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The thesis studies two refinements. The first, due to Awodey, Frey, and Speight, restores dependent elimination into 0-types assuming Σ-types, identity types, and function extensionality; the second, by Ripoll Echeveste inspired by Shulman, enables elimination into the full impredicative universe. It then extends both approaches to higher inductive types, tests generalisation on established classes, and fully formalises the first encoding in Agda.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/higher-inductive-types-via-impredicative-encodings-master-of-logic-thesis/657565/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/higher-inductive-types-via-impredicative-encodings-master-of-logic-thesis/657565.png","ImageObject",300,407,{"name":92,"@type":93},"Mabel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why do System F-style encodings for inductive types fail in the impredicative setting?","Question",{"text":112,"@type":113},"They do not satisfy the dependent elimination rule in an impredicative universe, so the 􀀁-rule does not hold under the usual encodings.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What assumptions enable the first refinement to recover dependent elimination?",{"text":117,"@type":113},"Awodey, Frey, and Speight’s refinement assumes Σ-types, identity types, and function extensionality, and it recovers dependent elimination into 0-types.",{"name":119,"@type":110,"acceptedAnswer":120},"How is the first refined encoding of higher inductive types handled in the thesis?",{"text":121,"@type":113},"It is fully formalised in the Agda proof assistant after extending the encoding and testing the generalisations on specific higher inductive type classes.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657565,1792982837,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},7971474920318,"https://ap-avatar.wpscdn.com/avatar/74000ee537e6b0ca360?x-image-process=image/resize,m_fixed,w_180,h_180&k=1788430171632181674","Higher Inductive Types Via Impredicative Encodings  \nMSc Thesis (Afstudeerscriptie)  \nwritten by  \nStefano Volpe  \nunder the supervision of Dr Benno van den Berg, and submitted to the Examinations Board in partial fulfilment of the requirements for the degree of  \nMSc in Logic  \nat the Universiteit van Amsterdam.  \nDate of the public defence:  \nAugust 27, 2025  \nMembers of the Thesis Committee:  \nDr Benno van den Berg Dr Malvin Gattinger Prof Dr Herman Geuvers  \nDr Kristina Sojakova  \nIt would be so nice if something made sense for a change.  \n—Alice, from Alice’s Adventures in Wonderland by Lewis Carroll  \niii  \nAbstract  \nIn dependent type theory with an impredicative universe, the usual encodings for inductive types borrowed from System F do not satisfy the dependent elimination rule (equivalently, no form of the 􀀁-rule holds) . Two refinements of these encodings have been proposed to remedy this. Both apply to all 􀀂-types. The first is by S. Awodey, J. Frey, and S. Speight [4] . They manage to recover dependent elimination into 0-types of the impredicative universe by assuming the existence of Σ-types, identity types, and function extensionality. By also assuming the existence of a natural numbers type, X. Ripoll Echeveste [21], inspired by the ideas of M. Shulman [23], devises a second, distinct refinement for 􀀂-types. This time, dependent elimination is possible into the entire impredicative universe.  \nIn this thesis, we show how both encodings can be extended to higher inductive types. While 􀀂-types are a popular choice for a working definition of “inductive types”, there are multiple competing definitions of “higher inductive types” in the literature. We test our generalisation of the first refinement on the class of higher inductive types defined by H. Basold, H. Geuvers, and N. van der Weide [7], and our generalisation of the second on the 􀀂-suspensions by K. Sojakova [24] . Finally, we fully formalise the first refined encoding of higher inductive types in the Agda proof assistant [1] .  \nAcknowledgements  \nBenno van den Berg, my supervisor, has had the utmost patience when dealing with me during my two years at the Master of Logic, first as a lecturer, and then during my thesis semester. Similarly, my “unofficial” supervisor, Daniël Otten, had also acted as a great teaching assistant before my thesis. It takes a great deal of energy to teach a computer scientist some mathematical logic! For this, I am very grateful to both. Similarly, I also wish to thank the rest of the committee for the interest they showed in my work. It was a pleasure to discuss it with them. My family surely deserves a mention, too. Everything Ilaria, mom, and dad could have done to make this adventure work, they did. I am relieved to see their efforts brought me here: the Master of Logic is a special place. If you have come across it, you know. There are too many students, from inside and outside the MoL room, for me to thank them all. Please allow me to single out Lu(crezia), Klarise, Emma, Bas, Yoshi (Josje), and Alyssa. They have never given up on me.  \nHaving to say goodbye to all these people makes leaving Amsterdam a painful experience. Wherever I go, whomever I meet, I hope the good they have sowed in me is nurtured and shared.  \nContents  \n1. Background ................................................. 1  \n1.1. Higher Inductive Types .............................................. 1  \n1.1.1. Van der Weide’s Higher Inductive Types ............................. 2  \n1.1.2. W-Suspensions ................................................ 5  \n1.1.3. Examples ................................................... 6  \n1.2. Impredicative Encodings ............................................. 7  \n1.3. The System ...................................................... 7  \n1.4. Contributions ..................................................... 8  \n2 Set-Truncated Higher Inductive Types .............................. 9  \n2.1. Lawful Set Algebras ....","cbCaipuzAcmhgWXx","https://ap.wps.com/l/cbCaipuzAcmhgWXx","pdf",15766720,57,"English","# Background\n## Higher Inductive Types\n### Van der Weide’s Higher Inductive Types\n### W-Suspensions\n### Examples\n## Impredicative Encodings\n## The System\n## Contributions\n# Set-Truncated Higher Inductive Types\n## Lawful Set Algebras\n## Limits in the Lawful Set Algebras Category\n## Initial Lawful Set Algebra\n## Rules\n# General Higher Inductive Types\n## Path Algebra Tools\n## W-Suspension Algebras\n## Naturality\n## W-Suspension Algebras and Quasi-Idempotents\n## Initiality\n# Formalisation\n## Project Structure\n## Cubical Mode\n## Extensibility\n## Impredicativity\n# Conclusions\n## Related Work\n## Limitations\n## Future Work\n## References","[{\"question\":\"Why do System F-style encodings for inductive types fail in the impredicative setting?\",\"answer\":\"They do not satisfy the dependent elimination rule in an impredicative universe, so the 􀀁-rule does not hold under the usual encodings.\"},{\"question\":\"What assumptions enable the first refinement to recover dependent elimination?\",\"answer\":\"Awodey, Frey, and Speight’s refinement assumes Σ-types, identity types, and function extensionality, and it recovers dependent elimination into 0-types.\"},{\"question\":\"How is the first refined encoding of higher inductive types handled in the thesis?\",\"answer\":\"It is fully formalised in the Agda proof assistant after extending the encoding and testing the generalisations on specific higher inductive type classes.\"}]","Higher Inductive Types Via Impredicative Encodings - Master of Logic Thesis | PDF",144]