Related papers: Formalizing May's Theorem
This paper contributes to a burgeoning area of investigation, the ambiguity inherent in mathematics and the implications for physics of this ambiguity. To display the mathematical form of equations of quantum theory used to describe…
Data analysis requires translating higher level questions and hypotheses into computable statistical models. We present a mixed-methods study aimed at identifying the steps, considerations, and challenges involved in operationalizing…
An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory…
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal…
We describe the first results of a project of analyzing in which theories formal proofs can be ex- pressed. We use this analysis as the basis of interoperability between proof systems.
The purpose of this lecture is to describe the KAM theorem in its most basic form and to give a complete and detailed proof. This proof essentially follows the traditional lines laid out by the inventors of this theory, and the emphasis is…
In this paper, we describe the formalization of the axiom of choice and several of its famous equivalent theorems in Morse-Kelley set theory. These theorems include Tukey's lemma, the Hausdorff maximal principle, the maximal principle,…
Inspired by the work of Feynman, Deutsch, We formally propose the theory of physical computability and accordingly, the physical complexity theory. To achieve this, a framework that can evaluate almost all forms of computation using various…
The use of formal methods provides confidence in the correctness of developments. Yet one may argue about the actual level of confidence obtained when the method itself -- or its implementation -- is not formally checked. We address this…
Quantifier-elimination or model-completeness of the affine part of some classical first order theories are proved.
LLM-generated explanations can make technical content more accessible, but there is a ceiling on what they can support interactively. Because LLM outputs are static text, they cannot be executed or stepped through. We argue that grounding…
This essay examines how automation has reconfigured mathematical proof and labor, and what might happen in the future. It discusses practical standards of proof, distinguishes between prominent forms of automation in research, provides…
We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that…
PyLog is a minimal experimental proof assistant based on linearised natural deduction for intuitionistic and classical first-order logic extended with a comprehension operator. PyLog is interesting as a tool to be used in conjunction with…
The goal of this project is to (i) accumulate annotated informal/formal mathematical corpora suitable for training semi-automated translation between informal and formal mathematics by statistical machine-translation methods, (ii) to…
We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple…
The continuous functional calculus is perhaps the most fundamental construction in the theory of operator algebras, especially $C^{*}$-algebras. Here we document our formalization of the continuous functional calculus in Lean, which…
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…
In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special…
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…