4,862 research outputs found
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
Mathematics and language
This essay considers the special character of mathematical reasoning, and
draws on observations from interactive theorem proving and the history of
mathematics to clarify the nature of formal and informal mathematical language.
It proposes that we view mathematics as a system of conventions and norms that
is designed to help us make sense of the world and reason efficiently. Like any
designed system, it can perform well or poorly, and the philosophy of
mathematics has a role to play in helping us understand the general principles
by which it serves its purposes well
Beyond Notations: Hygienic Macro Expansion for Theorem Proving Languages
In interactive theorem provers (ITPs), extensible syntax is not only crucial
to lower the cognitive burden of manipulating complex mathematical objects, but
plays a critical role in developing reusable abstractions in libraries. Most
ITPs support such extensions in the form of restrictive "syntax sugar"
substitutions and other ad hoc mechanisms, which are too rudimentary to support
many desirable abstractions. As a result, libraries are littered with
unnecessary redundancy. Tactic languages in these systems are plagued by a
seemingly unrelated issue: accidental name capture, which often produces
unexpected and counterintuitive behavior. We take ideas from the Scheme family
of programming languages and solve these two problems simultaneously by
proposing a novel hygienic macro system custom-built for ITPs. We further
describe how our approach can be extended to cover type-directed macro
expansion resulting in a single, uniform system offering multiple abstraction
levels that range from supporting simplest syntax sugars to elaboration of
formerly baked-in syntax. We have implemented our new macro system and
integrated it into the upcoming version (v4) of the Lean theorem prover.
Despite its expressivity, the macro system is simple enough that it can easily
be integrated into other systems.Comment: accepted to IJCAR 202
Gauge Consistent Wilson Renormalization Group I: Abelian Case
A version of the Wilson Renormalization Group Equation consistent with gauge
symmetry is presented. A perturbative renormalizability proof is established. A
wilsonian derivation of the Callan-Symanzik equation is given.Comment: Latex2e, 39 pages, 3 eps figures. Revised version to appear in Int.
J. Mod. Phy
Man and machine thinking about the smooth 4-dimensional Poincar\'e conjecture
While topologists have had possession of possible counterexamples to the
smooth 4-dimensional Poincar\'{e} conjecture (SPC4) for over 30 years, until
recently no invariant has existed which could potentially distinguish these
examples from the standard 4-sphere. Rasmussen's s-invariant, a slice
obstruction within the general framework of Khovanov homology, changes this
state of affairs. We studied a class of knots K for which nonzero s(K) would
yield a counterexample to SPC4. Computations are extremely costly and we had
only completed two tests for those K, with the computations showing that s was
0, when a landmark posting of Akbulut (arXiv:0907.0136) altered the terrain.
His posting, appearing only six days after our initial posting, proved that the
family of ``Cappell--Shaneson'' homotopy spheres that we had geared up to study
were in fact all standard. The method we describe remains viable but will have
to be applied to other examples. Akbulut's work makes SPC4 seem more plausible,
and in another section of this paper we explain that SPC4 is equivalent to an
appropriate generalization of Property R (``in S^3, only an unknot can yield
S^1 x S^2 under surgery''). We hope that this observation, and the rich
relations between Property R and ideas such as taut foliations, contact
geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to
look at SPC4.Comment: 37 pages; changes reflecting that the integer family of
Cappell-Shaneson spheres are now known to be standard (arXiv:0907.0136
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