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The need for formal definition of the very basis of mathematics arose in the last century. The scale and complexity of mathematics, along with discovered paradoxes, revealed the danger of accumulating errors across theories. Although,…
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…
This extended abstract reports on current progress of SMTCoq, a communication tool between the Coq proof assistant and external SAT and SMT solvers. Based on a checker for generic first-order certificates implemented and proved correct in…
The interoperability of proof assistants and the integration of their libraries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof…
We present a formalization of convex polyhedra in the proof assistant Coq. The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination. This allows us to define…
This work is a part of an ongoing effort to prove the correctness of invertibility conditions for the theory of fixed-width bit-vectors, which are used to solve quantified bit-vector formulas in the Satisfiability Modulo Theories (SMT)…
We propose a type-theoretic framework for describing and proving properties of quantum computations, in particular those presented as quantum circuits. Our proposal is based on an observation that, in the polymorphic type system of Coq,…
Reasoning over knowledge graphs is traditionally built upon a hierarchy of languages in the Semantic Web Stack. Starting from the Resource Description Framework (RDF) for knowledge graphs, more advanced constructs have been introduced…
We introduce functorial language models: a principled way to compute probability distributions over word sequences given a monoidal functor from grammar to meaning. This yields a method for training categorical compositional distributional…
Instruction sets, from families like x86 and ARM, are at the center of many ambitious formal-methods projects. Many verification, synthesis, programming, and debugging tools rely on formal semantics of instruction sets, but different tools…
The Agora system is a prototypical Wiki for formal mathematics: a web-based system for collaborating on formal mathematics, intended to support informal documentation of formal developments. This system requires a reusable proof editor…
Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in…
We present a new way of embedding functional languages into the Coq proof assistant by using meta-programming. This allows us to develop the meta-theory of the language using the deep embedding and provides a convenient way for reasoning…
We present formalized proofs verifying that the first-order unification algorithm defined over lists of satisfiable constraints generates a most general unifier (MGU), which also happens to be idempotent. All of our proofs have been…
Good tools can bring mechanical verification to programs written in mainstream functional languages. We use hs-to-coq to translate significant portions of Haskell's containers library into Coq, and verify it against specifications that we…
We give an algebraic characterization of the syntax and semantics of a class of languages with variable binding. We introduce a notion of 2-signature: such a signature specifies not only the terms of a language, but also reduction rules on…
In this paper, we revisit Moggi's celebrated calculus of computational effects from the perspective of logic of monoidal action (actegory). Our development takes the following steps. Firstly, we perform proof-theoretic reconstruction of…
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler…
This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization…