José A. Alonso

Lean Metaprogramming Etudes: Execution is Elaboration

I think I’ve kind of been hitting a watershed of understanding of lean metaprogramming.

Calculemus

Reto 6: Si aₙ y cₙ convergen a L y aₙ ≤ bₙ ≤ cₙ para todo n, entonces

El reto de esta semana consiste en demostrar en Lean 4 el teorema del emparedado; es decir que si aₙ y cₙ convergen a L y aₙ ≤ bₙ ≤ cₙ para todo n, entonces bₙ converge a L. Para ello, completar la si

Proofs and Prompts

A beginning for mathematics

“Three years ago, AI systems could not reliably add two numbers. A year ago, internal models at OpenAI and DeepMind received the equivalent of a gold-medal score on the IMO. Now, these systems are …

Happy, those able to know the causes of things

Calculemus

Reto 8: Sucesiones con infinitos términos grandes no convergen a límit

El reto de esta semana consiste en demostrar en Lean 4 que que una sucesión que posee infinitos términos con valor absoluto superior a 10 no puede converger a un límite cuyo valor absoluto sea menor q

Calculemus

Reto 7: La composición de funciones inyectivas es inyectiva

El reto de esta semana consiste en demostrar en Lean 4 que la composición de funciones inyectivas es inyectiva. Para ello, completar la siguiente teoría de Lean 4: import Mathlib.Tactic open Function

arXiv.org

Mechanizing Gödel's Incompleteness Theorems and Provability Logic

We mechanized proof of Gödel's first and second incompleteness theorems, Solovay's arithmetical completeness theorem of \mathbf{GL}, and related results in the Lean 4 theorem prover.

arXiv.org

A machine-checked proof of the Dong-Yang classification of optimal (n,4) binary codes for BSCs

We present a machine-checked Lean~4 formalization of Dong and Yang's classification of optimal finite-length $(n,4)$ binary block codes for binary symmetric channels. The formalization was developed mainly by feeding the paper's proofs to an AI tool. To establish correctness, the authors verified the main theorem statements in Lean and the accepted axioms. This note discusses the corrections and simplifications made to the AI-generated formalization, and records discrepancies found in the paper

arXiv.org

Stable Singularity of the Euler Equations on $\mathbb{R}^3$

We provide evidence of a stable finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile suggests that linear damping can be established throughout the domain, providing strong evidence for the overall stab

arXiv.org

Henstock--Kurzweil Gauge Integral in the Non--Gaussian Regime: A Machine--Verified Construction

We develop a machine-checked construction of non-Gaussian functional integrals using the Henstock--Kurzweil gauge integral and Chernoff product approximations. The central object is a finite family of bosonic modes with action S(phi) = (1/2) phi^T A phi + lambda * sum_i phi_i^4, where A is positive definite. We prove that the one-mode integral I(omega, j, lambda) is finite, strictly positive, monotone and infinitely differentiable in the coupling lambda on [0, infinity). Its derivatives are give

geoconuk.github.io

MiscMath.Analysis.KolmogorovArnold

arXiv.org

A Four-Valued Graph Model for Conflict Resolution: Core Framework and a Machine-Checked Formalization in Lean 4

This note consolidates the core of the Quasi-Closed World Graph Model for Conflict Resolution (QCW-GMCR), which extends the standard Graph Model for Conflict Resolution with Belnap's four-valued logic to represent option-level epistemic ambiguity, and pairs the framework with a machine-checked Lean 4 formalization. QCW-GMCR combines: (1) FOUR-valued option assignments with compositional propagation to state-level feasibility; (2) graded reachability (definite, credible, possible) based on an FDE

Introduction - Metaprogramming in Lean 4

Vibe Coding Reconsidered

A blog about computers, functional languages, Lisp, and Scheme.

