Add OntoMathPro section and update invariants section
This commit is contained in:
Nicolas Méric
parent
94baf69f25
commit
b2aac7288d
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@ -20,6 +20,44 @@
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note = {Part of the Isabelle distribution.}
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}
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@TechReport{Parsia:12:OWO,
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author = "Bijan Parsia and Boris Motik and Peter Patel-Schneider",
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title = "{OWL} 2 Web Ontology Language Structural Specification and Functional-Style Syntax (Second Edition)",
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month = dec,
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note = "https://www.w3.org/TR/2012/REC-owl2-syntax-20121211/",
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year = "2012",
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bibsource = "https://w2.syronex.com/jmr/w3c-biblio",
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type = "{W3C} Recommendation",
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institution = "W3C",
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}
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@ARTICLE{1654194,
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author={Brachman},
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journal={Computer},
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title={What IS-A Is and Isn't: An Analysis of Taxonomic Links in Semantic Networks},
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year={1983},
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volume={16},
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number={10},
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pages={30-36},
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doi={10.1109/MC.1983.1654194}}
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@article{DBLP:journals/corr/NevzorovaZKL14,
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author = {Olga Nevzorova and
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Nikita Zhiltsov and
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Alexander Kirillovich and
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Evgeny K. Lipachev},
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title = {{\textdollar}OntoMath{\^{}}\{PRO\}{\textdollar} Ontology: {A} Linked
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Data Hub for Mathematics},
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journal = {CoRR},
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volume = {abs/1407.4833},
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year = {2014},
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url = {http://arxiv.org/abs/1407.4833},
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eprinttype = {arXiv},
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eprint = {1407.4833},
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timestamp = {Fri, 21 Aug 2020 16:53:16 +0200},
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biburl = {https://dblp.org/rec/journals/corr/NevzorovaZKL14.bib},
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bibsource = {dblp computer science bibliography, https://dblp.org}
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}
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@TechReport{ bsi:50128:2014,
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type = {Standard},
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@ -231,10 +231,10 @@ text\<open>As novel contribution, this work extends prior versions of \<^dof> by
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\<close>
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text\<open> For example, the \<^dof> evaluation command taking a HOL-expression:
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@{theory_text [display,indent=6, margin=70]
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\<open> value*[ass::Assertion, relvce=2::int] \<open>mapfilter (\<lambda> \<sigma>. relvce \<sigma> > 2) @{instance_of \<open>Assertion\<close>}\<close>\<close>
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\<open> value*[ass::Assertion, relvce=2::int] \<open>filter (\<lambda> \<sigma>. relvce \<sigma> > 2) @{Assertion-instances}\<close>\<close>
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}
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where \<^dof> command \<open>value*\<close> type-checks, expands in an own validation phase
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the \<open>instance_of\<close>-term antiquotation, and evaluates the resulting HOL expression
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the \<open>Assertion-instances\<close>-term antiquotation, and evaluates the resulting HOL expression
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above. Assuming an ontology providing the class \<open>Assertion\<close> having at least the
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integer attribute \<open>relvce\<close>, the command finally creates an instance of \<open>Assertion\<close> and
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binds this to the label \<open>ass\<close> for further use.
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@ -451,6 +451,8 @@ section*[invariants::technical,main_author="Some(@{docitem ''nic''}::author)"]
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\<open>Term-Context Support, Invariants and Queries in DOF\<close>
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(*<*)
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(* Ontology example for mathematical papers *)
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doc_class myauthor =
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email :: "string" <= "''''"
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doc_class mytext_section =
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@ -498,12 +500,12 @@ declare[[invariants_checking_with_tactics = false]]
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text\<open>
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To offer a smooth integration into the \<^emph>\<open>formal\<close> theory development process,
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Isabelle/HOL should be able to dynamically interpret the source document.
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But the specific antiquotations introduced by Isabelle/DOF are not directly recognized
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by Isabelle/HOL, and the process of term checking and evaluation must be enriched.
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\<^isabelle> should be able to dynamically interpret the source document.
