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Computer-assisted proof

A computer-assisted proof, also referred to as a machine-assisted proof, is a mathematical proof that leverages computational tools to execute lengthy or exhaustive verifications, symbolic manipulations, or formal checks that would be infeasible for humans to perform manually, thereby establishing the validity of theorems through a combination of human insight and algorithmic rigor.^[1] The origins of computer-assisted proofs trace back to mid-20th-century computational aids, but they achieved widespread recognition with the 1976 proof of the Four Color Theorem by Kenneth Appel and Wolfgang Haken, which used a computer program to analyze 1,478 unavoidable configurations of planar maps, confirming that four colors suffice to color any map without adjacent regions sharing the same color.^[2] This landmark result, published in the Bulletin of the American Mathematical Society, marked the first major theorem whose proof depended heavily on machine computation, sparking philosophical debates about the acceptability of non-surveyable proofs and the role of computers in mathematics.^[1] The proof's reliance on custom software written in assembly language raised concerns over potential errors, prompting later refinements, including a 1997 version by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas that reduced the cases to 633 and used higher-level C code.^[3] Subsequent advancements addressed these verification challenges through formal methods, exemplified by Georges Gonthier's 2005 machine-checked proof of the Four Color Theorem in the Coq proof assistant, which translated the argument into a fully verifiable logical framework without trusting external computations. Another pivotal example is the 1998 proof of the Kepler conjecture by Thomas Hales and Samuel Ferguson, which employed nonlinear optimization and linear programming to examine approximately 5,000 cases, demonstrating that the face-centered cubic lattice achieves the densest packing of equal spheres in three-dimensional space with a density of \pi / \sqrt{18}.^[4] Published in full in 2006 after extensive review, this proof faced similar scrutiny over its computational scale, leading to the Flyspeck project, which culminated in a 2014 formal verification using multiple proof assistants like Isabelle and HOL Light.^[1] Key methods in computer-assisted proofs include interval arithmetic for self-validating numerical computations, which guarantees enclosures of solutions using rigorous error bounds as per the IEEE 754 floating-point standard, and computer algebra systems like Mathematica for symbolic simplification of expressions.^[5] These techniques have enabled breakthroughs in diverse areas, such as the 2016 proof of the Boolean Pythagorean triples theorem via SAT solvers, which required four CPU-years of computation to partition the natural numbers into two sets excluding Pythagorean triples.^[1] Despite their power, such proofs raise ongoing questions about reproducibility, bug detection, and the balance between human understanding and machine reliance, with recent trends incorporating large language models and integrated proof assistants to enhance accessibility and reduce errors.^[3]

Definition and Overview

Core Concept

A computer-assisted proof is a mathematical proof that employs algorithms and software to conduct exhaustive enumerations, case analyses, or logical verifications that exceed practical human capabilities. These proofs leverage computational power to handle vast search spaces or intricate calculations, ensuring the validity of mathematical statements through mechanized rigor.^[1]^[5] Key characteristics of computer-assisted proofs include their reliance on non-trivial computational resources, such as processing millions of configurations or performing symbolic manipulations over large datasets, which would be prohibitively time-consuming manually. They are inherently hybrid endeavors, integrating human-directed strategies—like formulating reductions or selecting invariants—with machine-executed tasks, such as numerical validations using interval arithmetic or graph reductions. This symbiosis enables proofs of complex results that blend theoretical insight with verifiable computation.^[3]^[5] Unlike computer-generated conjectures, which may emerge from machine learning patterns in data without formal assurance, computer-assisted proofs prioritize rigorous verification to establish truth beyond doubt, often building on human conjectures through exhaustive or self-validating methods. The concept was implicitly introduced with the first major such proof in 1976, marking a shift toward computational augmentation in mathematical argumentation.^[1]

Role in Modern Mathematics

Computer-assisted proofs have become indispensable in modern mathematics for addressing theorems that involve vast case analyses, particularly in combinatorics and geometry, where human verification alone is infeasible due to the sheer scale of possibilities.^[6] For instance, these methods enable exhaustive enumeration of configurations in problems like sphere packings or manifold classifications, transforming intractable manual checks into feasible computational tasks. The Four Color Theorem represents an early landmark in this regard, demonstrating how computers can rigorously confirm results that elude traditional pen-and-paper approaches.^[6]^[1] These proofs enhance reliability by providing exhaustive computational verification, minimizing human error in complex derivations and ensuring reproducibility across independent implementations.^[7] They also accelerate research progress in diverse fields, such as number theory—where computational approximations supported the proof of results like the weak Goldbach theorem—and topology, facilitating explorations of knot invariants and low-dimensional structures, such as the computer-assisted classification of hyperbolic 3-manifolds using software like SnapPea, that would otherwise stall due to analytical barriers.^[6]^[5] By automating routine verifications, mathematicians can focus on conceptual innovations, fostering faster advancements in theoretical frameworks.^[1] Despite these advantages, computer-assisted proofs face challenges stemming from dependencies on software correctness and hardware limitations, which can introduce subtle errors if not rigorously addressed through independent validations.^[8] Numerical instabilities or implementation bugs may undermine results, necessitating transparent algorithms and multiple verification layers to build trust.^[1] Hardware constraints further limit scalability for ultra-large computations, though ongoing improvements in open-source tools mitigate these issues over time.^[8] By 2025, the impact is evident in the proliferation of such proofs, with an analysis of 2.6 million arXiv preprints from 1986 to 2024 identifying 2,652 containing computer-assisted proofs overall and 695 in mathematics categories alone, reflecting a rising trend with annual growth rates exceeding 7-8% in probability of inclusion.^[9] This surge underscores their integration into mainstream mathematical practice, with thousands of theorems now verified computationally.^[9]

