$\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \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{\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}{\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{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{#1[] \vdash #2 \lra #3}} \newcommand{\WEVT}[3]{\mbox{#1[] \vdash #2 \lra}\\ \mbox{ #3}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{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}]} \end{split}$

# The Coq commands¶

There are three Coq commands:

• coqtop: the Coq toplevel (interactive mode);

• coqc: the Coq compiler (batch compilation);

• coqchk: the Coq checker (validation of compiled libraries).

The options are (basically) the same for the first two commands, and roughly described below. You can also look at the man pages of coqtop and coqc for more details.

## Interactive use (coqtop)¶

In the interactive mode, also known as the Coq toplevel, the user can develop his theories and proofs step by step. The Coq toplevel is run by the command coqtop.

There are two different binary images of Coq: the byte-code one and the native-code one (if OCaml provides a native-code compiler for your platform, which is supposed in the following). By default, coqtop executes the native-code version; run coqtop.byte to get the byte-code version.

The byte-code toplevel is based on an OCaml toplevel (to allow dynamic linking of tactics). You can switch to the OCaml toplevel with the command Drop., and come back to the Coq toplevel with the command Coqloop.loop();;.

Flag Coqtop Exit On Error

This flag, off by default, causes coqtop to exit with status code 1 if a command produces an error instead of recovering from it.

## Batch compilation (coqc)¶

The coqc command takes a name file as argument. Then it looks for a file named file.v, and tries to compile it into a file.vo file (See Compiled files).

Caution

The name file should be a regular Coq identifier as defined in Section Lexical conventions. It should contain only letters, digits or underscores (_). For example /bar/foo/toto.v is valid, but /bar/foo/to-to.v is not.

## Customization at launch time¶

When Coq is launched, with either coqtop or coqc, the resource file $XDG_CONFIG_HOME/coq/coqrc.xxx, if it exists, will be implicitly prepended to any document read by Coq, whether it is an interactive session or a file to compile. Here, $XDG_CONFIG_HOME is the configuration directory of the user (by default it's ~/.config) and xxx is the version number (e.g. 8.8). If this file is not found, then the file $XDG_CONFIG_HOME/coqrc is searched. If not found, it is the file ~/.coqrc.xxx which is searched, and, if still not found, the file ~/.coqrc. If the latter is also absent, no resource file is loaded. You can also specify an arbitrary name for the resource file (see option -init-file below). The resource file may contain, for instance, Add LoadPath commands to add directories to the load path of Coq. It is possible to skip the loading of the resource file with the option -q. ### By environment variables¶ $COQPATH can be used to specify the load path. It is a list of directories separated by : (; on Windows). Coq will also honor $XDG_DATA_HOME and $XDG_DATA_DIRS (see Section Libraries and filesystem).

Some Coq commands call other Coq commands. In this case, they look for the commands in directory specified by $COQBIN. If this variable is not set, they look for the commands in the executable path. $COQ_COLORS can be used to specify the set of colors used by coqtop to highlight its output. It uses the same syntax as the $LS_COLORS variable from GNU’s ls, that is, a colon-separated list of assignments of the form name=attr*; where name is the name of the corresponding highlight tag and each attr is an ANSI escape code. The list of highlight tags can be retrieved with the -list-tags command-line option of coqtop. The string uses ANSI escape codes to represent attributes. For example: export COQ_COLORS=”diff.added=4;48;2;0;0;240:diff.removed=41” sets the highlights for added text in diffs to underlined (the 4) with a background RGB color (0, 0, 240) and for removed text in diffs to a red background. Note that if you specify COQ_COLORS, the predefined attributes are ignored. $OCAMLRUNPARAM, described here, can be used to specify certain runtime and memory usage parameters. In most cases, experimenting with these settings will likely not cause a significant performance difference and should be harmless.

If the variable is not set, Coq uses the default values, except that space_overhead is set to 120 and minor_heap_size is set to 32Mwords (256MB with 64-bit executables or 128MB with 32-bit executables).

### By command line options¶

The following command-line options are recognized by the commands coqc and coqtop, unless stated otherwise:

-I directory, -include directory

Names of libraries and the command Declare ML Module Section Compiled files.

-Q directory dirpath

Add physical path directory to the list of directories where Coq looks for a file and bind it to the logical directory dirpath. The subdirectory structure of directory is recursively available from Coq using absolute names (extending the dirpath prefix) (see Section Qualified identifiers). Note that only those subdirectories and files which obey the lexical conventions of what is an ident are taken into account. Conversely, the underlying file systems or operating systems may be more restrictive than Coq. While Linux’s ext4 file system supports any Coq recursive layout (within the limit of 255 bytes per filename), the default on NTFS (Windows) or HFS+ (MacOS X) file systems is on the contrary to disallow two files differing only in the case in the same directory.