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But the specific antiquotations introduced by \<^dof> are not directly recognized
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by \<^isabelle>, and the process of term checking and evaluation must be enriched.
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Previous works~@{cite "brucker.ea:isabelle-ontologies:2018" and "brucker.ea:isabelledof:2019"}
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added a validation step for the SML and semi-formal text contexts.
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To support Isabelle/DOF antiquotations in the term contexts, the validation step should
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To support \<^dof> antiquotations in the term contexts, the validation step should
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be improved and a new step, which we call \<^emph>\<open>elaboration\<close> must be added to allow
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these antiquotations in \<open>\<lambda>\<close>-terms.
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The resulting process encompasses the following steps:
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@ -511,16 +513,16 @@ text\<open>
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\<^item> Infer the type of the term;
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\<^item> Certify the term;
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\<^item> Pass on the information to PIDE;
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\<^item> Validate the term: the Isabelle/DOF antiquotations terms are parsed and type-checked;
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\<^item> Elaborate: the Isabelle/DOF antiquotations terms are expanded.
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The antiquotations are replaced by the object in HOL they reference
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i.e. a \(\lambda\)-calculus term;
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\<^item> Validate the term: the \<^dof> antiquotations terms are parsed and type-checked;
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\<^item> Elaborate: the \<^dof> antiquotations terms are expanded.
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The antiquotations are replaced by the object in \<^hol> they reference
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i.e. a \<open>\<lambda>\<close>-calculus term;
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\<^item> Code generation:
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\<^item> Evaluation of HOL expressions with ontological annotations,
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\<^item> Evaluation of \<^hol> expressions with ontological annotations,
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\<^item> Generation of ontological invariants processed simultaneously
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with the generation of the document (a theory in HOL).
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with the generation of the document (a theory in \<^hol>).
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Isabelle/HOL provides commands to type-check and print terms (the command \<^theory_text>\<open>term\<close>)
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\<^isabelle> provides commands to type-check and print terms (the command \<^theory_text>\<open>term\<close>)
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and evaluate and print a term (the command \<^theory_text>\<open>value\<close>).
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We provide the equivalent commands, respectively \<^theory_text>\<open>term*\<close> and \<^theory_text>\<open>value*\<close>, which support
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the validation and elaboration phase.
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@ -565,7 +567,7 @@ doc_class myconclusion = text_section +
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\caption{Excerpt of an ontology example for mathematical papers.}
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\label{fig-ontology-example}
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\end{figure}
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we add up some class instances with the \<^theory_text>\<open>text*\<close> command, as in \autoref{fig-instances-example}.
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we define some class instances with the \<^theory_text>\<open>text*\<close> command, as in \autoref{fig-instances-example}.
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\begin{figure}
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@{boxed_theory_text [display] \<open>
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text*[church::myauthor, email="\<open>church@lambda.org\<close>"]\<open>\<close>
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@ -582,8 +584,8 @@ text*[claimNotion::myclaim, authored_by = "{@{myauthor \<open>church\<close>}}",
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\caption{Some instances of the classes of the ontology in the \autoref{fig-ontology-example}.}
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\label{fig-instances-example}
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\end{figure}
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In the instance \<^theory_text>\<open>introduction1\<close>, \<^theory_text>\<open>@{author \<open>church\<close>}\<close> denotes
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the instance \<^theory_text>\<open>church\<close> of the class \<^theory_text>\<open>author\<close>.
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In the instance \<^theory_text>\<open>introduction1\<close>, \<^term>\<open>@{myauthor \<open>church\<close>}\<close> denotes
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the instance \<^theory_text>\<open>church\<close> of the class \<^term>\<open>myauthor\<close>.
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One can now reference a class instance in a \<^theory_text>\<open>term*\<close> command.
