This is a brief description of a project that has already autoformalized a large portion of the general topology from the Munkres textbook (which has in total 241 pages in 7 chapters and 39 sections). The project has been running since November 21, 2025 and has as of January 4, 2026, produced 160k lines of formalized topology. Most of it (about 130k lines) have been done in two weeks, from December 22 to January 4, for an LLM subscription cost of about $100. This includes a 3k-line proof of Urysohn’s lemma, a 2k-line proof of Urysohn’s Metrization theorem, over 10k-line proof of the Tietze extension theorem, and many more (in total over 1.5k lemmas/theorems). The approach is quite simple and cheap: build a long-running feedback loop between an LLM and a reasonably fast proof checker equipped with a core foundational library. The LLM is now instantiated as ChatGPT (mostly 5.2) or Claude Sonnet (4.5) run through the respective Codex or Claude Code command line interfaces. The proof checker is Chad Brown’s higher-order set theory system Megalodon, and the core library is Brown’s formalization of basic set theory and surreal numbers (including reals, etc). The rest is some prompt engineering and technical choices which we describe here. Based on the fast progress, low cost, virtually unknown ITP/library, and the simple setup available to everyone, we believe that (auto)formalization may become quite easy and ubiquitous in 2026, regardless of which proof assistant is used.

In this work we considerably improve the real-time performance of state-of-the-art SMT solving on first-order quantified problems by efficient machine learning guidance of quantifier selection. Quantifiers represent a significant challenge for SMT and are technically a source of undecidability. In our approach, we train an efficient machine learning model that informs the solver which quantifiers should be instantiated and which not. Each quantifier may be instantiated multiple times and the set of the currently active quantifiers changes as the solving progresses. Therefore, we invoke the ML predictor many times, during the whole run of the solver. To make this efficient, we use fast ML models based on gradient boosted decision trees. We integrate our approach into the state-of-the-art cvc5 SMT solver and show a considerable increase of the system’s holdout-set performance after training it on large sets of first-order problems. The method is tested in several ways, using both single-strategy and portfolio approaches. The evaluation is done on two large formal verification corpora: first-order problems created from the Mizar Mathematical Library, and first-order problems created from the HOL4 standard library.

Abstract We develop a self-learning approach for conjecturing of induction predicates on a dataset of 14,005 problems derived from the OEIS. These problems are hard for today’s SMT and ATP systems because they require a combination of inductive and arithmetical reasoning. Starting from scratch, our approach consists of a feedback loop that iterates between (i) training a neural translator to learn the correspondence between the problems solved so far and the induction predicates useful for them, (ii) using the trained neural system to generate many new induction predicates for the problems, (iii) fast runs of the Z3 prover attempting to prove the problems using the generated predicates, (iv) using heuristics such as predicate size and solution speed on the proved problems to choose the best predicates for the next iteration of training. The algorithm discovers on its own many interesting induction predicates, ultimately solving 3,590 problems, compared to 835 problems solved by CVC5, Vampire or Z3 in 60 s.

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In this work we considerably improve the state-of-the-art SMT solving on first-order quantified problems by efficient machine learning guidance of quantifier selection. Quantifiers represent a significant challenge for SMT and are technically a source of undecidability. In our approach, we train an efficient machine learning model that informs the solver which quantifiers should be instantiated and which not. Each quantifier may be instantiated multiple times and the set of the active quantifiers changes as the solving progresses. Therefore, we invoke the ML predictor many times, during the whole run of the solver. To make this efficient, we use fast ML models based on gradient boosted decision trees. We integrate our approach into the state-of-the-art cvc5 SMT solver and show a considerable increase of the system’s holdout-set performance after training it on a large set of first-order problems collected from the Mizar Mathematical Library.

The cvc5 solver is today one of the strongest systems for solving first order problems with theories but also without them. In this work we equip its enumeration-based instan- tiation with a neural network that guides the choice of the quantified formulas and their instances. For that we develop a relatively fast graph neural network that repeatedly scores all available instantiation options with respect to the available formulas. The network runs directly on a CPU without the need for any special hardware. We train the neural guidance on a large set of proofs generated by the e-matching instantiation strategy and evaluate its performance on a set of previously unseen problems.

