natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
The computer software Coq “runs” the formal foundations-language dependent type theory and serves in particular as a formal proof management system. It provides a formal language to write mathematical definitions, executable programs and theorems together with an environment for semi-interactive development of machine-checked proofs, i.e. for certified programming.
Coq is named after Thierry Coquand, and follows a tradition of naming languages after animals (compare OCaml).
Computer systems such as Coq and Agda have been used to give machine-assisted and machine-verified proofs of extraordinary length, such as of the four-colour theorem and the Kepler conjecture.
More generally, they are being used to formalise and machine-verify large parts of mathematics as such, see the section Formalization of set-based mathematics below.
One striking insight by Vladimir Voevodsky was that Coq naturally lends itself also to a formalization of higher mathematics that is founded not on sets, but on higher category theory and homotopy theory. For this see below the section Homotopy type theory.
Projects include
Major theorems whose proofs have been fully formalized in Coq include
For Coq-projects in homotopy type theory see the section Code.
Coq uses the Gallina specification language for specifying theories.
It uses a version of the calculus of constructions to implement natural deduction.
based on plain type theory/set theory:
based on dependent type theory/homotopy type theory:
For monoidal category theory:
projects for formalization of mathematics with proof assistants:
Archive of Formal Proofs (using Isabelle)
ForMath project (using Coq)
UniMath project (using Coq)
Xena project (using Lean)
Historical projects that died out:
JsCoq, a ready-to-use online Coq environment
Cocorico, the Coq wiki
Russell O’Connor, Believing In-Consistency
(On whether the consistency of Coq is believable, given that ordinal analysis techniques are nowhere near able to handle such strong systems.)
To get an idea how to use Coq from Emacs, there are Andrej Bauer‘s Video tutorials for the Coq proof assistant (web).
Yet properly learning Coq can be quite daunting, luckily the right material can help a lot:
Benjamin Pierce‘s Software Foundations is probably the most elementary introduction to Coq and functional progamming. The book is written in Coq so you can directly open the source files in CoqIDE and step through them to see what is going on and solve the exercises.
In a similar style, Andrej Bauer and Peter LeFanu Lumsdaine wrote a nice Coq tutorial (pdf) on homotopy type theory. See also Oberwolfach HoTT-Coq tutorial.
Adam Chlipala‘s trimmed down version of Certified Programming with Dependent Types explains more advanced Coq techniques.
On ForMath:
Thierry Coquand, ForMath: Formalisation of Mathematics research project (web)
Eelis van der Weegen, Bas Spitters, Robbert Krebbers, Matthieu Sozeau, Tom Prince, Jelle Herold,
Math Classes, Coq Library for basic mathematical structures (web)
Formalization of synthetic Euclidean geometry:
Formalization of internal categories in homotopy type theory: