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+}\r
+\r
+STRINGS { s t }\r
+\r
+FUNCTION {output.nonnull}\r
+{ 's :=\r
+ output.state mid.sentence =\r
+ { ", " * write$ }\r
+ { output.state after.block =\r
+ { add.period$ write$\r
+ newline$\r
+ "\newblock " write$\r
+ }\r
+ { output.state before.all =\r
+ 'write$\r
+ { add.period$ " " * write$ }\r
+ if$\r
+ }\r
+ if$\r
+ mid.sentence 'output.state :=\r
+ }\r
+ if$\r
+ s\r
+}\r
+\r
+FUNCTION {output}\r
+{ duplicate$ empty$\r
+ 'pop$\r
+ 'output.nonnull\r
+ if$\r
+}\r
+\r
+FUNCTION {output.check}\r
+{ 't :=\r
+ duplicate$ empty$\r
+ { pop$ "empty " t * " in " * cite$ * warning$ }\r
+ 'output.nonnull\r
+ if$\r
+}\r
+\r
+FUNCTION {output.bibitem}\r
+{ newline$\r
+ "\bibitem{" write$\r
+ cite$ write$\r
+ "}" write$\r
+ newline$\r
+ ""\r
+ before.all 'output.state :=\r
+}\r
+\r
+FUNCTION {fin.entry}\r
+{ add.period$\r
+ write$\r
+ newline$\r
+}\r
+\r
+FUNCTION {new.block}\r
+{ output.state before.all =\r
+ 'skip$\r
+ { after.block 'output.state := }\r
+ if$\r
+}\r
+\r
+FUNCTION {new.sentence}\r
+{ output.state after.block =\r
+ 'skip$\r
+ { output.state before.all =\r
+ 'skip$\r
+ { after.sentence 'output.state := }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {not}\r
+{ { #0 }\r
+ { #1 }\r
+ if$\r
+}\r
+\r
+FUNCTION {and}\r
+{ 'skip$\r
+ { pop$ #0 }\r
+ if$\r
+}\r
+\r
+FUNCTION {or}\r
+{ { pop$ #1 }\r
+ 'skip$\r
+ if$\r
+}\r
+\r
+FUNCTION {new.block.checka}\r
+{ empty$\r
+ 'skip$\r
+ 'new.block\r
+ if$\r
+}\r
+\r
+FUNCTION {new.block.checkb}\r
+{ empty$\r
+ swap$ empty$\r
+ and\r
+ 'skip$\r
+ 'new.block\r
+ if$\r
+}\r
+\r
+FUNCTION {new.sentence.checka}\r
+{ empty$\r
+ 'skip$\r
+ 'new.sentence\r
+ if$\r
+}\r
+\r
+FUNCTION {new.sentence.checkb}\r
+{ empty$\r
+ swap$ empty$\r
+ and\r
+ 'skip$\r
+ 'new.sentence\r
+ if$\r
+}\r
+\r
+FUNCTION {field.or.null}\r
+{ duplicate$ empty$\r
+ { pop$ "" }\r
+ 'skip$\r
+ if$\r
+}\r
+\r
+FUNCTION {emphasize}\r
+{ duplicate$ empty$\r
+ { pop$ "" }\r
+ { "{\em " swap$ * "}" * }\r
+ if$\r
+}\r
+\r
+INTEGERS { nameptr namesleft numnames }\r
+\r
+FUNCTION {format.names}\r
+{ 's :=\r
+ #1 'nameptr :=\r
+ s num.names$ 'numnames :=\r
+ numnames 'namesleft :=\r
+ { namesleft #0 > }\r
+ { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=\r
+ nameptr #1 >\r
+ { namesleft #1 >\r
+ { ", " * t * }\r
+ { numnames #2 >\r
+ { "," * }\r
+ 'skip$\r
+ if$\r
+ t "others" =\r
+ { " et~al." * }\r
+ { " and " * t * }\r
+ if$\r
+ }\r
+ if$\r
+ }\r
+ 't\r
+ if$\r
+ nameptr #1 + 'nameptr :=\r
+\r
+ namesleft #1 - 'namesleft :=\r
+ }\r
+ while$\r
+}\r
+\r
+FUNCTION {format.authors}\r
+{ author empty$\r
+ { "" }\r
+ { author format.names }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.editors}\r
+{ editor empty$\r
+ { "" }\r
+ { editor format.names\r
+ editor num.names$ #1 >\r
+ { ", editors" * }\r
+ { ", editor" * }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.title}\r
+{ title empty$\r
+ { "" }\r
+ { title "t" change.case$ }\r
+ if$\r
+}\r
+\r
+FUNCTION {n.dashify}\r
+{ 't :=\r
+ ""\r
+ { t empty$ not }\r
+ { t #1 #1 substring$ "-" =\r
+ { t #1 #2 substring$ "--" = not\r
+ { "--" *\r
+ t #2 global.max$ substring$ 't :=\r
+ }\r
+ { { t #1 #1 substring$ "-" = }\r
+ { "-" *\r
+ t #2 global.max$ substring$ 't :=\r
+ }\r
+ while$\r
+ }\r
+ if$\r
+ }\r
+ { t #1 #1 substring$ *\r
+ t #2 global.max$ substring$ 't :=\r
+ }\r
+ if$\r
+ }\r
+ while$\r
+}\r
+\r
+FUNCTION {format.date}\r
+{ year empty$\r
+ { month empty$\r
+ { "" }\r
+ { "there's a month but no year in " cite$ * warning$\r
+ month\r
+ }\r
+ if$\r
+ }\r
+ { month empty$\r
+ 'year\r
+ { month " " * year * }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.btitle}\r
+{ title emphasize\r
+}\r
+\r
+FUNCTION {tie.or.space.connect}\r
+{ duplicate$ text.length$ #3 <\r
+ { "~" }\r
+ { " " }\r
+ if$\r
+ swap$ * *\r
+}\r
+\r
+FUNCTION {either.or.check}\r
+{ empty$\r
+ 'pop$\r
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.bvolume}\r
+{ volume empty$\r
+ { "" }\r
+ { "volume" volume tie.or.space.connect\r
+ series empty$\r
+ 'skip$\r
+ { " of " * series emphasize * }\r
+ if$\r
+ "volume and number" number either.or.check\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.number.series}\r
+{ volume empty$\r
+ { number empty$\r
+ { series field.or.null }\r
+ { output.state mid.sentence =\r
+ { "number" }\r
+ { "Number" }\r
+ if$\r
+ number tie.or.space.connect\r
+ series empty$\r
+ { "there's a number but no series in " cite$ * warning$ }\r
+ { " in " * series * }\r
+ if$\r
+ }\r
+ if$\r
+ }\r
+ { "" }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.edition}\r
+{ edition empty$\r
+ { "" }\r
+ { output.state mid.sentence =\r
+ { edition "l" change.case$ " edition" * }\r
+ { edition "t" change.case$ " edition" * }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+INTEGERS { multiresult }\r
+\r
+FUNCTION {multi.page.check}\r
+{ 't :=\r
+ #0 'multiresult :=\r
+ { multiresult not\r
+ t empty$ not\r
+ and\r
+ }\r
+ { t #1 #1 substring$\r
+ duplicate$ "-" =\r
+ swap$ duplicate$ "," =\r
+ swap$ "+" =\r
+ or or\r
+ { #1 'multiresult := }\r
+ { t #2 global.max$ substring$ 't := }\r
+ if$\r
+ }\r
+ while$\r
+ multiresult\r
+}\r
+\r
+FUNCTION {format.pages}\r
+{ pages empty$\r
+ { "" }\r
+ { pages multi.page.check\r
+ { "pages" pages n.dashify tie.or.space.connect }\r
+ { "page" pages tie.or.space.connect }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.vol.num.pages}\r
+{ volume field.or.null\r
+ number empty$\r
+ 'skip$\r
+ { "(" number * ")" * *\r
+ volume empty$\r
+ { "there's a number but no volume in " cite$ * warning$ }\r
+ 'skip$\r
+ if$\r
+ }\r
+ if$\r
+ pages empty$\r
+ 'skip$\r
+ { duplicate$ empty$\r
+ { pop$ format.pages }\r
+ { ":" * pages n.dashify * }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.chapter.pages}\r
+{ chapter empty$\r
+ 'format.pages\r
+ { type empty$\r
+ { "chapter" }\r
+ { type "l" change.case$ }\r
+ if$\r
+ chapter tie.or.space.connect\r
+ pages empty$\r
+ 'skip$\r