We present a self-learning approach for synthesizing programs from integer sequences. Our method relies on a tree search guided by a learned policy. Our system is tested on the On-Line Encyclopedia of Integer Sequences. There, it discovers, on its own, solutions for 27987 sequences starting from basic operators and without human-written training examples.

As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60% of the Mizar theorems in the hammer setting. We also automatically prove 75% of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically.

In this work we extend an instantiation-based theorem prover iProver with machine learning (ML) guidance based on graph neural networks. For this we implement an interactive mode in iProver, which allows communication with an external agent via network sockets. The external (ML-based) agent guides the proof search by scoring generated clauses in the given clause loop. Our evaluation on a large set of Mizar problems shows that the ML guidance outperforms iProver’s standard human-programmed priority queues, solving more than twice as many problems in the same time. To our knowledge, this is the first time the performance of a state-of-the-art instantiation-based system is doubled by ML guidance.

We present a benchmark of 29687 problems derived from the On-Line Encyclopedia of Integer Sequences (OEIS). Each problem expresses the equivalence of two syntactically different programs generating the same OEIS sequence. Such programs were conjectured by a learning-guided synthesis system using a language with looping operators. The operators implement recursion, and thus many of the proofs require induction on natural numbers. The benchmark contains problems of varying difficulty from a wide area of mathematical domains. We believe that these characteristics will make it an effective judge for the progress of inductive theorem provers in this domain for years to come.

We significantly improve the performance of the E automated theorem prover on the Isabelle Sledgehammer problems by combining learning and theorem proving in several ways. In particular, we develop targeted versions of the ENIGMA guidance for the Isabelle problems, targeted versions of neural premise selection, and targeted strategies for E. The methods are trained in several iterations over hundreds of thousands untyped and typed first-order problems extracted from Isabelle. Our final best single-strategy ENIGMA and premise selection system improves the best previous version of E by 25.3% in 15 seconds, outperforming also all other previous ATP and SMT systems.

Connection calculi allow for very compact implementations of goal-directed proof search. We give an overview of our work related to connection tableaux calculi: first, we show optimised functional implementations of connection tableaux proof search, including a consistent Skolemisation procedure for machine learning. Then, we show two guidance methods based on machine learning, namely reordering of proof steps with Naive Bayesian probabilities, and expansion of a proof search tree with Monte Carlo Tree Search.

In this paper we share several experiments trying to automatically translate informal mathematics into formal mathematics. In our context informal mathematics refers to human-written mathematical sentences in the LaTeX format; and formal mathematics refers to statements in the Mizar language. We conducted our experiments against three established neural network-based machine translation models that are known to deliver competitive results on translating between natural languages. To train these models we also prepared four informal-to-formal datasets. We compare and analyze our results according to whether the model is supervised or unsupervised. In order to augment the data available for auto-formalization and improve the results, we develop a custom type-elaboration mechanism and integrate it in the supervised translation.

Automated reasoning and theorem proving have recently become major challenges for machine learning. In other domains, representations that are able to abstract over unimportant transformations, such as abstraction over translations and rotations in vision, are becoming more common. Standard methods of embedding mathematical formulas for learning theorem proving are however yet unable to handle many important transformations. In particular, embedding previously unseen labels, that often arise in definitional encodings and in Skolemization, has been very weak so far. Similar problems appear when transferring knowledge between known symbols. We propose a novel encoding of formulas that extends existing graph neural network models. This encoding represents symbols only by nodes in the graph, without giving the network any knowledge of the original labels. We provide additional links between such nodes that allow the network to recover the meaning and therefore correctly embed such nodes irrespective of the given labels. We test the proposed encoding in an automated theorem prover based on the tableaux connection calculus, and show that it improves on the best characterizations used so far. The encoding is further evaluated on the premise selection task and a newly introduced symbol guessing task, and shown to correctly predict 65% of the symbol names.