Historical Development

Pioneering Examples

In the 1950s and 1960s, computers began to assist in mathematical verifications by performing exhaustive checks on small-scale instances of problems, marking the initial forays into automated reasoning. One pioneering effort was the Logic Theorist program, developed by Allen Newell, J.C. Shaw, and Herbert A. Simon, which successfully proved 38 of the first 52 theorems in Bertrand Russell and Alfred North Whitehead's Principia Mathematica using heuristic search methods to mimic human deduction. This system demonstrated the potential for machines to verify logical propositions, though limited to propositional calculus without deeper mathematical structure. Similarly, Martin Davis implemented a decision procedure for Presburger arithmetic—a decidable fragment of first-order logic involving linear Diophantine equations—in the mid-1950s, allowing computers to systematically check solvability for integer solutions in additive constraints. By the 1960s, such computations extended to graph theory, where early programs enumerated colorings for small planar maps to test conjectures like the four-color theorem. For instance, in 1965, researchers at the University of Hannover ran initial tests on a CDC 1604A computer to explore reducible configurations in graph colorings, verifying that no counterexamples existed for maps with fewer than a certain number of regions.^[10] These efforts relied on brute-force enumeration, highlighting computers' utility for tedious case analysis but also their limitations in handling complexity without human-guided simplification. Concurrently, automated theorem provers like Hao Wang's system for first-order predicate calculus (1960) and the Davis-Putnam procedure (1960) tackled broader logical verifications, including attempts at open problems in algebra and logic, laying groundwork for more ambitious proofs.^[11] A key precursor to larger-scale applications was the exploration of automated proving for algebraic conjectures using early programs at Argonne National Laboratory in the 1960s, where researchers like Larry Wos applied resolution-based methods to test axioms and lemmas related to structures such as Boolean rings, foreshadowing later work on problems like the Robbins conjecture. These systems, though not resolving major open questions, verified intermediate results and generated proofs for specialized cases, building confidence in machine-assisted logic. The first major milestone came in 1976 with the proof of the four-color theorem by Kenneth Appel and Wolfgang Haken, who showed that every planar graph is four-colorable. Their approach reduced the problem to checking 1,482 specific reducible configurations, requiring approximately 1,200 hours of computation on the ILLIAC IV supercomputer to exhaustively verify that no minimal counterexample existed. This proof combined human insight in identifying unavoidable sets with machine enumeration, representing a paradigm shift by integrating computation as an essential component of the argument. The reception was marked by significant skepticism within the mathematical community, primarily due to the opacity of the computer code and the impracticality of manual verification of the exhaustive cases. Critics argued that the proof's reliance on unverifiable software undermined its rigor, prompting calls for independent checks and more transparent methods. This controversy spurred subsequent efforts, including a 1997 simplification by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas that reduced the cases to 633.

Key Milestones and Evolution

The development of computer-assisted proofs in the 1980s and 1990s marked a shift toward interactive proof assistants and numerical methods for handling complex verifications. In 1984, Gérard Huet and Thierry Coquand initiated the first implementation of the Calculus of Constructions (CoC), which evolved into the Coq proof assistant, enabling formalization of mathematical statements in a dependently typed functional programming language.^[12] This tool facilitated early efforts in formal verification, such as encoding logical systems and small-scale proofs, laying groundwork for larger mathematical formalizations. A pivotal milestone came in 1998 when Thomas Hales announced a proof of the Kepler conjecture, asserting that the densest packing of equal spheres in three-dimensional space is achieved by the face-centered cubic lattice; the proof relied on exhaustive case analysis using interval arithmetic to bound optimization problems over thousands of configurations.^[4] The 2000s saw increased emphasis on formal verification of landmark theorems using proof assistants. Beginning in 2005, Georges Gonthier led a project to formalize the 1976 proof of the Four Color Theorem in Coq, translating the original graph-theoretic arguments into verifiable code; this effort, spanning 2005 to 2013 with ongoing refinements, culminated in a fully machine-checked proof by 2008.^[13] This work demonstrated the feasibility of scaling proof assistants to intricate combinatorial proofs, reducing reliance on manual error-prone checks and inspiring similar formalizations in other areas. In the 2010s, advancements in automated methods, particularly SAT solvers, enabled breakthroughs in combinatorial number theory. In 2018, Marijn Heule computed the Schur number S(5)=160—the largest integer n such that the set {1, ..., n} can be 5-colored without monochromatic solutions to x + y = z—via a massive parallel search encoded as a Boolean satisfiability problem, requiring about 14 CPU-years on thousands of cores.^[14] Concurrently, computer-assisted computations advanced knowledge of van der Waerden numbers, which denote the minimal length guaranteeing monochromatic arithmetic progressions in colorings; for instance, 2011 efforts using SAT-based techniques determined w(2;4,9)=309 and established new bounds for w(2;3,t) up to t=12, highlighting the power of constraint satisfaction for exhaustive enumeration.^[15]^[16] The 2020s have witnessed the rise of more user-friendly proof assistants and the integration of machine learning for proof discovery. The Lean theorem prover, initially developed in 2013 by Leonardo de Moura and soon adopted widely, gained prominence through its mathlib library, which by the mid-2020s formalized thousands of theorems across algebra, topology, and analysis, supporting collaborative verification via an accessible syntax and integration with formal libraries.^[17] In early 2025, Terence Tao published an article in the Notices of the American Mathematical Society surveying machine-assisted proofs, emphasizing their role in verifying large-scale computations and exploring AI's potential to generate proof sketches for human refinement.^[1] This era also features AI-driven conjecture generation, where large language models propose hypotheses that are subsequently formalized and verified in systems like Lean; for example, tools such as LeanDojo have accelerated premise selection and tactic synthesis, leading to verified proofs of novel results in algebra and geometry.^[18]

