Formalization of PAL⋅\cdotS5 in Proof Assistant

As an experiment to the application of proof assistant for logic research, we formalize the model and proof system for multi-agent modal logic S5 with PAL-style dynamic modality in Lean theorem prover. We provide a formal proof for the reduction axiom of public announcement, and the soundness and completeness of modal logic S5, which can be typechecked with Lean 3.19.0. The complete proof is now available at Github.Comment: For proof codes, see https://github.com/ljt12138/Formalization-PA

Homotopy Type Theory: Univalent Foundations of Mathematics

Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.Comment: 465 pages. arXiv v1: first-edition-257-g5561b73, formatted for online reading. The most recent version, copies formatted for printing, and bound copies, are available at http://homotopytypetheory.org/book