From 7e1cce5938e0ef4c2f87df5ae2b11f51e6c00442 Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 12:59:04 +0200 Subject: Extract lexical conventions and attributes from Gallina chapter. --- .../language/gallina-specification-language.rst | 1586 -------------------- 1 file changed, 1586 deletions(-) (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index 186a23897d..4fadc8da02 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -1,43 +1,3 @@ -.. _gallinaspecificationlanguage: - ------------------------------------- - The Gallina specification language ------------------------------------- - -This chapter describes Gallina, the specification language of Coq. It allows -developing mathematical theories and to prove specifications of programs. The -theories are built from axioms, hypotheses, parameters, lemmas, theorems and -definitions of constants, functions, predicates and sets. The syntax of logical -objects involved in theories is described in Section :ref:`term`. The -language of commands, called *The Vernacular* is described in Section -:ref:`vernacular`. - -In Coq, logical objects are typed to ensure their logical correctness. The -rules implemented by the typing algorithm are described in Chapter :ref:`calculusofinductiveconstructions`. - - -.. About the grammars in the manual - ================================ - - Grammars are presented in Backus-Naur form (BNF). Terminal symbols are - set in black ``typewriter font``. In addition, there are special notations for - regular expressions. - - An expression enclosed in square brackets ``[…]`` means at most one - occurrence of this expression (this corresponds to an optional - component). - - The notation “``entry sep … sep entry``” stands for a non empty sequence - of expressions parsed by entry and separated by the literal “``sep``” [1]_. - - Similarly, the notation “``entry … entry``” stands for a non empty - sequence of expressions parsed by the “``entry``” entry, without any - separator between. - - At the end, the notation “``[entry sep … sep entry]``” stands for a - possibly empty sequence of expressions parsed by the “``entry``” entry, - separated by the literal “``sep``”. - .. _lexical-conventions: Lexical conventions @@ -127,1518 +87,6 @@ Other tokens ``~~`` generate different tokens, whereas if `~~` is not defined, then the two inputs are equivalent. -.. _term: - -Terms -===== - -Syntax of terms ---------------- - -The following grammars describe the basic syntax of the terms of the -*Calculus of Inductive Constructions* (also called Cic). The formal -presentation of Cic is given in Chapter :ref:`calculusofinductiveconstructions`. Extensions of this syntax -are given in Chapter :ref:`extensionsofgallina`. How to customize the syntax -is described in Chapter :ref:`syntaxextensionsandnotationscopes`. - -.. insertprodn term field_def - -.. prodn:: - term ::= forall @open_binders , @term - | fun @open_binders => @term - | @term_let - | if @term {? {? as @name } return @term100 } then @term else @term - | @term_fix - | @term_cofix - | @term100 - term100 ::= @term_cast - | @term10 - term10 ::= @term1 {+ @arg } - | @ @qualid {? @univ_annot } {* @term1 } - | @term1 - arg ::= ( @ident := @term ) - | @term1 - one_term ::= @term1 - | @ @qualid {? @univ_annot } - term1 ::= @term_projection - | @term0 % @scope_key - | @term0 - term0 ::= @qualid {? @univ_annot } - | @sort - | @numeral - | @string - | _ - | @term_evar - | @term_match - | ( @term ) - | %{%| {* @field_def } %|%} - | `%{ @term %} - | `( @term ) - | ltac : ( @ltac_expr ) - field_def ::= @qualid {* @binder } := @term - -.. note:: - - Many commands and tactics use :n:`@one_term` rather than :n:`@term`. - The former need to be enclosed in parentheses unless they're very - simple, such as a single identifier. This avoids confusing a space-separated - list of terms with a :n:`@term1` applied to a list of arguments. - -.. _types: - -Types ------ - -.. prodn:: - type ::= @term - -:n:`@type`\s are a subset of :n:`@term`\s; not every :n:`@term` is a :n:`@type`. -Every term has an associated type, which -can be determined by applying the :ref:`typing-rules`. Distinct terms -may share the same type, for example 0 and 1 are both of type `nat`, the -natural numbers. - -.. _gallina-identifiers: - -Qualified identifiers and simple identifiers --------------------------------------------- - -.. insertprodn qualid field_ident - -.. prodn:: - qualid ::= @ident {* @field_ident } - field_ident ::= .@ident - -*Qualified identifiers* (:n:`@qualid`) denote *global constants* -(definitions, lemmas, theorems, remarks or facts), *global variables* -(parameters or axioms), *inductive types* or *constructors of inductive -types*. *Simple identifiers* (or shortly :n:`@ident`) are a syntactic subset -of qualified identifiers. Identifiers may also denote *local variables*, -while qualified identifiers do not. - -Field identifiers, written :n:`@field_ident`, are identifiers prefixed by -`.` (dot) with no blank between the dot and the identifier. - - -Numerals and strings --------------------- - -Numerals and strings have no predefined semantics in the calculus. They are -merely notations that can be bound to objects through the notation mechanism -(see Chapter :ref:`syntaxextensionsandnotationscopes` for details). -Initially, numerals are bound to Peano’s representation of natural -numbers (see :ref:`datatypes`). - -.. note:: - - Negative integers are not at the same level as :n:`@num`, for this - would make precedence unnatural. - -.. index:: - single: Set (sort) - single: SProp - single: Prop - single: Type - -Sorts ------ - -.. insertprodn sort univ_constraint - -.. prodn:: - sort ::= Set - | Prop - | SProp - | Type - | Type @%{ _ %} - | Type @%{ @universe %} - universe ::= max ( {+, @universe_expr } ) - | @universe_expr - universe_expr ::= @universe_name {? + @num } - universe_name ::= @qualid - | Set - | Prop - univ_annot ::= @%{ {* @universe_level } %} - universe_level ::= Set - | Prop - | Type - | _ - | @qualid - univ_decl ::= @%{ {* @ident } {? + } {? %| {*, @univ_constraint } {? + } } %} - univ_constraint ::= @universe_name {| < | = | <= } @universe_name - -There are four sorts :g:`SProp`, :g:`Prop`, :g:`Set` and :g:`Type`. - -- :g:`SProp` is the universe of *definitionally irrelevant - propositions* (also called *strict propositions*). - -- :g:`Prop` is the universe of *logical propositions*. The logical propositions - themselves are typing the proofs. We denote propositions by :n:`@form`. - This constitutes a semantic subclass of the syntactic class :n:`@term`. - -- :g:`Set` is the universe of *program types* or *specifications*. The - specifications themselves are typing the programs. We denote - specifications by :n:`@specif`. This constitutes a semantic subclass of - the syntactic class :n:`@term`. - -- :g:`Type` is the type of sorts. - -More on sorts can be found in Section :ref:`sorts`. - -.. _binders: - -Binders -------- - -.. insertprodn open_binders binder - -.. prodn:: - open_binders ::= {+ @name } : @term - | {+ @binder } - name ::= _ - | @ident - binder ::= @name - | ( {+ @name } : @type ) - | ( @name {? : @type } := @term ) - | @implicit_binders - | @generalizing_binder - | ( @name : @type %| @term ) - | ' @pattern0 - -Various constructions such as :g:`fun`, :g:`forall`, :g:`fix` and :g:`cofix` -*bind* variables. A binding is represented by an identifier. If the binding -variable is not used in the expression, the identifier can be replaced by the -symbol :g:`_`. When the type of a bound variable cannot be synthesized by the -system, it can be specified with the notation :n:`(@ident : @type)`. There is also -a notation for a sequence of binding variables sharing the same type: -:n:`({+ @ident} : @type)`. A -binder can also be any pattern prefixed by a quote, e.g. :g:`'(x,y)`. - -Some constructions allow the binding of a variable to value. This is -called a “let-binder”. The entry :n:`@binder` of the grammar accepts -either an assumption binder as defined above or a let-binder. The notation in -the latter case is :n:`(@ident := @term)`. In a let-binder, only one -variable can be introduced at the same time. It is also possible to give -the type of the variable as follows: -:n:`(@ident : @type := @term)`. - -Lists of :n:`@binder`\s are allowed. In the case of :g:`fun` and :g:`forall`, -it is intended that at least one binder of the list is an assumption otherwise -fun and forall gets identical. Moreover, parentheses can be omitted in -the case of a single sequence of bindings sharing the same type (e.g.: -:g:`fun (x y z : A) => t` can be shortened in :g:`fun x y z : A => t`). - -.. index:: fun ... => ... - -Abstractions: fun ------------------ - -The expression :n:`fun @ident : @type => @term` defines the -*abstraction* of the variable :n:`@ident`, of type :n:`@type`, over the term -:n:`@term`. It denotes a function of the variable :n:`@ident` that evaluates to -the expression :n:`@term` (e.g. :g:`fun x : A => x` denotes the identity -function on type :g:`A`). The keyword :g:`fun` can be followed by several -binders as given in Section :ref:`binders`. Functions over -several variables are equivalent to an iteration of one-variable -functions. For instance the expression -:n:`fun {+ @ident__i } : @type => @term` -denotes the same function as :n:`{+ fun @ident__i : @type => } @term`. If -a let-binder occurs in -the list of binders, it is expanded to a let-in definition (see -Section :ref:`let-in`). - -.. index:: forall - -Products: forall ----------------- - -The expression :n:`forall @ident : @type, @term` denotes the -*product* of the variable :n:`@ident` of type :n:`@type`, over the term :n:`@term`. -As for abstractions, :g:`forall` is followed by a binder list, and products -over several variables are equivalent to an iteration of one-variable -products. Note that :n:`@term` is intended to be a type. - -If the variable :n:`@ident` occurs in :n:`@term`, the product is called -*dependent product*. The intention behind a dependent product -:g:`forall x : A, B` is twofold. It denotes either -the universal quantification of the variable :g:`x` of type :g:`A` -in the proposition :g:`B` or the functional dependent product from -:g:`A` to :g:`B` (a construction usually written -:math:`\Pi_{x:A}.B` in set theory). - -Non dependent product types have a special notation: :g:`A -> B` stands for -:g:`forall _ : A, B`. The *non dependent product* is used both to denote -the propositional implication and function types. - -Applications ------------- - -:n:`@term__fun @term` denotes applying the function :n:`@term__fun` to :token:`term`. - -:n:`@term__fun {+ @term__i }` denotes applying -:n:`@term__fun` to the arguments :n:`@term__i`. It is -equivalent to :n:`( … ( @term__fun @term__1 ) … ) @term__n`: -associativity is to the left. - -The notation :n:`(@ident := @term)` for arguments is used for making -explicit the value of implicit arguments (see -Section :ref:`explicit-applications`). - -.. index:: - single: ... : ... (type cast) - single: ... <: ... - single: ... <<: ... - -Type cast ---------- - -.. insertprodn term_cast term_cast - -.. prodn:: - term_cast ::= @term10 <: @term - | @term10 <<: @term - | @term10 : @term - | @term10 :> - -The expression :n:`@term : @type` is a type cast expression. It enforces -the type of :n:`@term` to be :n:`@type`. - -:n:`@term <: @type` locally sets up the virtual machine for checking that -:n:`@term` has type :n:`@type`. - -:n:`@term <<: @type` uses native compilation for checking that :n:`@term` -has type :n:`@type`. - -.. index:: _ - -Inferable subterms ------------------- - -Expressions often contain redundant pieces of information. Subterms that can be -automatically inferred by Coq can be replaced by the symbol ``_`` and Coq will -guess the missing piece of information. - -.. index:: let ... := ... (term) - -.. _let-in: - -Let-in definitions ------------------- - -.. insertprodn term_let term_let - -.. prodn:: - term_let ::= let @name {? : @type } := @term in @term - | let @name {+ @binder } {? : @type } := @term in @term - | let ( {*, @name } ) {? {? as @name } return @term100 } := @term in @term - | let ' @pattern := @term {? return @term100 } in @term - | let ' @pattern in @pattern := @term return @term100 in @term - -:n:`let @ident := @term in @term’` -denotes the local binding of :n:`@term` to the variable -:n:`@ident` in :n:`@term`’. There is a syntactic sugar for let-in -definition of functions: :n:`let @ident {+ @binder} := @term in @term’` -stands for :n:`let @ident := fun {+ @binder} => @term in @term’`. - -.. index:: match ... with ... - -Definition by cases: match --------------------------- - -.. insertprodn term_match pattern0 - -.. prodn:: - term_match ::= match {+, @case_item } {? return @term100 } with {? %| } {*| @eqn } end - case_item ::= @term100 {? as @name } {? in @pattern } - eqn ::= {+| {+, @pattern } } => @term - pattern ::= @pattern10 : @term - | @pattern10 - pattern10 ::= @pattern1 as @name - | @pattern1 {* @pattern1 } - | @ @qualid {* @pattern1 } - pattern1 ::= @pattern0 % @scope_key - | @pattern0 - pattern0 ::= @qualid - | %{%| {* @qualid := @pattern } %|%} - | _ - | ( {+| @pattern } ) - | @numeral - | @string - -Objects of inductive types can be destructured by a case-analysis -construction called *pattern matching* expression. A pattern matching -expression is used to analyze the structure of an inductive object and -to apply specific treatments accordingly. - -This paragraph describes the basic form of pattern matching. See -Section :ref:`Mult-match` and Chapter :ref:`extendedpatternmatching` for the description -of the general form. The basic form of pattern matching is characterized -by a single :n:`@case_item` expression, an :n:`@eqn` restricted to a -single :n:`@pattern` and :n:`@pattern` restricted to the form -:n:`@qualid {* @ident}`. - -The expression -:n:`match @term {? return @term100 } with {+| @pattern__i => @term__i } end` denotes a -*pattern matching* over the term :n:`@term` (expected to be -of an inductive type :math:`I`). The :n:`@term__i` -are the *branches* of the pattern matching -expression. Each :n:`@pattern__i` has the form :n:`@qualid @ident` -where :n:`@qualid` must denote a constructor. There should be -exactly one branch for every constructor of :math:`I`. - -The :n:`return @term100` clause gives the type returned by the whole match -expression. There are several cases. In the *non dependent* case, all -branches have the same type, and the :n:`return @term100` specifies that type. -In this case, :n:`return @term100` can usually be omitted as it can be -inferred from the type of the branches [1]_. - -In the *dependent* case, there are three subcases. In the first subcase, -the type in each branch may depend on the exact value being matched in -the branch. In this case, the whole pattern matching itself depends on -the term being matched. This dependency of the term being matched in the -return type is expressed with an :n:`@ident` clause where :n:`@ident` -is dependent in the return type. For instance, in the following example: - -.. coqtop:: in - - Inductive bool : Type := true : bool | false : bool. - Inductive eq (A:Type) (x:A) : A -> Prop := eq_refl : eq A x x. - Inductive or (A:Prop) (B:Prop) : Prop := - | or_introl : A -> or A B - | or_intror : B -> or A B. - - Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) := - match b as x return or (eq bool x true) (eq bool x false) with - | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) - | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) - end. - -the branches have respective types ":g:`or (eq bool true true) (eq bool true false)`" -and ":g:`or (eq bool false true) (eq bool false false)`" while the whole -pattern matching expression has type ":g:`or (eq bool b true) (eq bool b false)`", -the identifier :g:`b` being used to represent the dependency. - -.. note:: - - When the term being matched is a variable, the ``as`` clause can be - omitted and the term being matched can serve itself as binding name in - the return type. For instance, the following alternative definition is - accepted and has the same meaning as the previous one. - - .. coqtop:: none - - Reset bool_case. - - .. coqtop:: in - - Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) := - match b return or (eq bool b true) (eq bool b false) with - | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) - | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) - end. - -The second subcase is only relevant for annotated inductive types such -as the equality predicate (see Section :ref:`coq-equality`), -the order predicate on natural numbers or the type of lists of a given -length (see Section :ref:`matching-dependent`). In this configuration, the -type of each branch can depend on the type dependencies specific to the -branch and the whole pattern matching expression has a type determined -by the specific dependencies in the type of the term being matched. This -dependency of the return type in the annotations of the inductive type -is expressed with a clause in the form -:n:`in @qualid {+ _ } {+ @pattern }`, where - -- :n:`@qualid` is the inductive type of the term being matched; - -- the holes :n:`_` match the parameters of the inductive type: the - return type is not dependent on them. - -- each :n:`@pattern` matches the annotations of the - inductive type: the return type is dependent on them - -- in the basic case which we describe below, each :n:`@pattern` - is a name :n:`@ident`; see :ref:`match-in-patterns` for the - general case - -For instance, in the following example: - -.. coqtop:: in - - Definition eq_sym (A:Type) (x y:A) (H:eq A x y) : eq A y x := - match H in eq _ _ z return eq A z x with - | eq_refl _ _ => eq_refl A x - end. - -the type of the branch is :g:`eq A x x` because the third argument of -:g:`eq` is :g:`x` in the type of the pattern :g:`eq_refl`. On the contrary, the -type of the whole pattern matching expression has type :g:`eq A y x` because the -third argument of eq is y in the type of H. This dependency of the case analysis -in the third argument of :g:`eq` is expressed by the identifier :g:`z` in the -return type. - -Finally, the third subcase is a combination of the first and second -subcase. In particular, it only applies to pattern matching on terms in -a type with annotations. For this third subcase, both the clauses ``as`` and -``in`` are available. - -There are specific notations for case analysis on types with one or two -constructors: ``if … then … else …`` and ``let (…,…) := … in …`` (see -Sections :ref:`if-then-else` and :ref:`irrefutable-patterns`). - -.. index:: - single: fix - single: cofix - -Recursive and co-recursive functions: fix and cofix ---------------------------------------------------- - -.. insertprodn term_fix fixannot - -.. prodn:: - term_fix ::= let fix @fix_body in @term - | fix @fix_body {? {+ with @fix_body } for @ident } - fix_body ::= @ident {* @binder } {? @fixannot } {? : @type } := @term - fixannot ::= %{ struct @ident %} - | %{ wf @one_term @ident %} - | %{ measure @one_term {? @ident } {? @one_term } %} - - -The expression ":n:`fix @ident__1 @binder__1 : @type__1 := @term__1 with … with @ident__n @binder__n : @type__n := @term__n for @ident__i`" denotes the -:math:`i`-th component of a block of functions defined by mutual structural -recursion. It is the local counterpart of the :cmd:`Fixpoint` command. When -:math:`n=1`, the ":n:`for @ident__i`" clause is omitted. - -The association of a single fixpoint and a local definition have a special -syntax: :n:`let fix @ident {* @binder } := @term in` stands for -:n:`let @ident := fix @ident {* @binder } := @term in`. The same applies for co-fixpoints. - -Some options of :n:`@fixannot` are only supported in specific constructs. :n:`fix` and :n:`let fix` -only support the :n:`struct` option, while :n:`wf` and :n:`measure` are only supported in -commands such as :cmd:`Function` and :cmd:`Program Fixpoint`. - -.. insertprodn term_cofix cofix_body - -.. prodn:: - term_cofix ::= let cofix @cofix_body in @term - | cofix @cofix_body {? {+ with @cofix_body } for @ident } - cofix_body ::= @ident {* @binder } {? : @type } := @term - -The expression -":n:`cofix @ident__1 @binder__1 : @type__1 with … with @ident__n @binder__n : @type__n for @ident__i`" -denotes the :math:`i`-th component of a block of terms defined by a mutual guarded -co-recursion. It is the local counterpart of the :cmd:`CoFixpoint` command. When -:math:`n=1`, the ":n:`for @ident__i`" clause is omitted. - -.. _vernacular: - -The Vernacular -============== - -.. insertprodn vernacular sentence - -.. prodn:: - vernacular ::= {* @sentence } - sentence ::= {? @all_attrs } @command . - | {? @all_attrs } {? @num : } @query_command . - | {? @all_attrs } {? @toplevel_selector } @ltac_expr {| . | ... } - | @control_command - -The top-level input to |Coq| is a series of :n:`@sentence`\s, -which are :production:`tactic`\s or :production:`command`\s, -generally terminated with a period -and optionally decorated with :ref:`gallina-attributes`. :n:`@ltac_expr` syntax supports both simple -and compound tactics. For example: ``split`` is a simple tactic while ``split; auto`` combines two -simple tactics. - -Tactics specify how to transform the current proof state as a step in creating a proof. They -are syntactically valid only when |Coq| is in proof mode, such as after a :cmd:`Theorem` command -and before any subsequent proof-terminating command such as :cmd:`Qed`. See :ref:`proofhandling` for more -on proof mode. - -By convention, command names begin with uppercase letters, while -tactic names begin with lowercase letters. Commands appear in the -HTML documentation in blue boxes after the label "Command". In the pdf, they appear -after the boldface label "Command:". Commands are listed in the :ref:`command_index`. - -Similarly, tactics appear after the label "Tactic". Tactics are listed in the :ref:`tactic_index`. - -.. _gallina-assumptions: - -Assumptions ------------ - -Assumptions extend the environment with axioms, parameters, hypotheses -or variables. An assumption binds an :n:`@ident` to a :n:`@type`. It is accepted -by Coq if and only if this :n:`@type` is a correct type in the environment -preexisting the declaration and if :n:`@ident` was not previously defined in -the same module. This :n:`@type` is considered to be the type (or -specification, or statement) assumed by :n:`@ident` and we say that :n:`@ident` -has type :n:`@type`. - -.. _Axiom: - -.. cmd:: @assumption_token {? Inline {? ( @num ) } } {| {+ ( @assumpt ) } | @assumpt } - :name: Axiom; Axioms; Conjecture; Conjectures; Hypothesis; Hypotheses; Parameter; Parameters; Variable; Variables - - .. insertprodn assumption_token of_type - - .. prodn:: - assumption_token ::= {| Axiom | Axioms } - | {| Conjecture | Conjectures } - | {| Parameter | Parameters } - | {| Hypothesis | Hypotheses } - | {| Variable | Variables } - assumpt ::= {+ @ident_decl } @of_type - ident_decl ::= @ident {? @univ_decl } - of_type ::= {| : | :> | :>> } @type - - These commands bind one or more :n:`@ident`\(s) to specified :n:`@type`\(s) as their specifications in - the global context. The fact asserted by the :n:`@type` (or, equivalently, the existence - of an object of this type) is accepted as a postulate. - - :cmd:`Axiom`, :cmd:`Conjecture`, :cmd:`Parameter` and their plural forms - are equivalent. They can take the :attr:`local` attribute (see :ref:`gallina-attributes`), - which makes the defined :n:`@ident`\s accessible by :cmd:`Import` and its variants - only through their fully qualified names. - - Similarly, :cmd:`Hypothesis`, :cmd:`Variable` and their plural forms are equivalent. Outside - of a section, these are equivalent to :n:`Local Parameter`. Inside a section, the - :n:`@ident`\s defined are only accessible within the section. When the current section - is closed, the :n:`@ident`\(s) become undefined and every object depending on them will be explicitly - parameterized (i.e., the variables are *discharged*). See Section :ref:`section-mechanism`. - - The :n:`Inline` clause is only relevant inside functors. See :cmd:`Module`. - -.. example:: Simple assumptions - - .. coqtop:: reset in - - Parameter X Y : Set. - Parameter (R : X -> Y -> Prop) (S : Y -> X -> Prop). - Axiom R_S_inv : forall x y, R x y <-> S y x. - -.. exn:: @ident already exists. - :name: @ident already exists. (Axiom) - :undocumented: - -.. warn:: @ident is declared as a local axiom - - Warning generated when using :cmd:`Variable` or its equivalent - instead of :n:`Local Parameter` or its equivalent. - -.. note:: - We advise using the commands :cmd:`Axiom`, :cmd:`Conjecture` and - :cmd:`Hypothesis` (and their plural forms) for logical postulates (i.e. when - the assertion :n:`@type` is of sort :g:`Prop`), and to use the commands - :cmd:`Parameter` and :cmd:`Variable` (and their plural forms) in other cases - (corresponding to the declaration of an abstract object of the given type). - -.. _gallina-definitions: - -Definitions ------------ - -Definitions extend the environment with associations of names to terms. -A definition can be seen as a way to give a meaning to a name or as a -way to abbreviate a term. In any case, the name can later be replaced at -any time by its definition. - -The operation of unfolding a name into its definition is called -:math:`\delta`-conversion (see Section :ref:`delta-reduction`). A -definition is accepted by the system if and only if the defined term is -well-typed in the current context of the definition and if the name is -not already used. The name defined by the definition is called a -*constant* and the term it refers to is its *body*. A definition has a -type which is the type of its body. - -A formal presentation of constants and environments is given in -Section :ref:`typing-rules`. - -.. cmd:: {| Definition | Example } @ident_decl @def_body - :name: Definition; Example - - .. insertprodn def_body def_body - - .. prodn:: - def_body ::= {* @binder } {? : @type } := {? @reduce } @term - | {* @binder } : @type - - These commands bind :n:`@term` to the name :n:`@ident` in the environment, - provided that :n:`@term` is well-typed. They can take the :attr:`local` attribute (see :ref:`gallina-attributes`), - which makes the defined :n:`@ident` accessible by :cmd:`Import` and its variants - only through their fully qualified names. - If :n:`@reduce` is present then :n:`@ident` is bound to the result of the specified - computation on :n:`@term`. - - These commands also support the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`program` and - :attr:`canonical` attributes. - - If :n:`@term` is omitted, :n:`@type` is required and Coq enters proof editing mode. - This can be used to define a term incrementally, in particular by relying on the :tacn:`refine` tactic. - In this case, the proof should be terminated with :cmd:`Defined` in order to define a constant - for which the computational behavior is relevant. See :ref:`proof-editing-mode`. - - The form :n:`Definition @ident : @type := @term` checks that the type of :n:`@term` - is definitionally equal to :n:`@type`, and registers :n:`@ident` as being of type - :n:`@type`, and bound to value :n:`@term`. - - The form :n:`Definition @ident {* @binder } : @type := @term` is equivalent to - :n:`Definition @ident : forall {* @binder }, @type := fun {* @binder } => @term`. - - .. seealso:: :cmd:`Opaque`, :cmd:`Transparent`, :tacn:`unfold`. - - .. exn:: @ident already exists. - :name: @ident already exists. (Definition) - :undocumented: - - .. exn:: The term @term has type @type while it is expected to have type @type'. - :undocumented: - -.. _gallina-inductive-definitions: - -Inductive types ---------------- - -.. cmd:: Inductive @inductive_definition {* with @inductive_definition } - - .. insertprodn inductive_definition constructor - - .. prodn:: - inductive_definition ::= {? > } @ident_decl {* @binder } {? %| {* @binder } } {? : @type } {? := {? @constructors_or_record } } {? @decl_notations } - constructors_or_record ::= {? %| } {+| @constructor } - | {? @ident } %{ {*; @record_field } %} - constructor ::= @ident {* @binder } {? @of_type } - - This command defines one or more - inductive types and its constructors. Coq generates destructors - depending on the universe that the inductive type belongs to. - - The