Methods and Techniques

Automated Theorem Proving

Automated theorem proving encompasses symbolic methods in which computers systematically search for mathematical proofs by applying logical inference rules to explore the space of possible derivations exhaustively. Core techniques include resolution, which derives new clauses by unifying and resolving complementary literals from existing ones; semantic tableaux, which construct proof trees by branching on formula subcases to detect contradictions; and SAT solvers, which determine propositional satisfiability through conflict-driven clause learning and backtracking. These approaches operate primarily in first-order logic or its propositional fragments, enabling automated discovery of proofs without human intervention beyond problem specification.^[19] The typical process begins with a human providing a set of axioms formalizing the relevant mathematical theory and a goal conjecture to prove, often by refutation—negating the goal and deriving an empty clause from inconsistency. The prover then generates a proof structure, such as a resolution derivation or tableaux tree, by iteratively applying inference rules to expand sets of clauses, potentially yielding proofs with millions of clauses in demanding instances due to the combinatorial explosion of search spaces. This exhaustive enumeration ensures completeness within decidable fragments but relies on heuristics like ordering and literal selection to manage efficiency.^[19]^[20] Key implementations include Prover9, an automated prover using resolution and paramodulation inference rules for theorems in first-order and equational logic, capable of outputting detailed proof traces. Another leading tool is Vampire, a high-performance saturation-based prover that excels in first-order theorem proving through superposition calculi and supports equational reasoning via dedicated strategies for equality handling. Both tools facilitate applications in logic-based domains, such as verifying algebraic identities or exploring deductive closures.^[21]^[22]^[23] A representative workflow in combinatorics involves encoding the problem—such as constraints on colorings to avoid monochromatic solutions—as a propositional formula in conjunctive normal form, generating clauses that capture the axioms and negated goal. A SAT solver, like Glucose, then exhaustively searches for an unsatisfying assignment, producing a verifiable proof in LRAT format; for example, the proof that the Schur number S(5) equals 160 required solving an instance with over 130,000 initial clauses across 10 million subproblems, resulting in a 2.18-petabyte compressed proof reflecting vast clause additions and resolutions. This method highlights the scalability of automated search for finite but intricate existence problems.^[24]

Formal Verification and Proof Assistants

Formal verification in the context of computer-assisted proofs involves interactive theorem proving systems, known as proof assistants, where mathematicians guide the construction of machine-checkable proofs expressed in formal logical languages. These systems enable the rigorous encoding of mathematical statements and their derivations, ensuring that every step adheres to predefined inference rules without hidden assumptions.^[25]^[26] The core technique underlying most modern proof assistants relies on foundations such as dependent type theory or higher-order logic, where proofs are represented as structured scripts that the system compiles and verifies for correctness against a small, trusted kernel. In dependent type theory-based systems, propositions are treated as types, and proofs as terms inhabiting those types, leveraging the Curry-Howard correspondence to unify computation and deduction. Higher-order logic variants extend classical logic by allowing quantification over functions and predicates, providing expressive power for mathematical reasoning while maintaining decidable type-checking. These scripts, once verified, serve as unambiguous certificates of proof validity.^[27]^[28]^[26] Prominent proof assistants include Coq, which uses the Gallina specification language for defining terms, theorems, and proofs; Lean, supported by the community-maintained mathlib library that formalizes extensive mathematical content; and Isabelle/HOL, built on higher-order logic with integrated automation tools. These tools trace their lineage to precursors like de Bruijn's Automath project from the late 1960s, an early system for encoding and checking mathematical texts in a typed lambda calculus framework.^[29]^[30]^[31]^[32] The proof development process in these systems proceeds through step-by-step application of tactics—predefined commands that manipulate proof states—allowing users to decompose goals into subgoals and discharge them via lemmas or automation. Full formalization requires verifying every intermediate lemma and assumption, as exemplified by Georges Gonthier's Coq proof of the Four Color Theorem, which spans approximately 60,000 lines and encodes both the combinatorial core and supporting graph theory. This interactive approach contrasts with purely numerical methods by emphasizing discrete, symbolic verification over approximate computations.^[26]^[13]^[33] A key advantage of proof assistants is their ability to generate extractable proof certificates, which can be independently verified by separate checkers, enhancing trustworthiness beyond the originating system. Additionally, by operating entirely within exact logical frameworks, they avoid reliance on floating-point arithmetic, eliminating round-off errors that plague numerical validations.^[34]^[1]

Numerical Validation Methods

Numerical validation methods in computer-assisted proofs rely on techniques that provide rigorous bounds for continuous mathematical objects, such as real numbers and functions, by enclosing them within intervals that account for all possible rounding errors and uncertainties. The core technique is interval arithmetic, which represents real numbers as closed intervals [a, b] where a ≤ b, and defines arithmetic operations on these intervals to ensure that the result contains all possible values arising from the operands. This approach guarantees containment without overflow or underestimation of errors, as the operations are designed to be inclusion-monotonic: if x ∈ X and y ∈ Y, then x ⊕ y ∈ X ⊕ Y for any operation ⊕. For instance, basic interval addition is defined as [a, b] + [c, d] = [a + c, b + d], providing an exact enclosure for the sum of any values within the intervals.^[5]^[35] To handle floating-point computations accurately, interval arithmetic employs directed rounding modes, which force results to round outward—toward positive or negative infinity as needed—to ensure the computed interval fully encloses the true mathematical value. This prevents the dependency problem common in naive implementations, where repeated operations on correlated variables can lead to overly wide enclosures, by maintaining tight bounds through hardware-supported rounding in standards like IEEE 754. Advanced error bounding often incorporates Taylor models, which represent functions as a multivariate Taylor polynomial plus an interval remainder term, allowing precise control over higher-order dependencies and reducing the wrapping effect in enclosures. The remainder bound in a Taylor model of order n for a function f around a point c is typically estimated using the Lagrange form, ensuring the model encloses f(x) for x in a validation region.^[36]^[37] Practical implementations of these methods are supported by specialized software tools. INTLAB, a MATLAB toolbox, provides efficient interval arithmetic for vectors, matrices, and sparse structures, along with self-validating numerical methods for tasks like eigenvalue computation and optimization. Similarly, the Arb library offers arbitrary-precision ball arithmetic, enabling high-accuracy enclosures for complex functions and dynamical systems with automatic error tracking. These tools facilitate the verification of inequalities and existence results by computing enclosures that rigorously confirm mathematical statements.^[38]^[39] In optimization problems, numerical validation proceeds by iteratively computing enclosures for the objective function over a domain, using subdivision to isolate regions where minima or bounds can be certified. For example, in proving the Kepler conjecture on sphere packing, interval arithmetic in three dimensions was used to enclose volumes and densities, certifying that the optimal packing achieves a density greater than 0.7405 by verifying nonlinear inequalities across case configurations. Error propagation in such computations, particularly for integrals, is handled by enclosing the integrand with an interval function and bounding the integral via the mean value theorem: for a continuous f on [a, b], the integral ∫_a^b f(x) dx is enclosed in [(b - a) · inf f, (b - a) · sup f], with tighter bounds obtained through Taylor expansions or adaptive quadrature that propagate interval remainders. This process ensures that the computed bounds are mathematically rigorous, supporting proofs of global properties like convergence or stability.^[4]^[40]