+ { ", " * format.pages * }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.in.ed.booktitle}\r
+{ booktitle empty$\r
+ { "" }\r
+ { editor empty$\r
+ { "In " booktitle emphasize * }\r
+ { "In " format.editors * ", " * booktitle emphasize * }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {empty.misc.check}\r
+\r
+{ author empty$ title empty$ howpublished empty$\r
+ month empty$ year empty$ note empty$\r
+ and and and and and\r
+ key empty$ not and\r
+ { "all relevant fields are empty in " cite$ * warning$ }\r
+ 'skip$\r
+ if$\r
+}\r
+\r
+FUNCTION {format.thesis.type}\r
+{ type empty$\r
+ 'skip$\r
+ { pop$\r
+ type "t" change.case$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.tr.number}\r
+{ type empty$\r
+ { "Technical Report" }\r
+ 'type\r
+ if$\r
+ number empty$\r
+ { "t" change.case$ }\r
+ { number tie.or.space.connect }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.article.crossref}\r
+{ key empty$\r
+ { journal empty$\r
+ { "need key or journal for " cite$ * " to crossref " * crossref *\r
+ warning$\r
+ ""\r
+ }\r
+ { "In {\em " journal * "\/}" * }\r
+ if$\r
+ }\r
+ { "In " key * }\r
+ if$\r
+ " \cite{" * crossref * "}" *\r
+}\r
+\r
+FUNCTION {format.crossref.editor}\r
+{ editor #1 "{vv~}{ll}" format.name$\r
+ editor num.names$ duplicate$\r
+ #2 >\r
+ { pop$ " et~al." * }\r
+ { #2 <\r
+ 'skip$\r
+ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =\r
+ { " et~al." * }\r
+ { " and " * editor #2 "{vv~}{ll}" format.name$ * }\r
+ if$\r
+ }\r
+ if$\r
+ }\r
+ if$\r
+}\r
+\r
+FUNCTION {format.book.crossref}\r
+{ volume empty$\r
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$\r
+ "In "\r
+ }\r
+ { "Volume" volume tie.or.space.connect\r
+ " of " *\r
+ }\r
+ if$\r
+ editor empty$\r
+ editor field.or.null author field.or.null =\r
+ or\r
+ { key empty$\r
+ { series empty$\r
+ { "need editor, key, or series for " cite$ * " to crossref " *\r
+ crossref * warning$\r
+ "" *\r
+ }\r
+ { "{\em " * series * "\/}" * }\r
+ if$\r
+ }\r
+ { key * }\r
+ if$\r
+ }\r
+ { format.crossref.editor * }\r
+ if$\r
+ " \cite{" * crossref * "}" *\r
+}\r
+\r
+FUNCTION {format.incoll.inproc.crossref}\r
+{ editor empty$\r
+ editor field.or.null author field.or.null =\r
+ or\r
+ { key empty$\r
+ { booktitle empty$\r
+ { "need editor, key, or booktitle for " cite$ * " to crossref " *\r
+ crossref * warning$\r
+ ""\r
+ }\r
+ { "In {\em " booktitle * "\/}" * }\r
+ if$\r
+ }\r
+ { "In " key * }\r
+ if$\r
+ }\r
+ { "In " format.crossref.editor * }\r
+ if$\r
+ " \cite{" * crossref * "}" *\r
+}\r
+\r
+FUNCTION {article}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.title "title" output.check\r
+ new.block\r
+ crossref missing$\r
+ { journal emphasize "journal" output.check\r
+ format.vol.num.pages output\r
+ format.date "year" output.check\r
+ }\r
+ { format.article.crossref output.nonnull\r
+ format.pages output\r
+ }\r
+ if$\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {book}\r
+{ output.bibitem\r
+ author empty$\r
+ { format.editors "author and editor" output.check }\r
+ { format.authors output.nonnull\r
+ crossref missing$\r
+ { "author and editor" editor either.or.check }\r
+ 'skip$\r
+ if$\r
+ }\r
+ if$\r
+ new.block\r
+ format.btitle "title" output.check\r
+ crossref missing$\r
+ { format.bvolume output\r
+ new.block\r
+ format.number.series output\r
+ new.sentence\r
+ publisher "publisher" output.check\r
+ address output\r
+ }\r
+ { new.block\r
+ format.book.crossref output.nonnull\r
+ }\r
+ if$\r
+ format.edition output\r
+ format.date "year" output.check\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {booklet}\r
+{ output.bibitem\r
+ format.authors output\r
+ new.block\r
+ format.title "title" output.check\r
+ howpublished address new.block.checkb\r
+ howpublished output\r
+ address output\r
+ format.date output\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {inbook}\r
+{ output.bibitem\r
+ author empty$\r
+ { format.editors "author and editor" output.check }\r
+ { format.authors output.nonnull\r
+\r
+ crossref missing$\r
+ { "author and editor" editor either.or.check }\r
+ 'skip$\r
+ if$\r
+ }\r
+ if$\r
+ new.block\r
+ format.btitle "title" output.check\r
+ crossref missing$\r
+ { format.bvolume output\r
+ format.chapter.pages "chapter and pages" output.check\r
+ new.block\r
+ format.number.series output\r
+ new.sentence\r
+ publisher "publisher" output.check\r
+ address output\r
+ }\r
+ { format.chapter.pages "chapter and pages" output.check\r
+ new.block\r
+ format.book.crossref output.nonnull\r
+ }\r
+ if$\r
+ format.edition output\r
+ format.date "year" output.check\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {incollection}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.title "title" output.check\r
+ new.block\r
+ crossref missing$\r
+ { format.in.ed.booktitle "booktitle" output.check\r
+ format.bvolume output\r
+ format.number.series output\r
+ format.chapter.pages output\r
+ new.sentence\r
+ publisher "publisher" output.check\r
+ address output\r
+ format.edition output\r
+ format.date "year" output.check\r
+ }\r
+ { format.incoll.inproc.crossref output.nonnull\r
+ format.chapter.pages output\r
+ }\r
+ if$\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {inproceedings}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.title "title" output.check\r
+ new.block\r
+ crossref missing$\r
+ { format.in.ed.booktitle "booktitle" output.check\r
+ format.bvolume output\r
+ format.number.series output\r
+ format.pages output\r
+ address empty$\r
+ { organization publisher new.sentence.checkb\r
+ organization output\r
+ publisher output\r
+ format.date "year" output.check\r
+ }\r
+ { address output.nonnull\r
+ format.date "year" output.check\r
+ new.sentence\r
+ organization output\r
+ publisher output\r
+ }\r
+ if$\r
+ }\r
+ { format.incoll.inproc.crossref output.nonnull\r
+ format.pages output\r
+ }\r
+ if$\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {conference} { inproceedings }\r
+\r
+FUNCTION {manual}\r
+{ output.bibitem\r
+ author empty$\r
