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Title: Subtyping in signatures
Author: Lungu, Georgiana
ISNI:       0000 0004 8500 304X
Awarding Body: Royal Holloway, University of London
Current Institution: Royal Holloway, University of London
Date of Award: 2018
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Type theories with canonical objects like Martin Lof's Type Theory or Luo's UTT have increasingly gained popularity in the last decades due to their usage in proof assistants, formal semantics of natural language and formalization of mathematics. The main purpose of this work is to explore a new way of introducing coercive subtyping in such type theories which facilitates the representation of some practical notions of subtyping. Introducing subtyping in dependent type theories is not straightforward when the preservation of properties like canonicity and subject reduction is also desired. Previous research already showed how such properties are affected by the usual notion of subsumptive subtyping and offered an alternative in the form of coercive subtyping introduced by enriching the system with a set of coercive subtyping judgements. Here I introduce a new way of adding coercive subtyping to type theory, specifically by annotating certain functions in assumptions, arguing that this is more handy to represent practical cases. This system is also closer to the programming model of proof assistants like Coq where coercions are annotated as such at the assumption level. Assumptions in Type Theory are represented as either contexts, which are sequences of membership entries for variables that bear abstraction and substitution or signatures, which are sequences of memberships entries for constants for which abstraction and substitution are not available. I shall use signatures as an environment for subtyping assumptions. I will prove that the system thus obtained is well behaved, in that it is only abbreviational to the original system, by considering its relation with the previous version of coercive subtyping which was already proved to be well behaved. To demonstrate the ability of the system to argue about practical situations, I will present three case studies. The first one studies the relationship between a subsumptive subtyping system and coercive subtyping. The second case study discusses how Russell-style universe inclusions, as found in Homotopy Type Theory, can be understood as coercions in a system with Tarski style hierarchy. And the last discussion is the need to treat injectivity as an assumption as well in order to capture faithfully some notions of subtyping which are based on or generalize inclusion.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: typr theory ; subtyping ; conservative extension ; canonical objects ; universes