Notable Theorems and Proofs

Graph Theory and Combinatorics

One of the landmark applications of computer-assisted proofs in graph theory is the resolution of the Four Color Theorem, which states that every planar graph is 4-colorable. In 1976, Kenneth Appel and Wolfgang Haken proved this by reducing the problem to showing that every planar graph contains at least one of 1,476 specific reducible configurations, where reducibility means that any 4-coloring of the configuration's boundary can be extended to the interior. A computer program exhaustively verified the reducibility of these configurations by checking thousands of potential colorings and subcases for each, a task that would have been infeasible by hand due to the combinatorial explosion. Progress on Hadwiger's conjecture, which posits that every graph without a complete minor of order t is (t-1)-colorable, has also relied on computational methods to rule out counterexamples for small values of t. Researchers have used algorithms, including bounded searches informed by structural graph theory, to verify that no minimal counterexamples exist for t ≤ 6, with explicit bounds on graph sizes allowing exhaustive enumeration. While SAT solvers have been employed in related graph coloring problems to search for satisfiable assignments representing potential counterexamples, their application to Hadwiger's conjecture has focused on encoding minor-avoidance constraints, confirming partial results through automated case analysis. In combinatorial enumeration, computer-assisted proofs have determined exact values for van der Waerden numbers, such as W(3,4)=293, the smallest integer n such that any 4-coloring of {1, 2, ..., n} contains a monochromatic arithmetic progression of length 3. This result was established in 2012 through an exhaustive computational search that verified the absence of such progressions in all 4-colorings of {1, 2, ..., 292} while confirming their inevitability at 293, involving the analysis of over 10^12 potential colorings via optimized backtracking and symmetry reductions. The process exemplifies case reduction techniques, where the problem is broken into unavoidable sets of configurations whose properties are checked algorithmically, highlighting the role of computational power in handling exponential complexity. These proofs in graph theory and combinatorics typically involve discharging methods or constraint satisfaction to identify unavoidable sets, followed by automated verification of finitely many cases, often requiring millions of graph or coloring instances to be processed—demonstrating how computers enable rigorous confirmation in domains intractable to manual proof.

Geometry and Packing Problems

Computer-assisted proofs have played a pivotal role in resolving longstanding problems in geometry and packing, particularly those involving optimal spatial configurations and density bounds. One of the most prominent examples is the Kepler conjecture, which posits that the face-centered cubic lattice achieves the maximum density for packing equal spheres in three-dimensional Euclidean space. In 1998, Thomas Hales announced a proof, later fully published in 2005, demonstrating that no packing exceeds this density of \pi / (3 \sqrt{2}) \approx 0.74048. The proof relied on exhaustive case analysis, classifying several thousand possible tame plane graphs arising from Voronoi decompositions of sphere centers, and verifying nonlinear inequalities using interval arithmetic to ensure numerical rigor without floating-point errors.^[4] A complementary landmark is the double bubble conjecture, resolved in 2000 through a proof establishing that the standard double bubble—consisting of three spherical caps meeting at 120-degree angles—minimizes surface area for enclosing and separating two given volumes in \mathbb{R}^3. This result built on earlier computer-assisted verification for the equal-volume case in 1995, where rigorous computational searches eliminated nonstandard configurations like nested toroidal bands by checking stability across a two-parameter family of potential minimizers. The full proof incorporated symmetry arguments, proving rotational symmetry about an axis for area-minimizing enclosures, alongside enclosure methods using convexity and decomposition to show that the larger region remains connected and the smaller has at most two components.^[41] These proofs exemplify broader techniques in computer-assisted geometry, such as global optimization via branch-and-bound methods to explore configuration spaces and certify local minima through variational analysis and numerical bounds. In the Kepler case, branch-and-bound was applied to maximize a scoring function over compact domains derived from decomposition stars, ensuring all potential high-density arrangements were ruled out. Similarly, for the double bubble, computational enclosure verified that deviations from the standard form increase area, confirming its global minimality. Such approaches combine algorithmic enumeration with certified numerics to handle the complexity of infinite-dimensional problems reduced to finite, verifiable cases.^[4]^[41] Subsequent efforts addressed concerns over the original Kepler proof's readability and verification, culminating in a formalization project completed in 2014 using the HOL Light theorem prover, with parts in Isabelle. This machine-checked version translated the informal proof into over 120,000 lines of verified code, rigorously confirming the conjecture without relying on human oversight for the exhaustive cases and arithmetic validations. The formal proof not only bolstered confidence in the result but also highlighted the feasibility of scaling computer assistance to complex geometric arguments.^[42]

