\[\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}\]

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 raw universe 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 raw universe 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:

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

Command Local | Global | Export? Set flag

Sets flag on. Scoping qualifiers are described here.

Command Local | Global | Export? Unset flag

Sets flag off. Scoping qualifiers are described here.

Command Test flag

Prints the current value of flag.

Command Local | Global | Export? Set option ( num | string )

Sets option to the specified value. Scoping qualifiers are described here.

Command Local | Global | Export? 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 ( string | qualid )

Adds the specified value to table.

Command Remove table ( string | qualid )

Removes the specified value from table.

Command Test table for ( string | qualid )

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

Local | Global | Export?

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 convtactic 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).

参考

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.

参考

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%key

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 key (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 … 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 … 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 … [-]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 z_syntax_plugin.cmxs ... done] [Loading ML file quote_plugin.cmxs ... done] [Loading ML file newring_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

バージョン 8.5 で非推奨.

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 コンパイルされたファイル).

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.

注釈

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.

注釈

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.

参考

Section Locating notations

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.

参考

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.

コンパイルされたファイル

この節ではコンパイルされたファイルのロードに使うコマンドについて説明します (どのようにファイルをコンパイルするかについての文書については The Coq commands の章を参照)。コンパイルされたファイルは ライブラリファイル と呼ばれるモジュールの特殊なケースです。

Command Require qualid

このコマンドはロードパスの中でモジュール qualid を含むファイルを探し、対応するモジュールを Coq の環境に追加します。他のライブラリファイルへの依存を持つライブラリファイルについても、コマンド Require qualid はモジュール qualid の依存する全てのライブラリファイルを再帰的に要求し、対応するモジュールを Coq の環境へ追加します。Coq はコンパイルされたファイルは正しい Coq コンパイラによって生成されたことを仮定し、従ってそれらの内容は再実行も再検証もされません。

ファイルシステム上でそのファイルの場所を探すため、qualid は dirpath.ident という形式に分解され、Coq のロードパスから論理パスである dirpath (節 Libraries and filesystem 参照) へマップされたファイルシステムの物理ディレクトリの中でファイル ident.vo が検索されます。そのファイルを要求した時点での物理ディレクトリと論理名の間のマッピングはそのファイルのコンパイルに使われるマッピングと一貫していなければなりません。幾つかのファイルがマッチした場合、そのうちの一つが不特定の方法で選択されます。

Variant Require Import qualid

これは qualid モジュールとその依存をロードし宣言し、そして ここ で説明されるように qualid の内容をインポートします。それは qualid が依存するモジュールを、これらのモジュールが以下で説明されるように、それら自身が Require Export を使う qualid モジュール内で要求されるか、または再帰的に Require Export の列を通して要求されない限りインポートしません。もし要求されたモジュールが既にロードされていたら、Require Import qualid は Import qualid がそうするように単にそれをインポートします。

Variant Require Export qualid

このコマンドは Require Import qualid のように振る舞いますが、さらなるモジュール(A と称する)がコマンド Require Export B を含む場合、コマンド Require Import A はモジュール B もインポートします。

Variant Require [Import | Export] qualid+

これは qualid の列により指名されたモジュールとそれらの再帰的な依存をロードします。Import か Export が与えられると、それはさらにこれらのモジュールと Export としてマークまたは推移的にマークされた全ての再帰的依存をインポートします。

Variant From dirpath Require qualid

このコマンドは Require のように振る舞いますが、その絶対名が幾つかの dirpath' について dirpath.dirpath’.qualid という形式であるようなライブラリを選別します。これは qualid ライブラリがその絶対ルートを明示することによって与えられたパッケージに由来することを保証するために有用です。

Error Cannot load qualid: no physical path bound to dirpath.
Error Cannot find library foo in loadpath.

そのコマンドはファイル foo.vo を見つけられませんでした。foo.v は存在するがコンパイルされていないか foo.vo はあなたの LoadPath 内には無い (節 Libraries and filesystem 参照) ディレクトリにあるかのどちらかです。

Error Compiled library ident.vo makes inconsistent assumptions over library qualid.

そのコマンドは qualid ライブラリのいくつかの特定のバージョンに依存するライブラリファイル ident.vo をロードしようとしましたが、それは現在の Coq セッションに既にロードされているものではありません。恐らく ident.v はモジュール qualid を含む最新のバージョンのファイルとともに正しく再コンパイルされていません。

Error Bad magic number.

ファイル ident.vo は見つかりましたが、それは Coq がコンパイルしたモジュールではないか、またはそれは互換性の無いバージョンの Coq でコンパイルされたのいずれかです。

Error The file `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.

このコマンドはモジュールまたは定義されているモジュール型の中では許可されません。それはコンパイル単位間の依存関係を記述することを意図されています。ただし、コマンド Import と Export 単独ではモジュール内部で使うことが出来ます (節 Import 参照)。

参考

章 The Coq commands

Command Print Libraries

このコマンドは現在の Coq セッションにロードされたライブラリファイルのリストを表示します。これらのライブラリそれぞれについて、それがインポートされているならばそれも知らせます。

Command Declare ML Module string+

このコマンドは OCaml がコンパイルしたファイルを string の列で与えられた名前でロードします (動的リンク)。それは主にタクティックを動的にロードするために使用されます。そのファイルは現在の OCaml の loadpath の中も検索されます (Libraries and filesystem 節の Add ML Path コマンドを参照)。OCaml ファイルの読み込みはバイトコード版の coqtop (i.e. coqtop が -byte オプションを伴って呼ばれた、The Coq commands の章を参照) かまたは、Coq がネイティブ Dynlink (≥ 3.11) をサポートする OCaml のバージョンによってコンパイルされたときのみ可能です。

Variant Local Declare ML Module string+

この派生版はたとえそれがセクションの外であっても、それらが現れたモジュールをインポートしたモジュールへエクスポートされません。

Error File not found on loadpath: string.
Error Loading of ML object file forbidden in a native Coq.
Command Print ML Modules

これは Declare ML Module で読み込まれた全ての OCaml モジュールの名前を表示します。これらのモジュールがどこから読み込まれたのかを知るためには、ユーザは Locate File コマンド (ここ を参照) を使うべきです。

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 as 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 コンパイルされたファイル).

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 コンパイルされたファイル).

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.

Variant Backtrack num num num

バージョン 8.4 で非推奨.

Backtrack is a deprecated form of BackTo which allows explicitly manipulating the proof environment. The three numbers represent the following:

  • first number : State label to reach, as for BackTo.

  • second number : Proof state number to unbury once aborts have been done. Coq will compute the number of Undo to perform (see Chapter Proof handling).

  • third number : Number of Abort to perform, i.e. the number of currently opened nested proofs that must be canceled (see Chapter Proof handling).

Error Invalid backtrack.

The destination state label is unknown.

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();;

警告

  1. It only works with the bytecode version of Coq (i.e. coqtop.byte, see Section interactive-use).

  2. 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, the Fail statement succeeds, without changing the proof state, and in interactive mode, the system prints a message confirming the failure. If the given command succeeds, the statement is an error, and it prints a message indicating that the failure did not occur.

Error The command has not failed!

Controlling display

Flag Silent

This option 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 option 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 option 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 option 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 option controls the printing of the “(dependent evars: …)” line when -emacs is passed.

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.

参考

Sections Performing computations, Automating, Switching on/off the proof editing mode

Error The reference qualid was not found in the current environment.

There is no constant referred by qualid in the environment. Nevertheless, if you asked Opaque foo bar and if bar does not exist, foo is set opaque.

Command Transparent qualid+

This command is the converse of Opaque and it applies on unfoldable constants to restore their unfoldability after an Opaque command.

Note in particular that constants defined by a proof ended by Qed are not unfoldable and Transparent has no effect on them. This is to keep with the usual mathematical practice of proof irrelevance: what matters in a mathematical development is the sequence of lemma statements, not their actual proofs. This distinguishes lemmas from the usual defined constants, whose actual values are of course relevant in general.

Variant Global Transparent qualid+

The scope of Transparent is limited to the current section, or current file, unless the variant Global Transparent is used.

Error The reference qualid was not found in the current environment.

There is no constant referred by qualid in the environment.

参考

Sections Performing computations, Automating, Switching on/off the proof editing mode

Command Strategy level [ qualid+ ]

This command generalizes the behavior of Opaque and Transparent commands. It is used to fine-tune the strategy for unfolding constants, both at the tactic level and at the kernel level. This command associates a level to the qualified names in the qualid sequence. Whenever two expressions with two distinct head constants are compared (for instance, this comparison can be triggered by a type cast), the one with lower level is expanded first. In case of a tie, the second one (appearing in the cast type) is expanded.

Levels can be one of the following (higher to lower):

  • opaque : level of opaque constants. They cannot be expanded by tactics (behaves like +∞, see next item).

  • num : levels indexed by an integer. Level 0 corresponds to the default behavior, which corresponds to transparent constants. This level can also be referred to as transparent. Negative levels correspond to constants to be expanded before normal transparent constants, while positive levels correspond to constants to be expanded after normal transparent constants.

  • expand : level of constants that should be expanded first (behaves like −∞)

Variant Local Strategy level [ qualid+ ]

These directives survive section and module closure, unless the command is prefixed by Local. In the latter case, the behavior regarding sections and modules is the same as for the Transparent and Opaque commands.

Command Print Strategy qualid

This command prints the strategy currently associated to qualid. It fails if qualid is not an unfoldable reference, that is, neither a variable nor a constant.

Error The reference is not unfoldable.
Variant Print Strategies

Print all the currently non-transparent strategies.

Command Declare Reduction ident := convtactic

This command allows giving a short name to a reduction expression, for instance lazy beta delta [foo bar]. This short name can then be used in Eval ident in ... or eval directives. This command accepts the Local modifier, for discarding this reduction name at the end of the file or module. For the moment the name cannot be qualified. In particular declaring the same name in several modules or in several functor applications will be refused if these declarations are not local. The name ident cannot be used directly as an Ltac tactic, but nothing prevents the user to also perform a Ltac ident := convtactic.

参考

Performing computations

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.