\[\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{\plus}{\mathsf{plus}} \newcommand{\Prod}{\textsf{prod}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\si}{\textsf{if}} \newcommand{\sinon}{\textsf{else}} \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{\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]{{\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}]} \newcommand{\zeros}{\textsf{zeros}} \end{split}\]

Vernacular commands¶

Displaying¶

Command Print qualid¶

This command displays on the screen information about the declared or defined object referred by qualid.

Error messages:

Error qualid not a defined object.¶

Error Universe instance should have length num.¶

Error This object does not support universe names.¶

Variant Print Term qualid¶: This is a synonym of Print qualid when qualid denotes a global constant.

Variant Print Term? qualid@name: This locally renames the polymorphic universes of qualid. An underscore means the usual name is printed.

Command About qualid¶

This displays various information about the object denoted by qualid: its kind (module, constant, assumption, inductive, constructor, abbreviation, …), long name, type, implicit arguments and argument scopes. It does not print the body of definitions or proofs.

Variant About qualid@name: This locally renames the polymorphic universes of qualid. An underscore means the usual name is printed.

Command Print All¶

This command displays information about the current state of the environment, including sections and modules.

Variant Inspect num¶: This command displays the num last objects of the current environment, including sections and modules.

Variant Print Section ident: The name ident should correspond to a currently open section, this command displays the objects defined since the beginning of this section.

Flags, Options and Tables¶

Coq has many settings to control its behavior. Setting types include flags, options and tables:

A flag has a boolean value, such as Asymmetric Patterns.
An option generally has a numeric or string value, such as Firstorder Depth.
A table contains a set of strings or qualids.
In addition, some commands provide settings, such as Extraction Language.

Flags, options and tables are identified by a series of identifiers, each with an initial capital letter.

Command LocalGlobalExport? Set flag¶: Sets flag on. Scoping qualifiers are described here.

Command LocalGlobalExport? Unset flag¶: Sets flag off. Scoping qualifiers are described here.

Command Test flag¶: Prints the current value of flag.

Command LocalGlobalExport? Set option numstring¶: Sets option to the specified value. Scoping qualifiers are described here.

Command LocalGlobalExport? Unset option¶: Sets option to its default value. Scoping qualifiers are described here.

Command Test option: Prints the current value of option.

Command Print Options¶: Prints the current value of all flags and options, and the names of all tables.

Command Add table stringqualid¶: Adds the specified value to table.

Command Remove table stringqualid¶: Removes the specified value from table.

Command Test table for stringqualid¶: Reports whether table contains the specified value.

Command Print Table table¶: Prints the values in table.

Command Test table: A synonym for Print Table @table.

Command Print Tables¶: A synonym for Print Options.

Scope qualifiers for `Set` and `Unset`¶

LocalGlobalExport?

Flag and option settings can be global in scope or local to nested scopes created by Module and Section commands. There are four alternatives:

no qualifier: the original setting is not restored at the end of the current module or section.
Local: the setting is applied within the current scope. The original value of the option or flag is restored at the end of the current module or section.
Global: similar to no qualifier, the original setting is not restored at the end of the current module or section. In addition, if the value is set in a file, then Require-ing the file sets the option.
Export: similar to Local, the original value of the option or flag is restored at the end of the current module or section. In addition, if the value is set in a file, then Import-ing the file sets the option.

Newly opened scopes inherit the current settings.

Requests to the environment¶

Command Check term¶

This command displays the type of term. When called in proof mode, the term is checked in the local context of the current subgoal.

Variant selector: Check term: This variant specifies on which subgoal to perform typing (see Section Invocation of tactics).

Command Eval redexpr in term¶: This command performs the specified reduction on term, and displays the resulting term with its type. The term to be reduced may depend on hypothesis introduced in the first subgoal (if a proof is in progress).

See also

Section Performing computations.

Command Compute term¶: This command performs a call-by-value evaluation of term by using the bytecode-based virtual machine. It is a shortcut for Eval vm_compute in term.

See also

Section Performing computations.

Command Print Assumptions qualid¶

This commands display all the assumptions (axioms, parameters and variables) a theorem or definition depends on. Especially, it informs on the assumptions with respect to which the validity of a theorem relies.