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In the command \<^theory_text>\<open>term*\<open>@{myauthor \<open>church\<close>}\<close>\<close>
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the term \<^term>\<open>@{myauthor \<open>church\<close>}\<close> is type-checked, \<^ie>, the command \<^theory_text>\<open>term*\<close> checks that
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@ -610,9 +612,9 @@ declare_reference*["evaluation-example"::side_by_side_figure]
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text\<open>
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The \<^theory_text>\<open>value*\<close> command \<^theory_text>\<open>value*\<open>email @{author \<open>church\<close>}\<close>\<close>
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type-checks the antiquotation \<^term>\<open>@{author \<open>church\<close>}\<close>,
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and then returns the value of the \<^theory_text>\<open>email\<close> attribute for the \<^theory_text>\<open>church\<close> instance,
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\<^ie> the HOL string \<^term>\<open>''church@lambda.org''\<close>
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type-checks the antiquotation \<^term>\<open>@{myauthor \<open>church\<close>}\<close>,
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and then returns the value of the \<^term>\<open>email\<close> attribute for the \<^theory_text>\<open>church\<close> instance,
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\<^ie> the \<^hol> string \<^term>\<open>''church@lambda.org''\<close>
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(see \<^side_by_side_figure>\<open>evaluation-example\<close>).
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\<close>
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@ -653,7 +655,7 @@ declare_reference*["term-context-equality-evaluation"::figure]
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text\<open>
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We can even compare class instances. \<^figure>\<open>term-context-equality-evaluation\<close>
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shows that the two class instances \<^theory_text>\<open>resultProof\<close> and \<^theory_text>\<open>resultProof2\<close> are not equivalent
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because their attribute \<^theory_text>\<open>property\<close> differ.
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because their attribute \<^term>\<open>property\<close> differ.
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\<close>
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figure*[
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@ -665,24 +667,25 @@ figure*[
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text\<open>
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A novel mechanism to specify constraints as invariants is implemented
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and can now be specified in common HOL syntax, using the keyword \<^theory_text>\<open>invariant\<close>
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and can now be specified in common \<^hol> syntax, using the keyword \<^theory_text>\<open>invariant\<close>
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in the class definition (recall \autoref{fig-ontology-example}).
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Following the constraints proposed in @{cite "brucker.ea:isabelle-ontologies:2018"},
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one can specify that any instance of a class \<^theory_text>\<open>myresult\<close>
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one can specify that any instance of a class \<^term>\<open>myresult\<close>
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finally has a non-empty property list, if its \<^typ>\<open>kind\<close> is \<^theory_text>\<open>"proof"\<close>
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(see the \<^theory_text>\<open>invariant has_property\<close>), or that
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the \<^theory_text>\<open>establish\<close> relation between \<^typ>\<open>myclaim\<close> and \<^typ>\<open>myresult\<close> must be defined when an instance
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of the class \<^theory_text>\<open>myconclusion\<close> is defined (see the \<^theory_text>\<open>invariant establish_defined\<close>).
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the \<^term>\<open>establish\<close> relation between \<^typ>\<open>myclaim\<close> and \<^typ>\<open>myresult\<close>
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must be defined when an instance
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of the class \<^term>\<open>myconclusion\<close> is defined (see the \<^theory_text>\<open>invariant establish_defined\<close>).
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In our example, the \<^theory_text>\<open>invariant author_finite\<close> of the class \<^theory_text>\<open>myintroduction\<close> enforces
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that the user sets the \<^theory_text>\<open>authored_by\<close> set.
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The \<^theory_text>\<open>\<sigma>\<close> symbol is reserved and references the future class instance.
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In our example, the \<^theory_text>\<open>invariant author_finite\<close> of the class \<^term>\<open>myintroduction\<close> enforces
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that the user sets the \<^term>\<open>authored_by\<close> set.
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The \<open>\<sigma>\<close> symbol is reserved and references the future class instance.
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By relying on the implementation of the Records
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in Isabelle/HOL~@{cite "wenzel:isabelle-isar:2020"},
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in \<^isabelle>~@{cite "wenzel:isabelle-isar:2020"},
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one can reference an attribute of an instance using its selector function.
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For example, \<^theory_text>\<open>establish \<sigma>\<close> denotes the value
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of the \<^theory_text>\<open>establish\<close> attribute
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of the future instance of the class \<^theory_text>\<open>myconclusion\<close>.