Recent Developments

In recent years, computer-assisted proofs have increasingly incorporated advanced computational techniques to tackle longstanding problems in Ramsey theory. A notable example is the determination of the Schur number S(5) = 160, achieved through a massive SAT-based search that generated a 2-petabyte proof certificate, formally verified using the ACL2 theorem prover to ensure correctness.^[43] This result, resolving a century-old question about the largest integer partitionable into five sum-free sets, exemplifies the power of satisfiability solvers in combinatorial number theory, with the formal certification providing rigorous assurance against computational errors.^[44] AI integration has accelerated progress in formal verification, particularly within proof assistants like Lean. In 2021, DeepMind advanced automated theorem proving by developing language models trained to generate proofs in Lean, enabling the formalization of complex functorial constructions in category theory and laying groundwork for AI-human collaboration in abstract mathematics. Building on this, in 2025, DeepSeek AI released DeepSeek-Prover-V2, a 671-billion-parameter model specialized for Lean 4 that verifies number theory lemmas with state-of-the-art accuracy, achieving 88.9% success on benchmarks like MiniF2F and supporting formal proofs of results in arithmetic progressions and Diophantine equations. From 2024 to 2025, machine learning has guided searches in related Diophantine problems, yielding progress on extensions of the Boolean Pythagorean triples theorem. While the original 2016 resolution showed no 2-coloring of natural numbers avoids monochromatic Pythagorean triples up to 7825, recent ML-enhanced SAT approaches have tightened bounds for higher-dimensional variants, using neural heuristics to prioritize promising branches in vast search spaces.^[45] Concurrently, mathematician Terence Tao's 2025 survey highlights the rise of hybrid human-AI workflows, where AI tools like language models assist in conjecture generation and proof sketching, as demonstrated in his own work on ergodic theory and partial differential equations.^[46] Overall trends indicate a surge in computer-assisted contributions, with analyses of arXiv preprints from 2020 to 2025 revealing over 100 new theorems formalized via proof assistants, particularly in algebraic geometry, where tools like Lean have enabled verifications of results on moduli spaces and sheaf cohomology. This growth reflects broader adoption of formal methods, subtly influencing philosophical views on proof reliability in the AI era.

Philosophical and Methodological Issues

Criticisms and Objections

One prominent criticism of computer-assisted proofs concerns their verifiability, as humans cannot manually inspect the vast computational steps involved, rendering the proofs quasi-empirical rather than purely a priori deductions. Philosopher Thomas Tymoczko argued in 1979 that such proofs, exemplified by the four-color theorem, depend on empirical trust in computer outputs, akin to scientific experiments, since no individual can survey the entire process.^[47] This raises risks from software bugs, as no foolproof method exists to guarantee program correctness without further computation.^[47] Critics also contend that computer-assisted proofs lack mathematical elegance and insight, often resembling brute-force enumerations rather than revealing deep structural understanding. In the case of the four-color theorem proof by Appel and Haken, the computational verification required over 1,000 hours on a mainframe computer to check thousands of cases, which some viewed as non-mathematical drudgery devoid of explanatory power.^[48]^[47] Practical and social objections highlight the dependency on specialized hardware and software, creating barriers to replication for mathematicians without access to equivalent resources. This reliance exacerbates inequalities, as proofs become tied to institutional privileges and may hinder independent verification by non-experts or those in under-resourced settings.^[49] A specific instance of these concerns arose with Thomas Hales' 1998 proof of the Kepler conjecture, which spanned approximately 300 pages of mathematical text and extensive code; the Annals of Mathematics accepted only an abridged version due to its length and complexity, sparking controversy over the full proof's publication and accessibility.^[4]^[50]

Responses and Justifications

One key response to early objections, such as Thomas Tymoczko's 1979 argument that computer-assisted proofs resemble empirical experiments rather than a priori deductions, emphasizes the feasibility of independent verification to ensure reliability.^[47] For the Four Color Theorem, initial concerns in the early 1980s about potential flaws were addressed through multiple independent checks by teams, including examinations by Ulrich Schmidt and others using different programs and hardware, who discovered and led to the correction of minor errors, ultimately confirming the proof's correctness.^[51] More robustly, modern formal verifications, such as Georges Gonthier's 2005 implementation in the Coq proof assistant, produce extractable certificates that allow machine-checked reproducibility, transforming opaque computations into verifiable logical steps.^[52] Similarly, for the Kepler conjecture, Thomas Hales' Flyspeck project culminated in a 2017 formal proof in HOL Light (with ongoing efforts in Lean), yielding certificates that encapsulate the entire argument for independent scrutiny.^[53] Proponents argue that computer-assisted proofs preserve mathematical insight by leveraging human expertise in high-level strategy while delegating routine computations to machines, thereby enhancing rather than diminishing understanding. In Hales' proof of the Kepler conjecture, humans devised the core approach—reducing the problem to optimizing decomposition stars via a scoring function and classifying tame plane graphs—while computers exhaustively handled the tedium of evaluating thousands of cases through linear programming and branch-and-bound algorithms, allowing mathematicians to focus on conceptual innovations.^[4] This division of labor, as Hales described in reflections on the project, clarifies geometric intuitions that manual checks could obscure, turning a sprawling verification into a structured exploration of packing densities.^[54] By 2025, mathematical norms have evolved to embrace hybrid proofs, where formalization in tools like Coq and Lean is increasingly required for peer-reviewed acceptance of complex results, reflecting a consensus on machine verification as a standard for rigor. Terence Tao, in his 2025 analysis of machine-assisted proofs, advocates for this hybrid model, arguing that it combines human creativity in conjecture and strategy with machine precision in verification and case analysis, yielding proofs that are both insightful and error-free at scale.^[55] Such approaches, exemplified by collaborative formalizations, have normalized computer assistance as an extension of traditional methods rather than a departure. The empirical track record further bolsters confidence, with no known errors persisting in major computer-assisted proofs following thorough verification; for instance, the Flyspeck formalization of the Kepler conjecture corrected hundreds of minor issues in the original but affirmed the theorem's validity, while the Four Color Theorem's Coq version has withstood subsequent audits without flaws.^[54] This success underscores how verification processes, including re-runs by independent teams and formal certificates, mitigate risks and uphold mathematical standards.

Current Advances and Applications

Integration with AI and Machine Learning

The integration of artificial intelligence (AI) and machine learning (ML) into computer-assisted proofs has transformed the process by automating conjecture generation, optimizing search strategies, and suggesting proof tactics, thereby accelerating formal verification in theorem provers. AI systems leverage large language models (LLMs) trained on vast corpora of mathematical proofs to predict and guide proof steps, reducing the manual effort required in traditional interactive theorem proving. This synergy allows for tackling complex problems that were previously computationally infeasible, while enhancing the efficiency of proof construction in systems like Lean and Coq.^[56] A prominent example is AlphaGeometry, developed by DeepMind in 2024, which combines neural language models with symbolic deduction engines to solve Olympiad-level geometry problems. Trained on synthetic data and human proofs, AlphaGeometry achieved a silver-medal standard at the International Mathematical Olympiad (IMO) by solving 25 out of 30 recent geometry problems, outperforming prior methods that solved only 10. Its successor, AlphaGeometry 2 (2025), further improved performance to gold-medal level on IMO geometry problems from 2000–2024, solving 88% of a benchmark set through enhanced search and language modeling. In proof assistants like Lean, neural networks provide tactic suggestions by predicting the next proof step based on the current state, as implemented in tools like Lean Copilot and LeanProgress, which integrate LLMs to generate and evaluate multi-step proof outlines.^[56] A more recent advancement is AlphaProof, introduced by DeepMind and published on November 12, 2025, which employs an AlphaZero-inspired agent using reinforcement learning to find formal proofs. Trained on millions of auto-formalized problems, AlphaProof, in combination with AlphaGeometry 2, achieved a silver-medal standard at the IMO 2024 by solving 4 out of 6 problems, marking a breakthrough in AI's ability to handle diverse mathematical reasoning tasks at competition level.^[57] Machine learning processes typically involve training on proof corpora extracted from repositories such as Lean's mathlib, where models learn to predict tactics or subgoals from partial proofs. For instance, DeepSeek-Prover-V2 (2025), an open-source LLM for formal theorem proving in Lean 4, uses reinforcement learning for subgoal decomposition and achieves an 88.9% pass rate on the MiniF2F benchmark, which includes algebraic problems requiring identity verification and proof generation. This model iteratively refines proofs by breaking them into verifiable subcomponents, demonstrating high success in formalizing algebraic identities and related theorems. An illustrative application is the 2023 use of FunSearch, an AI-driven evolutionary search method, to discover new lower bounds for the cap set problem in additive combinatorics, yielding the largest known cap sets in dimensions up to 8 through program evolution and verification, surpassing prior computational limits.^[58]^[59] Despite these advances, AI-assisted proofs necessitate human oversight to ensure rigor and correctness, as automated systems may produce plausible but unverifiable outputs or overlook edge cases in formal settings. Current limitations include dependency on high-quality training data and the challenge of scaling to unstructured or novel mathematical domains without extensive fine-tuning.^[58]^[59]

Broader Impacts and Future Prospects

Computer-assisted proofs have extended beyond pure mathematics into interdisciplinary fields, enabling rigorous verification in domains where traditional manual methods are impractical. In physics, formal verification tools have been applied to quantum algorithms, such as the CoqQ framework, which uses the Coq proof assistant to reason about quantum programs and ensure their foundational correctness.^[60] Similarly, in engineering, automated polynomial formal verification techniques generate human-readable proofs for circuit correctness, facilitating the validation of complex hardware designs like multipliers.^[61] In cryptography, computer-aided methods, including the EasyCrypt tool, elaborate security proofs for protocols like public-key encryption and multiparty computation, bridging game-based and computational models to confirm protocol robustness.^[62] These advancements democratize access to complex proofs through open-source tools, allowing broader collaboration among mathematicians and scientists without requiring specialized expertise in formal methods. For instance, the Lean theorem prover's mathlib library supports community-driven formalization of mathematical knowledge, breaking down intricate proofs into modular, verifiable components that can be contributed to and built upon globally.^[63] This accessibility accelerates discoveries in longstanding problems; computational verifications have confirmed the Riemann hypothesis for the first 10 trillion non-trivial zeros of the zeta function, providing strong partial evidence and guiding theoretical progress.^[64] Looking ahead, quantum computing holds promise for enhancing computer-assisted proofs by accelerating exhaustive enumerations in proof searches, such as through Grover's algorithm, which offers a quadratic speedup for unstructured search problems relevant to case analysis in formal verification. The 2025 Strachey Lecture by Kevin Buzzard anticipates full AI autonomy in proving research-level theorems by the 2030s, through integrating language models with proof assistants like Lean to handle complex, original mathematics.^[65] However, this evolution raises ethical concerns regarding proof ownership and authorship, as AI-generated content challenges traditional notions of human contribution and intellectual property rights in mathematical works.^[66] Terence Tao's analysis in the AMS Notices suggests that machine-assisted tools will increasingly automate routine tasks and suggest novel ideas, fostering new collaborative paradigms in mathematical research, though full autonomy remains a long-term goal.^[55]

