This document is the Reference Manual of version 8.7.0 of the Coq proof assistant. A companion volume, the Coq Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A book  on practical uses of the Coq system was published in 2004 and is a good support for both the beginner and the advanced user.
The Coq system is designed to develop mathematical proofs, and especially to write formal specifications, programs and to verify that programs are correct with respect to their specification. It provides a specification language named Gallina. Terms of Gallina can represent programs as well as properties of these programs and proofs of these properties. Using the so-called Curry-Howard isomorphism, programs, properties and proofs are formalized in the same language called Calculus of Inductive Constructions, that is a λ-calculus with a rich type system. All logical judgments in Coq are typing judgments. The very heart of the Coq system is the type-checking algorithm that checks the correctness of proofs, in other words that checks that a program complies to its specification. Coq also provides an interactive proof assistant to build proofs using specific programs called tactics.
All services of the Coq proof assistant are accessible by interpretation of a command language called the vernacular.
Coq has an interactive mode in which commands are interpreted as the user types them in from the keyboard and a compiled mode where commands are processed from a file.
- The interactive mode may be used as a debugging mode in which the user can develop his theories and proofs step by step, backtracking if needed and so on. The interactive mode is run with the coqtop command from the operating system (which we shall assume to be some variety of UNIX in the rest of this document).
- The compiled mode acts as a proof checker taking a file containing a whole development in order to ensure its correctness. Moreover, Coq’s compiler provides an output file containing a compact representation of its input. The compiled mode is run with the coqc command from the operating system.
These two modes are documented in Chapter 14.
Other modes of interaction with Coq are possible: through an emacs shell window, an emacs generic user-interface for proof assistant (Proof General ) or through a customized interface (PCoq ). These facilities are not documented here. There is also a Coq Integrated Development Environment described in Chapter 16.
How to read this book
This is a Reference Manual, not a User Manual, so it is not made for a continuous reading. However, it has some structure that is explained below.
- The first part describes the specification language, Gallina. Chapters 1 and 2 describe the concrete syntax as well as the meaning of programs, theorems and proofs in the Calculus of Inductive Constructions. Chapter 3 describes the standard library of Coq. Chapter 4 is a mathematical description of the formalism. Chapter 5 describes the module system.
- The second part describes the proof engine. It is divided in five chapters. Chapter 6 presents all commands (we call them vernacular commands) that are not directly related to interactive proving: requests to the environment, complete or partial evaluation, loading and compiling files. How to start and stop proofs, do multiple proofs in parallel is explained in Chapter 7. In Chapter 8, all commands that realize one or more steps of the proof are presented: we call them tactics. The language to combine these tactics into complex proof strategies is given in Chapter 9. Examples of tactics are described in Chapter 10.
- The third part describes how to extend the syntax of Coq. It corresponds to the Chapter 12.
- In the fourth part more practical tools are documented. First in Chapter 14, the usage of coqc (batch mode) and coqtop (interactive mode) with their options is described. Then, in Chapter 15, various utilities that come with the Coq distribution are presented. Finally, Chapter 16 describes the Coq integrated development environment.
- The fifth part documents a number of advanced features, including coercions, canonical structures, typeclasses, program extraction, and specialized solvers and tactics. See the table of contents for a complete list.
At the end of the document, after the global index, the user can find specific indexes for tactics, vernacular commands, and error messages.
List of additional documentation
This manual does not contain all the documentation the user may need about Coq. Various informations can be found in the following documents:
- A companion volume to this reference manual, the Coq Tutorial, is aimed at gently introducing new users to developing proofs in Coq without assuming prior knowledge of type theory. In a second step, the user can read also the tutorial on recursive types (document RecTutorial.ps).
- A text file INSTALL that comes with the sources explains how to install Coq.
- The Coq standard library
- A commented version of sources of the Coq standard library (including only the specifications, the proofs are removed) is given in the additional document Library.ps.