnstile> (Set{\<two>,\<one>}->includes\<^sub>S\<^sub>e\<^sub>t(\<one>))"
Assert "\<not> (\<tau> \<Turnstile> (Set{\<two>}->includes\<^sub>S\<^sub>e\<^sub>t(\<one>)))"
Assert "\<not> (\<tau> \<Turnstile> (Set{\<two>,\<one>}->includes\<^sub>S\<^sub>e\<^sub>t(null)))"
Assert    "\<tau> \<Turnstile> (Set{\<two>,null}->includes\<^sub>S\<^sub>e\<^sub>t(null))"
Assert    "\<tau> \<Turnstile> (Set{null,\<two>}->includes\<^sub>S\<^sub>e\<^sub>t(null))"

Assert    "\<tau> \<Turnstile> ((Set{})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z))"

Assert    "\<tau> \<Turnstile> ((Set{\<two>,\<one>})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z))"
Assert "\<not> (\<tau> \<Turnstile> ((Set{\<two>,\<one>})->exists\<^sub>S\<^sub>e\<^sub>t(z | z <\<^sub>i\<^sub>n\<^sub>t \<zero> )))"
Assert "\<not> (\<tau> \<Turnstile> (\<delta>(Set{\<two>,null})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z)))"
Assert "\<not> (\<tau> \<Turnstile> ((Set{\<two>,null})->forAll\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z)))"
Assert    "\<tau> \<Turnstile> ((Set{\<two>,null})->exists\<^sub>S\<^sub>e\<^sub>t(z | \<zero> <\<^sub>i\<^sub>n\<^sub>t z))"


Assert "\<not> (\<tau> \<Turnstile> (Set{null::'a Boolean} \<doteq> Set{}))"
Assert "\<not> (\<tau> \<Turnstile> (Set{null::'a Integer} \<doteq> Set{}))"

Assert "\<not> (\<tau> \<Turnstile> (Set{true} \<doteq> Set{false}))"
Assert "\<not> (\<tau> \<Turnstile> (Set{true,true} \<doteq> Set{false}))"
Assert "\<not> (\<tau> \<Turnstile> (Set{\<two>} \<doteq> Set{\<one>}))"
Assert    "\<tau> \<Turnstile> (Set{\<two>,null,\<two>} \<doteq> Set{null,\<two>})"
Assert    "\<tau> \<Turnstile> (Set{\<one>,null,\<two>} <> Set{null,\<two>})"
Assert    "\<tau> \<Turnstile> (Set{Set{\<two>,null}} \<doteq> Set{Set{null,\<two>}})"
Assert    "\<tau> \<Turnstile> (Set{Set{\<two>,null}} <> Set{Set{null,\<two>},null})"
Assert    "\<tau> \<Turnstile> (Set{null}->select\<^sub>S\<^sub>e\<^sub>t(x | not x) \<doteq> Set{null})"
Assert    "\<tau> \<Turnstile> (Set{null}->reject\<^sub>S\<^sub>e\<^sub>t(x | not x) \<doteq> Set{null})"

lemma     "const (Set{Set{\<two>,null}, invalid})" by(simp add: const_ss)


text\<open>Elementary computations on Sequences.\<close>

Assert "\<not> (\<tau> \<Turnstile> \<upsilon>(invalid::('\<AA>,'\<alpha>::null) Sequence))"
Assert    "\<tau> \<Turnstile> \<upsilon>(null::('\<AA>,'\<alpha>::null) Sequence)"
Assert "\<not> (\<tau> \<Turnstile> \<delta>(null::('\<AA>,'\<alpha>::null) Sequence))"
Assert    "\<tau> \<Turnstile> \<upsilon>(Sequence{})"

lemma     "const (Sequence{Sequence{\<two>,null}, invalid})" by(simp add: const_ss)

end