Related papers: A formalization of Borel determinacy in Lean
n infinite two-player zero-sum game with a Borel winning set, in which the opponent's actions are monitored eventually but not necessarily immediately after they are played, is determined. The proof relies on a representation of the game as…
First proved my Donald Martin in 1975, the result of Borel determinacy has been the subject of multiple revised proofs. Following Martin's book [1], we present a recent streamlined proof which implements ideas of Martin, Moschovakis, and…
We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.
Blackwell games are infinite games of imperfect information. The two players simultaneously make their moves, and are then informed of each other's moves. Payoff is determined by a Borel measurable function $f$ on the set of possible…
It is shown that Borel games of length $\omega^2$ are determined if, and only if, for every countable ordinal $\alpha$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $\alpha$-many iterated…
Formalizing mathematical proofs using computerized verification languages like Lean 4 has the potential to significantly impact the field of mathematics, it offers prominent capabilities for advancing mathematical reasoning. However,…
We give an elementary proof that in a Borel family of games, the set of games for which player II has a winning strategy is Baire measurable, universally measurable, and completely Ramsey in the case where $X = [\mathbb{N}]^{\aleph_0}$.
We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather…
We prove that the determinacy of Gale-Stewart games whose winning sets are infinitary rational relations accepted by 2-tape B\"uchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a…
There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper, we…
Game theory is usually considered applied mathematics, but a few game-theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e. the…
Formal verification guarantees proof validity but not formalization faithfulness. For natural-language logical reasoning, where models construct axiom systems from scratch without library constraints, this gap between valid proofs and…
We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (2001). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN…
Chemical theory can be made more rigorous using the Lean theorem prover, an interactive theorem prover for complex mathematics. We formalize the Langmuir and BET theories of adsorption, making each scientific premise clear and every step of…
The ABC conjecture implies many conjectures and theorems in number theory, including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a function field analogue of the ABC conjecture that admits a much more elementary proof…
We formalize a proof of the irrationality of $\zeta(3)$ in Lean 4, using Beukers' method. To support this, we extend the Lean mathematical library (Mathlib) by formalizing shifted Legendre polynomials and important results in analytic…
The Kolmogorov complexity function K can be relativized using any oracle A, and most properties of K remain true for relativized versions. In section 1 we provide an explanation for this observation by giving a game-theoretic interpretation…
We introduce a new type of game on natural numbers of variable countable length, which can be regarded as a diagonalization of all games of fixed countable length on natural numbers. Building on previous work by Trang and Woodin, we show…
The principle of open determinacy for class games---two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals---is not provable in…
Applying Gr\"obner basis theory to concrete problems in Lean 4 remains difficult since the current formalization of multivariate polynomials is based on a non-computable representation and is therefore not suitable for efficient symbolic…