Section Names of libraries.

-R directory dirpath

Do as -Q directory dirpath but make the subdirectory structure of directory recursively visible so that the recursive contents of physical directory is available from Coq using short or partially qualified names.

Section Names of libraries.

-top dirpath

Set the toplevel module name to dirpath instead of Top. Not valid for coqc as the toplevel module name is inferred from the name of the output file.

-exclude-dir directory

Exclude any subdirectory named directory while processing options such as -R and -Q. By default, only the conventional version control management directories named CVS and_darcs are excluded.

-nois, -noinit

Start from an empty state instead of loading the Init.Prelude module.

-init-file file

-q

Do not to load the default resource file.

Load and execute the Coq script from file.v.

Load and execute the Coq script from file.v. Write its contents to the standard output as it is executed.

Load Coq compiled library qualid. This is equivalent to running Require qualid.

Note

Note that the relative order of this command-line option and its variants (-rfrom, -ri, -re, etc.) and of the -set and -unset options matters since the various Require, Require Import, Require Export, Set and Unset commands will be executed in the order specified on the command-line.

-rfrom dirpath qualid

Load Coq compiled library qualid. This is equivalent to running From dirpath Require qualid. See the note above regarding the order of command-line options.

-ri qualid, -require-import qualid

Load Coq compiled library qualid and import it. This is equivalent to running Require Import qualid. See the note above regarding the order of command-line options.

-re qualid, -require-export qualid

Load Coq compiled library qualid and transitively import it. This is equivalent to running Require Export qualid. See the note above regarding the order of command-line options.

-rifrom dirpath qualid, -require-import-from dirpath qualid

Load Coq compiled library qualid and import it. This is equivalent to running From dirpath Require Import qualid. See the note above regarding the order of command-line options.

-refrom dirpath qualid, -require-export-from dirpath qualid

Load Coq compiled library qualid and transitively import it. This is equivalent to running From dirpath Require Export qualid. See the note above regarding the order of command-line options.

-batch

Exit just after argument parsing. Available for coqtop only.

-verbose

Output the content of the input file as it is compiled. This option is available for coqc only.

-native-compiler (yes|no|ondemand)

Enable the native_compute reduction machine and precompilation to .cmxs files for future use by native_compute. Setting yes enables native_compute; it also causes Coq to precompile the native code for future use; all dependencies need to have been precompiled beforehand. Setting no disables native_compute which defaults back to vm_compute; no files are precompiled. Setting ondemand enables native_compute but disables precompilation; all missing dependencies will be recompiled every time native_compute is called.

Changed in version 8.13: The default value is set at configure time, -config can be used to retrieve it. All this can be summarized in the following table:

configure

coqc

native_compute

outcome

requirements

yes

yes (default)

native_compute

.cmxs

.cmxs of deps

yes

no

vm_compute

none

none

yes

ondemand

native_compute

none

none

no

yes, no, ondemand

vm_compute

none

none

ondemand

yes

native_compute

.cmxs

.cmxs of deps

ondemand

no

vm_compute

none

none

ondemand

ondemand (default)

native_compute

none

none

-native-output-dir

Set the directory in which to put the aforementioned .cmxs for native_compute. Defaults to .coq-native.

-vos

Indicate Coq to skip the processing of opaque proofs (i.e., proofs ending with Qed or Admitted), output a .vos files instead of a .vo file, and to load .vos files instead of .vo files when interpreting Require commands.

-vok

Indicate Coq to check a file completely, to load .vos files instead of .vo files when interpreting Require commands, and to output an empty .vok files upon success instead of writing a .vo file.

-w (all|none|w₁,…,wₙ)

Configure the display of warnings. This option expects all, none or a comma-separated list of warning names or categories (see Section Controlling display).

-color (on|off|auto)

Coqtop only. Enable or disable color output. Default is auto, meaning color is shown only if the output channel supports ANSI escape sequences.

-diffs (on|off|removed)

Coqtop only. Controls highlighting of differences between proof steps. on highlights added tokens, removed highlights both added and removed tokens. Requires that -color is enabled. (see Section Showing differences between proof steps).

-beautify

Pretty-print each command to file.beautified when compiling file.v, in order to get old-fashioned syntax/definitions/notations.

-emacs, -ide-slave

Start a special toplevel to communicate with a specific IDE.

-impredicative-set

Change the logical theory of Coq by declaring the sort Set impredicative.

Warning

This is known to be inconsistent with some standard axioms of classical mathematics such as the functional axiom of choice or the principle of description.