destructors are named :n:`@ident`\ ``_rect``, :n:`@ident`\ ``_ind``, - :n:`@ident`\ ``_rec`` and :n:`@ident`\ ``_sind``, which - respectively correspond to elimination principles on :g:`Type`, :g:`Prop`, - :g:`Set` and :g:`SProp`. The type of the destructors - expresses structural induction/recursion principles over objects of - type :n:`@ident`. The constant :n:`@ident`\ ``_ind`` is always - generated, whereas :n:`@ident`\ ``_rec`` and :n:`@ident`\ ``_rect`` - may be impossible to derive (for example, when :n:`@ident` is a - proposition). - - This command supports the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`universes(template)`, - :attr:`universes(notemplate)`, :attr:`universes(cumulative)`, - :attr:`universes(noncumulative)` and :attr:`private(matching)` - attributes. - - Mutually inductive types can be defined by including multiple :n:`@inductive_definition`\s. - The :n:`@ident`\s are simultaneously added to the environment before the types of constructors are checked. - Each :n:`@ident` can be used independently thereafter. - See :ref:`mutually_inductive_types`. - - If the entire inductive definition is parameterized with :n:`@binder`\s, the parameters correspond - to a local context in which the entire set of inductive declarations is interpreted. - For this reason, the parameters must be strictly the same for each inductive type. - See :ref:`parametrized-inductive-types`. - - Constructor :n:`@ident`\s can come with :n:`@binder`\s, in which case - the actual type of the constructor is :n:`forall {* @binder }, @type`. - - .. exn:: Non strictly positive occurrence of @ident in @type. - - The types of the constructors have to satisfy a *positivity condition* - (see Section :ref:`positivity`). This condition ensures the soundness of - the inductive definition. The positivity checking can be disabled using - the :flag:`Positivity Checking` flag (see :ref:`controlling-typing-flags`). - - .. exn:: The conclusion of @type is not valid; it must be built from @ident. - - The conclusion of the type of the constructors must be the inductive type - :n:`@ident` being defined (or :n:`@ident` applied to arguments in - the case of annotated inductive types — cf. next section). - -The following subsections show examples of simple inductive types, -simple annotated inductive types, simple parametric inductive types, -mutually inductive types and private (matching) inductive types. - -.. _simple-inductive-types: - -Simple inductive types -~~~~~~~~~~~~~~~~~~~~~~ - -A simple inductive type belongs to a universe that is a simple :n:`@sort`. - -.. example:: - - The set of natural numbers is defined as: - - .. coqtop:: reset all - - Inductive nat : Set := - | O : nat - | S : nat -> nat. - - The type nat is defined as the least :g:`Set` containing :g:`O` and closed by - the :g:`S` constructor. The names :g:`nat`, :g:`O` and :g:`S` are added to the - environment. - - This definition generates four elimination principles: - :g:`nat_rect`, :g:`nat_ind`, :g:`nat_rec` and :g:`nat_sind`. The type of :g:`nat_ind` is: - - .. coqtop:: all - - Check nat_ind. - - This is the well known structural induction principle over natural - numbers, i.e. the second-order form of Peano’s induction principle. It - allows proving universal properties of natural numbers (:g:`forall - n:nat, P n`) by induction on :g:`n`. - - The types of :g:`nat_rect`, :g:`nat_rec` and :g:`nat_sind` are similar, except that they - apply to, respectively, :g:`(P:nat->Type)`, :g:`(P:nat->Set)` and :g:`(P:nat->SProp)`. They correspond to - primitive induction principles (allowing dependent types) respectively - over sorts ```Type``, ``Set`` and ``SProp``. - -In the case where inductive types don't have annotations (the next section -gives an example of annotations), a constructor can be defined -by giving the type of its arguments alone. - -.. example:: - - .. coqtop:: reset none - - Reset nat. - - .. coqtop:: in - - Inductive nat : Set := O | S (_:nat). - -Simple annotated inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -In annotated inductive types, the universe where the inductive type -is defined is no longer a simple :n:`@sort`, but what is called an arity, -which is a type whose conclusion is a :n:`@sort`. - -.. example:: - - As an example of annotated inductive types, let us define the - :g:`even` predicate: - - .. coqtop:: all - - Inductive even : nat -> Prop := - | even_0 : even O - | even_SS : forall n:nat, even n -> even (S (S n)). - - The type :g:`nat->Prop` means that :g:`even` is a unary predicate (inductively - defined) over natural numbers. The type of its two constructors are the - defining clauses of the predicate :g:`even`. The type of :g:`even_ind` is: - - .. coqtop:: all - - Check even_ind. - - From a mathematical point of view, this asserts that the natural numbers satisfying - the predicate :g:`even` are exactly in the smallest set of naturals satisfying the - clauses :g:`even_0` or :g:`even_SS`. This is why, when we want to prove any - predicate :g:`P` over elements of :g:`even`, it is enough to prove it for :g:`O` - and to prove that if any natural number :g:`n` satisfies :g:`P` its double - successor :g:`(S (S n))` satisfies also :g:`P`. This is analogous to the - structural induction principle we got for :g:`nat`. - -.. _parametrized-inductive-types: - -Parameterized inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -In the previous example, each constructor introduces a different -instance of the predicate :g:`even`. In some cases, all the constructors -introduce the same generic instance of the inductive definition, in -which case, instead of an annotation, we use a context of parameters -which are :n:`@binder`\s shared by all the constructors of the definition. - -Parameters differ from inductive type annotations in that the -conclusion of each type of constructor invokes the inductive type with -the same parameter values of its specification. - -.. example:: - - A typical example is the definition of polymorphic lists: - - .. coqtop:: all - - Inductive list (A:Set) : Set := - | nil : list A - | cons : A -> list A -> list A. - - In the type of :g:`nil` and :g:`cons`, we write ":g:`list A`" and not - just ":g:`list`". The constructors :g:`nil` and :g:`cons` have these types: - - .. coqtop:: all - - Check nil. - Check cons. - - Observe that the destructors are also quantified with :g:`(A:Set)`, for example: - - .. coqtop:: all - - Check list_ind. - - Once again, the types of the constructor arguments and of the conclusion can be omitted: - - .. coqtop:: none - - Reset list. - - .. coqtop:: in - - Inductive list (A:Set) : Set := nil | cons (_:A) (_:list A). - -.. note:: - + The constructor type can - recursively invoke the inductive definition on an argument which is not - the parameter itself. - - One can define : - - .. coqtop:: all - - Inductive list2 (A:Set) : Set := - | nil2 : list2 A - | cons2 : A -> list2 (A*A) -> list2 A. - - that can also be written by specifying only the type of the arguments: - - .. coqtop:: all reset - - Inductive list2 (A:Set) : Set := - | nil2 - | cons2 (_:A) (_:list2 (A*A)). - - But the following definition will give an error: - - .. coqtop:: all - - Fail Inductive listw (A:Set) : Set := - | nilw : listw (A*A) - | consw : A -> listw (A*A) -> listw (A*A). - - because the conclusion of the type of constructors should be :g:`listw A` - in both cases. - - + A parameterized inductive definition can be defined using annotations - instead of parameters but it will sometimes give a different (bigger) - sort for the inductive definition and will produce a less convenient - rule for case elimination. - -.. flag:: Uniform Inductive Parameters - - When this flag is set (it is off by default), - inductive definitions are abstracted over their parameters - before type checking constructors, allowing to write: - - .. coqtop:: all - - Set Uniform Inductive Parameters. - Inductive list3 (A:Set) : Set := - | nil3 : list3 - | cons3 : A -> list3 -> list3. - - This behavior is essentially equivalent to starting a new section - and using :cmd:`Context` to give the uniform parameters, like so - (cf. :ref:`section-mechanism`): - - .. coqtop:: all reset - - Section list3. - Context (A:Set). - Inductive list3 : Set := - | nil3 : list3 - | cons3 : A -> list3 -> list3. - End list3. - - For finer control, you can use a ``|`` between the uniform and - the non-uniform parameters: - - .. coqtop:: in reset - - Inductive Acc {A:Type} (R:A->A->Prop) | (x:A) : Prop - := Acc_in : (forall y, R y x -> Acc y) -> Acc x. - - The flag can then be seen as deciding whether the ``|`` is at the - beginning (when the flag is unset) or at the end (when it is set) - of the parameters when not explicitly given. - -.. seealso:: - Section :ref:`inductive-definitions` and the :tacn:`induction` tactic. - -.. _mutually_inductive_types: - -Mutually defined inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. example:: Mutually defined inductive types - - A typical example of mutually inductive data types is trees and - forests. We assume two types :g:`A` and :g:`B` that are given as variables. The types can - be declared like this: - - .. coqtop:: in - - Parameters A B : Set. - - Inductive tree : Set := node : A -> forest -> tree - - with forest : Set := - | leaf : B -> forest - | cons : tree -> forest -> forest. - - This declaration automatically generates eight induction principles. They are not the most - general principles, but they correspond to each inductive part seen as a single inductive definition. - - To illustrate this point on our example, here are the types of :g:`tree_rec` - and :g:`forest_rec`. - - .. coqtop:: all - - Check tree_rec. - - Check forest_rec. - - Assume we want to parameterize our mutual inductive definitions with the - two type variables :g:`A` and :g:`B`, the declaration should be - done as follows: - - .. coqdoc:: - - Inductive tree (A B:Set) : Set := node : A -> forest A B -> tree A B - - with forest (A B:Set) : Set := - | leaf : B -> forest A B - | cons : tree A B -> forest A B -> forest A B. - - Assume we define an inductive definition inside a section - (cf. :ref:`section-mechanism`). When the section is closed, the variables - declared in the section and occurring free in the declaration are added as - parameters to the inductive definition. - -.. seealso:: - A generic command :cmd:`Scheme` is useful to build automatically various - mutual induction principles. - -Private (matching) inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. attr:: private(matching) - - This attribute can be used to forbid the use of the :g:`match` - construct on objects of this inductive type outside of the module - where it is defined. There is also a legacy syntax using the - ``Private`` prefix (cf. :n:`@legacy_attr`). - - The main use case of private (matching) inductive types is to emulate - quotient types / higher-order inductive types in projects such as - the `HoTT library `_. - -.. example:: - - .. coqtop:: all - - Module Foo. - #[ private(matching) ] Inductive my_nat := my_O : my_nat | my_S : my_nat -> my_nat. - Check (fun x : my_nat => match x with my_O => true | my_S _ => false end). - End Foo. - Import Foo. - Fail Check (fun x : my_nat => match x with my_O => true | my_S _ => false end). - -Variants -~~~~~~~~ - -.. cmd:: Variant @variant_definition {* with @variant_definition } - - .. insertprodn variant_definition variant_definition - - .. prodn:: - variant_definition ::= @ident_decl {* @binder } {? %| {* @binder } } {? : @type } := {? %| } {+| @constructor } {? @decl_notations } - - The :cmd:`Variant` command is similar to the :cmd:`Inductive` command, except - that it disallows recursive definition of types (for instance, lists cannot - be defined using :cmd:`Variant`). No induction scheme is generated for - this variant, unless the :flag:`Nonrecursive Elimination Schemes` flag is on. - - This command supports the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`universes(template)`, - :attr:`universes(notemplate)`, :attr:`universes(cumulative)`, - :attr:`universes(noncumulative)` and :attr:`private(matching)` - attributes. - - .. exn:: The @num th argument of @ident must be @ident in @type. - :undocumented: - -.. _coinductive-types: - -Co-inductive types ------------------- - -The objects of an inductive type are well-founded with respect to the -constructors of the type. In other words, such objects contain only a -*finite* number of constructors. Co-inductive types arise from relaxing -this condition, and admitting types whose objects contain an infinity of -constructors. Infinite objects are introduced by a non-ending (but -effective) process of construction, defined in terms of the constructors -of the type. - -.. cmd:: CoInductive @inductive_definition {* with @inductive_definition } - - This command introduces a co-inductive type. - The syntax of the command is the same as the command :cmd:`Inductive`. - No principle of induction is derived from the definition of a co-inductive - type, since such principles only make sense for inductive types. - For co-inductive types, the only elimination principle is case analysis. - - This command supports the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`universes(template)`, - :attr:`universes(notemplate)`, :attr:`universes(cumulative)`, - :attr:`universes(noncumulative)` and :attr:`private(matching)` - attributes. - -.. example:: - - The type of infinite sequences of natural numbers, usually called streams, - is an example of a co-inductive type. - - .. coqtop:: in - - CoInductive Stream : Set := Seq : nat -> Stream -> Stream. - - The usual destructors on streams :g:`hd:Stream->nat` and :g:`tl:Str->Str` - can be defined as follows: - - .. coqtop:: in - - Definition hd (x:Stream) := let (a,s) := x in a. - Definition tl (x:Stream) := let (a,s) := x in s. - -Definitions of co-inductive predicates and blocks of mutually -co-inductive definitions are also allowed. - -.. example:: - - The extensional equality on streams is an example of a co-inductive type: - - .. coqtop:: in - - CoInductive EqSt : Stream -> Stream -> Prop := - eqst : forall s1 s2:Stream, - hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2. - - In