6 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
Experience Implementing a Performant Category-Theory Library in Coq
We describe our experience implementing a broad category-theory library in
Coq. Category theory and computational performance are not usually mentioned in
the same breath, but we have needed substantial engineering effort to teach Coq
to cope with large categorical constructions without slowing proof script
processing unacceptably. In this paper, we share the lessons we have learned
about how to represent very abstract mathematical objects and arguments in Coq
and how future proof assistants might be designed to better support such
reasoning. One particular encoding trick to which we draw attention allows
category-theoretic arguments involving duality to be internalized in Coq's
logic with definitional equality. Ours may be the largest Coq development to
date that uses the relatively new Coq version developed by homotopy type
theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This
version includes a full bibliography which does not fit in the Springer
version; other than the more complete references, this is the version
submitted as a final copy to ITP 201
Large Formal Wikis: Issues and Solutions
We present several steps towards large formal mathematical wikis. The Coq
proof assistant together with the CoRN repository are added to the pool of
systems handled by the general wiki system described in
\cite{DBLP:conf/aisc/UrbanARG10}. A smart re-verification scheme for the large
formal libraries in the wiki is suggested for Mizar/MML and Coq/CoRN, based on
recently developed precise tracking of mathematical dependencies. We propose to
use features of state-of-the-art filesystems to allow real-time cloning and
sandboxing of the entire libraries, allowing also to extend the wiki to a true
multi-user collaborative area. A number of related issues are discussed.Comment: To appear in The Conference of Intelligent Computer Mathematics: CICM
201
Developing the algebraic hierarchy with type classes in Coq
Contains fulltext :
83812.pdf (publisher's version ) (Closed access