Variant Print Opaque Dependencies qualid¶: Displays the set of opaque constants qualid relies on in addition to the assumptions.

Variant Print Transparent Dependencies qualid¶: Displays the set of transparent constants qualid relies on in addition to the assumptions.

Variant Print All Dependencies qualid¶: Displays all assumptions and constants qualid relies on.

Command Search qualid¶

This command displays the name and type of all objects (hypothesis of the current goal, theorems, axioms, etc) of the current context whose statement contains qualid. This command is useful to remind the user of the name of library lemmas.

Error The reference qualid was not found in the current environment.¶: There is no constant in the environment named qualid.

Variant Search string: If string is a valid identifier, this command displays the name and type of all objects (theorems, axioms, etc) of the current context whose name contains string. If string is a notation’s string denoting some reference qualid (referred to by its main symbol as in "+" or by its notation’s string as in "_ + _" or "_ 'U' _", see Section Notations), the command works like Search qualid.

Variant Search string%ident: The string string must be a notation or the main symbol of a notation which is then interpreted in the scope bound to the delimiting key ident (see Section Local interpretation rules for notations).

Variant Search term_pattern: This searches for all statements or types of definition that contains a subterm that matches the pattern term_pattern (holes of the pattern are either denoted by _ or by ?ident when non linear patterns are expected).

Variant Search -?term_pattern_string+: where term_pattern_string is a term_pattern, a string, or a string followed by a scope delimiting key %key. This generalization of Search searches for all objects whose statement or type contains a subterm matching term_pattern (or qualid if string is the notation for a reference qualid) and whose name contains all string of the request that correspond to valid identifiers. If a term_pattern or a string is prefixed by -, the search excludes the objects that mention that term_pattern or that string.

Variant Search -?term_pattern_string+ inside qualid+: This restricts the search to constructions defined in the modules named by the given qualid sequence.

Variant Search -?term_pattern_string+ outside qualid+: This restricts the search to constructions not defined in the modules named by the given qualid sequence.

Variant selector: Search -?term_pattern_string+

This specifies the goal on which to search hypothesis (see Section Invocation of tactics). By default the 1st goal is searched. This variant can be combined with other variants presented here.

Example

Require Import ZArith.: [Loading ML file newring_plugin.cmxs ... done] [Loading ML file zify_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done]

Search Z.mul Z.add "distr".: Z.mul_add_distr_l: forall n m p : Z, (n * (m + p))%Z = (n * m + n * p)%Z Z.mul_add_distr_r: forall n m p : Z, ((n + m) * p)%Z = (n * p + m * p)%Z fast_Zmult_plus_distr_l: forall (n m p : Z) (P : Z -> Prop), P (n * p + m * p)%Z -> P ((n + m) * p)%Z
Search "+"%Z "*"%Z "distr" -positive -Prop.: Z.mul_add_distr_l: forall n m p : Z, (n * (m + p))%Z = (n * m + n * p)%Z Z.mul_add_distr_r: forall n m p : Z, ((n + m) * p)%Z = (n * p + m * p)%Z
Search (?x * _ + ?x * _)%Z outside OmegaLemmas.: Z.mul_add_distr_l: forall n m p : Z, (n * (m + p))%Z = (n * m + n * p)%Z

Variant SearchAbout¶: Deprecated since version 8.5.

Up to Coq version 8.4, Search had the behavior of current SearchHead and the behavior of current Search was obtained with command SearchAbout. For compatibility, the deprecated name SearchAbout can still be used as a synonym of Search. For compatibility, the list of objects to search when using SearchAbout may also be enclosed by optional [ ] delimiters.

Command SearchHead term¶

This command displays the name and type of all hypothesis of the current goal (if any) and theorems of the current context whose statement’s conclusion has the form (term t1 .. tn). This command is useful to remind the user of the name of library lemmas.