We present a system that utilizes machine learning for tactic proof search in the Coq Proof Assistant. In a similar vein as the TacticToe project for HOL4, our system predicts appropriate tactics and finds proofs in the form of tactic scripts. To do this, it learns from previous tactic scripts and how they are applied to proof states. The performance of the system is evaluated on the Coq Standard Library. Currently, our predictor can identify the correct tactic to be applied to a proof state 23.4% of the time. Our proof searcher can fully automatically prove 39.3% of the lemmas. When combined with the CoqHammer system, the two systems together prove 56.7% of the library’s lemmas.

In this work we develop a new learning-based method for selecting facts (premises) when proving new goals over large formal libraries. Unlike previous methods that choose sets of facts independently of each other by their rank, the new method uses the notion of state that is updated each time a choice of a fact is made. Our stateful architecture is based on recurrent neural networks which have been recently very successful in stateful tasks such as language translation. The new method is combined with data augmentation techniques, evaluated in several ways on a standard large-theory benchmark and compared to state-of-the-art premise approach based on gradient boosted trees. It is shown to perform significantly better and to solve many new problems.

We describe a very large improvement of existing hammer-style proof automation over large ITP libraries by combining learning and theorem proving. In particular, we have integrated state-of-the-art machine learners into the E automated theorem prover, and developed methods that allow learning and efficient internal guidance of E over the whole Mizar library. The resulting trained system improves the real-time performance of E on the Mizar library by 70% in a single-strategy setting.

We introduce a theorem proving algorithm that uses practically no domain heuristics for guiding its connection-style proof search. Instead, it runs many Monte-Carlo simulations guided by reinforcement learning from previous proof attempts. We produce several versions of the prover, parameterized by different learning and guiding algorithms. The strongest version of the system is trained on a large corpus of mathematical problems and evaluated on previously unseen problems. The trained system solves within the same number of inferences over 40% more problems than a baseline prover, which is an unusually high improvement in this hard AI domain. To our knowledge this is the first time reinforcement learning has been convincingly applied to solving general mathematical problems on a large scale.

Inventing targeted proof search strategies for specific problem sets is a difficult task. State-of-the-art automated theorem provers (ATPs) such as E allow a large number of user-specified proof search strategies described in a rich domain specific language. Several machine learning methods that invent strategies automatically for ATPs were proposed previously. One of them is the Blind Strategymaker (BliStr), a system for automated invention of ATP strategies.

Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such “hammer” techniques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences combined with an optimized small-scale hammering approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39% of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32% in the same amount of time.

In this paper we explore the possibility of emulating the Mizar environment as close as possible inside the Isabelle logical framework. We introduce adaptations to the Isabelle/FOL object logic that correspond to the logic of Mizar, as well as Isar inner syntax notations that correspond to these of the Mizar language. We show how Isabelle types can be used to differentiate between the syntactic categories of the Mizar language, such as sets and Mizar types including modes and attributes, and show how they interact with the basic constructs of the Tarski-Grothendieck set theory. We discuss Mizar definitions and provide simple abbreviations that allow the introduction of Mizar predicates, functions, attributes and modes using the Isabelle/Pure language elements for introducing definitions and theorems. We finally consider the definite and indefinite description operators in Mizar and their use to introduce definitions by “means” and “equals”. We demonstrate the usability of the environment on a sample Mizar-style formalization, with cluster inferences and “by” steps performed manually.

We study the effectiveness of neural sequence models for premise selection in automated theorem proving, a key bottleneck for progress in formalized mathematics. We propose a two stage approach for this task that yields good results for the premise selection task on the Mizar corpus while avoiding the hand-engineered features of existing state-of-the-art models. To our knowledge, this is the first time deep learning has been applied theorem proving on a large scale.

In recent years, the Metis prover based on ordered paramodulation and model elimination has replaced the earlier built-in methods for general-purpose proof automation in HOL4 and Isabelle/HOL. In the annual CASC competition, the leanCoP system based on connection tableaux has however performed better than Metis. In this paper we show how the leanCoP's core algorithm can be implemented inside HOL Light. leanCoP's flagship feature, namely its minimalistic core, results in a very simple proof system. This plays a crucial role in extending the MESON proof reconstruction mechanism to connection tableaux proofs, providing an implementation of leanCoP that certifies its proofs. We discuss the differences between our direct implementation using an explicit Prolog stack,to the continuation passing implementation of MESON present in HOL Light and compare their performance on all core HOL Light goals. The resulting prover can be also used as a general purpose TPTP prover. We compare its performance against the resolution based Metis on TPTP and other interesting datasets.

