Type theory in type theory using quotient inductive types
Altenkirch, Thorsten and Kaposi, Ambrus (2016) Type theory in type theory using quotient inductive types. In: POPL '16 The 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, 20-22 January 2016, St Petersburg, Florida, United States of America.
Official URL: http://dl.acm.org/citation.cfm?doid=2837614.2837638
We present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids referring to preterms or a typability relation but defines directly well typed objects by an inductive definition. We use the elimination principle to define the set-theoretic and logical predicate interpretation. The work has been formalized using the Agda system extended with QITs using postulates.
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