Example

SearchHead le.: le_n: forall n : nat, n <= n le_0_n: forall n : nat, 0 <= n le_S: forall n m : nat, n <= m -> n <= S m le_pred: forall n m : nat, n <= m -> Nat.pred n <= Nat.pred m le_n_S: forall n m : nat, n <= m -> S n <= S m le_S_n: forall n m : nat, S n <= S m -> n <= m
SearchHead (@eq bool).: andb_true_intro: forall b1 b2 : bool, b1 = true /\ b2 = true -> (b1 && b2)%bool = true

Variant SearchHead term inside qualid+: This restricts the search to constructions defined in the modules named by the given qualid sequence.

Variant SearchHead term outside qualid+: This restricts the search to constructions not defined in the modules named by the given qualid sequence.

Error Module/section qualid not found.¶: No module qualid has been required (see Section Compiled files).

Variant selector: SearchHead term: This specifies the goal on which to search hypothesis (see Section Invocation of tactics). By default the 1st goal is searched. This variant can be combined with other variants presented here.

Note

Up to Coq version 8.4, SearchHead was named Search.

Command SearchPattern term¶

This command displays the name and type of all hypothesis of the current goal (if any) and theorems of the current context whose statement’s conclusion or last hypothesis and conclusion matches the expressionterm where holes in the latter are denoted by _. It is a variant of Search term_pattern that does not look for subterms but searches for statements whose conclusion has exactly the expected form, or whose statement finishes by the given series of hypothesis/conclusion.

Example

Require Import Arith.

SearchPattern (_ + _ = _ + _).: Nat.add_comm: forall n m : nat, n + m = m + n plus_Snm_nSm: forall n m : nat, S n + m = n + S m Nat.add_succ_comm: forall n m : nat, S n + m = n + S m Nat.add_shuffle3: forall n m p : nat, n + (m + p) = m + (n + p) plus_assoc_reverse: forall n m p : nat, n + m + p = n + (m + p) Nat.add_assoc: forall n m p : nat, n + (m + p) = n + m + p Nat.add_shuffle0: forall n m p : nat, n + m + p = n + p + m f_equal2_plus: forall x1 y1 x2 y2 : nat, x1 = y1 -> x2 = y2 -> x1 + x2 = y1 + y2 Nat.add_shuffle2: forall n m p q : nat, n + m + (p + q) = n + q + (m + p) Nat.add_shuffle1: forall n m p q : nat, n + m + (p + q) = n + p + (m + q)
SearchPattern (nat -> bool).: Nat.odd: nat -> bool Init.Nat.odd: nat -> bool Nat.even: nat -> bool Init.Nat.even: nat -> bool Init.Nat.testbit: nat -> nat -> bool Nat.leb: nat -> nat -> bool Nat.eqb: nat -> nat -> bool Init.Nat.eqb: nat -> nat -> bool Nat.ltb: nat -> nat -> bool Nat.testbit: nat -> nat -> bool Init.Nat.leb: nat -> nat -> bool Init.Nat.ltb: nat -> nat -> bool BinNat.N.testbit_nat: BinNums.N -> nat -> bool BinPosDef.Pos.testbit_nat: BinNums.positive -> nat -> bool BinPos.Pos.testbit_nat: BinNums.positive -> nat -> bool BinNatDef.N.testbit_nat: BinNums.N -> nat -> bool
SearchPattern (forall l : list _, _ l l).: List.incl_refl: forall (A : Type) (l : list A), List.incl l l List.lel_refl: forall (A : Type) (l : list A), List.lel l l

Patterns need not be linear: you can express that the same expression must occur in two places by using pattern variables ?ident.

Example

SearchPattern (?X1 + _ = _ + ?X1).: Nat.add_comm: forall n m : nat, n + m = m + n

Variant SearchPattern term inside qualid+: This restricts the search to constructions defined in the modules named by the given qualid sequence.

Variant SearchPattern term outside qualid+: This restricts the search to constructions not defined in the modules named by the given qualid sequence.

Variant selector: SearchPattern term: This specifies the goal on which to search hypothesis (see Section Invocation of tactics). By default the 1st goal is searched. This variant can be combined with other variants presented here.

Command SearchRewrite term¶

This command displays the name and type of all hypothesis of the current goal (if any) and theorems of the current context whose statement’s conclusion is an equality of which one side matches the expression term. Holes in term are denoted by “_”.

Example

Require Import Arith.

SearchRewrite (_ + _ + _).: Nat.add_shuffle0: forall n m p : nat, n + m + p = n + p + m plus_assoc_reverse: forall n m p : nat, n + m + p = n + (m + p) Nat.add_assoc: forall n m p : nat, n + (m + p) = n + m + p Nat.add_shuffle1: forall n m p q : nat, n + m + (p + q) = n + p + (m + q) Nat.add_shuffle2: forall n m p q : nat, n + m + (p + q) = n + q + (m + p) Nat.add_carry_div2: forall (a b : nat) (c0 : bool), (a + b + Nat.b2n c0) / 2 = a / 2 + b / 2 + Nat.b2n (Nat.testbit a 0 && Nat.testbit b 0 || c0 && (Nat.testbit a 0 || Nat.testbit b 0))

Variant SearchRewrite term inside qualid+: This restricts the search to constructions defined in the modules named by the given qualid sequence.

Variant SearchRewrite term outside qualid+: This restricts the search to constructions not defined in the modules named by the given qualid sequence.

Variant selector: SearchRewrite term: This specifies the goal on which to search hypothesis (see Section Invocation of tactics). By default the 1st goal is searched. This variant can be combined with other variants presented here.

Note

Table Search Blacklist string¶

Specifies a set of strings used to exclude lemmas from the results of Search, SearchHead, SearchPattern and SearchRewrite queries. A lemma whose fully-qualified name contains any of the strings will be excluded from the search results. The default blacklisted substrings are _subterm, _subproof and Private_.

Use the Add @table and Remove @table commands to update the set of blacklisted strings.

Command Locate qualid¶

This command displays the full name of objects whose name is a prefix of the qualified identifier qualid, and consequently the Coq module in which they are defined. It searches for objects from the different qualified namespaces of Coq: terms, modules, Ltac, etc.

Example

Locate nat.: Inductive Coq.Init.Datatypes.nat
Locate Datatypes.O.: Constructor Coq.Init.Datatypes.O (shorter name to refer to it in current context is O)
Locate Init.Datatypes.O.: Constructor Coq.Init.Datatypes.O (shorter name to refer to it in current context is O)
Locate Coq.Init.Datatypes.O.: Constructor Coq.Init.Datatypes.O (shorter name to refer to it in current context is O)
Locate I.Dont.Exist.: No object of suffix I.Dont.Exist

Variant Locate Term qualid: As Locate but restricted to terms.

Variant Locate Module qualid: As Locate but restricted to modules.

Variant Locate Ltac qualid: As Locate but restricted to tactics.

Printing flags¶

Flag Fast Name Printing¶: When turned on, Coq uses an asymptotically faster algorithm for the generation of unambiguous names of bound variables while printing terms. While faster, it is also less clever and results in a typically less elegant display, e.g. it will generate more names rather than reusing certain names across subterms. This flag is not enabled by default, because as Ltac observes bound names, turning it on can break existing proof scripts.

Loading files¶

Coq offers the possibility of loading different parts of a whole development stored in separate files. Their contents will be loaded as if they were entered from the keyboard. This means that the loaded files are ASCII files containing sequences of commands for Coq’s toplevel. This kind of file is called a script for Coq. The standard (and default) extension of Coq’s script files is .v.

Command Load ident¶

This command loads the file named ident.v, searching successively in each of the directories specified in the loadpath. (see Section Libraries and filesystem)

Files loaded this way cannot leave proofs open, and the Load command cannot be used inside a proof either.

Variant Load string: Loads the file denoted by the string string, where string is any complete filename. Then the ~ and .. abbreviations are allowed as well as shell variables. If no extension is specified, Coq will use the default extension .v.

Variant Load Verbose ident
Variant Load Verbose string: Display, while loading, the answers of Coq to each command (including tactics) contained in the loaded file.

See also

Section Controlling display.

Error Can’t find file ident on loadpath.¶

Error Load is not supported inside proofs.¶

Error Files processed by Load cannot leave open proofs.¶

Compiled files¶

This section describes the commands used to load compiled files (see Chapter The Coq commands for documentation on how to compile a file). A compiled file is a particular case of module called library file.

Command Require qualid¶

This command looks in the loadpath for a file containing module qualid and adds the corresponding module to the environment of Coq. As library files have dependencies in other library files, the command Require qualid recursively requires all library files the module qualid depends on and adds the corresponding modules to the environment of Coq too. Coq assumes that the compiled files have been produced by a valid Coq compiler and their contents are then not replayed nor rechecked.

To locate the file in the file system, qualid is decomposed under the form dirpath.ident and the file ident.vo is searched in the physical directory of the file system that is mapped in Coq loadpath to the logical path dirpath (see Section Libraries and filesystem). The mapping between physical directories and logical names at the time of requiring the file must be consistent with the mapping used to compile the file. If several files match, one of them is picked in an unspecified fashion.

Variant Require Import qualid¶: This loads and declares the module qualid and its dependencies then imports the contents of qualid as described here. It does not import the modules on which qualid depends unless these modules were themselves required in module qualid using Require Export, as described below, or recursively required through a sequence of Require Export. If the module required has already been loaded, Require Import qualid simply imports it, as Import qualid would.

Variant Require Export qualid¶: This command acts as Require Import qualid, but if a further module, say A, contains a command Require Export B, then the command Require Import A also imports the module B.

Variant Require ImportExport qualid+: This loads the modules named by the qualid sequence and their recursive dependencies. If Import or Export is given, it also imports these modules and all the recursive dependencies that were marked or transitively marked as Export.

Variant From dirpath Require qualid¶: This command acts as Require, but picks any library whose absolute name is of the form dirpath.dirpath’.qualid for some dirpath’. This is useful to ensure that the qualid library comes from a given package by making explicit its absolute root.

Error Cannot load qualid: no physical path bound to dirpath.¶

Error Cannot find library foo in loadpath.¶: The command did not find the file foo.vo. Either foo.v exists but is not compiled or foo.vo is in a directory which is not in your LoadPath (see Section Libraries and filesystem).

Error Compiled library ident.vo makes inconsistent assumptions over library qualid.¶: The command tried to load library file ident.vo that depends on some specific version of library qualid which is not the one already loaded in the current Coq session. Probably ident.v was not properly recompiled with the last version of the file containing module qualid.

Error Bad magic number.¶: The file ident.vo was found but either it is not a Coq compiled module, or it was compiled with an incompatible version of Coq.

Error The file :n:`ident.vo` contains library dirpath and not library dirpath’.¶: The library file dirpath’ is indirectly required by the Require command but it is bound in the current loadpath to the file ident.vo which was bound to a different library name dirpath at the time it was compiled.

Error Require is not allowed inside a module or a module type.¶: This command is not allowed inside a module or a module type being defined. It is meant to describe a dependency between compilation units. Note however that the commands Import and Export alone can be used inside modules (see Section Import).

See also

Chapter The Coq commands

Command Print Libraries¶: This command displays the list of library files loaded in the current Coq session.

Command Declare ML Module string+¶

This commands loads the OCaml compiled files with names given by the string sequence (dynamic link). It is mainly used to load tactics dynamically. The files are searched into the current OCaml loadpath (see the command Add ML Path). Loading of OCaml files is only possible under the bytecode version of coqtop (i.e. coqtop called with option -byte, see chapter The Coq commands), or when Coq has been compiled with a version of OCaml that supports native Dynlink (≥ 3.11).

Variant Local Declare ML Module string+: This variant is not exported to the modules that import the module where they occur, even if outside a section.

Error File not found on loadpath: string.¶

Error Loading of ML object file forbidden in a native Coq.¶

Command Print ML Modules¶: This prints the name of all OCaml modules loaded with Declare ML Module. To know from where these module were loaded, the user should use the command Locate File.

Loadpath¶

Loadpaths are preferably managed using Coq command line options (see Section libraries-and-filesystem) but there remain vernacular commands to manage them for practical purposes. Such commands are only meant to be issued in the toplevel, and using them in source files is discouraged.

Command Pwd¶: This command displays the current working directory.

Command Cd string¶

This command changes the current directory according to string which can be any valid path.

Variant Cd: Is equivalent to Pwd.

Command Add LoadPath string as dirpath¶

This command is equivalent to the command line option -Q string dirpath. It adds the physical directory string to the current Coq loadpath and maps it to the logical directory dirpath.

Variant Add LoadPath string: Performs as Add LoadPath string dirpath but for the empty directory path.

Command Add Rec LoadPath string as dirpath¶

This command is equivalent to the command line option -R string dirpath. It adds the physical directory string and all its subdirectories to the current Coq loadpath.

Variant Add Rec LoadPath string: Works as Add Rec LoadPath string as dirpath but for the empty logical directory path.

Command Remove LoadPath string¶: This command removes the path string from the current Coq loadpath.

Command Print LoadPath¶

This command displays the current Coq loadpath.

Variant Print LoadPath dirpath: Works as Print LoadPath but displays only the paths that extend the dirpath prefix.

Command Add ML Path string¶: This command adds the path string to the current OCaml loadpath (see the command Declare ML Module` in Section Compiled files).

Command Add Rec ML Path string¶: This command adds the directory string and all its subdirectories to the current OCaml loadpath (see the command Declare ML Module).

Command Print ML Path string¶: This command displays the current OCaml loadpath. This command makes sense only under the bytecode version of coqtop, i.e. using option -byte (see the command Declare ML Module in Section Compiled files).

Command Locate File string¶: This command displays the location of file string in the current loadpath. Typically, string is a .cmo or .vo or .v file.

Command Locate Library dirpath¶: This command gives the status of the Coq module dirpath. It tells if the module is loaded and if not searches in the load path for a module of logical name dirpath.

Backtracking¶

The backtracking commands described in this section can only be used interactively, they cannot be part of a vernacular file loaded via Load or compiled by coqc.

Command Reset ident¶

This command removes all the objects in the environment since ident was introduced, including ident. ident may be the name of a defined or declared object as well as the name of a section. One cannot reset over the name of a module or of an object inside a module.

Error ident: no such entry.¶

Variant Reset Initial: Goes back to the initial state, just after the start of the interactive session.

Command Back¶

This command undoes all the effects of the last vernacular command. Commands read from a vernacular file via a Load are considered as a single command. Proof management commands are also handled by this command (see Chapter Proof handling). For that, Back may have to undo more than one command in order to reach a state where the proof management information is available. For instance, when the last command is a Qed, the management information about the closed proof has been discarded. In this case, Back will then undo all the proof steps up to the statement of this proof.

Variant Back num: Undo num vernacular commands. As for Back, some extra commands may be undone in order to reach an adequate state. For instance Back num will not re-enter a closed proof, but rather go just before that proof.

Error Invalid backtrack.¶: The user wants to undo more commands than available in the history.

Command BackTo num¶: This command brings back the system to the state labeled num, forgetting the effect of all commands executed after this state. The state label is an integer which grows after each successful command. It is displayed in the prompt when in -emacs mode. Just as Back (see above), the BackTo command now handles proof states. For that, it may have to undo some extra commands and end on a state num′ ≤ num if necessary.

Quitting and debugging¶

Command Quit¶: This command permits to quit Coq.

Command Drop¶

This is used mostly as a debug facility by Coq’s implementers and does not concern the casual user. This command permits to leave Coq temporarily and enter the OCaml toplevel. The OCaml command:

#use "include";;

adds the right loadpaths and loads some toplevel printers for all abstract types of Coq- section_path, identifiers, terms, judgments, …. You can also use the file base_include instead, that loads only the pretty-printers for section_paths and identifiers. You can return back to Coq with the command:

go();;

Warning

It only works with the bytecode version of Coq (i.e. coqtop.byte, see Section interactive-use).
You must have compiled Coq from the source package and set the environment variable COQTOP to the root of your copy of the sources (see Section customization-by-environment-variables).

Command Time command¶: This command executes the vernacular command command and displays the time needed to execute it.

Command Redirect string command¶: This command executes the vernacular command command, redirecting its output to "string.out".

Command Timeout num command¶

This command executes the vernacular command command. If the command has not terminated after the time specified by the num (time expressed in seconds), then it is interrupted and an error message is displayed.

Option Default Timeout num¶: This option controls a default timeout for subsequent commands, as if they were passed to a Timeout command. Commands already starting by a Timeout are unaffected.

Command Fail command¶

For debugging scripts, sometimes it is desirable to know whether a command or a tactic fails. If the given command fails, then Fail command succeeds (excepts in the case of critical errors, like a "stack overflow"), without changing the proof state, and in interactive mode, the system prints a message confirming the failure.

Error The command has not failed!¶: If the given command succeeds, then Fail command fails with this error message.

Controlling display¶

Flag Silent¶: This flag controls the normal displaying.

Option Warnings "-+? ident+,"¶: This option configures the display of warnings. It is experimental, and expects, between quotes, a comma-separated list of warning names or categories. Adding - in front of a warning or category disables it, adding + makes it an error. It is possible to use the special categories all and default, the latter containing the warnings enabled by default. The flags are interpreted from left to right, so in case of an overlap, the flags on the right have higher priority, meaning that A,-A is equivalent to -A.

Flag Search Output Name Only¶: This flag restricts the output of search commands to identifier names; turning it on causes invocations of Search, SearchHead, SearchPattern, SearchRewrite etc. to omit types from their output, printing only identifiers.

Option Printing Width num¶: This command sets which left-aligned part of the width of the screen is used for display. At the time of writing this documentation, the default value is 78.

Option Printing Depth num¶: This option controls the nesting depth of the formatter used for pretty- printing. Beyond this depth, display of subterms is replaced by dots. At the time of writing this documentation, the default value is 50.

Flag Printing Compact Contexts¶: This flag controls the compact display mode for goals contexts. When on, the printer tries to reduce the vertical size of goals contexts by putting several variables (even if of different types) on the same line provided it does not exceed the printing width (see Printing Width). At the time of writing this documentation, it is off by default.

Flag Printing Unfocused¶: This flag controls whether unfocused goals are displayed. Such goals are created by focusing other goals with bullets (see Bullets or curly braces). It is off by default.

Flag Printing Dependent Evars Line¶: This flag controls the printing of the “(dependent evars: …)” information after each tactic. The information is used by the Prooftree tool in Proof General. (https://askra.de/software/prooftree)

Controlling the reduction strategies and the conversion algorithm¶

Coq provides reduction strategies that the tactics can invoke and two different algorithms to check the convertibility of types. The first conversion algorithm lazily compares applicative terms while the other is a brute-force but efficient algorithm that first normalizes the terms before comparing them. The second algorithm is based on a bytecode representation of terms similar to the bytecode representation used in the ZINC virtual machine [Ler90]. It is especially useful for intensive computation of algebraic values, such as numbers, and for reflection-based tactics. The commands to fine- tune the reduction strategies and the lazy conversion algorithm are described first.

Command Opaque qualid+¶

This command has an effect on unfoldable constants, i.e. on constants defined by Definition or Let (with an explicit body), or by a command assimilated to a definition such as Fixpoint, Program Definition, etc, or by a proof ended by Defined. The command tells not to unfold the constants in the qualid sequence in tactics using δ-conversion (unfolding a constant is replacing it by its definition).

Opaque has also an effect on the conversion algorithm of Coq, telling it to delay the unfolding of a constant as much as possible when Coq has to check the conversion (see Section Conversion rules) of two distinct applied constants.

Variant Global Opaque qualid+¶: The scope of Opaque is limited to the current section, or current file, unless the variant Global Opaque is used.

Controlling the locality of commands¶

Command Local command¶

Command Global command¶

Some commands support a Local or Global prefix modifier to control the scope of their effect. There are four kinds of commands:

Commands whose default is to extend their effect both outside the section and the module or library file they occur in. For these commands, the Local modifier limits the effect of the command to the current section or module it occurs in. As an example, the Coercion and Strategy commands belong to this category.
Commands whose default behavior is to stop their effect at the end of the section they occur in but to extend their effect outside the module or library file they occur in. For these commands, the Local modifier limits the effect of the command to the current module if the command does not occur in a section and the Global modifier extends the effect outside the current sections and current module if the command occurs in a section. As an example, the Arguments, Ltac or Notation commands belong to this category. Notice that a subclass of these commands do not support extension of their scope outside sections at all and the Global modifier is not applicable to them.
Commands whose default behavior is to stop their effect at the end of the section or module they occur in. For these commands, the Global modifier extends their effect outside the sections and modules they occur in. The Transparent and Opaque (see Section Controlling the reduction strategies and the conversion algorithm) commands belong to this category.
Commands whose default behavior is to extend their effect outside sections but not outside modules when they occur in a section and to extend their effect outside the module or library file they occur in when no section contains them. For these commands, the Local modifier limits the effect to the current section or module while the Global modifier extends the effect outside the module even when the command occurs in a section. The Set and Unset commands belong to this category.

Controlling Typing Flags¶

Flag Guard Checking¶: This flag can be used to enable/disable the guard checking of fixpoints. Warning: this can break the consistency of the system, use at your own risk. Decreasing argument can still be specified: the decrease is not checked anymore but it still affects the reduction of the term. Unchecked fixpoints are printed by Print Assumptions.

Flag Positivity Checking¶: This flag can be used to enable/disable the positivity checking of inductive types and the productivity checking of coinductive types. Warning: this can break the consistency of the system, use at your own risk. Unchecked (co)inductive types are printed by Print Assumptions.

Flag Universe Checking¶: This flag can be used to enable/disable the checking of universes, providing a form of "type in type". Warning: this breaks the consistency of the system, use at your own risk. Constants relying on "type in type" are printed by Print Assumptions. It has the same effect as -type-in-type command line argument (see By command line options).

Command Print Typing Flags¶: Print the status of the three typing flags: guard checking, positivity checking and universe checking.

Example

Unset Guard Checking.
Print Typing Flags.: check_guarded: false check_positive: true check_universes: true
Fixpoint f (n : nat) : False := f n.: f is defined f is recursively defined (decreasing on 1st argument)
Fixpoint ackermann (m n : nat) {struct m} : nat := match m with | 0 => S n | S m => match n with | 0 => ackermann m 1 | S n => ackermann m (ackermann (S m) n) end end.: ackermann is defined ackermann is recursively defined (decreasing on 1st argument)
Print Assumptions ackermann.: Axioms: ackermann is assumed to be guarded.

Note that the proper way to define the Ackermann function is to use an inner fixpoint:

Fixpoint ack m := fix ackm n := match m with | 0 => S n | S m' => match n with | 0 => ack m' 1 | S n' => ack m' (ackm n') end end.: ack is defined ack is recursively defined (decreasing on 1st argument)

Internal registration commands¶

Due to their internal nature, the commands that are presented in this section are not for general use. They are meant to appear only in standard libraries and in support libraries of plug-ins.

Exposing constants to OCaml libraries¶

Command Register qualid₁ as qualid₂¶

This command exposes the constant qualid₁ to OCaml libraries under the name qualid₂. This constant can then be dynamically located calling Coqlib.lib_ref "qualid₂"; i.e., there is no need to known where is the constant defined (file, module, library, etc.).

As a special case, when the first segment of qualid₂ is kernel, the constant is exposed to the kernel. For instance, the Int63 module features the following declaration:

Register bool as kernel.ind_bool.

This makes the kernel aware of what is the type of boolean values. This information is used for instance to define the return type of the #int63_eq primitive.

Inlining hints for the fast reduction machines¶

Command Register Inline qualid¶: This command gives as a hint to the reduction machines (VM and native) that the body of the constant qualid should be inlined in the generated code.

Registering primitive operations¶

Command Primitive ident₁ := #ident₂.¶: Declares ident₁ as the primitive operator #ident₂. When running this command, the type of the primitive should be already known by the kernel (this is achieved through this command for primitive types and through the Register command with the kernel name-space for other types).

Vernacular commands¶

Displaying¶

Flags, Options and Tables¶

Scope qualifiers for Set and Unset¶

Requests to the environment¶

Printing flags¶

Loading files¶

Compiled files¶

Loadpath¶

Backtracking¶

Quitting and debugging¶

Controlling display¶

Controlling the reduction strategies and the conversion algorithm¶

Controlling the locality of commands¶

Controlling Typing Flags¶

Internal registration commands¶

Exposing constants to OCaml libraries¶

Inlining hints for the fast reduction machines¶

Registering primitive operations¶

Scope qualifiers for `Set` and `Unset`¶