As a present to Mizar on its 40th anniversary, we develop an AI/ATP system that in 30 seconds of real time on a 14-CPU machine automatically proves 40 % of the theorems in the latest official version of the Mizar Mathematical Library (MML). This is a considerable improvement over previous performance of large-theory AI/ATP methods measured on the whole MML. To achieve that, a large suite of AI/ATP methods is employed and further developed. We implement the most useful methods efficiently, to scale them to the 150000 formulas in MML. This reduces the training times over the corpus to 1–3 seconds, allowing a simple practical deployment of the methods in the online automated reasoning service for the Mizar users (Miz $\mathbb {A}\mathbb {R}$ ).

MaLeS is an automatic tuning framework for automated theorem provers. It provides solutions for both the strategy finding as well as the strategy scheduling problem. This paper describes the tool and the methods used in it, and evaluates its performance on three automated theorem provers: E, LEO-II and Satallax. On a representative subset of the TPTP library a MaLeS-tuned prover solves on average 8.67 % more problems than the prover with its default settings.

Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of the lemmas in the HOL Light and Flyspeck libraries, adding up to millions of the best lemmas to the pool of statements that can be re-used in later proofs. We show that in combination with learning-based relevance filtering, such methods significantly strengthen automated theorem proving of new conjectures over large formal mathematical libraries such as Flyspeck.

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the Flyspeck proofs, producing an AI system capable of proving a wide range of mathematical conjectures automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39 % of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided.

Learning reasoning techniques from previous knowledge is a largely underdeveloped area of automated reasoning. As large bodies of formal knowledge are becoming available to automated reasoners, state-of-the-art machine learning methods can provide powerful heuristics for problem-specific detection of relevant knowledge contained in the libraries. In this paper we develop a semantic graph kernel suitable for learning in structured mathematical domains. Our kernel incorporates contextual information about the features and unlike “random walk”-based graph kernels it is also applicable to sparse graphs. We evaluate the proposed semantic graph kernel on a subset of the large formal Mizar mathematical library. Our empirical evaluation demonstrates that graph kernels in general are particularly suitable for the automated reasoning domain and that in many cases our semantic graph kernel leads to improvement in performance compared to linear, Gaussian, latent semantic, and geometric graph kernels.

The tremendous growth rates of the Internet as well as the area of mobile communications give rise to the chance that the mobile Internet is most promising by combining both the Internet and mobile communications. These prospects are the motivation for the European research project BRAIN (Broadband Radio Access for IP-based Networks), which is developing an open architecture for a broadband wireless mobile access network offering an integrated communication platform across heterogeneous networks and, thus, goes beyond current third generation systems and towards the mobile Internet. The project covers three major technical areas: support of seamless service provision in a mobile environment; the design of an IP-based access network that will support non-cellular technologies such as wireless LANs; and requirements of a broadband air interface suitable for hot spots. BRAIN is going to integrate HIPERLAN/2 with UMTS by means of an IP access network. The work is guided by a user-centric top-down approach ensuring that user functionality is the key driver of the project. This article will focus on that part of the BRAIN work which specifies the main interfaces of the BRAIN architecture and deals with aspects related to the support of Quality of Service and mobility. Copyright © 2001 John Wiley & Sons, Ltd.

Second generation digital mobile radio systems have been very successful for voice communications and are now beginning to offer support for data services. Third generation mobile radio systems are currently being standardized worldwide to be initially deployed starting in 2001 providing support for multimedia applications with a flexible air interface and higher bandwidths. Wireless LAN technology is complementary to 3G systems and could be used to provide high bandwidth hot spot coverage, for example in railway stations and offices, for the emerging video and broadband services that will begin to emerge on fixed networks. The IST BRAIN project, which is partly funded by the European Commission, has been formed to solve the problems of providing seamless service for broadband users in these hot spots. This paper describes these problems in greater detail as well as outlining how the BRAIN project is tackling them.