We present an Isabelle/HOL formalisation of the foundations of analytic number theory related to modular forms. We begin by refactoring and extending the existing library on elliptic functions, adding the theorem that every elliptic function can be written in terms of the Weierstraß elliptic function ℘ and the addition theorem for ℘, which links complex lattices to elliptic curves. Next, we develop an extensive library on the Jacobi theta functions, including well-known results such as the Jacobi triple product, the Pentagonal Number Theorem, and the Rogers-Ramanujan identities. Finally, we apply this library to the study of the Dedekind η function and "forbidden" Eisenstein series G₂. In all of this, we aim for short and clean proofs, building a library of reusable lemmas.

Real root isolation is a fundamental subroutine in computer algebra, with applications ranging from algebraic number arithmetic to solving polynomial systems. Modern implementations typically employ subdivision methods based on root counting via Descartes' rule of signs. In contrast, most existing formally verified root isolation procedures rely on Sturm's theorem for root counting, leading to a noticeable gap between practical implementations and formally verified approaches. We take an initial step towards efficient verified real root isolation by formally verifying two simple algorithms based on Descartes' rule of signs: a classical bisection procedure and a Newton-accelerated variant. In this paper, we describe the algorithms, present formal proofs of termination, soundness, and completeness, and discuss our code-generation efforts. Brief experiments show promising performance improvements over existing formally verified algorithms in Isabelle/HOL.

Large language models (LLMs) present a promising pathway for advancing the domain of formal theorem proving. In this paper, we aim to improve the performance of LLMs in formal theorem proving by thoroughly examining the structure and organization of demonstrative in-context examples. We introduce a subgoal-based demonstration learning framework, specifically designed to enhance the efficiency of proof search in LLMs. First, drawing upon the insights of subgoal learning from reinforcement learning and robotics, we propose the construction of distinct subgoals for each demonstration example and refine these subgoals in accordance with the pertinent theories of subgoal learning. Second, we build upon recent advances in diffusion models to predict the optimal organization, simultaneously addressing two intricate issues that persist within the domain of demonstration organization: subset selection and order determination. Our integration of subgoal-based learning has notably increased proof accuracy from 38.9% to 44.1% on the miniF2F benchmark. Furthermore, the adoption of diffusion models for demonstration organization can lead to an additional enhancement in accuracy to 45.5%, or a $5\times$ improvement in sampling efficiency compared to previously established methods.

In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.

Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on the classical Sturm theorem, which only counts distinct roots.

We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates, separating solving from verifying: efficient external tools perform expensive real algebraic computations, producing evidence that is formally checked within Isabelle’s logic. This allows us to exploit highly-tuned computer algebra systems like Mathematica to guide our procedure without impacting the correctness of its results. We present experiments demonstrating the efficacy of this approach, in many cases yielding orders of magnitude improvements over previous methods.

This paper presents a construction of the real algebraic numbers with executable arithmetic operations in Isabelle/HOL. Instead of verified resultants, arithmetic operations on real algebraic numbers are based on a decision procedure to decide the sign of a bivariate polynomial (with rational coefficients) at a real algebraic point. The modular design allows the safe use of fast external code. This work can be the basis for decision procedures that rely on real algebraic numbers.