Experience Implementing a Performant Category-Theory Library in Coq

@inproceedings{Gross2014ExperienceIA,
  title={Experience Implementing a Performant Category-Theory Library in Coq},
  author={Jason Gross and Adam Chlipala and David I. Spivak},
  booktitle={ITP},
  year={2014}
}
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… 
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References

SHOWING 1-10 OF 60 REFERENCES
Verified computing in homological algebra
TLDR
It is shown that Coq is not always up to its promises and that theoretical works will be necessary to understand how these limits can be relaxed.
Categorical semantics of programming languages ( in C OQ )
TLDR
The first goal is to build a library of category theory in the proof assistent COQ, and the second is to apply this library to formalize a specific theory of programming languages.
Developing the Algebraic Hierarchy with Type Classes in Coq
We present a new formalization of the algebraic hierarchy in Coq, exploiting its new type class mechanism to make practical a solution formerly thought infeasible. Our approach addresses both
CATEGORY THEORY as an extension of Martin-Lflf Type Theory
Category theory has tong been widely recognised as being conveniently formalisable In constructive mathematics. We describe a computer implementation of its basic concepts, as an extension of the
Packaging Mathematical Structures
TLDR
This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system, and presents a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.
A taste of category theory for computer scientists
Category theory is a field that impinges more and more frequently on the awareness of many computer scientists, especially those with an interest in programming languages and formal specifications.
Univalent categories and the Rezk completion
TLDR
A definition of ‘category’ for which equality and equivalence of categories agree is proposed, and it is shown that any category is weakly equivalent to a univalent one in a universal way.
Category Theory
TLDR
The main result that has been formalized is that the Yoneda functor is a full and faithful embedding of many sorted monadic equational logic.
The Implicit Calculus of Constructions
In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Curry-style variant of the Calculus of Constructions that we extend by adding an intersection type
Towards Observational Type Theory
Observational Type Theory (OTT) combines beneficial aspects of Intensional and Extensional Type Theory (ITT/ETT). It separates definitional equality, decidable as in ITT, and a substitutive
...
...