+ { organization empty$\r
+ 'skip$\r
+ { organization output.nonnull\r
+ address output\r
+ }\r
+ if$\r
+ }\r
+ { format.authors output.nonnull }\r
+ if$\r
+ new.block\r
+ format.btitle "title" output.check\r
+ author empty$\r
+ { organization empty$\r
+ { address new.block.checka\r
+ address output\r
+ }\r
+ 'skip$\r
+ if$\r
+ }\r
+ { organization address new.block.checkb\r
+ organization output\r
+ address output\r
+ }\r
+ if$\r
+ format.edition output\r
+ format.date output\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {mastersthesis}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.title "title" output.check\r
+ new.block\r
+ "Master's thesis" format.thesis.type output.nonnull\r
+ school "school" output.check\r
+ address output\r
+ format.date "year" output.check\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {misc}\r
+{ output.bibitem\r
+ format.authors output\r
+ title howpublished new.block.checkb\r
+ format.title output\r
+ howpublished new.block.checka\r
+ howpublished output\r
+ format.date output\r
+ new.block\r
+ note output\r
+ fin.entry\r
+ empty.misc.check\r
+}\r
+\r
+FUNCTION {phdthesis}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.btitle "title" output.check\r
+ new.block\r
+ "PhD thesis" format.thesis.type output.nonnull\r
+ school "school" output.check\r
+ address output\r
+ format.date "year" output.check\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {proceedings}\r
+{ output.bibitem\r
+ editor empty$\r
+ { organization output }\r
+ { format.editors output.nonnull }\r
+\r
+ if$\r
+ new.block\r
+ format.btitle "title" output.check\r
+ format.bvolume output\r
+ format.number.series output\r
+ address empty$\r
+ { editor empty$\r
+ { publisher new.sentence.checka }\r
+ { organization publisher new.sentence.checkb\r
+ organization output\r
+ }\r
+ if$\r
+ publisher output\r
+ format.date "year" output.check\r
+ }\r
+ { address output.nonnull\r
+ format.date "year" output.check\r
+ new.sentence\r
+ editor empty$\r
+ 'skip$\r
+ { organization output }\r
+ if$\r
+ publisher output\r
+ }\r
+ if$\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {techreport}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.title "title" output.check\r
+ new.block\r
+ format.tr.number output.nonnull\r
+ institution "institution" output.check\r
+ address output\r
+ format.date "year" output.check\r
+ new.block\r
+ note output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {unpublished}\r
+{ output.bibitem\r
+ format.authors "author" output.check\r
+ new.block\r
+ format.title "title" output.check\r
+ new.block\r
+ note "note" output.check\r
+ format.date output\r
+ fin.entry\r
+}\r
+\r
+FUNCTION {default.type} { misc }\r
+\r
+MACRO {jan} {"Jan."}\r
+\r
+MACRO {feb} {"Feb."}\r
+\r
+MACRO {mar} {"Mar."}\r
+\r
+MACRO {apr} {"Apr."}\r
+\r
+MACRO {may} {"May"}\r
+\r
+MACRO {jun} {"June"}\r
+\r
+MACRO {jul} {"July"}\r
+\r
+MACRO {aug} {"Aug."}\r
+\r
+MACRO {sep} {"Sept."}\r
+\r
+MACRO {oct} {"Oct."}\r
+\r
+MACRO {nov} {"Nov."}\r
+\r
+MACRO {dec} {"Dec."}\r
+\r
+MACRO {acmcs} {"ACM Comput. Surv."}\r
+\r
+MACRO {acta} {"Acta Inf."}\r
+\r
+MACRO {cacm} {"Commun. ACM"}\r
+\r
+MACRO {ibmjrd} {"IBM J. Res. Dev."}\r
+\r
+MACRO {ibmsj} {"IBM Syst.~J."}\r
+\r
+MACRO {ieeese} {"IEEE Trans. Softw. Eng."}\r
+\r
+MACRO {ieeetc} {"IEEE Trans. Comput."}\r
+\r
+MACRO {ieeetcad}\r
+ {"IEEE Trans. Comput.-Aided Design Integrated Circuits"}\r
+\r
+MACRO {ipl} {"Inf. Process. Lett."}\r
+\r
+MACRO {jacm} {"J.~ACM"}\r
+\r
+MACRO {jcss} {"J.~Comput. Syst. Sci."}\r
+\r
+MACRO {scp} {"Sci. Comput. Programming"}\r
+\r
+MACRO {sicomp} {"SIAM J. Comput."}\r
+\r
+MACRO {tocs} {"ACM Trans. Comput. Syst."}\r
+\r
+MACRO {tods} {"ACM Trans. Database Syst."}\r
+\r
+MACRO {tog} {"ACM Trans. Gr."}\r
+\r
+MACRO {toms} {"ACM Trans. Math. Softw."}\r
+\r
+MACRO {toois} {"ACM Trans. Office Inf. Syst."}\r
+\r
+MACRO {toplas} {"ACM Trans. Prog. Lang. Syst."}\r
+\r
+MACRO {tcs} {"Theoretical Comput. Sci."}\r
+\r
+READ\r
+\r
+FUNCTION {sortify}\r
+{ purify$\r
+ "l" change.case$\r
+}\r
+\r
+INTEGERS { len }\r
+\r
+FUNCTION {chop.word}\r
+{ 's :=\r
+ 'len :=\r
+ s #1 len substring$ =\r
+ { s len #1 + global.max$ substring$ }\r
+ 's\r
+ if$\r
+}\r
+\r
+FUNCTION {sort.format.names}\r
+{ 's :=\r
+ #1 'nameptr :=\r
+ ""\r
+ s num.names$ 'numnames :=\r
+ numnames 'namesleft :=\r
+ { namesleft #0 > }\r
+ { nameptr #1 >\r
+ { " " * }\r
+ 'skip$\r
+ if$\r
+ s nameptr "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}" format.name$ 't :=\r
+ nameptr numnames = t "others" = and\r
+ { "et al" * }\r
+ { t sortify * }\r
+ if$\r
+ nameptr #1 + 'nameptr :=\r
+ namesleft #1 - 'namesleft :=\r
+ }\r
+ while$\r
+}\r
+\r
+FUNCTION {sort.format.title}\r
+{ 't :=\r
+ "A " #2\r
+ "An " #3\r
+ "The " #4 t chop.word\r
+ chop.word\r
+ chop.word\r
+ sortify\r
+ #1 global.max$ substring$\r
+}\r
+\r
+FUNCTION {author.sort}\r
+{ author empty$\r
+ { key empty$\r
+ { "to sort, need author or key in " cite$ * warning$\r
+ ""\r
+ }\r
+ { key sortify }\r
+ if$\r
+ }\r
+ { author sort.format.names }\r
+ if$\r
+}\r
+\r
+FUNCTION {author.editor.sort}\r
+{ author empty$\r
+ { editor empty$\r
+ { key empty$\r
+ { "to sort, need author, editor, or key in " cite$ * warning$\r
+ ""\r
+ }\r
+ { key sortify }\r
+ if$\r
+ }\r
+ { editor sort.format.names }\r
+ if$\r
+ }\r
+ { author sort.format.names }\r
+ if$\r
+}\r
+\r
+FUNCTION {author.organization.sort}\r
+{ author empty$\r
+\r
+ { organization empty$\r
+ { key empty$\r
+ { "to sort, need author, organization, or key in " cite$ * warning$\r
+ ""\r
+ }\r
+ { key sortify }\r
+ if$\r
+ }\r
+ { "The " #4 organization chop.word sortify }\r
+ if$\r
+ }\r
+ { author sort.format.names }\r
+ if$\r
+}\r
+\r
+FUNCTION {editor.organization.sort}\r
+{ editor empty$\r
+ { organization empty$\r
+ { key empty$\r
+ { "to sort, need editor, organization, or key in " cite$ * warning$\r
+ ""\r
+ }\r
+ { key sortify }\r
+ if$\r
+ }\r
+ { "The " #4 organization chop.word sortify }\r
+ if$\r