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For example, \<^term>\<open>establish \<sigma>\<close> denotes the value
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of the \<^term>\<open>establish\<close> attribute
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of the future instance of the class \<^term>\<open>myconclusion\<close>.
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\<close>
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(*<*)
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@ -693,16 +696,16 @@ text\<open>
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The value of each attribute defined for the instances is checked at run-time
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against their class invariants.
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The \<^theory_text>\<open>resultProof\<close> instance respects the \<^theory_text>\<open>invariant has_property\<close>,
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because we specify its attribute \<^theory_text>\<open>evidence\<close> to the \<^theory_text>\<open>kind\<close> \<^theory_text>\<open>"proof"\<close>,
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we also specify its attribute \<^theory_text>\<open>property\<close> with a suited value
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as a list of \<^theory_text>\<open>"thm"\<close>.
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because we specify its attribute \<^term>\<open>evidence\<close> to the \<^typ>\<open>kind\<close> \<^theory_text>\<open>"proof"\<close>,
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we also specify its attribute \<^term>\<open>property\<close> with a suited value
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as a list of \<^typ>\<open>thm\<close>.
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In \<^figure>\<open>invariant-checking-figure\<close>,
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we try to specify a new instance \<^theory_text>\<open>introduction1\<close> of the class \<^theory_text>\<open>myintroduction\<close>.
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we try to specify a new instance \<^theory_text>\<open>introduction1\<close> of the class \<^term>\<open>myintroduction\<close>.
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But an invariant checking error is triggered because we do not respect the
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constraint specified in the \<^theory_text>\<open>invariant force_level\<close>,
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when we specify the \<^theory_text>\<open>level\<close> attribute of \<^theory_text>\<open>introduction1\<close> to \<^theory_text>\<open>Some 0\<close>.
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when we specify the \<^term>\<open>level\<close> attribute of \<^theory_text>\<open>introduction1\<close> to \<^term>\<open>Some 0\<close>.
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The \<^theory_text>\<open>invariant force_level\<close> forces the value of the argument
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of the attribute \<^theory_text>\<open>level\<close> to be greater than 1.
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of the attribute \<^term>\<open>level\<close> to be greater than 1.
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\<close>
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figure*[
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@ -718,8 +721,8 @@ declare_reference*["inherited-invariant-checking-figure"::figure]
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text\<open>
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Classes inherit the invariants from their superclass.
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As the class \<^theory_text>\<open>myclaim\<close> is a subsclass
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of the class \<^theory_text>\<open>myintroduction\<close>, it inherits the \<^theory_text>\<open>myintroduction\<close> invariants.
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As the class \<^term>\<open>myclaim\<close> is a subsclass
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of the class \<^term>\<open>myintroduction\<close>, it inherits the \<^term>\<open>myintroduction\<close> invariants.
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Hence the \<^theory_text>\<open>invariant force_level\<close> is checked
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when the instance \<^theory_text>\<open>claimNotion\<close> is defined,
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like in \<^figure>\<open>inherited-invariant-checking-figure\<close>.
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@ -743,21 +746,22 @@ value*\<open>filter (\<lambda>\<sigma>. myresult.evidence \<sigma> = argument) @
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(*>*)
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text\<open>
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A new mechanism to make query on instances is available and uses the HOL implementation of lists.
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A new mechanism to make query on instances is available and uses
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the \<^hol> implementation of lists.
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Complex queries can then be defined using functions over the instances list.
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To get the list of the properties of the instances of the class \<^theory_text>\<open>myresult\<close>,
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and the list of the authors of the instances of the class \<^theory_text>\<open>myintroduction\<close>,
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To get the list of the properties of the instances of the class \<^term>\<open>myresult\<close>,
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and the list of the authors of the instances of the class \<^term>\<open>myintroduction\<close>,
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one can use the command \<^theory_text>\<open>value*\<close>:
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@{theory_text [display,indent=10, margin=70] \<open>
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value*\<open>map (myresult.property) @{myresult-instances}\<close>
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value*\<open>map (mytext_section.authored_by) @{myintroduction-instances}\<close>
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\<close>}
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To get the list of the instances of the class \<^theory_text>\<open>myresult\<close> whose \<^theory_text>\<open>evidence\<close> is a \<^theory_text>\<open>"proof"\<close>,
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on can use the command:
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To get the list of the instances of the class \<^term>\<open>myresult\<close>
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whose \<^term>\<open>evidence\<close> is a \<^theory_text>\<open>"proof"\<close>, on can use the command:
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@{theory_text [display,indent=10, margin=70] \<open>
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value*\<open>filter (\<lambda>\<sigma>. myresult.evidence \<sigma> = proof) @{myresult-instances}\<close>
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\<close>}
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To get the list of the instances of the class \<^theory_text>\<open>myintroduction\<close> whose \<^theory_text>\<open>level\<close> > 1,
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To get the list of the instances of the class \<^term>\<open>myintroduction\<close> whose \<^term>\<open>level\<close> > 1,
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on can use the command:
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@{theory_text [display,indent=10, margin=70] \<open>
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value*\<open>filter (\<lambda>\<sigma>. the (mytext_section.level \<sigma>) > 1) @{myintroduction-instances}\<close>
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@ -770,8 +774,176 @@ section*[ontoexample::text_section,main_author="Some(@{docitem ''idir''}::author
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subsection\<open>Engineering Example : An Extract from PLib\<close>
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subsection\<open>Mathematics Example : An Extract from OntoMathPro\<close>
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subsection\<open>Mathematics Example : An Extract from OntoMath\textsuperscript{PRO}\<close>
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(*<*)
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(* OntoMathPro_Ontology example *)
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datatype dc = creator string | title string
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datatype owl =
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backwardCompatibleWith string
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| deprecated string
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| incompatibleWith string
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| priorVersion string
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| versionInfo string
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datatype rdfs =
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comment string
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| isDefinedBy string
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| label string
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| seeAlso string
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datatype annotation = DC dc | OWL owl | RDFS rdfs
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onto_class Top =
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Annotations :: "annotation list"
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onto_class Field_of_mathematics =
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Annotations :: "annotation list"
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invariant restrict_annotation_F ::"set (Annotations \<sigma>) \<subseteq>
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range (RDFS o label) \<union> range (RDFS o comment)"
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onto_class Mathematical_knowledge_object =
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Annotations :: "annotation list"
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invariant restrict_annotation_M ::"set (Annotations \<sigma>) \<subseteq>
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range (RDFS o label) \<union> range (RDFS o comment)"
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onto_class assoc_F_M =
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contains:: "(Field_of_mathematics \<times> Mathematical_knowledge_object) set"
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invariant contains_defined :: "\<forall> x. x \<in> Domain (contains \<sigma>)
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\<longrightarrow> (\<exists> y \<in> Range (contains \<sigma>). (x, y) \<in> contains \<sigma>)"
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onto_class assoc_M_F =
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belongs_to :: "(Mathematical_knowledge_object \<times> Field_of_mathematics) set"
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invariant belong_to_defined :: "\<forall> x. x \<in> Domain (belongs_to \<sigma>)
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\<longrightarrow> (\<exists> y \<in> Range (belongs_to \<sigma>). (x, y) \<in> belongs_to \<sigma>)"
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onto_class assoc_M_M =
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is_defined_by :: "(Mathematical_knowledge_object \<times> Mathematical_knowledge_object) set"
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invariant is_defined_by_defined :: "\<forall> x. x \<in> Domain (is_defined_by \<sigma>)
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\<longrightarrow> (\<exists> y \<in> Range (is_defined_by \<sigma>). (x, y) \<in> is_defined_by \<sigma>)"
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(*typedef 'a A'_scheme =
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"{x :: 'a A_scheme. "
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*)
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onto_class assoc_M_M' =
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"defines" :: "(Mathematical_knowledge_object \<times> Mathematical_knowledge_object) set"
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invariant defines_defined :: "\<forall> x. x \<in> Domain (defines \<sigma>)
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\<longrightarrow> (\<exists> y \<in> Range (defines \<sigma>). (x, y) \<in> defines \<sigma>)"
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onto_class assoc_M_M_see_also =
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see_also :: "(Mathematical_knowledge_object rel) set"
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||||
invariant see_also_trans :: "\<forall> r \<in> (see_also \<sigma>). trans r"
|
||||
invariant see_also_sym :: "\<forall> r \<in> (see_also \<sigma>). sym r"
|
||||
|
||||
onto_class Problem = Mathematical_knowledge_object +
|
||||
Annotations :: "annotation list"
|
||||
|
||||
onto_class Method = Mathematical_knowledge_object +
|
||||
Annotations :: "annotation list"
|
||||
|
||||
onto_class assoc_Problem_Method =
|
||||
is_solved_by :: "(Problem \<times> Method) set"
|
||||
invariant is_solved_by_defined :: "\<forall> x. x \<in> Domain (is_solved_by \<sigma>)
|
||||
\<longrightarrow> (\<exists> y \<in> Range (is_solved_by \<sigma>). (x, y) \<in> is_solved_by \<sigma>)"
|
||||
|
||||
onto_class assoc_Method_Problem =
|
||||
solves :: "(Method \<times> Problem) set"
|
||||
invariant solves_defined :: "\<forall> x. x \<in> Domain (solves \<sigma>)
|
||||
\<longrightarrow> (\<exists> y \<in> Range (solves \<sigma>). (x, y) \<in> solves \<sigma>)"
|
||||
|
||||
(*>*)
|
||||
|
||||
text\<open>
|
||||
The \<^emph>\<open>OntoMath\textsuperscript{PRO}\<close> ontology @{cite "DBLP:journals/corr/NevzorovaZKL14"}
|
||||
is an OWL ontology of mathematical knowledge concepts.
|
||||
It posits the ISA semantics @{cite "1654194"} for hierarchies of mathematical knowledge objects,
|
||||
and defines these objects as two hierarchies of classes:
|
||||
a taxonomy of the fields of mathematics and a taxonomy of mathematical knowledge objects.
|
||||
It defines two main type of relations for these two taxonomies:
|
||||
directed relations between elements of the two hierarchies
|
||||
like \<^term>\<open>belongs_to\<close>, \<^term>\<open>contains\<close>, \<^term>\<open>defines\<close>, \<^term>\<open>is_defined_by\<close>, \<^term>\<open>solves\<close>,
|
||||
\<^term>\<open>is_solved_by\<close>, and a symmetric transitive associative relation \<^term>\<open>see_also\<close>
|
||||
between two mathematical knowledge objects.
|
||||
It also represents links with external resources such as DBpedia
|
||||
with annotations properties @{cite "Parsia:12:OWO"}.
|
||||
\<^dof> covers a wide range of the OWL concepts used by the \<^emph>\<open>OntoMath\textsuperscript{PRO}\<close> ontology.
|
||||
We can represent the annotations properties as datatypes and
|
||||
then attach them as an attributes list to a class.
|
||||
For example the declaration:
|
||||
@{theory_text [display,indent=10, margin=70] \<open>
|
||||
datatype dc = creator string | title string
|
||||
datatype owl = backwardCompatibleWith string
|
||||
| deprecated string
|
||||
| incompatibleWith string
|
||||
| priorVersion string
|
||||
| versionInfo string
|
||||
datatype rdfs = comment string
|
||||
| isDefinedBy string
|
||||
| label string
|
||||
| seeAlso string
|
||||
datatype annotation = DC dc | OWL owl | RDFS rdfs
|
||||
|
||||
onto_class Field_of_mathematics =
|
||||
Annotations :: "annotation list"
|
||||
invariant restrict_annotation_F ::"set (Annotations \<sigma>) \<subseteq>
|
||||
range (RDFS o label) \<union> range (RDFS o comment)"
|
||||
|
||||
\<close>}
|
||||
defines the class \<^term>\<open>Field_of_mathematics\<close> with an attribute \<^term>\<open>Annotations\<close>
|
||||
which is a list of \<^typ>\<open>annotation\<close>s.
