Related papers: Certifying rings of integers in number fields
In this paper we present a new "external checker" for the Lean theorem prover, written in Lean itself. This is the first complete typechecker for Lean 4 other than the reference implementation in C++ used by Lean itself, and our new checker…
The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
This paper presents a formalization of the theory of amicable numbers in the Lean~4 proof assistant. Two positive integers $m$ and $n$ are called an amicable pair if the sum of proper divisors of $m$ equals $n$ and the sum of proper…
The certification of intrinsic randomness is foundational to quantum information theory and central in many practical applications thereof, such as in the generation of unquestionably random numbers and in cryptographic protocols.…
We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…
{log} (read 'setlog') was born as a Constraint Logic Programming (CLP) language where sets and binary relations are first-class citizens, thus fostering set programming. Internally, {log} is a constraint satisfiability solver implementing…
Verifying mathematical proofs is difficult, but can be automated with the assistance of a computer. Autoformalization is the task of automatically translating natural language mathematics into a formal language that can be verified by a…
The present paper introduces a practical protocol for provably secure, outsourced computation. Our protocol minimizes overhead for verification by requiring solutions to withstand an interactive game between a prover and challenger. For…
The physics community relies on index notation to effectively manipulate types of tensors. This paper introduces the first formally verified implementation of index notation in the interactive theorem prover Lean 4. By integrating index…
Insurance claims processing involves multi-domain entities and multi-source data, along with a number of human-agent interactions. Use of Blockchain technology-based platform can significantly improve scalability and response time for…
A compact metric ring containing the set of positive integers as dense subset is studied. It is proven that tis ring is isomorph with a ring of reminder classes of ring of polyadic integers.
The power of quantum computers relies on the capability of their components to maintain faithfully and process accurately quantum information. Since this property eludes classical certification methods, fundamentally new protocols are…
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every…
In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
We give a new theoretical solution to a leading-edge experimental challenge, namely to the verification of quantum computations in the regime of high computational complexity. Our results are given in the language of quantum interactive…
Software engineering concepts and processes are worthy of formal study; and yet we seldom formalize them. This "research ideas" article explores what a theory of software engineering could and should look like. Software engineering research…
This paper aims to develop a verification method for procedural programs via a transformation into Logically Constrained Term Rewriting Systems (LCTRSs). To this end, we extend transformation methods based on integer TRSs to handle…
We present necessary and sufficient conditions for the termination of linear homogeneous programs. We also develop a complete method to check termination for this class of programs. Our complete characterization of termination for such…