Coq implements a program specification and mathematical higher-level language called Gallina that is based on an expressive formal language called the Calculus of Inductive Constructions that itself combines both a higher-order logic and a richly-typed functional programming language. Through a vernacular language of commands, Coq allows:

• to define functions or predicates, that can be evaluated efficiently;
• to state mathematical theorems and software specifications;
• to interactively develop formal proofs of these theorems;
• to machine-check these proofs by a relatively small certification "kernel";
• to extract certified programs to languages like OCaml, Haskell or Scheme.

As a proof development system, Coq provides interactive proof methods, decision and semi-decision algorithms, and a tactic language for letting the user define its own proof methods. Connection with external computer algebra system or theorem provers is available.

As a platform for the formalization of mathematics or the development of programs, Coq provides support for high-level notations, implicit contents and various other useful kinds of macros.