order to prove the extensional equality of two streams :g:`s1` and :g:`s2` - we have to construct an infinite proof of equality, that is, an infinite - object of type :g:`(EqSt s1 s2)`. We will see how to introduce infinite - objects in Section :ref:`cofixpoint`. - -Caveat -~~~~~~ - -The ability to define co-inductive types by constructors, hereafter called -*positive co-inductive types*, is known to break subject reduction. The story is -a bit long: this is due to dependent pattern-matching which implies -propositional η-equality, which itself would require full η-conversion for -subject reduction to hold, but full η-conversion is not acceptable as it would -make type checking undecidable. - -Since the introduction of primitive records in Coq 8.5, an alternative -presentation is available, called *negative co-inductive types*. This consists -in defining a co-inductive type as a primitive record type through its -projections. Such a technique is akin to the *co-pattern* style that can be -found in e.g. Agda, and preserves subject reduction. - -The above example can be rewritten in the following way. - -.. coqtop:: none - - Reset Stream. - -.. coqtop:: all - - Set Primitive Projections. - CoInductive Stream : Set := Seq { hd : nat; tl : Stream }. - CoInductive EqSt (s1 s2: Stream) : Prop := eqst { - eqst_hd : hd s1 = hd s2; - eqst_tl : EqSt (tl s1) (tl s2); - }. - -Some properties that hold over positive streams are lost when going to the -negative presentation, typically when they imply equality over streams. -For instance, propositional η-equality is lost when going to the negative -presentation. It is nonetheless logically consistent to recover it through an -axiom. - -.. coqtop:: all - - Axiom Stream_eta : forall s: Stream, s = Seq (hd s) (tl s). - -More generally, as in the case of positive coinductive types, it is consistent -to further identify extensional equality of coinductive types with propositional -equality: - -.. coqtop:: all - - Axiom Stream_ext : forall (s1 s2: Stream), EqSt s1 s2 -> s1 = s2. - -As of Coq 8.9, it is now advised to use negative co-inductive types rather than -their positive counterparts. - -.. seealso:: - :ref:`primitive_projections` for more information about negative - records and primitive projections. - - -Definition of recursive functions ---------------------------------- - -Definition of functions by recursion over inductive objects -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -This section describes the primitive form of definition by recursion over -inductive objects. See the :cmd:`Function` command for more advanced -constructions. - -.. _Fixpoint: - -.. cmd:: Fixpoint @fix_definition {* with @fix_definition } - - .. insertprodn fix_definition fix_definition - - .. prodn:: - fix_definition ::= @ident_decl {* @binder } {? @fixannot } {? : @type } {? := @term } {? @decl_notations } - - This command allows defining functions by pattern matching over inductive - objects using a fixed point construction. The meaning of this declaration is - to define :n:`@ident` as a recursive function with arguments specified by - the :n:`@binder`\s such that :n:`@ident` applied to arguments - corresponding to these :n:`@binder`\s has type :n:`@type`, and is - equivalent to the expression :n:`@term`. The type of :n:`@ident` is - consequently :n:`forall {* @binder }, @type` and its value is equivalent - to :n:`fun {* @binder } => @term`. - - To be accepted, a :cmd:`Fixpoint` definition has to satisfy syntactical - constraints on a special argument called the decreasing argument. They - are needed to ensure that the :cmd:`Fixpoint` definition always terminates. - The point of the :n:`{struct @ident}` annotation (see :n:`@fixannot`) is to - let the user tell the system which argument decreases along the recursive calls. - - The :n:`{struct @ident}` annotation may be left implicit, in which case the - system successively tries arguments from left to right until it finds one - that satisfies the decreasing condition. - - :cmd:`Fixpoint` without the :attr:`program` attribute does not support the - :n:`wf` or :n:`measure` clauses of :n:`@fixannot`. - - The :n:`with` clause allows simultaneously defining several mutual fixpoints. - It is especially useful when defining functions over mutually defined - inductive types. Example: :ref:`Mutual Fixpoints`. - - If :n:`@term` is omitted, :n:`@type` is required and Coq enters proof editing mode. - This can be used to define a term incrementally, in particular by relying on the :tacn:`refine` tactic. - In this case, the proof should be terminated with :cmd:`Defined` in order to define a constant - for which the computational behavior is relevant. See :ref:`proof-editing-mode`. - - .. note:: - - + Some fixpoints may have several arguments that fit as decreasing - arguments, and this choice influences the reduction of the fixpoint. - Hence an explicit annotation must be used if the leftmost decreasing - argument is not the desired one. Writing explicit annotations can also - speed up type checking of large mutual fixpoints. - - + In order to keep the strong normalization property, the fixed point - reduction will only be performed when the argument in position of the - decreasing argument (which type should be in an inductive definition) - starts with a constructor. - - - .. example:: - - One can define the addition function as : - - .. coqtop:: all - - Fixpoint add (n m:nat) {struct n} : nat := - match n with - | O => m - | S p => S (add p m) - end. - - The match operator matches a value (here :g:`n`) with the various - constructors of its (inductive) type. The remaining arguments give the - respective values to be returned, as functions of the parameters of the - corresponding constructor. Thus here when :g:`n` equals :g:`O` we return - :g:`m`, and when :g:`n` equals :g:`(S p)` we return :g:`(S (add p m))`. - - The match operator is formally described in - Section :ref:`match-construction`. - The system recognizes that in the inductive call :g:`(add p m)` the first - argument actually decreases because it is a *pattern variable* coming - from :g:`match n with`. - - .. example:: - - The following definition is not correct and generates an error message: - - .. coqtop:: all - - Fail Fixpoint wrongplus (n m:nat) {struct n} : nat := - match m with - | O => n - | S p => S (wrongplus n p) - end. - - because the declared decreasing argument :g:`n` does not actually - decrease in the recursive call. The function computing the addition over - the second argument should rather be written: - - .. coqtop:: all - - Fixpoint plus (n m:nat) {struct m} : nat := - match m with - | O => n - | S p => S (plus n p) - end. - - .. example:: - - The recursive call may not only be on direct subterms of the recursive - variable :g:`n` but also on a deeper subterm and we can directly write - the function :g:`mod2` which gives the remainder modulo 2 of a natural - number. - - .. coqtop:: all - - Fixpoint mod2 (n:nat) : nat := - match n with - | O => O - | S p => match p with - | O => S O - | S q => mod2 q - end - end. - -.. _example_mutual_fixpoints: - - .. example:: Mutual fixpoints - - The size of trees and forests can be defined the following way: - - .. coqtop:: all - - Fixpoint tree_size (t:tree) : nat := - match t with - | node a f => S (forest_size f) - end - with forest_size (f:forest) : nat := - match f with - | leaf b => 1 - | cons t f' => (tree_size t + forest_size f') - end. - -.. _cofixpoint: - -Definitions of recursive objects in co-inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. cmd:: CoFixpoint @cofix_definition {* with @cofix_definition } - - .. insertprodn cofix_definition cofix_definition - - .. prodn:: - cofix_definition ::= @ident_decl {* @binder } {? : @type } {? := @term } {? @decl_notations } - - This command introduces a method for constructing an infinite object of a - coinductive type. For example, the stream containing all natural numbers can - be introduced applying the following method to the number :g:`O` (see - Section :ref:`coinductive-types` for the definition of :g:`Stream`, :g:`hd` - and :g:`tl`): - - .. coqtop:: all - - CoFixpoint from (n:nat) : Stream := Seq n (from (S n)). - - Unlike recursive definitions, there is no decreasing argument in a - co-recursive definition. To be admissible, a method of construction must - provide at least one extra constructor of the infinite object for each - iteration. A syntactical guard condition is imposed on co-recursive - definitions in order to ensure this: each recursive call in the - definition must be protected by at least one constructor, and only by - constructors. That is the case in the former definition, where the single - recursive call of :g:`from` is guarded by an application of :g:`Seq`. - On the contrary, the following recursive function does not satisfy the - guard condition: - - .. coqtop:: all - - Fail CoFixpoint filter (p:nat -> bool) (s:Stream) : Stream := - if p (hd s) then Seq (hd s) (filter p (tl s)) else filter p (tl s). - - The elimination of co-recursive definition is done lazily, i.e. the - definition is expanded only when it occurs at the head of an application - which is the argument of a case analysis expression. In any other - context, it is considered as a canonical expression which is completely - evaluated. We can test this using the command :cmd:`Eval`, which computes - the normal forms of a term: - - .. coqtop:: all - - Eval compute in (from 0). - Eval compute in (hd (from 0)). - Eval compute in (tl (from 0)). - - As in the :cmd:`Fixpoint` command, the :n:`with` clause allows simultaneously - defining several mutual cofixpoints. - - If :n:`@term` is omitted, :n:`@type` is required and Coq enters proof editing mode. - This can be used to define a term incrementally, in particular by relying on the :tacn:`refine` tactic. - In this case, the proof should be terminated with :cmd:`Defined` in order to define a constant - for which the computational behavior is relevant. See :ref:`proof-editing-mode`. - -.. _Computations: - -Computations ------------- - -.. insertprodn reduce pattern_occ - -.. prodn:: - reduce ::= Eval @red_expr in - red_expr ::= red - | hnf - | simpl {? @delta_flag } {? @ref_or_pattern_occ } - | cbv {? @strategy_flag } - | cbn {? @strategy_flag } - | lazy {? @strategy_flag } - | compute {? @delta_flag } - | vm_compute {? @ref_or_pattern_occ } - | native_compute {? @ref_or_pattern_occ } - | unfold {+, @unfold_occ } - | fold {+ @one_term } - | pattern {+, @pattern_occ } - | @ident - delta_flag ::= {? - } [ {+ @smart_qualid } ] - strategy_flag ::= {+ @red_flags } - | @delta_flag - red_flags ::= beta - | iota - | match - | fix - | cofix - | zeta - | delta {? @delta_flag } - ref_or_pattern_occ ::= @smart_qualid {? at @occs_nums } - | @one_term {? at @occs_nums } - occs_nums ::= {+ {| @num | @ident } } - | - {| @num | @ident } {* @int_or_var } - int_or_var ::= @int - | @ident - unfold_occ ::= @smart_qualid {? at @occs_nums } - pattern_occ ::= @one_term {? at @occs_nums } - -See :ref:`Conversion-rules`. - -.. todo:: Add text here - -.. _Assertions: - -Assertions and proofs ---------------------- - -An assertion states a proposition (or a type) of which the proof (or an -inhabitant of the type) is interactively built using tactics. The interactive -proof mode is described in Chapter :ref:`proofhandling` and the tactics in -Chapter :ref:`Tactics`. The basic assertion command is: - -.. cmd:: @thm_token @ident_decl {* @binder } : @type {* with @ident_decl {* @binder } : @type } - :name: Theorem; Lemma; Fact; Remark; Corollary; Proposition; Property - - .. insertprodn thm_token thm_token - - .. prodn:: - thm_token ::= Theorem - | Lemma - | Fact - | Remark - | Corollary - | Proposition - | Property - - After the statement is asserted, Coq needs a proof. Once a proof of - :n:`@type` under the assumptions represented by :n:`@binder`\s is given and - validated, the proof is generalized into a proof of :n:`forall {* @binder }, @type` and - the theorem is bound to the name :n:`@ident` in the environment. - - Forms using the :n:`with` clause are useful for theorems that are proved by simultaneous induction - over a mutually inductive assumption, or that assert mutually dependent - statements in some mutual co-inductive type. It is equivalent to - :cmd:`Fixpoint` or :cmd:`CoFixpoint` but using tactics to build the proof of - the statements (or the body of the specification, depending on the point of - view). The inductive or co-inductive types on which the induction or - coinduction has to be done is assumed to be non ambiguous and is guessed by - the system. - - Like in a :cmd:`Fixpoint` or :cmd:`CoFixpoint` definition, the induction hypotheses - have to be used on *structurally smaller* arguments (for a :cmd:`Fixpoint`) or - be *guarded by a constructor* (for a :cmd:`CoFixpoint`). The verification that - recursive proof arguments are correct is done only at the time of registering - the lemma in the environment. To know if the use of induction hypotheses is - correct at some time of the interactive development of a proof, use the - command :cmd:`Guarded`. - - .. exn:: The term @term has type @type which should be Set, Prop or Type. - :undocumented: - - .. exn:: @ident already exists. - :name: @ident already exists. (Theorem) - - The name you provided is already defined. You have then to choose - another name. - - .. exn:: Nested proofs are not allowed unless you turn the Nested Proofs Allowed flag on. - - You are asserting a new statement while already being in proof editing mode. - This feature, called nested proofs, is disabled by default. - To activate it, turn the :flag:`Nested Proofs Allowed` flag on. - -Proofs start with the keyword :cmd:`Proof`. Then Coq enters the proof editing mode -until the proof is completed. In proof editing mode, the user primarily enters -tactics, which are described in chapter :ref:`Tactics`. The user may also enter -commands to manage the proof editing mode. They are described in Chapter -:ref:`proofhandling`. - -When the proof is complete, use the :cmd:`Qed` command so the kernel verifies -the proof and adds it to the