[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84321-en":3,"doc-seo-84321-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84321,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Directed Proof-Relevant Logical Relations in Simplicial HoTT","The paper develops a directed version of proof-relevant logical relations for simplicial homotopy type theory, addressing how reduction direction is handled in logical-relations arguments. It internalizes reductions as inequality types and builds an object syntax as a directed quotient inductive type. Contravariant families in simplicial type theory yield the proof-relevant closure under expansion needed to transport computability evidence backward along reductions, with functoriality and universal properties built in. It proves unary directed Boolean canonicity and extends to dependent types, universes, and binary relations via a comonadic flat modality and parametricity separation.","arXiv :2607 .08 154v 1 [ cs .LO] 9 Jul 2026  \nDirected proof-relevant logical relations in simplicial HoTT  \nRUNMING LI, Carnegie Mellon University, USA HARRISON GRODIN, Carnegie Mellon University, USA ROBERT HARPER, Carnegie Mellon University, USA  \nIntrinsically-typed presentations of type theory often use equality in the meta-language to represent objectlanguage judgmental equality. In such equational syntax, proof-relevant logical relations define computability predicates on judgmental equivalence classes of types and terms. This approach, however, does not directly account for reduction, which is directed and plays a central role in many logical-relations arguments. This paper develops a directed version of proof-relevant logical relations in simplicial homotopy type theory, where reductions are internalized as inequality types. We construct object syntax as a directed quotient inductive type. The central observation is that contravariant families in simplicial type theory provide exactly the proof-relevant form of closure under expansion for logical relations: computability evidence can be transported backward along reductions, with the required functoriality and universal property built in. Using this observation, we construct a unary logical relations model with contravariant computability predicatesand prove directed Boolean canonicity: every closed Boolean term reduces to either true or false. We then extend the construction to dependent types and universes, where a comonadic flat modality provides the discreteness needed for type conversion and universe predicates. Finally, we adapt the method to binary logical relations, separating vertical reduction from horizontal parametricity and obtaining a proof-relevant account of representation independence.  \nCCS Concepts: • Theory of computation → Type theory; Constructive mathematics; Categorical semantics.  \n1 Introduction  \nLogical relations begin with a simple idea: interpret each type by a family of computable terms, and interpret each type former by its action on such families. This is the pattern behind Tait’s computability method [Tait 1967] . A product is computable when its projections are computable; a function is computable when it takes computable arguments to computable results. The fundamental theorem then states that every well-typed term is computable at its type. This method has become one of the standard tools of programming-language semantics, used to prove properties such as normalization, contextual equivalence, representation independence, and noninterference. The same idea also shapes foundational accounts of type theory itself. In the NuPRL tradition, a computational semantics of types—closely related to PER/logical-relations models—forms part of the basis on which the type system is justified [Allen 1987; Constable et al. 1986] . Related methods are also central to the separation-logic framework Iris [Jung et al. 2018; Timany et al. 2024] .  \nIn programming-language semantics, logical relations are often formulated relative to a reduction relation → and its reflexive-transitive closure → ∗ . In that setting, computation has a direction: a term steps to, or reduces to, another term. A logical relation is then a family of predicates defined by induction on types; for each type 􀀖, the predicate 􀀖• is the logical interpretation of 􀀖:  \n(−)• (−) : (􀀖 : Ty) → Tm 􀀖 → Prop  \nBool• (􀀢) ≔ (􀀢 → ∗ true) ∨ (􀀢 → ∗ false)  \n(􀀖 × 􀀗)• (􀀥) ≔ 􀀖• (fst 􀀥) ∧ 􀀗• (snd 􀀥)  \n(􀀖 → 􀀗)• (􀀛 ) ≔ (􀀢 : 􀀖) → 􀀖• (􀀢) → 􀀗• (app 􀀛 􀀢) .  \nAuthors’ Contact Information: Runming Li, [runmingl@cs.cmu.edu](runmingl@cs.cmu.edu), Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA; Harrison Grodin, [hgrodin@cs.cmu.edu](hgrodin@cs.cmu.edu), Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA; Robert Harper, [rwh@cs.cmu.edu](rwh@cs.cmu.edu), Carnegie Mellon University, Computer Science Department, Pittsburgh, PA, USA.  \n2 Runming Li, Harrison G","cbCaidE1WVGRxbrR","https://ap.wps.com/l/cbCaidE1WVGRxbrR","pdf",940877,1,37,"English","en",105,"# Introduction\n## Logical relations and directed reduction\n## Proof-relevant categorical gluing and related semantics\n## Expansion closure lemma and computability\n## Inductive definitions for logical relation predicates\n# Core construction overview\n## Directed quotient inductive type\n## Contravariant families and backward transport of evidence\n## Unary model and directed Boolean canonicity\n## Extension to dependent types and universes\n# Beyond unary: binary relations and representation independence","[{\"question\":\"What problem does the paper address with proof-relevant logical relations?\",\"answer\":\"Equational syntaxes for proof-relevant logical relations use equality in the meta-language, but they do not directly account for directed reduction. The paper targets this gap by internalizing reduction direction into the logical-relations framework.\"},{\"question\":\"How are reductions represented in the directed proof-relevant framework?\",\"answer\":\"Reductions are internalized as inequality types within the theory. This allows directed computation steps to be handled internally rather than only via an external transition system.\"},{\"question\":\"What is proved about Boolean terms in the unary directed model?\",\"answer\":\"The paper proves directed Boolean canonicity: every closed Boolean term reduces to either true or false, establishing a canonicity result in the directed setting.\"}]",1784194814,93,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"directed-proof-relevant-logical-relations-in-simplicial-hott","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/directed-proof-relevant-logical-relations-in-simplicial-hott/84321/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address with proof-relevant logical relations?","Question",{"text":75,"@type":76},"Equational syntaxes for proof-relevant logical relations use equality in the meta-language, but they do not directly account for directed reduction. The paper targets this gap by internalizing reduction direction into the logical-relations framework.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are reductions represented in the directed proof-relevant framework?",{"text":80,"@type":76},"Reductions are internalized as inequality types within the theory. This allows directed computation steps to be handled internally rather than only via an external transition system.",{"name":82,"@type":73,"acceptedAnswer":83},"What is proved about Boolean terms in the unary directed model?",{"text":84,"@type":76},"The paper proves directed Boolean canonicity: every closed Boolean term reduces to either true or false, establishing a canonicity result in the directed setting.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]