Reseña: Windmills of the minds: an algorithm for Fermat’s two squares theorem

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en HOL4 sobre teoría de números titulado Windmills of the minds: an algorithm for Fermat’s two squares theorem.

Sus autor es Hing Lun Chan (de la Australian National University en Canberra, Australia).

Su resumen es

The two squares theorem of Fermat is a gem in number theory, with a spectacular one-sentence “proof from the Book“. Here is a formalisation of this proof, with an interpretation using windmill patterns. The theory behind involves involutions on a finite set, especially the parity of the number of fixed points in the involutions. Starting as an existence proof that is non-constructive, there is an ingenious way to turn it into a constructive one. This gives an algorithm to compute the two squares by iterating the two involutions alternatively from a known fixed point.

El trabajo se presentará en el Certified Programs and Proofs (CPP) 2022 el 18 de enero de 2022.

El código de las correspondientes teorías se encuentra aquí.

Reseña: Formalizing ordinal partition relations using Isabelle/HOL

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Isabelle/HOL sobre combinatoria titulado Formalizing ordinal partition relations using Isabelle/HOL.

Sus autores son

Su resumen es

This is an overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős–Milner, Specker, Larson and Nash-Williams, leading to Larson’s proof of the unpublished result by E.C. Milner asserting that for all m ∈ ℕ, ω^ω → (ω^ω,m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalization process. This project is also a demonstration of working with Zermelo–Fraenkel set theory in higher-order logic.

El trabajo se ha publicado en Experimental Mathematics.

El código de las correspondientes teorías se encuentra aquí.

Reseña: A machine-checked direct proof of the Steiner-Lehmus theorem

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Nuprl sobre geometría titulado A machine-checked direct proof of the Steiner-Lehmus theorem.

Su autora es Ariel Kellison (de Cornell University).

Su resumen es

A direct proof of the Steiner-Lehmus theorem has eluded geometers for over 170 years. The challenge has been that a proof is only considered direct if it does not rely on reductio ad absurdum. Thus, any proof that claims to be direct must show, going back to the axioms, that all of the auxiliary theorems used are also proved directly. In this paper, we give a proof of the Steiner-Lehmus theorem that is guaranteed to be direct. The evidence for this claim is derived from our methodology: we have formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and have built the proof of the Steiner-Lehmus theorem on this constructive foundation.

El trabajo se presentará en el Certified Programs and Proofs (CPP) 2022 el 18 de enero de 2022.

El código de las correspondientes teorías se encuentra aquí.

Reseña: Completeness theorems for first-order logic analysed in constructive type theory

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Coq sobre lógica titulado Completeness theorems for first-order logic analysed in constructive type theory.

Sus autores son

Su resumen es

We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov’s Principle and Weak König’s Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.

El trabajo se ha presentado en el Logical Foundations of Computer Science (LFCS 2020) y publicado en el Journal of Logic and Computation.

El código de las correspondientes teorías se encuentra aquí.

Reseña: Formalising Lie algebras in Lean

PorJosé A. Alonso

Se ha publicado un artículo de razonamiento formalizado en Lean sobre álgebras de Lie titulado Formalising Lie algebras.

Su autor es Oliver Nash (Imperial College in London, U.K.).

Su resumen es

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 is assumed; the intention is that the overall themes will be accessible even to readers unfamiliar with Lie theory.

Particular attention is paid to the construction of the classical and exceptional Lie algebras. Thanks to these constructions, it is possible to state the classification theorem for finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero.

In addition to the focus on Lie theory, we also aim to highlight the unity of Mathlib. To this end, we include examples of achievements made possible only by leaning on several branches of the library simultaneously.

El código de las correspondientes teorías se encuentra aquí.

El trabajo se presentará en el Certified Programs and Proofs (CPP) 2022 el 18 de enero de 2022.