+ }\r
+ { editor sort.format.names }\r
+ if$\r
+}\r
+\r
+FUNCTION {presort}\r
+{ type$ "book" =\r
+ type$ "inbook" =\r
+ or\r
+ 'author.editor.sort\r
+ { type$ "proceedings" =\r
+ 'editor.organization.sort\r
+ { type$ "manual" =\r
+ 'author.organization.sort\r
+ 'author.sort\r
+ if$\r
+ }\r
+ if$\r
+ }\r
+ if$\r
+ " "\r
+ *\r
+ year field.or.null sortify\r
+ *\r
+ " "\r
+ *\r
+ title field.or.null\r
+ sort.format.title\r
+ *\r
+ #1 entry.max$ substring$\r
+ 'sort.key$ :=\r
+}\r
+\r
+ITERATE {presort}\r
+\r
+SORT\r
+\r
+STRINGS { longest.label }\r
+\r
+INTEGERS { number.label longest.label.width }\r
+\r
+FUNCTION {initialize.longest.label}\r
+{ "" 'longest.label :=\r
+ #1 'number.label :=\r
+ #0 'longest.label.width :=\r
+}\r
+\r
+FUNCTION {longest.label.pass}\r
+{ number.label int.to.str$ 'label :=\r
+ number.label #1 + 'number.label :=\r
+ label width$ longest.label.width >\r
+ { label 'longest.label :=\r
+ label width$ 'longest.label.width :=\r
+ }\r
+ 'skip$\r
+ if$\r
+}\r
+\r
+EXECUTE {initialize.longest.label}\r
+\r
+ITERATE {longest.label.pass}\r
+\r
+FUNCTION {begin.bib}\r
+{ preamble$ empty$\r
+ 'skip$\r
+ { preamble$ write$ newline$ }\r
+ if$\r
+ "\begin{thebibliography}{" longest.label * \r
+ "}\setlength{\itemsep}{-1ex}\small" * write$ newline$\r
+}\r
+\r
+EXECUTE {begin.bib}\r
+\r
+EXECUTE {init.state.consts}\r
+\r
+ITERATE {call.type$}\r
+\r
+FUNCTION {end.bib}\r
+{ newline$\r
+ "\end{thebibliography}" write$ newline$\r
+}\r
+\r
+EXECUTE {end.bib}\r
+\r
+% end of file latex8.bst\r
+% ---------------------------------------------------------------\r
+\r
+\r
+\r
--- /dev/null
+\r
+% --------------------------------------------------------------- \r
+%\r
+% $Id: latex8.sty,v 1.2 1995/09/15 15:31:13 ienne Exp $\r
+% \r
+% by Paolo.Ienne@di.epfl.ch \r
+%\r
+% --------------------------------------------------------------- \r
+%\r
+% no guarantee is given that the format corresponds perfectly to \r
+% IEEE 8.5" x 11" Proceedings, but most features should be ok.\r
+%\r
+% --------------------------------------------------------------- \r
+% with LaTeX2e:\r
+% =============\r
+%\r
+% use as \r
+% \documentclass[times,10pt,twocolumn]{article} \r
+% \usepackage{latex8}\r
+% \usepackage{times}\r
+%\r
+% --------------------------------------------------------------- \r
+\r
+% with LaTeX 2.09:\r
+% ================\r
+%\r
+% use as \r
+% \documentstyle[times,art10,twocolumn,latex8]{article}\r
+%\r
+% --------------------------------------------------------------- \r
+% with both versions:\r
+% ===================\r
+%\r
+% specify \pagestyle{empty} to omit page numbers in the final \r
+% version\r
+%\r
+% specify references as\r
+% \bibliographystyle{latex8}\r
+% \bibliography{...your files...}\r
+%\r
+% use Section{} and SubSection{} instead of standard section{} \r
+% and subsection{} to obtain headings in the form \r
+% "1.3. My heading"\r
+%\r
+% ---------------------------------------------------------------\r
+\r
+\typeout{IEEE 8.5 x 11-Inch Proceedings Style `latex8.sty'.}\r
+\r
+% ten point helvetica bold required for captions\r
+% in some sites the name of the helvetica bold font may differ, \r
+% change the name here:\r
+\font\tenhv = phvb at 10pt\r
+%\font\tenhv = phvb7t at 10pt\r
+\r
+% eleven point times bold required for second-order headings \r
+% \font\elvbf = cmbx10 scaled 1100\r
+\font\elvbf = ptmb scaled 1100\r
+\r
+% set dimensions of columns, gap between columns, and paragraph indent \r
+\setlength{\textheight}{8.875in}\r
+\setlength{\textwidth}{6.875in}\r
+\setlength{\columnsep}{0.3125in}\r
+\setlength{\topmargin}{0in}\r
+\setlength{\headheight}{0in}\r
+\setlength{\headsep}{0in}\r
+\setlength{\parindent}{1pc}\r
+\setlength{\oddsidemargin}{-.304in}\r
+\setlength{\evensidemargin}{-.304in}\r
+\r
+% memento from size10.clo\r
+% \normalsize{\@setfontsize\normalsize\@xpt\@xiipt} \r
+% \small{\@setfontsize\small\@ixpt{11}}\r
+% \footnotesize{\@setfontsize\footnotesize\@viiipt{9.5}} \r
+% \scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt}\r
+% \tiny{\@setfontsize\tiny\@vpt\@vipt}\r
+% \large{\@setfontsize\large\@xiipt{14}} \r
+% \Large{\@setfontsize\Large\@xivpt{18}} \r
+% \LARGE{\@setfontsize\LARGE\@xviipt{22}} \r
+% \huge{\@setfontsize\huge\@xxpt{25}}\r
+% \Huge{\@setfontsize\Huge\@xxvpt{30}}\r
+\r
+\def\@maketitle\r
+ {\r
+ \newpage\r
+ \null\r
+ \vskip .375in \r
+ \begin{center}\r
+ {\Large \bf \@title \par} \r
+ % additional two empty lines at the end of the title \r
+ \vspace*{24pt} \r
+ {\r
+ \large \r
+ \lineskip .5em\r
+ \begin{tabular}[t]{c}\r
+ \@author \r
+ \end{tabular}\r
+ \par\r
+ } \r
+ % additional small space at the end of the author name \r
+ \vskip .5em \r
+ {\r
+ \large \r
+ \begin{tabular}[t]{c}\r
+ \@affiliation \r
+ \end{tabular}\r
+ \par \r
+ \ifx \@empty \@email\r
+ \else\r
+ \begin{tabular}{r@{~}l}\r
+ E-mail: & {\tt \@email}\r
+ \end{tabular}\r
+ \par\r
+ \fi\r
+ }\r
+ % additional empty line at the end of the title block \r
+ \vspace*{12pt} \r
+ \end{center}\r
+ } \r
+\r
+\def\abstract\r
+ {%\r
+ \centerline{\large\bf Abstract}%\r
+ \vspace*{12pt}%\r
+ \it%\r
+ }\r
+\r
+\def\endabstract\r
+ {\r
+ % additional empty line at the end of the abstract \r
+ \vspace*{12pt} \r
+ }\r
+\r
+\def\affiliation#1{\gdef\@affiliation{#1}} \gdef\@affiliation{}\r
+\r
+\def\email#1{\gdef\@email{#1}}\r
+\gdef\@email{}\r
+\r
+\newlength{\@ctmp}\r
+\newlength{\@figindent}\r
+\setlength{\@figindent}{1pc}\r
+\r
+\long\def\@makecaption#1#2{\r
+ \vskip 10pt\r
+ \setbox\@tempboxa\hbox{\tenhv\noindent #1.~#2} \r
+ \setlength{\@ctmp}{\hsize}\r
+ \addtolength{\@ctmp}{-\@figindent}\addtolength{\@ctmp}{-\@figindent} \r
+ % IF longer than one indented paragraph line\r
+ \ifdim \wd\@tempboxa >\@ctmp\r
+ % THEN set as an indented paragraph\r
+ \begin{list}{}{\leftmargin\@figindent \rightmargin\leftmargin} \r
+ \item[]\tenhv #1.~#2\par\r
+ \end{list}\r
+ \else\r
+ % ELSE center\r
+ \hbox to\hsize{\hfil\box\@tempboxa\hfil} \r
+ \fi}\r
+\r
+% correct heading spacing and type\r
