\[\begin{split}\newcommand{\alors}{\textsf{then}} \newcommand{\alter}{\textsf{alter}} \newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\bool}{\textsf{bool}} \newcommand{\case}{\kw{case}} \newcommand{\conc}{\textsf{conc}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\conshl}{\textsf{cons\_hl}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\EqSt}{\textsf{EqSt}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\false}{\textsf{false}} \newcommand{\filter}{\textsf{filter}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\from}{\textsf{from}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\haslength}{\textsf{has\_length}} \newcommand{\hd}{\textsf{hd}} \newcommand{\ident}{\textsf{ident}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\lb}{\lambda} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\Nat}{\mathbb{N}} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \newcommand{\Sort}{\cal S} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\true}{\textsf{true}} \newcommand{\Type}{\textsf{Type}} \newcommand{\unfold}{\textsf{unfold}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\cal W\!F}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\cal W\!F}(#2)} \newcommand{\WFTWOLINES}[2]{{\cal W\!F}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \newcommand{\zeros}{\textsf{zeros}} \end{split}\]

Introduction

This document is the Reference Manual of the Coq proof assistant. To start using Coq, it is advised to first read a tutorial. Links to several tutorials can be found at https://coq.inria.fr/documentation and https://github.com/coq/coq/wiki#coq-tutorials

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 specifications. 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 \(\lambda\)-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.

  • In interactive mode, users can develop their theories and proofs step by step, and query the system for available theorems and definitions. The interactive mode is generally run with the help of an IDE, such as CoqIDE, documented in Coq Integrated Development Environment, Emacs with Proof-General [Asp00] [1], or jsCoq to run Coq in your browser (see https://github.com/ejgallego/jscoq). The coqtop read-eval-print-loop can also be used directly, for debugging purposes.
  • 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.

See also

The Coq commands.

How to read this book

This is a Reference Manual, so it is not intended for continuous reading. We recommend using the various indexes to quickly locate the documentation you are looking for. There is a global index, and a number of specific indexes for tactics, vernacular commands, and error messages and warnings. Nonetheless, the manual has some structure that is explained below.

  • The first part describes the specification language, Gallina. Chapters The Gallina specification language and Extensions of Gallina describe the concrete syntax as well as the meaning of programs, theorems and proofs in the Calculus of Inductive Constructions. Chapter The Coq library describes the standard library of Coq. Chapter Calculus of Inductive Constructions is a mathematical description of the formalism. Chapter The Module System describes the module system.
  • The second part describes the proof engine. It is divided in six chapters. Chapter Vernacular commands 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 Proof handling. In Chapter Tactics, 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 The tactic language. Examples of tactics are described in Chapter Detailed examples of tactics. Finally, the SSReflect proof language is presented in Chapter The SSReflect proof language.
  • The third part describes how to extend the syntax of Coq in Chapter Syntax extensions and interpretation scopes and how to define new induction principles in Chapter Proof schemes.
  • In the fourth part more practical tools are documented. First in Chapter The Coq commands, the usage of coqc (batch mode) and coqtop (interactive mode) with their options is described. Then, in Chapter Utilities, various utilities that come with the Coq distribution are presented. Finally, Chapter Coq Integrated Development Environment describes CoqIDE.
  • 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.

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:

Tutorial
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).
Installation
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 available at https://coq.inria.fr/stdlib/.

Table of contents

Indexes

Preamble

The language

The proof engine

User extensions

Practical tools

Addendum

Reference

License

This material (the Coq Reference Manual) may be distributed only subject to the terms and conditions set forth in the Open Publication License, v1.0 or later (the latest version is presently available at http://www.opencontent.org/openpub). Options A and B are not elected.

[1]Proof-General is available at https://proofgeneral.github.io/. Optionally, you can enhance it with the minor mode Company-Coq [PCC16] (see https://github.com/cpitclaudel/company-coq).