nLab Coq



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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) 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, with a reference as well to Coquand’s Calculus of Constructions (CoC). It 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 the section Code.

Related entries

proof assistants:

based on plain type theory/set theory:

based on dependent type theory/homotopy type theory:

based on cubical type theory:

based on modal type theory:

based on simplicial type theory:

For monoidal category theory:

For higher category theory:

projects for formalization of mathematics with proof assistants:

Other proof assistants

Historical projects that died out:



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

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:

category: software

Last revised on November 12, 2022 at 10:11:45. See the history of this page for a list of all contributions to it.