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Every planar map is four colorable - Project Euclid
Every planar map is four colorable. K. Appel, W. Haken. DOWNLOAD PDF + SAVE TO MY LIBRARY. Bull. Amer. Math. Soc. 82(5): 711-712 (September 1976).
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[PDF] COMPUTER ASSISTED PROOFS: - University of Washington
Mar 23, 2015 · Definition of “Proof”. • Proof: An argument or evidence ... Examples of Computer Proof. Next topic: The Four Color Theorem. Page 8 ...
[4]
[PDF] A proof of the Kepler conjecture - Annals of Mathematics
Remark 1.8. The Kepler conjecture is an optimization problem in an in- finite number of variables (the coordinates of the points of Λ). The maximiza- tion of σ ...
[5]
[PDF] Computer-assisted Proofs and Self-validating Methods - TUHH
In this chapter41 we discuss the possibility of computing validated answers to mathematical problems. Among such approaches are so-called computer-assisted ...
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[PDF] Computational Mathematics in computer assisted proofs
Computer assisted proofs are becoming a mainstay in modern mathematics, as there are an increasing number of famous conjectures and theorems that have recently ...
[7]
Mathematicians welcome computer-assisted proof in 'grand ... - Nature
Jun 18, 2021 · In this way, proof assistants can help to verify mathematical proofs that would otherwise be time-consuming and difficult, perhaps even ...
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[PDF] Computer-assisted proofs - Arnold Neumaier
This paper discusses the problem what makes a computer-assisted proof trustworthy, the quest for an al- gorithmic support system for computer-assisted proof, re ...
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None
### Key Findings on Computer-Assisted Proofs in arXiv Preprints (1986-2024)
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[PDF] A Historical Overview of the Four-Color Theorem - Adelphi University
May 17, 2004 · Appel and Haken presented their proof to a group of mathematicians at a meeting in Toronto.
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https://dl.acm.org/doi/10.1145/321033.321034
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Early history of Coq — Coq 8.19.0 documentation - Rocq Prover
A first implementation of CoC was started in 1984 by G. Huet and T. Coquand. Its implementation language was CAML, a functional programming language from the ML ...
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[PDF] A computer-checked proof of the Four Colour Theorem 1 The story
This report gives an account of a successful formalization of the proof of the Four. Colour Theorem, which was fully checked by the Coq v7.3.1 proof ...
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Schur Number Five | Proceedings of the AAAI Conference on ...
Apr 26, 2018 · We present the solution of a century-old problem known as Schur Number Five: What is the largest (natural) number n such that there exists a ...
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[PDF] a71 integers 11 (2011) on computation of exact van der waerden ...
Dec 13, 2011 · In this section, we report that the exact value of the previously unknown van der. Waerden number w(2; 4,9) is 309. We also provide new lower ...Missing: assisted | Show results with:assisted
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https://arxiv.org/pdf/1102.5433v1
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Lean - Microsoft Research
Lean is a functional programming language and interactive theorem prover. Our project strives to revolutionize mathematics by empowering anyone with an interest ...Downloads · Publications · People · GroupsMissing: 2020s | Show results with:2020s<|separator|>
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AI-Driven Formal Theorem Proving in the Lean Ecosystem
An integrated development environment specifically designed for Lean theorem proving, combining modern IDE features with AI-powered assistance to create the ...
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None
Nothing is retrieved...<|separator|>
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[PDF] Automated Theorem Proving - andrew.cmu.ed
Modern SAT solvers can handle tens of thousands of variables and millions of clauses. Page 17. First-order theorem proving. First-order logic adds relations r ...
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Prover9 and Mace4
- **Prover9 Description**:
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Vampire
although now it can do much more! Its main focus is in proving theorems in first-order ...Missing: equational | Show results with:equational
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First-Order Theorem Proving and Vampire - SpringerLink
In this paper we give a short introduction in first-order theorem proving and the use of the theorem prover Vampire. We discuss the superposition calculus ...
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None
### Summary of SAT Proof for Schur Number Five (arXiv:1711.08076)
[25]
[PDF] Theorem Proving in Lean
Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms.
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Reference Manual | The Coq Proof Assistant - Rocq
Coq is a proof assistant for higher-order logic, allowing the development of computer programs consistent with their formal specification. It is the result ...
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1. Introduction — Theorem Proving in Lean 3 (outdated) 3.23.0 ...
Formal verification involves the use of logical and computational methods to ... We will describe various methods to support this in dependent type theory.Missing: assistants | Show results with:assistants
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Overview - Isabelle
Feb 29, 2024 · Isabelle is a generic proof assistant. It allows mathematical formulas to be expressed in a formal language and provides tools for proving those formulas in a ...
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The Gallina specification language — Coq 8.9.1 documentation
This chapter describes Gallina, the specification language of Coq. It allows developing mathematical theories and to prove specifications of programs.
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Mathematics in mathlib - Lean community
A mathlib overview. The goal of this web page is to give a rough list of topics currently covered in mathlib, and provide pointers for exploration.
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Isabelle
Feb 3, 2025 · Isabelle is a generic proof assistant. It allows mathematical formulas to be expressed in a formal language and provides tools for proving those formulas in a ...Overview · Installation · Documentation
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N.G. de Bruijn (1918–2012) and his Road to Automath, the Earliest ...
Oct 20, 2012 · N.G. de Bruijn (1918–2012) and his Road to Automath, the Earliest Proof Checker. Article; Published: 20 October 2012. Volume 34, pages 4–11 ...Missing: precursor | Show results with:precursor
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[PDF] Computer-Aided Proofs and Their Significance - Sergei N. Artemov
Coq is widely used for formalization of mathematics: real analysis, construc- tive category theory, elements of constructive geometry, group theory, domain.
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Generating and Exploiting Automated Reasoning Proof Certificates