environment. - -.. note:: - - #. Several statements can be simultaneously asserted provided the - :flag:`Nested Proofs Allowed` flag was turned on. - - #. Not only other assertions but any vernacular command can be given - while in the process of proving a given assertion. In this case, the - command is understood as if it would have been given before the - statements still to be proved. Nonetheless, this practice is discouraged - and may stop working in future versions. - - #. Proofs ended by :cmd:`Qed` are declared opaque. Their content cannot be - unfolded (see :ref:`performingcomputations`), thus - realizing some form of *proof-irrelevance*. To be able to unfold a - proof, the proof should be ended by :cmd:`Defined`. - - #. :cmd:`Proof` is recommended but can currently be omitted. On the opposite - side, :cmd:`Qed` (or :cmd:`Defined`) is mandatory to validate a proof. - - #. One can also use :cmd:`Admitted` in place of :cmd:`Qed` to turn the - current asserted statement into an axiom and exit the proof editing mode. - .. _gallina-attributes: Attributes @@ -1680,41 +128,7 @@ Legacy attribute New attribute `Program` :attr:`program` ================ ================================ -.. attr:: deprecated ( {? since = @string , } {? note = @string } ) - :name: deprecated - - At least one of :n:`since` or :n:`note` must be present. If both are present, - either one may appear first and they must be separated by a comma. - - This attribute is supported by the following commands: :cmd:`Ltac`, - :cmd:`Tactic Notation`, :cmd:`Notation`, :cmd:`Infix`. - - It can trigger the following warnings: - - .. warn:: Tactic @qualid is deprecated since @string__since. @string__note. - Tactic Notation @qualid is deprecated since @string__since. @string__note. - Notation @string is deprecated since @string__since. @string__note. - - :n:`@qualid` or :n:`@string` is the notation, :n:`@string__since` is the version number, - :n:`@string__note` is the note (usually explains the replacement). - - .. example:: - - .. coqtop:: all reset warn - - #[deprecated(since="8.9.0", note="Use idtac instead.")] - Ltac foo := idtac. - - Goal True. - Proof. - now foo. - Abort. - .. warn:: Unsupported attribute This warning is an error by default. It is caused by using a command with some attribute it does not understand. - -.. [1] - Except if the inductive type is empty in which case there is no - equation that can be used to infer the return type. -- cgit v1.2.3 From 9eb788ebca0305fc72940c92b9ae35bbaca56c5c Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 13:00:52 +0200 Subject: Create section on basics with just lexical conventions and attributes. --- doc/sphinx/language/core/basic.rst | 134 +++++++++++++++++++++ .../language/gallina-specification-language.rst | 134 --------------------- 2 files changed, 134 insertions(+), 134 deletions(-) create mode 100644 doc/sphinx/language/core/basic.rst delete mode 100644 doc/sphinx/language/gallina-specification-language.rst (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/core/basic.rst b/doc/sphinx/language/core/basic.rst new file mode 100644 index 0000000000..4fadc8da02 --- /dev/null +++ b/doc/sphinx/language/core/basic.rst @@ -0,0 +1,134 @@ +.. _lexical-conventions: + +Lexical conventions +=================== + +Blanks + Space, newline and horizontal tab are considered blanks. + Blanks are ignored but they separate tokens. + +Comments + Comments are enclosed between ``(*`` and ``*)``. They can be nested. + They can contain any character. However, embedded :n:`@string` literals must be + correctly closed. Comments are treated as blanks. + +Identifiers + Identifiers, written :n:`@ident`, are sequences of letters, digits, ``_`` and + ``'``, that do not start with a digit or ``'``. That is, they are + recognized by the following grammar (except that the string ``_`` is reserved; + it is not a valid identifier): + + .. insertprodn ident subsequent_letter + + .. prodn:: + ident ::= @first_letter {* @subsequent_letter } + first_letter ::= {| a .. z | A .. Z | _ | @unicode_letter } + subsequent_letter ::= {| @first_letter | @digit | ' | @unicode_id_part } + + All characters are meaningful. In particular, identifiers are case-sensitive. + :production:`unicode_letter` non-exhaustively includes Latin, + Greek, Gothic, Cyrillic, Arabic, Hebrew, Georgian, Hangul, Hiragana + and Katakana characters, CJK ideographs, mathematical letter-like + symbols and non-breaking space. :production:`unicode_id_part` + non-exhaustively includes symbols for prime letters and subscripts. + +Numerals + Numerals are sequences of digits with an optional fractional part + and exponent, optionally preceded by a minus sign. :n:`@int` is an integer; + a numeral without fractional or exponent parts. :n:`@num` is a non-negative + integer. Underscores embedded in the digits are ignored, for example + ``1_000_000`` is the same as ``1000000``. + + .. insertprodn numeral digit + + .. prodn:: + numeral ::= {+ @digit } {? . {+ @digit } } {? {| e | E } {? {| + | - } } {+ @digit } } + int ::= {? - } {+ @digit } + num ::= {+ @digit } + digit ::= 0 .. 9 + +Strings + Strings begin and end with ``"`` (double quote). Use ``""`` to represent + a double quote character within a string. In the grammar, strings are + identified with :production:`string`. + +Keywords + The following character sequences are reserved keywords that cannot be + used as identifiers:: + + _ Axiom CoFixpoint Definition Fixpoint Hypothesis IF Parameter Prop + SProp Set Theorem Type Variable as at by cofix discriminated else + end exists exists2 fix for forall fun if in lazymatch let match + multimatch return then using where with + + Note that plugins may define additional keywords when they are loaded. + +Other tokens + The set of + tokens defined at any given time can vary because the :cmd:`Notation` + command can define new tokens. A :cmd:`Require` command may load more notation definitions, + while the end of a :cmd:`Section` may remove notations. Some notations + are defined in the basic library (see :ref:`thecoqlibrary`) and are normally + loaded automatically at startup time. + + Here are the character sequences that Coq directly defines as tokens + without using :cmd:`Notation` (omitting 25 specialized tokens that begin with + ``#int63_``):: + + ! #[ % & ' ( () (bfs) (dfs) ) * ** + , - -> + . .( .. ... / : ::= := :> :>> ; < <+ <- <: + <<: <= = => > >-> >= ? @ @{ [ [= ] _ + `( `{ { {| | |- || } + + When multiple tokens match the beginning of a sequence of characters, + the longest matching token is used. + Occasionally you may need to insert spaces to separate tokens. For example, + if ``~`` and ``~~`` are both defined as tokens, the inputs ``~ ~`` and + ``~~`` generate different tokens, whereas if `~~` is not defined, then the + two inputs are equivalent. + +.. _gallina-attributes: + +Attributes +----------- + +.. insertprodn all_attrs legacy_attr + +.. prodn:: + all_attrs ::= {* #[ {*, @attr } ] } {* @legacy_attr } + attr ::= @ident {? @attr_value } + attr_value ::= = @string + | ( {*, @attr } ) + legacy_attr ::= {| Local | Global } + | {| Polymorphic | Monomorphic } + | {| Cumulative | NonCumulative } + | Private + | Program + +Attributes modify the behavior of a command or tactic. +Syntactically, most commands and tactics can be decorated with attributes, but +attributes not supported by the command or tactic will be flagged as errors. + +The order of top-level attributes doesn't affect their meaning. ``#[foo,bar]``, ``#[bar,foo]``, +``#[foo]#[bar]`` and ``#[bar]#[foo]`` are equivalent. + +The legacy attributes (:n:`@legacy_attr`) provide an older, alternate syntax +for certain attributes. They are equivalent to new attributes as follows: + +================ ================================ +Legacy attribute New attribute +================ ================================ +`Local` :attr:`local` +`Global` :attr:`global` +`Polymorphic` :attr:`universes(polymorphic)` +`Monomorphic` :attr:`universes(monomorphic)` +`Cumulative` :attr:`universes(cumulative)` +`NonCumulative` :attr:`universes(noncumulative)` +`Private` :attr:`private(matching)` +`Program` :attr:`program` +================ ================================ + +.. warn:: Unsupported attribute + + This warning is an error by default. It is caused by using a + command with some attribute it does not understand. diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst deleted file mode 100644 index 4fadc8da02..0000000000 --- a/doc/sphinx/language/gallina-specification-language.rst +++ /dev/null @@ -1,134 +0,0 @@ -.. _lexical-conventions: - -Lexical conventions -=================== - -Blanks - Space, newline and horizontal tab are considered blanks. - Blanks are ignored but they separate tokens. - -Comments - Comments are enclosed between ``(*`` and ``*)``. They can be nested. - They can contain any character. However, embedded :n:`@string` literals must be - correctly closed. Comments are treated as blanks. - -Identifiers - Identifiers, written :n:`@ident`, are sequences of letters, digits, ``_`` and - ``'``, that do not start with a digit or ``'``. That is, they are - recognized by the following grammar (except that the string ``_`` is reserved; - it is not a valid identifier): - - .. insertprodn ident subsequent_letter - - .. prodn:: - ident ::= @first_letter {* @subsequent_letter } - first_letter ::= {| a .. z | A .. Z | _ | @unicode_letter } - subsequent_letter ::= {| @first_letter | @digit | ' | @unicode_id_part } - - All characters are meaningful. In particular, identifiers are case-sensitive. - :production:`unicode_letter` non-exhaustively includes Latin, - Greek, Gothic, Cyrillic, Arabic, Hebrew, Georgian, Hangul, Hiragana - and Katakana characters, CJK ideographs, mathematical letter-like - symbols and non-breaking space. :production:`unicode_id_part` - non-exhaustively includes symbols for prime letters and subscripts. - -Numerals - Numerals are sequences of digits with an optional fractional part - and exponent, optionally preceded by a minus sign. :n:`@int` is an integer; - a numeral without fractional or exponent parts. :n:`@num` is a non-negative - integer. Underscores embedded in the digits are ignored, for example - ``1_000_000`` is the same as ``1000000``. - - .. insertprodn numeral digit - - .. prodn:: - numeral ::= {+ @digit } {? . {+ @digit } } {? {| e | E } {? {| + | - } } {+ @digit } } - int ::= {? - } {+ @digit } - num ::= {+ @digit } - digit ::= 0 .. 9 - -Strings - Strings begin and end with ``"`` (double quote). Use ``""`` to represent - a double quote character within a string. In the grammar, strings are - identified with :production:`string`. - -Keywords - The following character sequences are reserved keywords that cannot be - used as identifiers:: - - _ Axiom CoFixpoint Definition Fixpoint Hypothesis IF Parameter Prop - SProp Set Theorem Type Variable as at by cofix discriminated else - end exists exists2 fix for forall fun if in lazymatch let match - multimatch return then using where with - - Note that plugins may define additional keywords when they are loaded. - -Other tokens - The set of - tokens defined at any given time can vary because the :cmd:`Notation` - command can define new tokens. A :cmd:`Require` command may load more notation definitions, - while the end of a :cmd:`Section` may remove notations. Some notations - are defined in the basic library (see :ref:`thecoqlibrary`) and are normally - loaded automatically at startup time. - - Here are the character sequences that Coq directly defines as tokens - without using :cmd:`Notation` (omitting 25 specialized tokens that begin with - ``#int63_``):: - - ! #[ % & ' ( () (bfs) (dfs) ) * ** + , - -> - . .( .. ... / : ::= := :> :>> ; < <+ <- <: - <<: <= = => > >-> >= ? @ @{ [ [= ] _ - `( `{ { {| | |- || } - - When multiple tokens match the beginning of a sequence of characters, - the longest matching token is used. - Occasionally you may need to insert spaces to separate tokens. For example, - if ``~`` and ``~~`` are both defined as tokens, the inputs ``~ ~`` and - ``~~`` generate different tokens, whereas if `~~` is not defined, then the - two inputs are equivalent. - -.. _gallina-attributes: - -Attributes ------------ - -.. insertprodn all_attrs legacy_attr - -.. prodn:: - all_attrs ::= {* #[ {*, @attr } ] } {* @legacy_attr } - attr ::= @ident {? @attr_value } - attr_value ::= = @string - | ( {*, @attr } ) - legacy_attr ::= {| Local | Global } - | {| Polymorphic | Monomorphic } - | {| Cumulative | NonCumulative } - | Private - | Program - -Attributes modify the behavior of a command or tactic. -Syntactically, most commands and tactics can be decorated with attributes, but -attributes not supported by the command or tactic will be flagged as errors. - -The order of top-level attributes doesn't affect their meaning. ``#[foo,bar]``, ``#[bar,foo]``, -``#[foo]#[bar]`` and ``#[bar]#[foo]`` are equivalent. - -The legacy attributes (:n:`@legacy_attr`) provide an older, alternate syntax -for certain attributes. They are equivalent to new attributes as follows: - -================ ================================ -Legacy attribute New attribute -================ ================================ -`Local` :attr:`local` -`Global` :attr:`global` -`Polymorphic` :attr:`universes(polymorphic)` -`Monomorphic` :attr:`universes(monomorphic)` -`Cumulative` :attr:`universes(cumulative)` -`NonCumulative` :attr:`universes(noncumulative)` -`Private` :attr:`private(matching)` -`Program` :attr:`program` -================ ================================ - -.. warn:: Unsupported attribute - - This warning is an error by default. It is caused by using a - command with some attribute it does not understand. -- cgit v1.2.3 From a42cf3402094e6f7ce019243edfc6b6137de011a Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 13:03:08 +0200 Subject: Create section on basics with just flags, options and tables. --- doc/sphinx/language/core/basic.rst | 127 +++++++++++++++++++++++++++++++++++++ 1 file changed, 127 insertions(+) create mode 100644 doc/sphinx/language/core/basic.rst (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/core/basic.rst b/doc/sphinx/language/core/basic.rst new file mode 