|
||||
We can even constraint the type of allowed \<^typ>\<open>annotation\<close>s with an invariant.
|
||||
Here the \<^theory_text>\<open>invariant restrict_annotation_F\<close> forces the \<^typ>\<open>annotation\<close>s to be
|
||||
a \<^term>\<open>label\<close> or a \<^term>\<open>comment\<close>.
|
||||
Subsumption relation is straightforward.
|
||||
The ontology \<^emph>\<open>OntoMath\textsuperscript{PRO}\<close> defines
|
||||
a \<^term>\<open>Problem\<close> and a \<^term>\<open>Method\<close>
|
||||
as subclasses of the class \<^term>\<open>Mathematical_knowledge_object\<close>.
|
||||
It gives in \<^dof>:
|
||||
@{theory_text [display,indent=10, margin=70] \<open>
|
||||
onto_class Problem = Mathematical_knowledge_object +
|
||||
Annotations :: "annotation list"
|
||||
|
||||
onto_class Method = Mathematical_knowledge_object +
|
||||
Annotations :: "annotation list"
|
||||
\<close>}
|
||||
We can express the relations between the two taxonomies
|
||||
with association \<^theory_text>\<open>onto_class\<close>es which specify
|
||||
the relation as an attribute and enforces the relation with an \<^theory_text>\<open>invariant\<close>.
|
||||
The two directed relations \<^term>\<open>is_solved_by\<close> and \<^term>\<open>solves\<close>
|
||||
between \<^term>\<open>Problem\<close> and a \<^term>\<open>Method\<close> can be represented in \<^dof> like this:
|
||||
@{theory_text [display,indent=10, margin=70] \<open>
|
||||
|
||||
onto_class assoc_M_M_see_also =
|
||||
see_also :: "(Mathematical_knowledge_object rel) set"
|
||||
invariant see_also_trans :: "\<forall> r \<in> (see_also \<sigma>). trans r"
|
||||
invariant see_also_sym :: "\<forall> r \<in> (see_also \<sigma>). sym r"
|
||||
|
||||
onto_class Method = Mathematical_knowledge_object +
|
||||
Annotations :: "annotation list"
|
||||
|
||||
onto_class assoc_Problem_Method =
|
||||
is_solved_by :: "(Problem \<times> Method) set"
|
||||
invariant is_solved_by_defined :: "\<forall> x. x \<in> Domain (is_solved_by \<sigma>)
|
||||
\<longrightarrow> (\<exists> y \<in> Range (is_solved_by \<sigma>). (x, y) \<in> is_solved_by \<sigma>)"
|
||||
|
||||
onto_class assoc_Method_Problem =
|
||||
solves :: "(Method \<times> Problem) set"
|
||||
invariant solves_defined :: "\<forall> x. x \<in> Domain (solves \<sigma>)
|
||||
\<longrightarrow> (\<exists> y \<in> Range (solves \<sigma>). (x, y) \<in> solves \<sigma>)"
|
||||
\<close>}
|
||||
where the attributes \<^term>\<open>is_solved_by\<close> and \<^term>\<open>solves\<close> define the relations
|
||||
of the classes and the invariants \<^theory_text>\<open>is_solved_by_defined\<close> and \<^theory_text>\<open>solves_defined\<close> enforce
|
||||
the existence of the relations when one define instances of the classes.
|
||||
|
||||
\<^dof> as a framework allows to define an ontology and specify constraints
|
||||
on its concepts, and dynamically checks at run-time the concepts and instances.
|
||||
It offers an environment to define and test ontologies in an integrated document,
|
||||
where all the knowledge and the proof-checking can be specified,
|
||||
and thus can be a solution to go over
|
||||
the trade-off between plain vocabularies and highly formalized models.
|
||||
\<close>
|
||||
|
||||
section*[concl::conclusion]\<open>Conclusion\<close>
|
||||
subsection*[rw::related_work]\<open>Related Works\<close>
|
||||
|
|
|
|||
Loading…
Reference in New Issue