{"id":"https://openalex.org/W7160508335","doi":"https://doi.org/10.48550/arxiv.2605.04472","title":"Automated Formal Proofs of Combinatorial Identities via Wilf-Zeilberger Guidance and LLMs","display_name":"Automated Formal Proofs of Combinatorial Identities via Wilf-Zeilberger Guidance and LLMs","publication_year":2026,"publication_date":"2026-05-06","ids":{"openalex":"https://openalex.org/W7160508335","doi":"https://doi.org/10.48550/arxiv.2605.04472"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.04472","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.04472","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.04472","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102999284","display_name":"Beibei Xiong","orcid":"https://orcid.org/0000-0003-2132-7136"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xiong, Beibei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135605855","display_name":"Hangyu Lv","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lv, Hangyu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135555232","display_name":"Junqi Liu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Junqi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135578682","display_name":"Yisen Wang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Yisen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070141631","display_name":"Shaoshi Chen","orcid":"https://orcid.org/0000-0001-8756-3006"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chen, Shaoshi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100736616","display_name":"Jianlin Wang","orcid":"https://orcid.org/0000-0003-3398-7967"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Jianlin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135577199","display_name":"Zhengfeng Yang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yang, Zhengfeng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5026679766","display_name":"Lihong Zhi","orcid":"https://orcid.org/0000-0001-9973-2217"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhi, Lihong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.6255000233650208,"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":0.6255000233650208,"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":0.05860000103712082,"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"}},{"id":"https://openalex.org/T10743","display_name":"Software Testing and Debugging Techniques","score":0.043699998408555984,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/mathematical-proof","display_name":"Mathematical proof","score":0.8672999739646912},{"id":"https://openalex.org/keywords/gas-meter-prover","display_name":"Gas meter prover","score":0.7062000036239624},{"id":"https://openalex.org/keywords/executable","display_name":"Executable","score":0.7038999795913696},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.6104000210762024},{"id":"https://openalex.org/keywords/proof-of-concept","display_name":"Proof of concept","score":0.49470001459121704},{"id":"https://openalex.org/keywords/combinatorial-explosion","display_name":"Combinatorial explosion","score":0.49140000343322754},{"id":"https://openalex.org/keywords/scope","display_name":"Scope (computer science)","score":0.47859999537467957},{"id":"https://openalex.org/keywords/bootstrapping","display_name":"Bootstrapping (finance)","score":0.4778999984264374},{"id":"https://openalex.org/keywords/formal-proof","display_name":"Formal proof","score":0.42739999294281006}],"concepts":[{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.8672999739646912},{"id":"https://openalex.org/C159718280","wikidata":"https://www.wikidata.org/wiki/Q5526353","display_name":"Gas meter prover","level":3,"score":0.7062000036239624},{"id":"https://openalex.org/C160145156","wikidata":"https://www.wikidata.org/wiki/Q778586","display_name":"Executable","level":2,"score":0.7038999795913696},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6319000124931335},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.6104000210762024},{"id":"https://openalex.org/C124978682","wikidata":"https://www.wikidata.org/wiki/Q1201019","display_name":"Proof of concept","level":2,"score":0.49470001459121704},{"id":"https://openalex.org/C115908005","wikidata":"https://www.wikidata.org/wiki/Q2668364","display_name":"Combinatorial explosion","level":2,"score":0.49140000343322754},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.48019999265670776},{"id":"https://openalex.org/C2778012447","wikidata":"https://www.wikidata.org/wiki/Q1034415","display_name":"Scope (computer science)","level":2,"score":0.47859999537467957},{"id":"https://openalex.org/C207609745","wikidata":"https://www.wikidata.org/wiki/Q4944086","display_name":"Bootstrapping (finance)","level":2,"score":0.4778999984264374},{"id":"https://openalex.org/C94461902","wikidata":"https://www.wikidata.org/wiki/Q2762418","display_name":"Formal proof","level":3,"score":0.42739999294281006},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.399399995803833},{"id":"https://openalex.org/C105605280","wikidata":"https://www.wikidata.org/wiki/Q7249999","display_name":"Proof complexity","level":3,"score":0.3959999978542328},{"id":"https://openalex.org/C124304363","wikidata":"https://www.wikidata.org/wiki/Q673661","display_name":"Abstraction","level":2,"score":0.3804999887943268},{"id":"https://openalex.org/C203265346","wikidata":"https://www.wikidata.org/wiki/Q11387554","display_name":"Proof assistant","level":3,"score":0.3709000051021576},{"id":"https://openalex.org/C206880738","wikidata":"https://www.wikidata.org/wiki/Q431667","display_name":"Automated theorem proving","level":2,"score":0.3296999931335449},{"id":"https://openalex.org/C195344581","wikidata":"https://www.wikidata.org/wiki/Q2555318","display_name":"Automated reasoning","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C2318724","wikidata":"https://www.wikidata.org/wiki/Q852732","display_name":"Proof theory","level":3,"score":0.28700000047683716},{"id":"https://openalex.org/C184670325","wikidata":"https://www.wikidata.org/wiki/Q512604","display_name":"Loop (graph theory)","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C13766981","wikidata":"https://www.wikidata.org/wiki/Q11387554","display_name":"Automated proof checking","level":3,"score":0.27720001339912415},{"id":"https://openalex.org/C161505775","wikidata":"https://www.wikidata.org/wiki/Q3240280","display_name":"Direct proof","level":2,"score":0.27480000257492065},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26840001344680786},{"id":"https://openalex.org/C39920170","wikidata":"https://www.wikidata.org/wiki/Q693083","display_name":"Soundness","level":2,"score":0.257099986076355},{"id":"https://openalex.org/C2777810591","wikidata":"https://www.wikidata.org/wiki/Q16861606","display_name":"Credential","level":2,"score":0.2540000081062317},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.2533000111579895}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.04472","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.04472","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.04472","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.04472","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Automating":[0],"formal":[1],"proofs":[2],"of":[3,35,148],"combinatorial":[4,36,112],"identities":[5,37,144],"is":[6,15],"challenging":[7],"for":[8,143],"LLM-based":[9,72],"provers,":[10],"as":[11,25,118],"long-horizon":[12],"proof":[13,34,60,64,105],"planning":[14],"required":[16],"and":[17,43,69,120,122,128,150],"unconstrained":[18],"search":[19],"quickly":[20],"explodes.":[21],"Symbolic":[22],"methods":[23],"such":[24,117],"the":[26,76,146],"Wilf-Zeilberger":[27],"(WZ)":[28],"method":[29],"can":[30],"achieve":[31],"a":[32,54,83,87,103,159],"mechanized":[33],"by":[38,95],"constructing":[39],"special":[40],"auxiliary":[41],"functions":[42],"demonstrating":[44],"that":[45,57,100,133],"they":[46],"satisfy":[47],"specific":[48],"recurrence":[49],"relations.":[50],"We":[51,80],"propose":[52],"WZ-LLM,":[53],"neuro-symbolic":[55],"framework":[56,135],"turns":[58],"WZ":[59,156],"plans":[61],"into":[62],"executable":[63],"sketches":[65,157],"in":[66],"Lean":[67],"4":[68],"uses":[70],"an":[71],"prover":[73],"to":[74],"discharge":[75],"resulting":[77],"machine-checkable":[78],"subgoals.":[79],"also":[81],"train":[82],"dedicated":[84],"WZ-Prover":[85],"via":[86],"Lean-kernel-verified":[88],"bootstrapping":[89],"loop":[90],"with":[91],"expert-verified":[92],"iteration,":[93],"followed":[94],"DAPO-based":[96],"refinement.":[97],"Experiments":[98],"show":[99],"WZ-LLM":[101],"achieves":[102],"34%":[104],"success":[106,154],"rate":[107],"on":[108,126],"LCI-Test":[109],"(100":[110],"classic":[111],"identities),":[113],"outperforming":[114],"strong":[115],"baselines":[116],"DeepSeek-V3":[119],"Goedel-Prover-V2,":[121],"delivering":[123],"consistent":[124],"gains":[125],"CombiBench":[127],"PutnamBench-Comb.":[129],"These":[130],"results":[131],"indicate":[132],"our":[134],"provides":[136],"two":[137],"complementary":[138],"strengths:":[139],"improved":[140],"direct":[141],"proving":[142],"beyond":[145],"scope":[147],"WZ,":[149],"substantially":[151],"higher":[152],"end-to-end":[153],"when":[155],"guide":[158],"specialized":[160],"prover.":[161]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-08T00:00:00"}