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A library of abstract interfaces for mathematical structures in Coq [[email protected]]

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Math Classes

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Math classes is a library of abstract interfaces for mathematical structures, such as:

  • Algebraic hierarchy (groups, rings, fields, …)
  • Relations, orders, …
  • Categories, functors, universal algebra, …
  • Numbers: N, Z, Q, …
  • Operations, (shift, power, abs, …)

It is heavily based on Coq’s new type classes in order to provide: structure inference, multiple inheritance/sharing, convenient algebraic manipulation (e.g. rewriting) and idiomatic use of notations.


Building and installation instructions

The easiest way to install the latest released version of Math Classes is via OPAM:

opam repo add coq-released
opam install coq-math-classes

To instead build and install manually, do:

git clone
cd math-classes
make   # or make -j 
make install

Directory structure


Proofs that certain structures form categories.



Definitions of abstract interfaces/structures.


Definitions of concrete data structures and algorithms, and proofs that they are instances of certain structures (i.e. implement certain interfaces).


Miscellaneous things.


Theory about orders on different structures.


Prototype implementation of type class based quoting. To be integrated.


Proofs of properties of structures.


Proofs that certain structures are varieties, and translation to/from type classes dedicated to these structures (defined in interfaces/).

The reason we treat categories and varieties differently from other structures (like groups and rings) is that they are like meta-interfaces whose implementations are not concrete data structures and algorithms but are themselves abstract structures.

To be able to distinguish the various arrows, we recommend using a variable width font.

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