-type-in-type

Collapse the universe hierarchy of Coq.

Warning

This makes the logic inconsistent.

-mangle-names ident

Experimental. Do not depend on this option. Replace Coq's auto-generated name scheme with names of the form ident0, ident1, etc. Within Coq, the Mangle Names flag turns this behavior on, and the Mangle Names Prefix option sets the prefix to use. This feature is intended to be used as a linter for developments that want to be robust to changes in the auto-generated name scheme. The options are provided to facilitate tracking down problems.

-set string

Enable flags and set options. string should be setting_name=value, the value is interpreted according to the type of the option. For flags setting_name is equivalent to setting_name=true. For instance -set "Universe Polymorphism" will enable Universe Polymorphism. Note that the quotes are shell syntax, Coq does not see them. See the note above regarding the order of command-line options.

-unset string

As -set but used to disable options and flags. string must be "setting_name". See the note above regarding the order of command-line options.

-compat version

Load a file that sets a few options to maintain partial backward-compatibility with a previous version. This is equivalent to Require Import Coq.Compat.CoqXXX with XXX one of the last three released versions (including the current version). Note that the explanations above regarding the order of command-line options apply, and this could be relevant if you are resetting some of the compatibility options.

-dump-glob file

Dump references for global names in file file (to be used by coqdoc, see Documenting Coq files with coqdoc). By default, if file.v is being compiled, file.glob is used.

-no-glob

Disable the dumping of references for global names.

-image file

Set the binary image to be used by coqc to be file instead of the standard one. Not of general use.

-bindir directory

Set the directory containing Coq binaries to be used by coqc. It is equivalent to doing export COQBIN= directory before launching coqc.

-where

Print the location of Coq’s standard library and exit.

-config

Print the locations of Coq’s binaries, dependencies, and libraries, then exit.

-filteropts

Print the list of command line arguments that coqtop has recognized as options and exit.

-v

Print Coq’s version and exit.

-list-tags

Print the highlight tags known by Coq as well as their currently associated color and exit.

-h, --help

Print a short usage and exit.

## Compiled interfaces (produced using -vos)¶

Compiled interfaces help saving time while developing Coq formalizations, by compiling the formal statements exported by a library independently of the proofs that it contains.

Warning

Compiled interfaces should only be used for development purposes. At the end of the day, one still needs to proof check all files by producing standard .vo files. (Technically, when using -vos, fewer universe constraints are collected.) Moreover, this feature is still experimental, it may be subject to change without prior notice.

Principle.

The compilation using coqc -vos foo.v produces a file called foo.vos, which is similar to foo.vo except that all opaque proofs are skipped in the compilation process.

The compilation using coqc -vok foo.v checks that the file foo.v correctly compiles, including all its opaque proofs. If the compilation succeeds, then the output is a file called foo.vok, with empty contents. This file is only a placeholder indicating that foo.v has been successfully compiled. (This placeholder is useful for build systems such as make.)

When compiling a file bar.v that depends on foo.v (for example via a Require Foo. command), if the compilation command is coqc -vos bar.v or coqc -vok bar.v, then the file foo.vos gets loaded (instead of foo.vo). A special case is if file foo.vos exists and has empty contents, and foo.vo exists, then foo.vo is loaded.

Appart from the aforementioned case where foo.vo can be loaded in place of foo.vos, in general the .vos and .vok files live totally independently from the .vo files.

Dependencies generated by coq_makefile.

The files foo.vos and foo.vok both depend on foo.v.

Furthermore, if a file foo.v requires bar.v, then foo.vos and foo.vok also depend on bar.vos.

Note, however, that foo.vok does not depend on bar.vok. Hence, as detailed further, parallel compilation of proofs is possible.

In addition, coq_makefile generates for a file foo.v a target foo.required_vos which depends on the list of .vos files that foo.vos depends upon (excluding foo.vos itself). As explained next, the purpose of this target is to be able to request the minimal working state for editing interactively the file foo.v.

Warning

When writing a custom build system, be aware that coqdep only produces dependencies related to .vos and .vok if the -vos command line flag is passed. This is to maintain compatibility with dune (see ocaml/dune#2642 on github).

Typical compilation of a set of file using a build system.

Assume a file foo.v that depends on two files f1.v and f2.v. The command make foo.required_vos will compile f1.v and f2.v using the option -vos to skip the proofs, producing f1.vos and f2.vos. At this point, one is ready to work interactively on the file foo.v, even though it was never needed to compile the proofs involved in the files f1.v and f2.v.