100644 index 0000000000..66a955e48d --- /dev/null +++ b/doc/sphinx/language/core/basic.rst @@ -0,0 +1,127 @@ +.. _flags-options-tables: + +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 :flag:`Asymmetric Patterns`. +* An *option* generally has a numeric or string value, such as :opt:`Firstorder Depth`. +* A *table* contains a set of strings or qualids. +* In addition, some commands provide settings, such as :cmd:`Extraction Language`. + +.. FIXME Convert "Extraction Language" to an option. + +Flags, options and tables are identified by a series of identifiers, each with an initial +capital letter. + +.. cmd:: Set @setting_name {? {| @int | @string } } + :name: Set + + .. insertprodn setting_name setting_name + + .. prodn:: + setting_name ::= {+ @ident } + + If :n:`@setting_name` is a flag, no value may be provided; the flag + is set to on. + If :n:`@setting_name` is an option, a value of the appropriate type + must be provided; the option is set to the specified value. + + This command supports the :attr:`local`, :attr:`global` and :attr:`export` attributes. + They are described :ref:`here `. + + .. warn:: There is no flag or option with this name: "@setting_name". + + This warning message can be raised by :cmd:`Set` and + :cmd:`Unset` when :n:`@setting_name` is unknown. It is a + warning rather than an error because this helps library authors + produce Coq code that is compatible with several Coq versions. + To preserve the same behavior, they may need to set some + compatibility flags or options that did not exist in previous + Coq versions. + +.. cmd:: Unset @setting_name + :name: Unset + + If :n:`@setting_name` is a flag, it is set to off. If :n:`@setting_name` is an option, it is + set to its default value. + + This command supports the :attr:`local`, :attr:`global` and :attr:`export` attributes. + They are described :ref:`here `. + +.. cmd:: Add @setting_name {+ {| @qualid | @string } } + + Adds the specified values to the table :n:`@setting_name`. + +.. cmd:: Remove @setting_name {+ {| @qualid | @string } } + + Removes the specified value from the table :n:`@setting_name`. + +.. cmd:: Test @setting_name {? for {+ {| @qualid | @string } } } + + If :n:`@setting_name` is a flag or option, prints its current value. + If :n:`@setting_name` is a table: if the `for` clause is specified, reports + whether the table contains each specified value, otherise this is equivalent to + :cmd:`Print Table`. The `for` clause is not valid for flags and options. + + .. exn:: There is no flag, option or table with this name: "@setting_name". + + This error message is raised when calling the :cmd:`Test` + command (without the `for` clause), or the :cmd:`Print Table` + command, for an unknown :n:`@setting_name`. + + .. exn:: There is no qualid-valued table with this name: "@setting_name". + There is no string-valued table with this name: "@setting_name". + + These error messages are raised when calling the :cmd:`Add` or + :cmd:`Remove` commands, or the :cmd:`Test` command with the + `for` clause, if :n:`@setting_name` is unknown or does not have + the right type. + +.. cmd:: Print Options + + Prints the current value of all flags and options, and the names of all tables. + +.. cmd:: Print Table @setting_name + + Prints the values in the table :n:`@setting_name`. + +.. cmd:: Print Tables + + A synonym for :cmd:`Print Options`. + +.. _set_unset_scope_qualifiers: + +Locality attributes supported by :cmd:`Set` and :cmd:`Unset` +```````````````````````````````````````````````````````````` + +The :cmd:`Set` and :cmd:`Unset` commands support the :attr:`local`, +:attr:`global` and :attr:`export` locality attributes: + +* no attribute: the original setting is *not* restored at the end of + the current module or section. +* :attr:`local` (an alternative syntax is to use the ``Local`` + prefix): the setting is applied within the current module or + section. The original value of the setting is restored at the end + of the current module or section. +* :attr:`export` (an alternative syntax is to use the ``Export`` + prefix): similar to :attr:`local`, the original value of the setting + is restored at the end of the current module or section. In + addition, if the value is set in a module, then :cmd:`Import`\-ing + the module sets the option or flag. +* :attr:`global` (an alternative syntax is to use the ``Global`` + prefix): 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 :cmd:`Require`\-ing the file sets the option. + +Newly opened modules and sections inherit the current settings. + +.. note:: + + The use of the :attr:`global` attribute with the :cmd:`Set` and + :cmd:`Unset` commands is discouraged. If your goal is to define + project-wide settings, you should rather use the command-line + arguments ``-set`` and ``-unset`` for setting flags and options + (cf. :ref:`command-line-options`). -- cgit v1.2.3 From 3d919e9d4d66925655812ddbd14f50ebfd995f28 Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 13:08:45 +0200 Subject: Remove lexical conventions and attributes from Gallina chapter. --- .../language/gallina-specification-language.rst | 165 --------------------- 1 file changed, 165 deletions(-) (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index 186a23897d..e43fa84e67 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -38,95 +38,6 @@ rules implemented by the typing algorithm are described in Chapter :ref:`calculu possibly empty sequence of expressions parsed by the “``entry``” entry, separated by the literal “``sep``”. -.. _lexical-conventions: - -Lexical conventions -=================== - -Blanks - Space, newline and horizontal tab are considered blanks. - Blanks are ignored but they separate tokens. - -Comments - Comments are enclosed between ``(*`` and ``*)``. They can be nested. - They can contain any character. However, embedded :n:`@string` literals must be - correctly closed. Comments are treated as blanks. - -Identifiers - Identifiers, written :n:`@ident`, are sequences of letters, digits, ``_`` and - ``'``, that do not start with a digit or ``'``. That is, they are - recognized by the following grammar (except that the string ``_`` is reserved; - it is not a valid identifier): - - .. insertprodn ident subsequent_letter - - .. prodn:: - ident ::= @first_letter {* @subsequent_letter } - first_letter ::= {| a .. z | A .. Z | _ | @unicode_letter } - subsequent_letter ::= {| @first_letter | @digit | ' | @unicode_id_part } - - All characters are meaningful. In particular, identifiers are case-sensitive. - :production:`unicode_letter` non-exhaustively includes Latin, - Greek, Gothic, Cyrillic, Arabic, Hebrew, Georgian, Hangul, Hiragana - and Katakana characters, CJK ideographs, mathematical letter-like - symbols and non-breaking space. :production:`unicode_id_part` - non-exhaustively includes symbols for prime letters and subscripts. - -Numerals - Numerals are sequences of digits with an optional fractional part - and exponent, optionally preceded by a minus sign. :n:`@int` is an integer; - a numeral without fractional or exponent parts. :n:`@num` is a non-negative - integer. Underscores embedded in the digits are ignored, for example - ``1_000_000`` is the same as ``1000000``. - - .. insertprodn numeral digit - - .. prodn:: - numeral ::= {+ @digit } {? . {+ @digit } } {? {| e | E } {? {| + | - } } {+ @digit } } - int ::= {? - } {+ @digit } - num ::= {+ @digit } - digit ::= 0 .. 9 - -Strings - Strings begin and end with ``"`` (double quote). Use ``""`` to represent - a double quote character within a string. In the grammar, strings are - identified with :production:`string`. - -Keywords - The following character sequences are reserved keywords that cannot be - used as identifiers:: - - _ Axiom CoFixpoint Definition Fixpoint Hypothesis IF Parameter Prop - SProp Set Theorem Type Variable as at by cofix discriminated else - end exists exists2 fix for forall fun if in lazymatch let match - multimatch return then using where with - - Note that plugins may define additional keywords when they are loaded. - -Other tokens - The set of - tokens defined at any given time can vary because the :cmd:`Notation` - command can define new tokens. A :cmd:`Require` command may load more notation definitions, - while the end of a :cmd:`Section` may remove notations. Some notations - are defined in the basic library (see :ref:`thecoqlibrary`) and are normally - loaded automatically at startup time. - - Here are the character sequences that Coq directly defines as tokens - without using :cmd:`Notation` (omitting 25 specialized tokens that begin with - ``#int63_``):: - - ! #[ % & ' ( () (bfs) (dfs) ) * ** + , - -> - . .( .. ... / : ::= := :> :>> ; < <+ <- <: - <<: <= = => > >-> >= ? @ @{ [ [= ] _ - `( `{ { {| | |- || } - - When multiple tokens match the beginning of a sequence of characters, - the longest matching token is used. - Occasionally you may need to insert spaces to separate tokens. For example, - if ``~`` and ``~~`` are both defined as tokens, the inputs ``~ ~`` and - ``~~`` generate different tokens, whereas if `~~` is not defined, then the - two inputs are equivalent. - .. _term: Terms @@ -1639,82 +1550,6 @@ the proof and adds it to the environment. #. One can also use :cmd:`Admitted` in place of :cmd:`Qed` to turn the current asserted statement into an axiom and exit the proof editing mode. -.. _gallina-attributes: - -Attributes ------------ - -.. insertprodn all_attrs legacy_attr - -.. prodn:: - all_attrs ::= {* #[ {*, @attr } ] } {* @legacy_attr } - attr ::= @ident {? @attr_value } - attr_value ::= = @string - | ( {*, @attr } ) - legacy_attr ::= {| Local | Global } - | {| Polymorphic | Monomorphic } - | {| Cumulative | NonCumulative } - | Private - | Program - -Attributes modify the behavior of a command or tactic. -Syntactically, most commands and tactics can be decorated with attributes, but -attributes not supported by the command or tactic will be flagged as errors. - -The order of top-level attributes doesn't affect their meaning. ``#[foo,bar]``, ``#[bar,foo]``, -``#[foo]#[bar]`` and ``#[bar]#[foo]`` are equivalent. - -The legacy attributes (:n:`@legacy_attr`) provide an older, alternate syntax -for certain attributes. They are equivalent to new attributes as follows: - -================ ================================ -Legacy attribute New attribute -================ ================================ -`Local` :attr:`local` -`Global` :attr:`global` -`Polymorphic` :attr:`universes(polymorphic)` -`Monomorphic` :attr:`universes(monomorphic)` -`Cumulative` :attr:`universes(cumulative)` -`NonCumulative` :attr:`universes(noncumulative)` -`Private` :attr:`private(matching)` -`Program` :attr:`program` -================ ================================ - -.. attr:: deprecated ( {? since = @string , } {? note = @string } ) - :name: deprecated - - At least one of :n:`since` or :n:`note` must be present. If both are present, - either one may appear first and they must be separated by a comma. - - This attribute is supported by the following commands: :cmd:`Ltac`, - :cmd:`Tactic Notation`, :cmd:`Notation`, :cmd:`Infix`. - - It can trigger the following warnings: - - .. warn:: Tactic @qualid is deprecated since @string__since. @string__note. - Tactic Notation @qualid is deprecated since @string__since. @string__note. - Notation @string is deprecated since @string__since. @string__note. - - :n:`@qualid` or :n:`@string` is the notation, :n:`@string__since` is the version number, - :n:`@string__note` is the note (usually explains the replacement). - - .. example:: - - .. coqtop:: all reset warn - - #[deprecated(since="8.9.0", note="Use idtac instead.")] - Ltac foo := idtac. - - Goal True. - Proof. - now foo. - Abort. - -.. warn:: Unsupported attribute - - This warning is an error by default. It is caused by using a - command with some attribute it does not understand. - .. [1] Except if the inductive type is empty in which case there is no equation that can be used to infer the return type. -- cgit v1.2.3 From 3762e15a4a5380e03b7d0e8d6bd451ce3bf9125d Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 13:12:14 +0200 Subject: Extract deprecated attribute from Gallina chapter. --- .../language/gallina-specification-language.rst | 1691 -------------------- 1 file changed, 1691 deletions(-) (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index 186a23897d..91634ea023 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -1,1685 +1,3 @@ -.. _gallinaspecificationlanguage: - ------------------------------------- - The Gallina specification language ------------------------------------- - -This chapter describes Gallina, the specification language of Coq. It allows -developing mathematical theories and to prove specifications of programs. The -theories are built from axioms, hypotheses, parameters, lemmas, theorems and -definitions of constants, functions, predicates and sets. The syntax of logical -objects involved in theories is described in Section :ref:`term`. The -language of commands, called *The Vernacular* is described in Section -:ref:`vernacular`. - -In Coq, logical objects are typed to ensure their logical correctness. The -rules implemented by the typing algorithm are described in Chapter :ref:`calculusofinductiveconstructions`. - - -.. About the grammars in the manual - ================================ - - Grammars are presented in Backus-Naur form (BNF). Terminal symbols are - set in black ``typewriter font``. In addition, there are special notations for - regular expressions. - - An expression enclosed in square brackets ``[…]`` means at most one - occurrence of this expression (this corresponds to an optional - component). - - The notation “``entry sep … sep entry``” stands for a non empty sequence - of expressions parsed by entry and separated by the literal “``sep``” [1]_. - - Similarly, the notation “``entry … entry``” stands for a non empty - sequence of expressions parsed by the “``entry``” entry, without any - separator between. - - At the end, the notation “``[entry sep … sep entry]``” stands for a - possibly empty sequence of expressions parsed by the “``entry``” entry, - separated by the literal “``sep``”. - -.. _lexical-conventions: - -Lexical conventions -=================== - -Blanks - Space, newline and horizontal tab are considered blanks. - Blanks are ignored but they separate tokens. - -Comments - Comments are enclosed between ``(*`` and ``*)``. They can be nested. - They can contain any character. However, embedded :n:`@string` literals must be - correctly closed. Comments are treated as blanks. - -Identifiers - Identifiers, written :n:`@ident`, are sequences of letters, digits, ``_`` and - ``'``, that do not start with a digit or ``'``. That is, they are - recognized by the following grammar (except that the string ``_`` is reserved; - it is not a valid identifier): - - .. insertprodn ident subsequent_letter - - .. prodn:: - ident ::= @first_letter {* @subsequent_letter } - first_letter ::= {| a .. z | A .. Z | _ | @unicode_letter } - subsequent_letter ::= {| @first_letter | @digit | ' | @unicode_id_part } - - All characters are meaningful. In particular, identifiers are case-sensitive. - :production:`unicode_letter` non-exhaustively includes Latin, - Greek, Gothic, Cyrillic, Arabic, Hebrew, Georgian, Hangul, Hiragana - and Katakana characters, CJK ideographs, mathematical letter-like - symbols and non-breaking space. :production:`unicode_id_part` - non-exhaustively includes symbols for prime letters and subscripts. - -Numerals - Numerals are sequences of digits with an optional fractional part - and exponent, optionally preceded by a minus sign. :n:`@int` is an integer; - a numeral without fractional or exponent parts. :n:`@num` is a non-negative - integer. Underscores embedded in the digits are ignored, for example - ``1_000_000`` is the same as ``1000000``. - - .. insertprodn numeral digit - - .. prodn:: - numeral ::= {+ @digit } {? . {+ @digit } } {? {| e | E } {? {| + | - } } {+ @digit } } - int ::= {? - } {+ @digit } - num ::= {+ @digit } - digit ::= 0 .. 