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Abstract
The term UniMath refers both to a formal system for mathematics, as well as a computer-checked library of mathematics formalized in that system. The UniMath system is a core dependent type theory, augmented by the univalence axiom. The system is kept as small as possible in order to ease verification of it—in particular, general inductive types are not part of the system. In this work, we partially remedy the lack of inductive types by constructing some set-level datatypes and their associated induction principles from other type constructors. This involves a formalization of a category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the datatypes obtained. We also connect this construction to a previous formalization of substitution for languages with variable binding. Altogether, we construct a framework that allows us to concisely specify, via a simple notion of binding signature, a language with variable binding. From such a specification we obtain the datatype of terms of that language, equipped with a certified monadic substitution operation and a suitable recursion scheme. Using this we formalize the untyped lambda calculus and the raw syntax of Martin-Löf type theory.
Highlights
The UniMath1 language is meant to be a core dependent type theory, making use of as few type constructors as possible
In this work we focus on a particular class of datatypes that represent languages with variable binding
Still within the target area of datatype-generic programming, but the datatypes we focus on in the present work are canonically equipped with a substitution operation—itself defined via a variant of the recursion principle associated to the datatypes
Summary
The UniMath language is meant to be a core dependent type theory, making use of as few type constructors as possible. We partially remedy the lack of general inductive types in UniMath by constructing datatypes as initial algebras. (iii) Instantiating the base category C of the previous section to Set, we construct the datatype, as an initial algebra of the endofunctor on endofunctors on the category of sets that is specified by a binding signature, using just the type constructors available in the UniMath language. We base the needed scheme of generalized iteration in Mendler-style on another theorem in [11] that is based on cocontinuity assumptions instead of the existence of right Kan extensions We apply this modified construction to the datatype constructed in (iii), where the base category is the category of sets.
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