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In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and…
We propose a new framework for integrating quantifiers with other logical connectives in a higher-categorical setting. Our method systematically incorporates key coherence conditions-including those akin to the Beck-Chevalley property-and…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
Proof assistants like Coq are increasingly popular to help mathematicians carry out proofs of the results they conjecture. However, formal proofs remain highly technical and are especially difficult to reuse. In this paper, we present a…
The well-quasi-orders (WQO) play an important role in various fields such as Computer Science, Logic or Graph Theory. Since the class of WQOs lacks closure under some important operations, the proof that a certain quasi-order is WQO…
We introduce a library which provides an abstract data type of environments, as a functor parameterized by a module defining variables, and a function which builds environments for such variables with any Type of type. Usual operations over…
Common Knowledge Logic is meant to describe situations of the real world where a group of agents is involved. These agents share knowledge and make strong statements on the knowledge of the other agents (the so called \emph{common…
With the success of deep learning techniques in a broad range of application domains, many deep learning software frameworks have been developed and are being updated frequently to adapt to new hardware features and software libraries,…
This paper develops a systematic framework for integrating local categories that model logical connectives using higher category theory. By extending these local categories into a unified two-category enriched with natural isomorphisms, the…
Quantum Information Processing, which is an exciting area of research at the intersection of physics and computer science, has great potential for influencing the future development of information processing systems. The building of…
In the context of interactive theorem provers based on a dependent type theory, automation tactics (dedicated decision procedures, call of automated solvers, ...) are often limited to goals which are exactly in some expected logical…
In mathematics, it is common practice to have several constructions for the same objects. Mathematicians will identify them modulo isomorphism and will not worry later on which construction they use, as theorems proved for one construction…
Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws,…
We present ViCAR, a library for working with monoidal categories in the Coq proof assistant. ViCAR provides definitions for categorical structures that users can instantiate with their own verification projects. Upon verifying relevant…
Distributed word representations have been demonstrated to be effective in capturing semantic and syntactic regularities. Unsupervised representation learning from large unlabeled corpora can learn similar representations for those words…
In this paper, we facilitate the reasoning about impure programming languages, by annotating terms with `decorations' that describe what computational (side) effect evaluation of a term may involve. In a point-free categorical…
Libraries of formalized mathematics use a possibly broad range of different representations for a same mathematical concept. Yet light to major manual input from users remains most often required for obtaining the corresponding variants of…
In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct…
Traditional category theory is typically based on set-theoretic principles and ideas, which are often non-constructive. An alternative approach to formalizing category theory is to use E-category theory, where hom sets become setoids. Our…
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic…