Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




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.

Formalization of set-based mathematics

Projects include

Formalized proofs

Major theorems whose proofs have been fully formalized in Coq include

Homotopy type theory

For Coq-projects in homotopy type theory see homotopy type theory.

Coq uses the Gallina specification language for specifying theories.

It uses a version of the calculus of constructions to implement natural deduction.

Dependent type theory softwares similar to Coq are Agda and Lean.

Similar but non-dependent type theory software includes Haskell.



A web-based version of Coq for HoTT and online documentation can be found here.

Learning Coq

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:

  1. 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.

  2. 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.

  3. Adam Chlipala’s trimmed down version of Certified Programming with Dependent Types explains more advanced Coq techniques.

Applications to formal mathematics

  • 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:

category: software

Last revised on May 9, 2017 at 09:41:11. See the history of this page for a list of all contributions to it.