{"id":"https://openalex.org/W2164688709","doi":"https://doi.org/10.1017/s0960129511000119","title":"Type classes for mathematics in type theory","display_name":"Type classes for mathematics in type theory","publication_year":2011,"publication_date":"2011-07-01","ids":{"openalex":"https://openalex.org/W2164688709","doi":"https://doi.org/10.1017/s0960129511000119","mag":"2164688709"},"language":"en","primary_location":{"id":"doi:10.1017/s0960129511000119","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0960129511000119","pdf_url":null,"source":{"id":"https://openalex.org/S101766102","display_name":"Mathematical Structures in Computer Science","issn_l":"0960-1295","issn":["0960-1295","1469-8072"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Structures in Computer Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043123968","display_name":"Bas Spitters","orcid":"https://orcid.org/0000-0002-2802-0973"},"institutions":[{"id":"https://openalex.org/I145872427","display_name":"Radboud University Nijmegen","ror":"https://ror.org/016xsfp80","country_code":"NL","type":"education","lineage":["https://openalex.org/I145872427"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"BAS SPITTERS","raw_affiliation_strings":["Radboud University Nijmegen, Nijmegen, The Netherlands Email:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Radboud University Nijmegen, Nijmegen, The Netherlands Email:","institution_ids":["https://openalex.org/I145872427"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041170857","display_name":"Eelis van der Weegen","orcid":null},"institutions":[{"id":"https://openalex.org/I145872427","display_name":"Radboud University Nijmegen","ror":"https://ror.org/016xsfp80","country_code":"NL","type":"education","lineage":["https://openalex.org/I145872427"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"EELIS VAN DER WEEGEN","raw_affiliation_strings":["Radboud University Nijmegen, Nijmegen, The Netherlands Email:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Radboud University Nijmegen, Nijmegen, The Netherlands Email:","institution_ids":["https://openalex.org/I145872427"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":14.5083,"has_fulltext":false,"cited_by_count":86,"citation_normalized_percentile":{"value":0.98970523,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"21","issue":"4","first_page":"795","last_page":"825"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9785000085830688,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.963699996471405,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/type-theory","display_name":"Type theory","score":0.6361692547798157},{"id":"https://openalex.org/keywords/universal-algebra","display_name":"Universal algebra","score":0.5940258502960205},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5854418873786926},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.5787962675094604},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5366482138633728},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5312768220901489},{"id":"https://openalex.org/keywords/categorical-variable","display_name":"Categorical variable","score":0.52993243932724},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.48008519411087036},{"id":"https://openalex.org/keywords/category-theory","display_name":"Category theory","score":0.47775593400001526},{"id":"https://openalex.org/keywords/syntax","display_name":"Syntax","score":0.47287532687187195},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.46294060349464417},{"id":"https://openalex.org/keywords/abstraction","display_name":"Abstraction","score":0.4620389938354492},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.44156670570373535},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.3892068862915039},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.37798362970352173},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3597673773765564},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22918596863746643},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.17917698621749878}],"concepts":[{"id":"https://openalex.org/C93682546","wikidata":"https://www.wikidata.org/wiki/Q1056428","display_name":"Type theory","level":3,"score":0.6361692547798157},{"id":"https://openalex.org/C188971069","wikidata":"https://www.wikidata.org/wiki/Q1636734","display_name":"Universal algebra","level":3,"score":0.5940258502960205},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5854418873786926},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5787962675094604},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5366482138633728},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5312768220901489},{"id":"https://openalex.org/C5274069","wikidata":"https://www.wikidata.org/wiki/Q2285707","display_name":"Categorical variable","level":2,"score":0.52993243932724},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.48008519411087036},{"id":"https://openalex.org/C54884031","wikidata":"https://www.wikidata.org/wiki/Q217413","display_name":"Category theory","level":2,"score":0.47775593400001526},{"id":"https://openalex.org/C60048249","wikidata":"https://www.wikidata.org/wiki/Q37437","display_name":"Syntax","level":2,"score":0.47287532687187195},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.46294060349464417},{"id":"https://openalex.org/C124304363","wikidata":"https://www.wikidata.org/wiki/Q673661","display_name":"Abstraction","level":2,"score":0.4620389938354492},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.44156670570373535},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.3892068862915039},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.37798362970352173},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3597673773765564},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22918596863746643},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.17917698621749878},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1017/s0960129511000119","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0960129511000119","pdf_url":null,"source":{"id":"https://openalex.org/S101766102","display_name":"Mathematical Structures in Computer Science","issn_l":"0960-1295","issn":["0960-1295","1469-8072"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Structures in Computer Science","raw_type":"journal-article"},{"id":"pmh:oai:repository.ubn.ru.nl:2066/92523","is_oa":false,"landing_page_url":"http://hdl.handle.net/2066/92523","pdf_url":null,"source":{"id":"https://openalex.org/S4306401067","display_name":"Radboud Repository (Radboud University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I145872427","host_organization_name":"Radboud University Nijmegen","host_organization_lineage":["https://openalex.org/I145872427"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematical Structures in Computer Science, 21, 4, pp. 795-825","raw_type":"Article / Letter to editor"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":44,"referenced_works":["https://openalex.org/W78059615","https://openalex.org/W91660649","https://openalex.org/W1481180394","https://openalex.org/W1486558830","https://openalex.org/W1498091397","https://openalex.org/W1508641834","https://openalex.org/W1516788060","https://openalex.org/W1553511155","https://openalex.org/W1558115254","https://openalex.org/W1560253649","https://openalex.org/W1561086741","https://openalex.org/W1561420154","https://openalex.org/W1569206238","https://openalex.org/W1581378641","https://openalex.org/W1653213632","https://openalex.org/W1795725592","https://openalex.org/W1833824395","https://openalex.org/W1945317924","https://openalex.org/W1976800160","https://openalex.org/W1986402635","https://openalex.org/W1988136938","https://openalex.org/W2002193175","https://openalex.org/W2013072751","https://openalex.org/W2013140497","https://openalex.org/W2059895047","https://openalex.org/W2067914884","https://openalex.org/W2078543558","https://openalex.org/W2097509104","https://openalex.org/W2106718208","https://openalex.org/W2120588128","https://openalex.org/W2120713972","https://openalex.org/W2124798629","https://openalex.org/W2136219937","https://openalex.org/W2143917322","https://openalex.org/W2144645833","https://openalex.org/W2574983198","https://openalex.org/W2610216665","https://openalex.org/W2756921402","https://openalex.org/W2913347770","https://openalex.org/W2914762105","https://openalex.org/W3099251155","https://openalex.org/W3104826616","https://openalex.org/W6603185079","https://openalex.org/W6636989977"],"related_works":["https://openalex.org/W3180855406","https://openalex.org/W2164569390","https://openalex.org/W3011960800","https://openalex.org/W4233993919","https://openalex.org/W2005380195","https://openalex.org/W223002628","https://openalex.org/W1974294638","https://openalex.org/W1607532628","https://openalex.org/W4366548651","https://openalex.org/W1607720949"],"abstract_inverted_index":{"The":[0,96,179],"introduction":[1],"of":[2,14,30,39,80,98,102,114,127,135,148,182,199],"first-class":[3],"type":[4,23,31,103,149,183],"classes":[5,32,104,150],"in":[6,22,83],"the":[7,15,173,197],"Coq":[8],"system":[9],"calls":[10],"for":[11,19,33,132],"a":[12,27,46,78,94,106,125,192],"re-examination":[13],"basic":[16],"interfaces":[17,131],"used":[18],"mathematical":[20],"formalisation":[21],"theory.":[24],"We":[25],"present":[26],"new":[28,67],"set":[29,126],"mathematics":[34],"and":[35,117,142,177],"take":[36],"full":[37],"advantage":[38],"their":[40],"unique":[41],"features":[42],"to":[43,54,73,93,189],"make":[44],"practical":[45],"particularly":[47],"flexible":[48],"approach":[49],"that":[50],"was":[51],"formerly":[52],"thought":[53],"be":[55],"unfeasible.":[56],"Thus,":[57],"we":[58,123],"address":[59],"traditional":[60],"proof":[61],"engineering":[62],"challenges":[63],"as":[64,66,110,112],"well":[65,111],"ones":[68],"resulting":[69],"from":[70],"our":[71,99],"ambition":[72],"build":[74,124],"upon":[75],"this":[76,121],"development":[77,100],"library":[79],"constructive":[81],"analysis":[82],"which":[84],"any":[85],"abstraction":[86],"penalties":[87],"inhibiting":[88],"efficient":[89,161],"computation":[90],"are":[91],"reduced":[92],"minimum.":[95],"basis":[97],"consists":[101],"representing":[105],"standard":[107],"algebraic":[108],"hierarchy,":[109],"portions":[113],"category":[115],"theory":[116],"universal":[118,143],"algebra.":[119],"On":[120],"foundation,":[122],"mathematically":[128],"sound":[129],"abstract":[130],"different":[133],"kinds":[134],"numbers,":[136],"succinctly":[137],"expressed":[138],"using":[139],"categorical":[140],"language":[141],"algebra":[144],"constructions.":[145],"Strategic":[146],"use":[147,198],"lets":[151],"us":[152,188],"support":[153],"these":[154],"high-level":[155],"theory-friendly":[156],"definitions,":[157],"while":[158],"still":[159],"enabling":[160],"implementations":[162],"unhindered":[163],"by":[164],"gratuitous":[165],"indirection,":[166],"conversion":[167],"or":[168],"projection.":[169],"Algebra":[170],"thrives":[171],"on":[172],"interplay":[174],"between":[175],"syntax":[176],"semantics.":[178],"Prolog-like":[180],"abilities":[181],"class":[184],"instance":[185],"resolution":[186],"allow":[187],"conveniently":[190],"define":[191],"quote":[193],"function,":[194],"thus":[195],"facilitating":[196],"reflective":[200],"techniques.":[201]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":12},{"year":2012,"cited_by_count":9}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}