|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-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|logical conjunction||product||product type|
|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, (idemponent) 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|
|synthetic mathematics||domain specific embedded programming language|
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.
Similar but non-dependent type theory software includes Haskell.
A web-based version of Coq for HoTT and online documentation can be found here.
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)