Related papers: Formalising Inductive and Coinductive Containers

We introduce a formal meta-language for probabilistic programming, capable of expressing both programs and the type systems in which they are embedded. We are motivated here by the desire to allow an AGI to learn not only relevant knowledge…

The expression problem describes a fundamental tradeoff between two types of extensibility: extending a type with new operations, such as by pattern matching on an algebraic data type in functional programming, and extending a type with new…

Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect…

This article presents a bidirectional type system for the Calculus of Inductive Constructions (CIC). It introduces a new judgement intermediate between the usual inference and checking, dubbed constrained inference, to handle the presence…

We present a novel dependent linear type theory in which the multiplicity of some variable-i.e., the number of times the variable can be used in a program-can depend on other variables. This allows us to give precise resource annotations to…

Based on Gandy's principles for models of computation we give category-theoretic axioms describing locally deterministic updates to finite objects. Rather than fixing a particular category of states, we describe what properties such a…

It is informally understood that the purpose of modal type constructors in programming calculi is to control the flow of information between types. In order to lend rigorous support to this idea, we study the category of classified sets, a…

Formal transformations somehow resembling the usual derivative are surprisingly common in computer science, with two notable examples being derivatives of regular expressions and derivatives of types. A newcomer to this list is the…

The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension…

A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets, and uniformly continuous maps is presented. In work of Berger it was shown how to extract certified algorithms working with the signed…

Variables are a crucial element in logic and are also addressed in institution theory, an effort to axiomatize logic. In institution theory, we typically use extensions (signature morphisms) obtained from variables instead of introducing…

We present a formalization of quasi-compact and quasi-separated schemes (qcqs-schemes) in the Cubical Agda proof assistant. We follow Grothendieck's functor of points approach, which defines schemes, the quintessential notion of modern…

We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We…

It is discussed a practical possibility of a provable programming of mathematics basing on intuitionism and the dependent types feature of a programming language.The principles of constructive mathematics and provable programming are…

In this paper a constructive formalization of quantifier elimination is presented, based on a classical formalization by Tobias Nipkow. The formalization is implemented and verified in the programming language/proof assistant Agda. It is…

Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…

We systematically develop the theory of definable functors between compactly generated triangulated categories. Such functors preserve pure triangles, pure injective objects, and definable subcategories, and as such appear in a wide range…

In previous work ("From signatures to monads in UniMath"), we described a category-theoretic construction of abstract syntax from a signature, mechanized in the UniMath library based on the Coq proof assistant. In the present work, we…

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal…

We present a method for generating possible proofs of a query with respect to a given Answer Set Programming (ASP) rule set using an abductive process where the space of abducibles is automatically constructed just from the input rules…