Related papers: The continuous functional calculus in Lean
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in…
Computability theory is traditionally conceived as the theoretical basis of informatics. Nevertheless, numerous proposals transcend computability theory, in particular by emphasizing interaction of modules, or components, parts,…
Distribution theory is a cornerstone of the theory of partial differential equations. We report on the progress of formalizing the theory of tempered distributions in the interactive proof assistant Lean, which is the first formalization in…
We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions in the calculus of inductive constructions. This widens the class of functions that can be modeled in type-theory based theorem proving tool to…
The main result of this paper is to prove the existence of a finite basis in the description logic ${\cal ALC}$. We show that the set of General Concept Inclusions (GCIs) holding in a finite model has always a finite basis, i.e. these GCIs…
In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in ${\rm L}^1$ of a certain sequence of step functions, and we prove…
We introduce a simple extension of the $\lambda$-calculus with pairs---called the distributive $\lambda$-calculus---obtained by adding a computational interpretation of the valid distributivity isomorphism $A \Rightarrow (B\wedge C)\ \…
We survey the use of extra-set-theoretic hypotheses, mainly the continuum hypothesis, in the C*-algebra literature. The Calkin algebra emerges as a basic object of interest.
We prove new results on generalized derivations on C$^*$-algebras. By considering the triple product $\{a,b,c\} =2^{-1} (a b^* c + c b^* a)$, we introduce the study of linear maps which are triple derivations or triple homomorphisms at a…
This report presents a formalization of May's theorem in the proof assistant Coq. It describes how the theorem statement is first translated into Coq definitions, and how it is subsequently proved. Various aspects of the proof and related…
When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow…
A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained. Various classical examples of this theorem, such as the Green's and…
The need for rigorous process composition is encountered in many situations pertaining to the development and analysis of complex systems. We discuss the use of Classical Linear Logic (CLL) for correct-by-construction resource-based process…
The work is devoted to the construction of a new type of intervals -- functional intervals. These intervals are built on the idea of expanding boundaries from numbers to functions. Functional intervals have shown themselves to be promising…
The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-accessible database cataloging interesting integer sequences and associated theorems. With more than 12,000 citations, the OEIS is one of the most highly cited resources in all…
We present an introduction to modern continuous model theory with an emphasis on its interactions with topics covered in this volume such as $C^*$-algebras and von Neumann algebras. The role of ultraproducts is highlighted and expositions…
Logical relations are one of the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…