Coq is a software that provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs.
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 Objective Caml, 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.
Coq comes with libraries for efficient arithmetics in N, Z and Q, libraries about lists, finite sets and finite maps, libraries on abstract sets, relations, classical analysis, etc.
Coq is released with:
* a graphical user interface based on gtk (CoqIde) (see screenshots),
* documentation tools (coqdoc and coq-tex) and a statistics tool (coqwc),
* dependency and makefile generation tools for Coq (coq_makefile and coqdep),
* a stand-alone proof verifier (coqchk, from Coq 8.2).
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