The fall of the theorem economy

How AI could destroy mathematics and barely touch it

La Ciencia de la Mula Francis

Un error en el núcleo de Lean aprovechado por una IA para refutar la conjetura de Collatz - La Ciencia de la Mula Francis

Las IA generativas encuentran errores de código (bugs) donde nadie los espera. Incluso en el núcleo de Lean (el verificador automático de demostraciones matemáticas más famoso). El 25 de julio […]

The Haskell Programming Language's blog

Quick tips for fast iteration in Haskell | The Haskell Programming Language's blog

Quick tips about tools and techniques for fast iteration when developing Haskell

Codeberg.org

real-world-haskell

An updated version of the Real World Haskell book.

arXiv.org

Learned Interventions in Lean 4 grind

Lean~4's \grind{} tactic combines congruence closure, \ematch{}ing, and case-splitting into a single automated solver, and like any such solver, it relies on hand-tuned heuristics to decide what to instantiate and where to case-split. These heuristics are tempting targets for learning, but there is a catch: because \grind{}'s search is non-monotone, a learned heuristic that helps one proof can break another, and an always-on replacement usually nets out near zero. We avoid this by invoking a lea

arXiv.org

The Ramanujan Challenge For AI

To help evaluate the mathematical skills of current AI systems, we present a set of formulas for fundamental mathematical constants. These problems are attractive for AI evaluation because they are concrete and can be checked numerically to arbitrary precision, yet proving them may require non-obvious mathematics. Mathematical constants such as $π$, $e$, Catalan's constant, and special values of the Riemann zeta function have fascinated mathematicians for centuries. The search for formulas evalu

Lean Playground

Try out Lean in your browser with the Lean Playground: an interactive live editor for testing Lean code.

Existentials on a leash

Article on using linear types for “naked” existential type variables.

haskell.foundation

2026 Haskell Workshop Videos Now Online

Couldn't make it to Rapperswil, or want to revisit a talk? The recordings from this year's Haskell Implementors' Workshop and Haskell Ecosystem Workshop are now available online. Both workshops were held on the lakeside campus of the University of Applied Sciences of Eastern Switzerland (OST) in Rapperswil on June 4th and 5th, alongside ZuriHac, and hosted by the Haskell Foundation.

New falsify release

arXiv.org

Formalizing Flag Algebras in Lean

Razborov's flag algebra method is a powerful tool for proving asymptotic inequalities in extremal graph theory, often reducing the task to finding a finite certificate by semidefinite programming. We present a machine-checked formalization of the method for finite simple graphs, together with a certificate-to-proof compiler that turns externally generated certificate data into algebraic proofs checked by Lean. The formalization covers the foundations of the method: partially labeled graphs, thei

GitHub - PierreSenellart/descriptive-complexity: A Lean library for descriptive complexity: NP-completeness and the polynomial hierarchy by first-order reductions, stronger than polynomial-time (Karp) reductions. Machine-free Cook–Levin, all 21 Karp probl

A Lean library for descriptive complexity: NP-completeness and the polynomial hierarchy by first-order reductions, stronger than polynomial-time (Karp) reductions. Machine-free Cook–Levin, all 21 K...

arXiv.org

Formalizing Wu-Ritt Method in Lean 4

We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their terminatio

arXiv.org

A quadratic form generalization of rational dinv

We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is e

Vestigium

Readings shared September 14, 2026

The readings shared in Mastodon on 14 September 2026 are: A Lean 4 formalization of Scott's continuous lattices (1972). ~ Lars Warren Ericson. #LeanProver #ITP #AI4Math A counterexample to a problem

notxor.nueva-actitud.org

gptel: Emacs y la IA

GitHub - mbrcic/ai-safety-formalization-atlas: The open workbench for AI safety, made formal. Turn safety questions into machine-checked Lean proofs — a shared launchpad where researchers and AI agents build provable safety together.

The open workbench for AI safety, made formal. Turn safety questions into machine-checked Lean proofs — a shared launchpad where researchers and AI agents build provable safety together. - mbrcic/a...

Proofs and Prompts

When inventing is not enough

“We are mathematicians: we seek answers to problems that, at present, have no solution. To do so, we study the work of other colleagues, adapt it to the context we are interested in and, starting f…

Navier-Stokes and Lean

I was working on this week's post on Lean after reading Kevin Hartnett's book  The Proof in the Code: How a Truth Machine Is Transforming Ma...

arXiv.org

Monadic Second-Order Logic in HOL: Deep and Shallow with Automated Faithfulness (Extended Preprint)

In Isabelle/HOL, we apply the deep-and-shallow embedding methodology of our prior work to monadic second-order logic (MSO). Three embeddings are developed side by side: a deep embedding (an inductive datatype with an explicit satisfaction relation); a maximal-shallow embedding that translates the connectives and quantifiers directly into HOL, carrying the interpretation and both assignments explicitly; and a minimal-shallow embedding -- a locale that fixes those parameters, collapsing the formul