+\def\section{\@startsection {section}{1}{\z@}\r
+ {14pt plus 2pt minus 2pt}{14pt plus 2pt minus 2pt} {\large\bf}} \r
+\def\subsection{\@startsection {subsection}{2}{\z@}\r
+ {13pt plus 2pt minus 2pt}{13pt plus 2pt minus 2pt} {\elvbf}}\r
+\r
+% add the period after section numbers \r
+\newcommand{\Section}[1]{\section{\hskip -1em.~#1}} \r
+\newcommand{\SubSection}[1]{\subsection{\hskip -1em.~#1}}\r
+\r
+% end of file latex8.sty\r
+% ---------------------------------------------------------------\r
--- /dev/null
+\r
+%\r
+% $Description: Author guidelines and sample document in LaTeX 2.09$ \r
+%\r
+% $Author: ienne $\r
+% $Date: 1995/09/15 15:20:59 $\r
+% $Revision: 1.4 $\r
+%\r
+\r
+\documentclass[times, 10pt,twocolumn]{article} \r
+\usepackage{latex8}\r
+\usepackage{times}\r
+\r
+%\documentstyle[times,art10,twocolumn,latex8]{article}\r
+\r
+%------------------------------------------------------------------------- \r
+% take the % away on next line to produce the final camera-ready version \r
+\pagestyle{empty}\r
+\r
+%------------------------------------------------------------------------- \r
+\begin{document}\r
+\r
+\title{\LaTeX\ Author Guidelines \r
+ for {\boldmath $8.5 \times 11$-Inch} Proceedings Manuscripts}\r
+\r
+\author{Paolo Ienne\\\r
+Swiss Federal Institute of Technology\\ Microcomputing Laboratory \\ IN-F \r
+Ecublens, 1015 Lausanne, Switzerland\\ Paolo.Ienne@di.epfl.ch\\\r
+% For a paper whose authors are all at the same institution, \r
+% omit the following lines up until the closing ``}''.\r
+% Additional authors and addresses can be added with ``\and'', \r
+% just like the second author.\r
+\and\r
+Second Author\\\r
+Institution2\\\r
+First line of institution2 address\\ Second line of institution2 address\\ \r
+SecondAuthor@institution2.com\\\r
+}\r
+\r
+\maketitle\r
+\thispagestyle{empty}\r
+\r
+\begin{abstract}\r
+ The ABSTRACT is to be in fully-justified italicized text, at the top \r
+ of the left-hand column, below the author and affiliation \r
+ information. Use the word ``Abstract'' as the title, in 12-point \r
+ Times, boldface type, centered relative to the column, initially \r
+ capitalized. The abstract is to be in 10-point, single-spaced type. \r
+ The abstract may be up to 3 inches (7.62 cm) long. Leave two blank \r
+ lines after the Abstract, then begin the main text. \r
+\end{abstract}\r
+\r
+\r
+\r
+%------------------------------------------------------------------------- \r
+\Section{Introduction}\r
+\r
+Please follow the steps outlined below when submitting your \r
+manuscript to the IEEE Computer Society Press. Note there have \r
+been some changes to the measurements from previous instructions. \r
+\r
+%------------------------------------------------------------------------- \r
+\Section{Instructions}\r
+\r
+Please read the following carefully.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Language}\r
+\r
+All manuscripts must be in English.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Printing your paper}\r
+\r
+Print your properly formatted text on high-quality, $8.5 \times 11$-inch \r
+white printer paper. A4 paper is also acceptable, but please leave the \r
+extra 0.5 inch (1.27 cm) at the BOTTOM of the page.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Margins and page numbering}\r
+\r
+All printed material, including text, illustrations, and charts, must be \r
+kept within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches \r
+(22.54 cm) high. Do not write or print anything outside the print area. \r
+Number your pages lightly, in pencil, on the upper right-hand corners of \r
+the BACKS of the pages (for example, 1/10, 2/10, or 1 of 10, 2 of 10, and \r
+so forth). Please do not write on the fronts of the pages, nor on the \r
+lower halves of the backs of the pages.\r
+\r
+\r
+%------------------------------------------------------------------------ \r
+\SubSection{Formatting your paper}\r
+\r
+All text must be in a two-column format. The total allowable width of \r
+the text area is 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm) \r
+high. Columns are to be 3-1/4 inches (8.25 cm) wide, with a 5/16 inch \r
+(0.8 cm) space between them. The main title (on the first page) should \r
+begin 1.0 inch (2.54 cm) from the top edge of the page. The second and \r
+following pages should begin 1.0 inch (2.54 cm) from the top edge. On \r
+all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the \r
+bottom edge of the page for $8.5 \times 11$-inch paper; for A4 paper, \r
+approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Type-style and fonts}\r
+\r
+Wherever Times is specified, Times Roman may also be used. If neither is \r
+available on your word processor, please use the font closest in \r
+appearance to Times that you have access to.\r
+\r
+MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of \r
+the first page. The title should be in Times 14-point, boldface type. \r
+Capitalize the first letter of nouns, pronouns, verbs, adjectives, and \r
+adverbs; do not capitalize articles, coordinate conjunctions, or \r
+prepositions (unless the title begins with such a word). Leave two blank \r
+lines after the title.\r
+\r
+AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title \r
+and printed in Times 12-point, non-boldface type. This information is to \r
+be followed by two blank lines.\r
+\r
+The ABSTRACT and MAIN TEXT are to be in a two-column format. \r
+\r
+MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use \r
+double-spacing. All paragraphs should be indented 1 pica (approx. 1/6 \r
+inch or 0.422 cm). Make sure your text is fully justified---that is, \r
+flush left and flush right. Please do not place any additional blank \r
+lines between paragraphs. Figure and table captions should be 10-point \r
+Helvetica boldface type as in\r
+\begin{figure}[h]\r
+ \caption{Example of caption.}\r
+\end{figure}\r
+\r
+\noindent Long captions should be set as in \r
+\begin{figure}[h] \r
+ \caption{Example of long caption requiring more than one line. It is \r
+ not typed centered but aligned on both sides and indented with an \r
+ additional margin on both sides of 1~pica.}\r
+\end{figure}\r
+\r
+\noindent Callouts should be 9-point Helvetica, non-boldface type. \r
+Initially capitalize only the first word of section titles and first-, \r
+second-, and third-order headings.\r
+\r
+FIRST-ORDER HEADINGS. (For example, {\large \bf 1. Introduction}) \r
+should be Times 12-point boldface, initially capitalized, flush left, \r
+with one blank line before, and one blank line after.\r
+\r
+SECOND-ORDER HEADINGS. (For example, {\elvbf 1.1. Database elements}) \r
+should be Times 11-point boldface, initially capitalized, flush left, \r
+with one blank line before, and one after. If you require a third-order \r
+heading (we discourage it), use 10-point Times, boldface, initially \r
+capitalized, flush left, preceded by one blank line, followed by a period \r
+and your text on the same line.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Footnotes}\r
+\r
+Please use footnotes sparingly%\r
+\footnote\r
+ {%\r
+ Or, better still, try to avoid footnotes altogether. To help your \r
+ readers, avoid using footnotes altogether and include necessary \r
+ peripheral observations in the text (within parentheses, if you \r
+ prefer, as in this sentence).\r
+ }\r
+and place them at the bottom of the column on the page on which they are \r
+referenced. Use Times 8-point type, single-spaced.\r
+\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{References}\r
+\r
+List and number all bibliographical references in 9-point Times, \r
+single-spaced, at the end of your paper. When referenced in the text, \r
+enclose the citation number in square brackets, for example~\cite{ex1}. \r
+Where appropriate, include the name(s) of editors of referenced books.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Illustrations, graphs, and photographs}\r
+\r
+All graphics should be centered. Your artwork must be in place in the \r
+article (preferably printed as part of the text rather than pasted up). \r
+If you are using photographs and are able to have halftones made at a \r
+print shop, use a 100- or 110-line screen. If you must use plain photos, \r
+they must be pasted onto your manuscript. Use rubber cement to affix the \r
+images in place. Black and white, clear, glossy-finish photos are \r
+preferable to color. Supply the best quality photographs and \r
+illustrations possible. Penciled lines and very fine lines do not \r
+reproduce well. Remember, the quality of the book cannot be better than \r
+the originals provided. Do NOT use tape on your pages!\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Color}\r
+\r
+The use of color on interior pages (that is, pages other\r
+than the cover) is prohibitively expensive. We publish interior pages in \r
+color only when it is specifically requested and budgeted for by the \r
+conference organizers. DO NOT SUBMIT COLOR IMAGES IN YOUR \r
+PAPERS UNLESS SPECIFICALLY INSTRUCTED TO DO SO.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Symbols}\r
+\r
+If your word processor or typewriter cannot produce Greek letters, \r
+mathematical symbols, or other graphical elements, please use \r
+pressure-sensitive (self-adhesive) rub-on symbols or letters (available \r
+in most stationery stores, art stores, or graphics shops).\r
+\r
+%------------------------------------------------------------------------ \r
+\SubSection{Copyright forms}\r
+\r
+You must include your signed IEEE copyright release form when you submit \r
+your finished paper. We MUST have this form before your paper can be \r
+published in the proceedings.\r
+\r
+%------------------------------------------------------------------------- \r
+\SubSection{Conclusions}\r
+\r
+Please direct any questions to the production editor in charge of these \r
+proceedings at the IEEE Computer Society Press: Phone (714) 821-8380, or \r
+Fax (714) 761-1784.\r
+\r
+%------------------------------------------------------------------------- \r
+\nocite{ex1,ex2}\r
+\bibliographystyle{latex8}\r
+\bibliography{latex8}\r
+\r
+\end{document}\r
+\r
-\documentclass[a4paper]{article}
+\documentclass[times, 10pt,twocolumn, draft]{article}
\pagestyle{headings}
%\usepackage{graphicx}
\usepackage{amssymb,amsmath,mathrsfs,stmaryrd,amsthm}
%\usepackage{hyperref}
%\usepackage{picins}
+\usepackage{latex8}
+\usepackage{times}
+
+\pagestyle{empty}
+
\newcommand{\semT}[1]{\ensuremath{\llbracket #1 \rrbracket}}
\newcommand{\sem}[1]{\llbracket \ensuremath{#1} \rrbracket}
\begin{document}
\maketitle
+\thispagestyle{empty}
\begin{abstract}
...
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Introduction}
+\Section{Introduction}
The characterization of the provable recursive functions of
Peano Arithmetic as the terms of system T is a well known
result of G\"odel \cite{Godel58,Godel90}. Although several authors acknowledge
This paper is the first step in this direction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{G\"odel system T}
+\Section{G\"odel system T}
We shall use a variant of system T with three atomic types $\N$ (natural
numbers), $\B$ (booleans) and $\one$ (a terminal object), and two binary
type constructors $\times$ (product) and $\to$ (arrow type).
\item $true: \B$, $false:\B$, $D:A\to A \to \B \to A$
\item $O:\N$, $S:\N \to \N$, $R:A \to (A \to \N \to A) \to \N \to A$,
\end{itemize}
-Redexes comprise $\beta$-reduction
-\[(\beta)~~ \lambda x:U.M ~ N \leadsto M[N/x]\]
-projections
-
-\[(\pi_1)~~fst \pair{M}{N} \leadsto M\\ \hspace{.6cm} (\pi_2)~~ snd \pair{M}{N}
-\leadsto N \]
-and the following type specific reductions:
-\[(D_{true})~~\\D~M~N~ true \leadsto M \hspace{.6cm}
- (D_{false})~~ D~M~N~false \leadsto N \]
-\[(R_0)~~\\R~M~F~ 0 \leadsto M \hspace{.6cm}
- (R_S)~~ R~M~F~(S~n) \leadsto F~n~(R~M~F~n) \]
-\[(*)~~ M \leadsto * \]
-where (*) holds for any $M$ of type $\one$.
