|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-level 1-type/h-prop|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
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).
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.
For Coq-projects in homotopy type theory see
Similar but non-dependent type theory software includes Haskell.
A web-based version of Coq is at
To start it, choose “Coq” from the menu “proof assistant” and Click on “guest login”. In the user interface that appears, enter Coq-code in the left window and hit the arrow-buttons to “run” it with output appearing in the right window. The guest account allows everything except saving files and loading libraries. But with copy-and-paste one can of course “include libraries” by hand.
(Notice, though, that the current version can for instance not read the HoTT libraries verbatim, since it does not understand implicit types yet.)
A tool for viewing proofs in static Coq files without loading them into Coq is
A proviola-enhanced version of the Coq-library for homotopy type theory is at
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.
Math Classes, Coq Library for basic mathematical structures (web)
For applications to homotopy type theory see the references listed there. Especially