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 :ref:`coqintegrateddevelopmentenvironment`, Emacs with Proof-General :cite:`Asp00` [#PG]_, 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.
.. seealso:: :ref:`thecoqcommands`.
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 :ref:`gallinaspecificationlanguage` and :ref:`extensionsofgallina` describe the concrete syntax as well as the meaning of programs, theorems and proofs in the Calculus of Inductive Constructions. Chapter :ref:`thecoqlibrary` describes the standard library of |Coq|. Chapter :ref:`calculusofinductiveconstructions` is a mathematical description of the formalism. Chapter :ref:`themodulesystem` describes the module system.
- The second part describes the proof engine. It is divided in six chapters. Chapter :ref:`vernacularcommands` 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 :ref:`proofhandling`. In Chapter :ref:`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 :ref:`ltac`. Examples of tactics are described in Chapter :ref:`detailedexamplesoftactics`. Finally, the |SSR| proof language is presented in Chapter :ref:`thessreflectprooflanguage`.
- The third part describes how to extend the syntax of |Coq| in Chapter :ref:`syntaxextensionsandinterpretationscopes` and how to define new induction principles in Chapter :ref:`proofschemes`.
- In the fourth part more practical tools are documented. First in Chapter :ref:`thecoqcommands`, the usage of coqc (batch mode) and coqtop (interactive mode) with their options is described. Then, in Chapter :ref:`utilities`, various utilities that come with the |Coq| distribution are presented. Finally, Chapter :ref:`coqintegrateddevelopmentenvironment` 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.
This manual does not contain all the documentation the user may need about |Coq|. Various informations can be found in the following documents:
- 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/.