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Most ATP benchmarks embed the final answer within the formal statement -- a convention we call "Easy Mode" -- a design that simplifies the task relative to what human competitors face and may lead to optimistic estimates of model capability. We call the stricter, more realistic setting "Hard Mode": the system must independently discover the answer before constructing a formal proof. To enable Hard Mode research, we make two contributions. First, we release MiniF2F-Hard and FIMO-Hard, expert-reannotated Hard Mode variants of two widely-used ATP benchmarks. Second, we introduce Discover And Prove (DAP), an agentic framework that uses LLM natural-language reasoning with explicit self-reflection to discover answers, then rewrites Hard Mode statements into Easy Mode ones for existing ATP provers. DAP sets the state of the art: on CombiBench it raises solved problems from 7 (previous SOTA, Pass@16) to 10; on PutnamBench it is the first system to formally prove 36 theorems in Hard Mode -- while simultaneously revealing that state-of-the-art LLMs exceed 80% answer accuracy on the same problems where formal provers manage under 10%, exposing a substantial gap that Hard Mode benchmarks are u

Cyber-physical systems are inherently complex due to their connection between software and the physical world. Iterative design reduces their complexity, but increases the need to repeatedly recheck their safety in full after every change. We introduce the refactoring-as-propositions principle in which refactorings are represented as propositions along with a method for proving that system refactorings preserve their required properties by transferring the proof along the respective modification. It is based on differential refinement logic (dRL), with which one can simultaneously and rigorously refer to properties of the systems and the relation between a refactored system and its original version. Refinements represent a uniform way of expressing different types of hybrid system refactorings, including those that introduce auxiliary variables. Furthermore, we show how these refactorings can be proved automatically, and/or reduce to a modular proof solely about the local change rather than about the whole system.

How can we trust the correctness of a learned model on a particular input of interest? Model accuracy is typically measured on average over a distribution of inputs, giving no guarantee for any fixed input. This paper proposes a theoretically-founded solution to this problem: to train Self-Proving models that prove the correctness of their output to a verification algorithm $V$ via an Interactive Proof. Self-Proving models satisfy that, with high probability over an input sampled from a given distribution, the model generates a correct output and successfully proves its correctness to $V$. The soundness property of $V$ guarantees that, for every input, no model can convince $V$ of the correctness of an incorrect output. Thus, a Self-Proving model proves correctness of most of its outputs, while all incorrect outputs (of any model) are detected by $V$. We devise and analyze two generic methods for learning Self-Proving models: Transcript Learning (TL) which relies on access to transcripts of accepting interactions, and Reinforcement Learning from Verifier Feedback (RLVF) which trains a model by emulating interactions with the verifier.

Recently Shekhar Suman [arXiv: 2407.07121v6 [math.GM] 3 Aug 2024] made an attempt to prove the irrationality of $ζ(5)$. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

The general-purpose interactive theorem-proving assistant called Prove-It was used to verify the Quantum Phase Estimation (QPE) algorithm, specifically claims about its outcome probabilities. Prove-It is unique in its ability to express sophisticated mathematical statements, including statements about quantum circuits, integrated firmly within its formal theorem-proving framework. We demonstrate our ability to follow a textbook proof to produce a formally certified proof, highlighting useful automation features to fill in obvious steps and make formal proving nearly as straightforward as informal theorem proving. Finally, we make comparisons with formal theorem-proving in other systems where similar claims about QPE have been proven.

We implement a automated tactical prover TacticToe on top of the HOL4 interactive theorem prover. TacticToe learns from human proofs which mathematical technique is suitable in each proof situation. This knowledge is then used in a Monte Carlo tree search algorithm to explore promising tactic-level proof paths. On a single CPU, with a time limit of 60 seconds, TacticToe proves 66.4 percent of the 7164 theorems in HOL4's standard library, whereas E prover with auto-schedule solves 34.5 percent. The success rate rises to 69.0 percent by combining the results of TacticToe and E prover.

Automatic theorem proving with deep learning methods has attracted attentions recently. In this paper, we construct an automatic proof system for trigonometric identities. We define the normalized form of trigonometric identities, design a set of rules for the proof and put forward a method which can generate theoretically infinite trigonometric identities. Our goal is not only to complete the proof, but to complete the proof in as few steps as possible. For this reason, we design a model to learn proof data generated by random BFS (rBFS), and it is proved theoretically and experimentally that the model can outperform rBFS after a simple imitation learning. After further improvement through reinforcement learning, we get AutoTrig, which can give proof steps for identities in almost as short steps as BFS (theoretically shortest method), with a time cost of only one-thousandth. In addition, AutoTrig also beats Sympy, Matlab and human in the synthetic dataset, and performs well in many generalization tasks.

One Monad to Prove Them All is a modern fairy tale about curiosity and perseverance, two important properties of a successful PhD student. We follow the PhD student Mona on her adventure of proving properties about Haskell programs in the proof assistant Coq. On the one hand, as a PhD student in computer science Mona observes an increasing demand for correct software products. In particular, because of the large amount of existing software, verifying existing software products becomes more important. Verifying programs in the functional programming language Haskell is no exception. On the other hand, Mona is delighted to see that communities in the area of theorem proving are becoming popular. Thus, Mona sets out to learn more about the interactive theorem prover Coq and verifying Haskell programs in Coq. To prove properties about a Haskell function in Coq, Mona has to translate the function into Coq code. As Coq programs have to be total and Haskell programs are often not, Mona has to model partiality explicitly in Coq. In her quest for a solution Mona finds an ancient manuscript that explains how properties about Haskell functions can be proven in the proof assistant Agda by tran

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to color and test planarity of planar graphs, which can be generalized to graphs containing $K_x(x>5)$ minor. There are four parts of this paper: Part-1: To Prove Four Color Theorem Part-2: An Equivalent Statement of Hadwiger Conjecture when $k=5$ Part-3: A New Proof of Wagner's Equivalence Theorem Part-4: A Geometric View of Outerplanar Graph

We show that it is impossible to prove that the outcome of a quantum measurement is random.

Recently GM Sofi & SA Shabir [arXive: 1903.01850v2 [math.GM] 6 Mar 2019] made an attempt to prove the Sendov's conjecture. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

The Riemann hypothesis is proved by quantum-extending the zeta Riemann function to a quantum mapping between quantum $1$-spheres with quantum algebra $A=\mathbb{C}$, in the sense of A. Prástaro \cite{PRAS01, PRAS02}. Algebraic topologic properties of quantum-complex manifolds and suitable bordism groups of morphisms in the category $\mathfrak{Q}_{\mathbb{C}}$ of quantum-complex manifolds are utilized.

A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.

The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At least one of these identities may be applied to prove the Riemann Hypothesis by induction. Additionally using this approach, the new series for Euler's constant gamma has been found.

In this paper we will propose a strategy to prove Goldbach's conjecture: every even integer greater than 2 can be written as the sum of two primes.

We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $λ$ with $ω\leqλ\leq\text{Ord}$ that there is a class function $F:\text{Ord}\toλ$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteqω\times\text{Ord}$, such that each section $A_n=\{α\mid (n,α)\in A\}$ contains a class club, but $\bigcap_n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $σ$-closed.