Assume a set of files f1.v ... fn.v with linear dependencies. The command make vos enables compiling the statements (i.e. excluding the proofs) in all the files. Next, make -j vok enables compiling all the proofs in parallel. Thus, calling make -j vok directly enables taking advantage of a maximal amount of parallelism during the compilation of the set of files.

Note that this comes at the cost of parsing and typechecking all definitions twice, once for the .vos file and once for the .vok file. However, if files contain nontrivial proofs, or if the files have many linear chains of dependencies, or if one has many cores available, compilation should be faster overall.

Need for Proof using

When a theorem is part of a section, typechecking the statement of this theorem might be insufficient for deducing the type of this statement as of at the end of the section. Indeed, the proof of the theorem could make use of section variables or section hypotheses that are not mentioned in the statement of the theorem.

For this reason, proofs inside section should begin with Proof using instead of Proof, where after the using clause one should provide the list of the names of the section variables that are required for the proof but are not involved in the typechecking of the statement. Note that it is safe to write Proof using. instead of Proof. also for proofs that are not within a section.

Warning You should use the “Proof using [...].” syntax instead of “Proof.” to enable skipping this proof which is located inside a section. Give as argument to “Proof using” the list of section variables that are not needed to typecheck the statement but that are required by the proof.

If Coq is invoked using the -vos option, whenever it finds the command Proof. inside a section, it will compile the proof, that is, refuse to skip it, and it will raise a warning. To disable the warning, one may pass the flag -w -proof-without-using-in-section.

Interaction with standard compilation

When compiling a file foo.v using coqc in the standard way (i.e., without -vos nor -vok), an empty file foo.vos and an empty file foo.vok are created in addition to the regular output file foo.vo. If coqc is subsequently invoked on some other file bar.v using option -vos or -vok, and that bar.v requires foo.v, if Coq finds an empty file foo.vos, then it will load foo.vo instead of foo.vos.

The purpose of this feature is to allow users to benefit from the -vos option even if they depend on libraries that were compiled in the traditional manner (i.e., never compiled using the -vos option).

## Compiled libraries checker (coqchk)¶

The coqchk command takes a list of library paths as argument, described either by their logical name or by their physical filename, which must end in .vo. The corresponding compiled libraries (.vo files) are searched in the path, recursively processing the libraries they depend on. The content of all these libraries is then type checked. The effect of coqchk is only to return with normal exit code in case of success, and with positive exit code if an error has been found. Error messages are not deemed to help the user understand what is wrong. In the current version, it does not modify the compiled libraries to mark them as successfully checked.

Note that non-logical information is not checked. By logical information, we mean the type and optional body associated with names. It excludes for instance anything related to the concrete syntax of objects (customized syntax rules, association between short and long names), implicit arguments, etc.

This tool can be used for several purposes. One is to check that a compiled library provided by a third-party has not been forged and that loading it cannot introduce inconsistencies 1. Another point is to get an even higher level of security. Since coqtop can be extended with custom tactics, possibly ill-typed code, it cannot be guaranteed that the produced compiled libraries are correct. coqchk is a standalone verifier, and thus it cannot be tainted by such malicious code.

Command-line options -Q, -R, -where and -impredicative-set are supported by coqchk and have the same meaning as for coqtop. As there is no notion of relative paths in object files -Q and -R have exactly the same meaning.

-norec module

Check module but do not check its dependencies.

Do not check module and any of its dependencies, unless explicitly required.

-o

At exit, print a summary about the context. List the names of all assumptions and variables (constants without body).

-silent

Do not write progress information to the standard output.

Environment variable \$COQLIB can be set to override the location of the standard library.

The algorithm for deciding which modules are checked or admitted is the following: assuming that coqchk is called with argument M, option -norec N, and -admit A. Let us write $$\overline{S}$$ for the set of reflexive transitive dependencies of set $$S$$. Then:

• Modules $$C = \overline{M} \backslash \overline{A} \cup M \cup N$$ are loaded and type checked before being added to the context.

• And $$M \cup N \backslash C$$ is the set of modules that are loaded and added to the context without type checking. Basic integrity checks (checksums) are nonetheless performed.

As a rule of thumb, -admit can be used to tell Coq that some libraries have already been checked. So coqchk A B can be split in coqchk A && coqchk B -admit A without type checking any definition twice. Of course, the latter is slightly slower since it makes more disk access. It is also less secure since an attacker might have replaced the compiled library A after it has been read by the first command, but before it has been read by the second command.

1

Ill-formed non-logical information might for instance bind Coq.Init.Logic.True to short name False, so apparently False is inhabited, but using fully qualified names, Coq.Init.Logic.False will always refer to the absurd proposition, what we guarantee is that there is no proof of this latter constant.