All Outputs (3)

Naive Type Theory (2019)
Book Chapter
Altenkirch, T. (2019). Naive Type Theory. In S. Centrone, D. Kant, & D. Sarikaya (Eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts (101-136). Springer

We introduce Type Theory, including Homotopy Type Theory, as an alternative to set theory as a foundation of Mathematics emphasising the intuitive and naive understanding of its concepts.

Setoid Type Theory—A Syntactic Translation (2019)
Conference Proceeding
Altenkirch, T., Boulier, S., Kaposi, A., & Tabereau, N. (2019). Setoid Type Theory—A Syntactic Translation. In Mathematics of Program Construction: 13th International Conference, MPC 2019, Porto, Portugal, October 7–9, 2019, Proceedings (155-196). https://doi.org/10.1007/978-3-030-33636-3_7

We introduce setoid type theory, an intensional type theory with a proof-irrelevant universe of propositions and an equality type satisfying functional extensionality and propositional extensionality. We justify the rules of setoid type theory by a s... Read More about Setoid Type Theory—A Syntactic Translation.

Constructing quotient inductive-inductive types (2019)
Conference Proceeding
Kaposi, A., Kovac, A., & Altenkirch, T. (2019). Constructing quotient inductive-inductive types. In Proceedings of the ACM on Programming Languages archive Volume 3 Issue POPL, January 2019 (1-24). https://doi.org/10.1145/3290315

Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In addition, equality constructors are also allowed. We work in a setting... Read More about Constructing quotient inductive-inductive types.