+Redexes comprise $\beta$-reduction, projections,
+and type specific reductions as reported in table \ref{tab:tredex}
+
+\begin{table}[!h]
+\hrule\vspace{0.1cm}
+\begin{tabular}{p{0.34\textwidth}r}
+ $\lambda x:U.M ~ N \leadsto M[N/x]$ & $(\beta)$ \\
+ $fst \pair{M}{N} \leadsto M$ & $(\pi_1)$ \\
+ $snd \pair{M}{N} \leadsto N$ & $(\pi_2)$ \\
+ $D~M~N~ true \leadsto M$ & $(D_{true})$ \\
+ $D~M~N~false \leadsto N$ & $(D_{false})$ \\
+ $R~M~F~ 0 \leadsto M$ & $(R_0)$ \\
+ $R~M~F~(S~n) \leadsto F~n~(R~M~F~n)$ & $(R_S)$ \\
+ $M \leadsto * ~~~~ $ for any M of type $\one$ & $(*)$
+\end{tabular}\vspace{0.1cm}
+\hrule
+\caption{\label{tab:tredex}Reduction rules for System T}
+\end{table}
Note that using the well known isomorpshims
$\one \to A \cong A$, $A \to \one \cong \one$
the above isomorphisms gives a simple way of just keeping the truly
informative part of the algorithms.
-
-
-\section{Heyting's arithmetics}
+\Section{Heyting's arithmetics}
{\bf Axioms}
{\bf Inference Rules}
say that ax:AX refers to the previous Axioms list...
-
-\[
- (Proj)\hspace{0.1cm} \Gamma, x:A, \Delta \vdash x:A
- \hspace{2cm}
- (Const)\hspace{0.1cm} \Gamma \vdash ax : AX
-\]
-
-\[
- (\to_i)\hspace{0.1cm}\frac{\Gamma,x:A \vdash M:Q}{\Gamma \vdash \lambda x:A.M: A \to Q} \hspace{2cm}
- (\to_e)\hspace{0.1cm}\frac{\Gamma \vdash M: A \to Q \hspace{1cm}\Gamma \vdash N: A}
- {\Gamma \vdash M N: Q}
-\]
+\begin{table}[!h]
+\hrule\vspace{0.1cm}
+\begin{tabular}{p{0.34\textwidth}r}
+ $\Gamma, x:A, \Delta \vdash x:A$ & $(Proj)$\\
+ $\Gamma \vdash ax : AX$ & $(Const)$\\
+ $\frac{\Gamma,x:A \vdash M:Q}{\Gamma \vdash \lambda x:A.M: A \to Q}$ &
+ $(\to_i)$\\
+ $\frac{\Gamma \vdash M: A \to Q \hspace{0.5cm}\Gamma \vdash N: A}
+ {\Gamma \vdash M N: Q}$ & $(\to_e)$\\
+ $\frac{\Gamma \vdash M:P}{\Gamma \vdash \lambda x:\N.M: \forall x.P}(*)$ &
+ $(\forall_i)$\\
+ $\frac{\Gamma \vdash M :\forall x.P}{\Gamma \vdash M t: P[t/x]}$ &
+ $(\forall_e)$
+\end{tabular}\vspace{0.1cm}
+\hrule
+\caption{\label{tab:HArules}Inference rules}
+\end{table}{lr}
%\[
% (\land_i)\frac{\Gamma \vdash M:A \hspace{1cm}\Gamma \vdash N:B}
% (\land_{er})\frac{\Gamma \vdash A \land B}{\Gamma \vdash B}
%\]
-\[
- (\forall_i)\hspace{0.1cm}\frac{\Gamma \vdash M:P}{\Gamma \vdash
- \lambda x:\N.M: \forall x.P}(*) \hspace{2cm}
- (\forall_e)\hspace{0.1cm}\frac{\Gamma \vdash M :\forall x.P}{\Gamma \vdash M t: P[t/x]}
-\]
-
%\[
% (\exists_i)\frac{\Gamma \vdash P[t/x]}{\Gamma \vdash \exists x.P}\hspace{2cm}
%{\Gamma \vdash Q}
%\]
-\section{Extraction}
+\Section{Extraction}
The formulae to types translation function $\sem{\cdot}$, see table
\ref{tab:formulae2types}, takes in input formulae in HA and returns
\begin{table}[!h]
\hrule\vspace{0.1cm}
-\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
+\begin{tabular}{p{0.21\textwidth}p{0.21\textwidth}}
$\sem{A} = \one$ if A is atomic &
$\sem{A \land B} = \sem{A}\times \sem{B}$ \\
$\sem{A \to B} = \sem{A}\to \sem{B}$ &
\begin{table}[!h]
\hrule\vspace{0.1cm}
-\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
+\begin{tabular}{p{0.20\textwidth}p{0.20\textwidth}}
$\semT{M N} = \semT{M} \semT{N}$ &
- $\semT{\lambda x:A.M} = \lambda x:\sem{A}.\semT{M}$ \\
- $\semT{\lambda x:\N.M} = \lambda x:\N.\semT{M}$ &
- $\semT{M t} = \semT{M} \semT{t}$
+ $\semT{M t} = \semT{M} \semT{t}$ \\
+ \multicolumn{2}{l}{$\semT{\lambda x:A.M} = \lambda x:\sem{A}.\semT{M}$} \\
+ \multicolumn{2}{l}{$\semT{\lambda x:\N.M} = \lambda x:\N.\semT{M}$}
\end{tabular}\vspace{0.1cm}
\hrule
\caption{\label{tab:structproof}Structured proofs}
\begin{table}[!h]
\hrule\vspace{0.1cm}
-\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
- $\sem{fst} = \pi_1$&
+\begin{tabular}{l}%{0.47\textwidth}p{0.47\textwidth}}
+ $\sem{fst} = \pi_1$\\
$\sem{snd} = \pi_2$\\
- $\sem{conj} = \lambda x:\sem{P}.\lambda y:\sem{Q}.\pair{x}{y}$&
+ $\sem{conj} = \lambda x:\sem{P}.\lambda y:\sem{Q}.\pair{x}{y}$\\
$\sem{false\_ind} = \canonical_{\sem{Q}}$\\
- $\sem{discriminate} = \lambda \_:\N.\lambda \_:\one.\star$&
+ $\sem{discriminate} = \lambda \_:\N.\lambda \_:\one.\star$\\
$\sem{injS}= \lambda \_:\N. \lambda \_:\N.\lambda \_:\one.\star$\\
- $\sem{plus\_O} = \sem{times\_O} = \lambda \_:\N.\star$&
+ $\sem{plus\_O} = \sem{times\_O} = \lambda \_:\N.\star$\\
$\sem{nat\_ind} = R$ \\
- \multicolumn{2}{l}{
- $\sem{plus\_S} = \sem{times\_S} = \lambda \_:\N. \lambda \_:\N.\star$
- }\\
- \multicolumn{2}{l}{
- $\sem{ex\_intro} = \lambda x:\N.\lambda f:\sem{P}.\pair{x}{f}$
- }\\
- \multicolumn{2}{l}{
- $\sem{ex\_ind} =
- \lambda f:(\N \to \sem{P} \to \sem{Q}).
- \lambda p:\N\times \sem{P}.f~(fst~p)~(snd~p)$.
