345 research outputs found

    Formalizing Norm Extensions and Applications to Number Theory

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    Epistemic systems and Flagg and Friedman's translation

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    In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation (⋅)□(\cdot)^\Box of Heyting arithmetic HA\bf HA to Shapiro's epistemic arithmetic EA\bf EA. In \S 2, we shall prove the faithfulness of (⋅)□(\cdot)^\Box without using stability, by introducing another translation from an epistemic system to corresponding intuitionistic system which we shall call \it the modified Rasiowa-Sikorski translation\rm . That is, this introduction of the new translation simplifies the original Flagg and Friedman's proof. In \S 3, we shall give some applications of the modified one for the disjunction property (DP\mathsf{DP}) and the numerical existence property (NEP\mathsf{NEP}) of Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule EMR\mathsf{EMR} in EA\bf EA is proved via HA\bf HA. So EA\bf EA ⊢EMR\vdash \mathsf{EMR} and HA\bf HA ⊢MR\vdash \mathsf{MR} are equivalent. In \S 5, we shall give some relations among the translations treated in the previous sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In \S 7, we shall propose several(modal-)epistemic versions of Markov's rule for Horsten's modal-epistemic arithmetic MEA\bf MEA. And, as in \S 4, we shall study some meta-implications among those versions of Markov's rules in MEA\bf MEA and one in HA\bf HA. Friedman and Sheard gave a modal analogue FS\mathsf{FS} (i.e. Theorem in \cite{fs}) of Friedman's theorem F\mathsf{F} (i.e. Theorem 1 in \cite {friedman}): \it Any recursively enumerable extension of HA\bf HA which has DP\mathsf{DP} also has NPE\mathsf{NPE}\rm . In \S 8, we shall give a proof of our \it Fundamental Conjecture \rm FC\mathsf{FC} proposed in Inou\'{e} \cite{ino90a} as follows: FC:FS⟹F.\mathsf{FC}: \enspace \mathsf{FS} \enspace \Longrightarrow \enspace \mathsf{F}. This is a new type of proofs. In \S 9, I shall give discussions.Comment: 33 page

    Making Presentation Math Computable

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    This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book

    Admissible types-to-PERs relativization in higher-order logic

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    Relativizing statements in Higher-Order Logic (HOL) from types to sets is useful for improving productivity when working with HOL-based interactive theorem provers such as HOL4, HOL Light and Isabelle/HOL. This paper provides the first comprehensive definition and study of types-to-sets relativization in HOL, done in the more general form of types-to-PERs (partial equivalence relations). We prove that, for a large practical fragment of HOL which includes container types such as datatypes and codatatypes, types-to-PERs relativization is admissible, in that the provability of the original, type-based statement implies the provability of its relativized, PER-based counterpart. Our results also imply the admissibility of a previously proposed axiomatic extension of HOL with local type definitions. We have implemented types-to-PERs relativization as an Isabelle tool that performs relativization of HOL theorems on demand

    Formalizing Chemical Physics using the Lean Theorem Prover

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    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 derivations explicit. Lean's math library, mathlib, provides formally verified theorems for infinite geometries series, which are central to BET theory. While writing these proofs, Lean prompts us to include mathematical constraints that were not originally reported. We also illustrate how Lean flexibly enables the reuse of proofs that build on more complex theories through the use of functions, definitions, and structures. Finally, we construct scientific frameworks for interoperable proofs, by creating structures for classical thermodynamics and kinematics, using them to formalize gas law relationships like Boyle's Law and equations of motion underlying Newtonian mechanics, respectively. This approach can be extended to other fields, enabling the formalization of rich and complex theories in science and engineering
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