Oct 1, 2023 · Having such tools produce formal proof certificates makes it possible to independently check their results, greatly improving trustworthiness.
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[PDF] COMPUTER ASSISTED PROOFS IN ANALYSIS Oscar E. Lanford III ...
Interval arithmetic. The techniques to be described here rest on a standard and elementary method of numerical analysis known as interval arithmetic. To explain ...
[36]
[PDF] Rounding of Floating Point Intervals Округление интервалов с ...
The directed rounding modes were introduced in the IEEE standard to allow interval arithmetic implementation at the user level. This is to guarantee that ...
[37]
[PDF] Taylor Models and Their Applications Martin Berz and Kyoko Makino
Proof. The proof for the binary operations follows directly from the definition of the remainder bounds for the binaries. Similarly, the proof for the ...
[38]
[PDF] intlab - interval laboratory - TUHH
INTLAB is a toolbox for Matlab supporting real and complex intervals, and vectors, full matrices and sparse matrices over those. It is designed to be very fast.Missing: assisted | Show results with:assisted
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[PDF] Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic
Abstract—Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic.
[40]
[PDF] Guaranteed Proofs Using Interval Arithmetic - HAL
This paper presents a set of tools that support mechani- cal proof checking of numerical bounds using interval arith- metic. The tools implement two techniques ...
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Hadwiger's conjecture is decidable - ACM Digital Library
May 31, 2009 · Every minimal counterexample to Hadwiger's conjecture for the case t has at most f(t) vertices for some explicit bound f(t). The bound f(t) is ...
[42]
Hadwiger's Conjecture with Certain Forbidden Induced Subgraphs
Nov 1, 2022 · We prove that \{\overline{K_3}, H\}-free graphs are not counterexamples to Hadwiger's Conjecture, where H is any one of 33 graphs on seven, eight, or nine ...
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[PDF] proof of the double bubble conjecture - Berkeley Math
Jul 17, 2000 · In 1995, Hass, Hutchings, and Schlafly [HHS] announced a computer-assisted proof for the case of equal volumes in R3. (See [M1], [HS1], [HS2] ...
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A FORMAL PROOF OF THE KEPLER CONJECTURE
May 29, 2017 · This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants.
[45]
[1711.08076] Schur Number Five - arXiv
Nov 21, 2017 · We present the solution of a century-old problem known as Schur Number Five: What is the largest (natural) number n such that there exists a five-coloring of ...
[46]
[PDF] Schur Number Five
Schur Number Five is the largest number n (n=160) where a five-coloring of numbers up to n exists without a monochromatic solution of a+b=c.
[47]
Solving and Verifying the boolean Pythagorean Triples problem via ...
May 3, 2016 · The boolean Pythagorean Triples problem asks if natural numbers can be divided into two parts, where no part contains a triple (a,b,c) with a^2 ...
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https://celebratio.org/Haken_W/article/794/
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[PDF] The Four-Color Problem and Its Philosophical Significance Thomas ...
Jan 9, 2008 · The present paper provides additional support for the thesis that mathematics is quasi- empirical. 5 For a simple account of the proof, see ...
[50]
Haken — Four-Color Solution - Celebratio Mathematica
The four-color problem asks if any map can be colored with four colors, where countries sharing a border have different colors. The solution was found in 1976.
[51]
[PDF] arXiv:2309.11457v1 [math.HO] 20 Sep 2023
Sep 20, 2023 · I suggest that past and present controversies about the status of computer-assisted proofs reflect a longstanding tension in modern mathematics, ...
[52]
Thomas Hales: The Proof of the Proof - Pittsburgh Quarterly
Using a proof assistant developed by Intel, Hales plans to check every computer calculation and logical step in his proof of the Kepler conjecture to put to ...Missing: interval arithmetic<|separator|>
[53]
Appel — Haken and 4C - Celebratio Mathematica
By the early 1980s, rumors were beginning to spread that there was a major error in Appel and Haken's proof of the four-color theorem. In view of its ...2. Enter Heesch And Haken · 3. Enter Appel · 4. Aftermath<|separator|>
[54]
Formalizing the Proof of the Kepler Conjecture - ResearchGate
Aug 7, 2025 · 1 Modern proof assistants-such as Coq, Lean, Isabelle and HOL Light-have enabled the formalization of large bodies of complex mathematical ...Missing: certificates | Show results with:certificates
[55]
Mathematicians deliver formal proof of Kepler Conjecture - Phys.org
Jun 16, 2017 · A team led by mathematician Thomas Hales has delivered a formal proof of the Kepler Conjecture, which is the definitive resolution of a problem that had gone ...
[56]
The Formal Proof of the Kepler Conjecture: a critical retrospective
Feb 12, 2024 · The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic ...
[57]
AMS :: Notices of the American Mathematical Society
### Summary of Machine-Assisted Proofs Future Prospects
[58]
Solving olympiad geometry without human demonstrations - Nature
Jan 17, 2024 · We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems ...
[59]
DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition
### Summary of DeepSeek-Prover-V2 Performance on Verifying Algebraic Identities or Math Proofs
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https://dl.acm.org/doi/10.1145/3571222
[61]
CoqQ: Foundational Verification of Quantum Programs
CoqQ is a framework for reasoning about quantum programs in the Coq proof assistant. Its main components are: a deeply embedded quantum programming language.Abstract · Information & Contributors · Cited By
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[PDF] Automated Polynomial Formal Verification: Human-Readable Proof ...
To prove the correctness of a circuit, formal verification techniques based on decision diagrams, e.g. Binary Decision Diagrams (BDDs) [1],. [2], Kronecker ...<|separator|>
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[PDF] Computer-Aided Security Proofs for the Working Cryptographer⋆
We present an automated tool for elaborating security proofs of cryptographic sys- tems from proof sketches—compact, formal representations of the essence of a ...
[64]
Lean enables correct, maintainable, and formally verified code
Lean is an open-source programming language and proof assistant that enables correct, maintainable, and formally verified code.Functional Programming in Lean · Install Lean · About Lean FRO · Lean APIMissing: 2020s | Show results with:2020s
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[PDF] arXiv:2004.09765v1 [math.NT] 21 Apr 2020 The Riemann ...
Apr 21, 2020 · In common with all modern partial verifications of the Riemann hypothesis, the algorithm computes values of the completed zeta function on the ...
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Strachey Lecture: Will Computers prove theorems? - YouTube
May 19, 2025 · Strachey Lecture: Will Computers prove theorems? Originally uploaded on the University of Oxford Podcast page 15/05/2025 Abstract of Kevin ...
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Authorship and Ownership Issues Raised by AI-Generated Works
Although AI systems cannot be considered legal authors under current law, the question of who can claim ownership among the humans involved remains open.