- }
+ $\sem{plus\_S} = \sem{times\_S} = \lambda \_:\N. \lambda \_:\N.\star$ \\
+ $\sem{ex\_intro} = \lambda x:\N.\lambda f:\sem{P}.\pair{x}{f}$ \\
+ $\sem{ex\_ind} = \lambda f:(\N \to \sem{P} \to \sem{Q}).$\\
+ $\qquad\lambda p:\N\times \sem{P}.f~(fst~p)~(snd~p)$
\end{tabular}\vspace{0.1cm}
\hrule
\caption{\label{tab:axioms}Axioms translation}
\begin{table}[!h]
\hrule\vspace{0.1cm}
-\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
- $\canonical_\one = \lambda x:\one.x$ &
+\begin{tabular}{l}%p{0.23\textwidth}p{0.23\textwidth}}
+ $\canonical_\one = \lambda x:\one.x$ \\
$\canonical_N = \lambda x:\one.0$ \\
$\canonical_{U\times V} = \lambda x:\one.\pair{\canonical_{U}
- x}{\canonical_{V} x}$ &
+ x}{\canonical_{V} x}$ \\
$\canonical_{U\to V} = \lambda x:\one.\lambda \_:U. \canonical_{V} x$
\end{tabular}\vspace{0.1cm}
\hrule
\end{table}
-\section{Realizability}
+\Section{Realizability}
The realizability relation is a relation $f \R P$ where $f: \sem{P}$, and
$P$ is a closed formula.
In particular:
Let us assume $h : \forall m.(\forall p. p < m \to P~p) \to P~m$.
We prove by induction on $n$ that $\forall q. q \le n \to P~q$.
For $n=0$, we get a proof of $P ~q$ by
-\[ B \equiv \lambda q.\lambda h_0:q \le 0. h ~q~
-(\lambda p.\lambda k:p < q. false\_ind ~(L~p~q~k~h_0)) \]
+\begin{eqnarray}
+B & \equiv & \lambda q.\lambda h_0:q \le 0. h ~q~ \nonumber\\
+& & \quad (\lambda p.\lambda k:p < q. false\_ind ~(L~p~q~k~h_0)) \nonumber
+\end{eqnarray}
In the inductive case, we must prove that, for any $n$,
\[(\forall q. q \le n \to P~q) \to (\forall q. q \le S n \to P~q)\]
Assume $h_1: \forall q. q \le n \to P q$ and
$h_1 ~p ~ (M~ p~ q~ n~ h_3~ h_2): P~p$.\\
In conclusion, the proof of the
inductive case is
-\[I \equiv \lambda n.\lambda h_1:\forall q. q \le n \to P~ q.\lambda q.\lambda h_2:q \le S n.
-h ~ q ~ (\lambda p.\lambda h_3:p < q.h_1 ~p~ (M~ p~ q~ n~ h_3~ h_2)) \]
+\begin{eqnarray}
+I & \equiv & \lambda n.\lambda h_1:\forall q. q \le n \to P~ q.\lambda q.\lambda h_2:q \le S n. \nonumber\\
+ & & \quad h ~ q ~ (\lambda p.\lambda h_3:p < q.h_1 ~p~ (M~ p~ q~ n~ h_3~ h_2))
+ \nonumber
+\end{eqnarray}
(where $h$ is free in I).
The full proof is
-\[ \lambda h: \forall m.(\forall p. p < m \to P~p) \to P~m.\lambda m.
-nat\_ind ~B ~ I ~m~m~ (le\_n ~ m) \]
+\begin{eqnarray}
+& \lambda h:\forall m.(\forall p. p < m\to P~p)\to P~m.\lambda m. &\nonumber\\
+& \quad nat\_ind ~B ~ I ~m~m~ (le\_n ~ m) & \nonumber
+\end{eqnarray}
+
where $le\_n$ is a proof that $\forall n. n \le n$, and the free $P$ in the definition of $nat_{ind}$ is instantiated with $\forall m.m \le m \to P~m$.\\
Form the previous proof,after stripping terminal objects,
and a bit of eta-contraction to make
$f: \lambda q.\lambda g.F[g]: N\to(N\to A) \to A$, such that
(morally) $h$ is the fixpoint of $f$. For instance,
in the case of the fibonacci function, $f$ is
-\[fibo \equiv \lambda q. \lambda g.
-if~ q = 0~then~ 1~ else~ if~ q = 1~ then~ 1~ else~ g (q-1)+g (q-2)\]
+\begin{eqnarray}
+fibo & \equiv & \lambda q. \lambda g. if~ q = 0~then~ 1~ else \nonumber\\
+& & \quad if~ q = 1~ then~ 1~ else ~ g (q-1)+g (q-2) \nonumber
+\end{eqnarray}
So $f$ build a new
approximation of $h$ from the previous approximation $h$ taken
se si spiega come array viene decente... forse. lunedi' provo a scrivere
meglio.
-\section{Inductive types}
+\Section{Inductive types}
The notation we will use is similar to the one used in
\cite{Werner} and \cite{Paulin89} but we prefer
giving a label to each constructor and use that label instead of the
$A \to B_1 \to \ldots \to B_n \to C$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Extensions to the logic framework}
+\SubSection{Extensions to the logic framework}
To talk about arbitrary inductive types (and not hard coded natural numbers) we
have to extend a bit our framework.
a generic inductive type $T$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Type definition}
+\SubSection{Type definition}
$$\ind\{c_1:C(X); \ldots ; c_n:C(X)\}$$
$$C(X) ::= X \| T \to C(X) \| X \to C(X)$$
In the second case we mean $T \neq X$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Induction principle}
+\SubSection{Induction principle}
The induction principle for an inductive type $X$ and a predicate $Q$
is a constant with the following type
$$\Xind:\vec{\triUP\{C(X), c\}} \to \forall t:X.Q(t)$$
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
-\subsection{Recursor}
-\subsubsection{Type}
+\SubSection{Recursor}
+%\SubSubSection{Type}
The type of the recursor $\Rx$ on an inductive type $X$ is
$$\Rx : \vec{\square\{C(X)\}} \to X \to \alpha$$
$\square$ is defined by recursion on the constructor type $C(X)$.
\square\{T \to C(X)\} & = & T \to \square\{C(X)\}\nonumber \\
\square\{X \to C(X)\} & = & X \to \alpha \to \square\{C(X)\}\nonumber
\end{eqnarray}
-\subsubsection{Reduction rules}
+%\SubSubSection{Reduction rules}
We say that
$$\Rx~\vec{f}~(c_i~\vec{m}) \leadsto
\triDOWN\{C(X)_i, f_i, \vec{m}\}$$
can omit $\vec{m}$ since it is an empty sequence.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Realizability of the induction principle}
+\SubSection{Realizability of the induction principle}
Once we have inductive types and their induction principle we want to show that
the recursor $\Rx$ realizes $\Xind$, that is that $\Rx$ has type
$\sem{\Xind}$ and is in relation $\R$ with $\Xind$.
Q(c_i~\vec{m}~\vec{t})$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Strong normalization of extended system T}
+\Section{Strong normalization of extended system T}
Strong normalization for system T is a well know result\cite{GLT}
that can be easily extended to System T with this kind of inductive
types. The first thing we have to do is to extend the definition of
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Improving inductive types}
+\Section{Improving inductive types}
It is possible to parametrise inductive types over other inductive types
without much difficulties since the type $T$ in $C(X)$ is free. Both the
recursor and the induction principle are schemas, parametric over $T$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\bibliographystyle{latex8}
+%\bibliography{latex8}
+
\begin{thebibliography}{}
\bibitem{AL91}A.Asperti, G.Longo. Categories, Types and Structures.
Foundations of Computing, Cambrdidge University press, 1991.
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