Formalization of Universal Algebra in Agda

In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

Signatures and Induction Principles for Higher Inductive-Inductive Types

Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalizing higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy real numbers and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a small type theory, named the theory of signatures. A context in this theory encodes a HIIT by listing the constructors. We also compute notions of induction and recursion for HIITs, by using variants of syntactic logical relation translations. Building full categorical semantics and constructing initial algebras is left for future work. The theory of HIIT signatures was formalised in Agda together with the syntactic translations. We also provide a Haskell implementation, which takes signatures as input and outputs translation results as valid Agda code

Unifying Cubical Models of Univalent Type Theory

We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure

Foundational Extensible Corecursion

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under well-behaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Experience Implementing a Performant Category-Theory Library in Coq

We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This version includes a full bibliography which does not fit in the Springer version; other than the more complete references, this is the version submitted as a final copy to ITP 201

Non-wellfounded trees in Homotopy Type Theory

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary files contain Agda files with formalized proof

Formalizing of Category Theory in Agda

The generality and pervasiness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a standard, working formalization of category theory. We document our work on solving this dilemma. The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked. In particular, we find that alternative definitions or alternative proofs from those found in standard textbooks can be advantageous, as well as "fit" Agda's type theory more smoothly. Some definitions regarded as equivalent in standard textbooks turn out to make different "universe level" assumptions, with some being more polymorphic than others. We also pay close attention to engineering issues so that the library integrates well with Agda's own standard library, as well as being compatible with as many of supported type theories in Agda as possible