9 - -Strings - Strings begin and end with ``"`` (double quote). Use ``""`` to represent - a double quote character within a string. In the grammar, strings are - identified with :production:`string`. - -Keywords - The following character sequences are reserved keywords that cannot be - used as identifiers:: - - _ Axiom CoFixpoint Definition Fixpoint Hypothesis IF Parameter Prop - SProp Set Theorem Type Variable as at by cofix discriminated else - end exists exists2 fix for forall fun if in lazymatch let match - multimatch return then using where with - - Note that plugins may define additional keywords when they are loaded. - -Other tokens - The set of - tokens defined at any given time can vary because the :cmd:`Notation` - command can define new tokens. A :cmd:`Require` command may load more notation definitions, - while the end of a :cmd:`Section` may remove notations. Some notations - are defined in the basic library (see :ref:`thecoqlibrary`) and are normally - loaded automatically at startup time. - - Here are the character sequences that Coq directly defines as tokens - without using :cmd:`Notation` (omitting 25 specialized tokens that begin with - ``#int63_``):: - - ! #[ % & ' ( () (bfs) (dfs) ) * ** + , - -> - . .( .. ... / : ::= := :> :>> ; < <+ <- <: - <<: <= = => > >-> >= ? @ @{ [ [= ] _ - `( `{ { {| | |- || } - - When multiple tokens match the beginning of a sequence of characters, - the longest matching token is used. - Occasionally you may need to insert spaces to separate tokens. For example, - if ``~`` and ``~~`` are both defined as tokens, the inputs ``~ ~`` and - ``~~`` generate different tokens, whereas if `~~` is not defined, then the - two inputs are equivalent. - -.. _term: - -Terms -===== - -Syntax of terms ---------------- - -The following grammars describe the basic syntax of the terms of the -*Calculus of Inductive Constructions* (also called Cic). The formal -presentation of Cic is given in Chapter :ref:`calculusofinductiveconstructions`. Extensions of this syntax -are given in Chapter :ref:`extensionsofgallina`. How to customize the syntax -is described in Chapter :ref:`syntaxextensionsandnotationscopes`. - -.. insertprodn term field_def - -.. prodn:: - term ::= forall @open_binders , @term - | fun @open_binders => @term - | @term_let - | if @term {? {? as @name } return @term100 } then @term else @term - | @term_fix - | @term_cofix - | @term100 - term100 ::= @term_cast - | @term10 - term10 ::= @term1 {+ @arg } - | @ @qualid {? @univ_annot } {* @term1 } - | @term1 - arg ::= ( @ident := @term ) - | @term1 - one_term ::= @term1 - | @ @qualid {? @univ_annot } - term1 ::= @term_projection - | @term0 % @scope_key - | @term0 - term0 ::= @qualid {? @univ_annot } - | @sort - | @numeral - | @string - | _ - | @term_evar - | @term_match - | ( @term ) - | %{%| {* @field_def } %|%} - | `%{ @term %} - | `( @term ) - | ltac : ( @ltac_expr ) - field_def ::= @qualid {* @binder } := @term - -.. note:: - - Many commands and tactics use :n:`@one_term` rather than :n:`@term`. - The former need to be enclosed in parentheses unless they're very - simple, such as a single identifier. This avoids confusing a space-separated - list of terms with a :n:`@term1` applied to a list of arguments. - -.. _types: - -Types ------ - -.. prodn:: - type ::= @term - -:n:`@type`\s are a subset of :n:`@term`\s; not every :n:`@term` is a :n:`@type`. -Every term has an associated type, which -can be determined by applying the :ref:`typing-rules`. Distinct terms -may share the same type, for example 0 and 1 are both of type `nat`, the -natural numbers. - -.. _gallina-identifiers: - -Qualified identifiers and simple identifiers --------------------------------------------- - -.. insertprodn qualid field_ident - -.. prodn:: - qualid ::= @ident {* @field_ident } - field_ident ::= .@ident - -*Qualified identifiers* (:n:`@qualid`) denote *global constants* -(definitions, lemmas, theorems, remarks or facts), *global variables* -(parameters or axioms), *inductive types* or *constructors of inductive -types*. *Simple identifiers* (or shortly :n:`@ident`) are a syntactic subset -of qualified identifiers. Identifiers may also denote *local variables*, -while qualified identifiers do not. - -Field identifiers, written :n:`@field_ident`, are identifiers prefixed by -`.` (dot) with no blank between the dot and the identifier. - - -Numerals and strings --------------------- - -Numerals and strings have no predefined semantics in the calculus. They are -merely notations that can be bound to objects through the notation mechanism -(see Chapter :ref:`syntaxextensionsandnotationscopes` for details). -Initially, numerals are bound to Peano’s representation of natural -numbers (see :ref:`datatypes`). - -.. note:: - - Negative integers are not at the same level as :n:`@num`, for this - would make precedence unnatural. - -.. index:: - single: Set (sort) - single: SProp - single: Prop - single: Type - -Sorts ------ - -.. insertprodn sort univ_constraint - -.. prodn:: - sort ::= Set - | Prop - | SProp - | Type - | Type @%{ _ %} - | Type @%{ @universe %} - universe ::= max ( {+, @universe_expr } ) - | @universe_expr - universe_expr ::= @universe_name {? + @num } - universe_name ::= @qualid - | Set - | Prop - univ_annot ::= @%{ {* @universe_level } %} - universe_level ::= Set - | Prop - | Type - | _ - | @qualid - univ_decl ::= @%{ {* @ident } {? + } {? %| {*, @univ_constraint } {? + } } %} - univ_constraint ::= @universe_name {| < | = | <= } @universe_name - -There are four sorts :g:`SProp`, :g:`Prop`, :g:`Set` and :g:`Type`. - -- :g:`SProp` is the universe of *definitionally irrelevant - propositions* (also called *strict propositions*). - -- :g:`Prop` is the universe of *logical propositions*. The logical propositions - themselves are typing the proofs. We denote propositions by :n:`@form`. - This constitutes a semantic subclass of the syntactic class :n:`@term`. - -- :g:`Set` is the universe of *program types* or *specifications*. The - specifications themselves are typing the programs. We denote - specifications by :n:`@specif`. This constitutes a semantic subclass of - the syntactic class :n:`@term`. - -- :g:`Type` is the type of sorts. - -More on sorts can be found in Section :ref:`sorts`. - -.. _binders: - -Binders -------- - -.. insertprodn open_binders binder - -.. prodn:: - open_binders ::= {+ @name } : @term - | {+ @binder } - name ::= _ - | @ident - binder ::= @name - | ( {+ @name } : @type ) - | ( @name {? : @type } := @term ) - | @implicit_binders - | @generalizing_binder - | ( @name : @type %| @term ) - | ' @pattern0 - -Various constructions such as :g:`fun`, :g:`forall`, :g:`fix` and :g:`cofix` -*bind* variables. A binding is represented by an identifier. If the binding -variable is not used in the expression, the identifier can be replaced by the -symbol :g:`_`. When the type of a bound variable cannot be synthesized by the -system, it can be specified with the notation :n:`(@ident : @type)`. There is also -a notation for a sequence of binding variables sharing the same type: -:n:`({+ @ident} : @type)`. A -binder can also be any pattern prefixed by a quote, e.g. :g:`'(x,y)`. - -Some constructions allow the binding of a variable to value. This is -called a “let-binder”. The entry :n:`@binder` of the grammar accepts -either an assumption binder as defined above or a let-binder. The notation in -the latter case is :n:`(@ident := @term)`. In a let-binder, only one -variable can be introduced at the same time. It is also possible to give -the type of the variable as follows: -:n:`(@ident : @type := @term)`. - -Lists of :n:`@binder`\s are allowed. In the case of :g:`fun` and :g:`forall`, -it is intended that at least one binder of the list is an assumption otherwise -fun and forall gets identical. Moreover, parentheses can be omitted in -the case of a single sequence of bindings sharing the same type (e.g.: -:g:`fun (x y z : A) => t` can be shortened in :g:`fun x y z : A => t`). - -.. index:: fun ... => ... - -Abstractions: fun ------------------ - -The expression :n:`fun @ident : @type => @term` defines the -*abstraction* of the variable :n:`@ident`, of type :n:`@type`, over the term -:n:`@term`. It denotes a function of the variable :n:`@ident` that evaluates to -the expression :n:`@term` (e.g. :g:`fun x : A => x` denotes the identity -function on type :g:`A`). The keyword :g:`fun` can be followed by several -binders as given in Section :ref:`binders`. Functions over -several variables are equivalent to an iteration of one-variable -functions. For instance the expression -:n:`fun {+ @ident__i } : @type => @term` -denotes the same function as :n:`{+ fun @ident__i : @type => } @term`. If -a let-binder occurs in -the list of binders, it is expanded to a let-in definition (see -Section :ref:`let-in`). - -.. index:: forall - -Products: forall ----------------- - -The expression :n:`forall @ident : @type, @term` denotes the -*product* of the variable :n:`@ident` of type :n:`@type`, over the term :n:`@term`. -As for abstractions, :g:`forall` is followed by a binder list, and products -over several variables are equivalent to an iteration of one-variable -products. Note that :n:`@term` is intended to be a type. - -If the variable :n:`@ident` occurs in :n:`@term`, the product is called -*dependent product*. The intention behind a dependent product -:g:`forall x : A, B` is twofold. It denotes either -the universal quantification of the variable :g:`x` of type :g:`A` -in the proposition :g:`B` or the functional dependent product from -:g:`A` to :g:`B` (a construction usually written -:math:`\Pi_{x:A}.B` in set theory). - -Non dependent product types have a special notation: :g:`A -> B` stands for -:g:`forall _ : A, B`. The *non dependent product* is used both to denote -the propositional implication and function types. - -Applications ------------- - -:n:`@term__fun @term` denotes applying the function :n:`@term__fun` to :token:`term`. - -:n:`@term__fun {+ @term__i }` denotes applying -:n:`@term__fun` to the arguments :n:`@term__i`. It is -equivalent to :n:`( … ( @term__fun @term__1 ) … ) @term__n`: -associativity is to the left. - -The notation :n:`(@ident := @term)` for arguments is used for making -explicit the value of implicit arguments (see -Section :ref:`explicit-applications`). - -.. index:: - single: ... : ... (type cast) - single: ... <: ... - single: ... <<: ... - -Type cast ---------- - -.. insertprodn term_cast term_cast - -.. prodn:: - term_cast ::= @term10 <: @term - | @term10 <<: @term - | @term10 : @term - | @term10 :> - -The expression :n:`@term : @type` is a type cast expression. It enforces -the type of :n:`@term` to be :n:`@type`. - -:n:`@term <: @type` locally sets up the virtual machine for checking that -:n:`@term` has type :n:`@type`. - -:n:`@term <<: @type` uses native compilation for checking that :n:`@term` -has type :n:`@type`. - -.. index:: _ - -Inferable subterms ------------------- - -Expressions often contain redundant pieces of information. Subterms that can be -automatically inferred by Coq can be replaced by the symbol ``_`` and Coq will -guess the missing piece of information. - -.. index:: let ... := ... (term) - -.. _let-in: - -Let-in definitions ------------------- - -.. insertprodn term_let term_let - -.. prodn:: - term_let ::= let @name {? : @type } := @term in @term - | let @name {+ @binder } {? : @type } := @term in @term - | let ( {*, @name } ) {? {? as @name } return @term100 } := @term in @term - | let ' @pattern := @term {? return @term100 } in @term - | let ' @pattern in @pattern := @term return @term100 in @term - -:n:`let @ident := @term in @term’` -denotes the local binding of :n:`@term` to the variable -:n:`@ident` in :n:`@term`’. There is a syntactic sugar for let-in -definition of functions: :n:`let @ident {+ @binder} := @term in @term’` -stands for :n:`let @ident := fun {+ @binder} => @term in @term’`. - -.. index:: match ... with ... - -Definition by cases: match --------------------------- - -.. insertprodn term_match pattern0 - -.. prodn:: - term_match ::= match {+, @case_item } {? return @term100 } with {? %| } {*| @eqn } end - case_item ::= @term100 {? as @name } {? in @pattern } - eqn ::= {+| {+, @pattern } } => @term - pattern ::= @pattern10 : @term - | @pattern10 - pattern10 ::= @pattern1 as @name - | @pattern1 {* @pattern1 } - | @ @qualid {* @pattern1 } - pattern1 ::= @pattern0 % @scope_key - | @pattern0 - pattern0 ::= @qualid - | %{%| {* @qualid := @pattern } %|%} - | _ - | ( {+| @pattern } ) - | @numeral - | @string - -Objects of inductive types can be destructured by a case-analysis -construction called *pattern matching* expression. A pattern matching -expression is used to analyze the structure of an inductive object and -to apply specific treatments accordingly. - -This paragraph describes the basic form of pattern matching. See -Section :ref:`Mult-match` and Chapter :ref:`extendedpatternmatching` for the description -of the general form. The basic form of pattern matching is characterized -by a single :n:`@case_item` expression, an :n:`@eqn` restricted to a -single :n:`@pattern` and :n:`@pattern` restricted to the form -:n:`@qualid {* @ident}`. - -The expression -:n:`match @term {? return @term100 } with {+| @pattern__i => @term__i } end` denotes a -*pattern matching* over the term :n:`@term` (expected to be -of an inductive type :math:`I`). The :n:`@term__i` -are the *branches* of the pattern matching -expression. Each :n:`@pattern__i` has the form :n:`@qualid @ident` -where :n:`@qualid` must denote a constructor. There should be -exactly one branch for every constructor of :math:`I`. - -The :n:`return @term100` clause gives the type returned by the whole match -expression. There are several cases. In the *non dependent* case, all -branches have the same type, and the :n:`return @term100` specifies that type. -In this case, :n:`return @term100` can usually be omitted as it can be -inferred from the type of the branches [1]_. - -In the *dependent* case, there are three subcases. In the first subcase, -the type in each branch may depend on the exact value being matched in -the branch. In this case, the whole pattern matching itself depends on -the term being matched. This dependency of the term being matched in the -return type is expressed with an :n:`@ident` clause where :n:`@ident` -is dependent in the return type. For instance, in the following example: - -.. coqtop:: in - - Inductive bool : Type := true : bool | false : bool. - Inductive eq (A:Type) (x:A) : A -> Prop := eq_refl : eq A x x. - Inductive or (A:Prop) (B:Prop) : Prop := - | or_introl : A -> or A B - | or_intror : B -> or A B. - - Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) := - match b as x return or (eq bool x true) (eq bool x false) with - | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) - | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) - end. - -the branches have respective types ":g:`or (eq bool true true) (eq bool true false)`" -and ":g:`or (eq bool false true) (eq bool false false)`" while the whole -pattern matching expression has type ":g:`or (eq bool b true) (eq bool b false)`", -the identifier :g:`b` being used to represent the dependency. - -.. note:: - - When the term being matched is a variable, the ``as`` clause can be - omitted and the term being matched can serve itself as binding name in - the return type. For instance, the following alternative definition is - accepted and has the same meaning as the previous one. - - .. coqtop:: none - - Reset bool_case. - - .. coqtop:: in - - Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) := - match b return or (eq bool b true) (eq bool b false) with - | true => or_introl (eq bool true true) (eq bool true false) (eq_refl bool true) - | false => or_intror (eq bool false true) (eq bool false false) (eq_refl bool false) - end. - -The second subcase is only relevant for annotated inductive types such -as the equality predicate (see Section :ref:`coq-equality`), -the order predicate on natural numbers or the type of lists of a given -length (see Section :ref:`matching-dependent`). In this configuration, the -type of each branch can depend on the type dependencies specific to the -branch and the whole pattern matching expression has a type determined -by the specific dependencies in the type of the term being matched. This -dependency of the return type in the annotations of the inductive type -is expressed with a clause in the form -:n:`in @qualid {+ _ } {+ @pattern }`, where - -- :n:`@qualid` is the inductive type of the term being matched; - -- the holes :n:`_` match the parameters of the inductive type: the - return type is not dependent on them. - -- each :n:`@pattern` matches the annotations of the - inductive type: the return type is dependent on them - -- in the basic case which we describe below, each :n:`@pattern` - is a name :n:`@ident`; see :ref:`match-in-patterns` for the - general case - -For instance, in the following example: - -.. coqtop:: in - - Definition eq_sym (A:Type) (x y:A) (H:eq A x y) : eq A y x := - match H in eq _ _ z return eq A z x with - | eq_refl _ _ => eq_refl A x - end. - -the type of the branch is :g:`eq A x x` because the third argument of -:g:`eq` is :g:`x` in the type of the pattern :g:`eq_refl`. On the contrary, the -type of the whole pattern matching expression has type :g:`eq A y x` because the -third argument of eq is y in the type of H. This dependency of the case analysis -in the third argument of :g:`eq` is expressed by the identifier :g:`z` in the -return type. - -Finally, the third subcase is a combination of the first and second -subcase. In particular, it only applies to pattern matching on terms in -a type with annotations. For this third subcase, both the clauses ``as`` and -``in`` are available. - -There are specific notations for case analysis on types with one or two -constructors: ``if … then … else …`` and ``let (…,…) := … in …`` (see -Sections :ref:`if-then-else` and :ref:`irrefutable-patterns`). - -.. index:: - single: fix - single: cofix - -Recursive and co-recursive functions: fix and cofix ---------------------------------------------------- - -.. insertprodn term_fix fixannot - -.. prodn:: - term_fix ::= let fix @fix_body in @term - | fix @fix_body {? {+ with @fix_body } for @ident } - fix_body ::= @ident {* @binder } {? @fixannot } {? : @type } := @term - fixannot ::= %{ struct @ident %} - | %{ wf @one_term @ident %} - | %{ measure @one_term {? @ident } {? @one_term } %} - - -The expression ":n:`fix @ident__1 @binder__1 : @type__1 := @term__1 with … with @ident__n @binder__n : @type__n := @term__n for @ident__i`" denotes the -:math:`i`-th component of a block of functions defined by mutual structural -recursion. It is the local counterpart of the :cmd:`Fixpoint` command. When -:math:`n=1`, the ":n:`for @ident__i`" clause is omitted. - -The association of a single fixpoint and a local definition have a special -syntax: :n:`let fix @ident {* @binder } := @term in` stands for -:n:`let @ident := fix @ident {* @binder } := @term in`. The same applies for co-fixpoints. - -Some options of :n:`@fixannot` are only supported in specific constructs. :n:`fix` and :n:`let fix` -only support the :n:`struct` option, while :n:`wf` and :n:`measure` are only supported in -commands such as :cmd:`Function` and :cmd:`Program Fixpoint`. - -.. insertprodn term_cofix cofix_body - -.. prodn:: - term_cofix ::= let cofix @cofix_body in @term - | cofix @cofix_body {? {+ with @cofix_body } for @ident } - cofix_body ::= @ident {* @binder } {? : @type } := @term - -The expression -":n:`cofix @ident__1 @binder__1 : @type__1 with … with @ident__n @binder__n : @type__n for @ident__i`" -denotes the :math:`i`-th component of a block of terms defined by a mutual guarded -co-recursion. It is the local counterpart of the :cmd:`CoFixpoint` command. When -:math:`n=1`, the ":n:`for @ident__i`" clause is omitted. - -.. _vernacular: - -The Vernacular -============== - -.. insertprodn vernacular sentence - -.. prodn:: - vernacular ::= {* @sentence } - sentence ::= {? @all_attrs } @command . - | {? @all_attrs } {? @num : } @query_command . - | {? @all_attrs } {? @toplevel_selector } @ltac_expr {| . | ... } - | @control_command - -The top-level input to |Coq| is a series of :n:`@sentence`\s, -which are :production:`tactic`\s or :production:`command`\s, -generally terminated with a period -and optionally decorated with :ref:`gallina-attributes`. :n:`@ltac_expr` syntax supports both simple -and compound tactics. For example: ``split`` is a simple tactic while ``split; auto`` combines two -simple tactics. - -Tactics specify how to transform the current proof state as a step in creating a proof. They -are syntactically valid only when |Coq| is in proof mode, such as after a :cmd:`Theorem` command -and before any subsequent proof-terminating command such as :cmd:`Qed`. See :ref:`proofhandling` for more -on proof mode. - -By convention, command names begin with uppercase letters, while -tactic names begin with lowercase letters. Commands appear in the -HTML documentation in blue boxes after the label "Command". In the pdf, they appear -after the boldface label "Command:". Commands are listed in the :ref:`command_index`. - -Similarly, tactics appear after the label "Tactic". Tactics are listed in the :ref:`tactic_index`. - -.. _gallina-assumptions: - -Assumptions ------------ - -Assumptions extend the environment with axioms, parameters, hypotheses -or variables. An assumption binds an :n:`@ident` to a :n:`@type`. It is accepted -by Coq if and only if this :n:`@type` is a correct type in the environment -preexisting the declaration and if :n:`@ident` was not previously defined in -the same module. This :n:`@type` is considered to be the type (or -specification, or statement) assumed by :n:`@ident` and we say that :n:`@ident` -has type :n:`@type`. - -.. _Axiom: - -.. cmd:: @assumption_token {? Inline {? ( @num ) } } {| {+ ( @assumpt ) } | @assumpt } - :name: Axiom; Axioms; Conjecture; Conjectures; Hypothesis; Hypotheses; Parameter; Parameters; Variable; Variables - - .. insertprodn assumption_token of_type - - .. prodn:: - assumption_token ::= {| Axiom | Axioms } - | {| Conjecture | Conjectures } - | {| Parameter | Parameters } - | {| Hypothesis | Hypotheses } - | {| Variable | Variables } - assumpt ::= {+ @ident_decl } @of_type - ident_decl ::= @ident {? @univ_decl } - of_type ::= {| : | :> | :>> } @type - - These commands bind one or more :n:`@ident`\(s) to specified :n:`@type`\(s) as their specifications in - the global context. The fact asserted by the :n:`@type` (or, equivalently, the existence - of an object of this type) is accepted as a postulate. - - :cmd:`Axiom`, :cmd:`Conjecture`, :cmd:`Parameter` and their plural forms - are equivalent. They can take the :attr:`local` attribute (see :ref:`gallina-attributes`), - which makes the defined :n:`@ident`\s accessible by :cmd:`Import` and its variants - only through their fully qualified names. - - Similarly, :cmd:`Hypothesis`, :cmd:`Variable` and their plural forms are equivalent. Outside - of a section, these are equivalent to :n:`Local Parameter`. Inside a section, the - :n:`@ident`\s defined are only accessible within the section. When the current section - is closed, the :n:`@ident`\(s) become undefined and every object depending on them will be explicitly - parameterized (i.e., the variables are *discharged*). See Section :ref:`section-mechanism`. - - The :n:`Inline` clause is only relevant inside functors. See :cmd:`Module`. - -.. example:: Simple assumptions - - .. coqtop:: reset in - - Parameter X Y : Set. - Parameter (R : X -> Y -> Prop) (S : Y -> X -> Prop). - Axiom R_S_inv : forall x y, R x y <-> S y x. - -.. exn:: @ident already exists. - :name: @ident already exists. (Axiom) - :undocumented: - -.. warn:: @ident is declared as a local axiom - - Warning generated when using :cmd:`Variable` or its equivalent - instead of :n:`Local Parameter` or its equivalent. - -.. note:: - We advise using the commands :cmd:`Axiom`, :cmd:`Conjecture` and - :cmd:`Hypothesis` (and their plural forms) for logical postulates (i.e. when - the assertion :n:`@type` is of sort :g:`Prop`), and to use the commands - :cmd:`Parameter` and :cmd:`Variable` (and their plural forms) in other cases - (corresponding to the declaration of an abstract object of the given type). - -.. _gallina-definitions: - -Definitions ------------ - -Definitions extend the environment with associations of names to terms. -A definition can be seen as a way to give a meaning to a name or as a -way to abbreviate a term. In any case, the name can later be replaced at -any time by its definition. - -The operation of unfolding a name into its definition is called -:math:`\delta`-conversion (see Section :ref:`delta-reduction`). A -definition is accepted by the system if and only if the defined term is -well-typed in the current context of the definition and if the name is -not already used. The name defined by the definition is called a -*constant* and the term it refers to is its *body*. A definition has a -type which is the type of its body. - -A formal presentation of constants and environments is given in -Section :ref:`typing-rules`. - -.. cmd:: {| Definition | Example } @ident_decl @def_body - :name: Definition; Example - - .. insertprodn def_body def_body - - .. prodn:: - def_body ::= {* @binder } {? : @type } := {? @reduce } @term - | {* @binder } : @type - - These commands bind :n:`@term` to the name :n:`@ident` in the environment, - provided that :n:`@term` is well-typed. They can take the :attr:`local` attribute (see :ref:`gallina-attributes`), - which makes the defined :n:`@ident` accessible by :cmd:`Import` and its variants - only through their fully qualified names. - If :n:`@reduce` is present then :n:`@ident` is bound to the result of the specified - computation on :n:`@term`. - - These commands also support the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`program` and - :attr:`canonical` attributes. - - If :n:`@term` is omitted, :n:`@type` is required and Coq enters proof editing mode. - This can be used to define a term incrementally, in particular by relying on the :tacn:`refine` tactic. - In this case, the proof should be terminated with :cmd:`Defined` in order to define a constant - for which the computational behavior is relevant. See :ref:`proof-editing-mode`. - - The form :n:`Definition @ident : @type := @term` checks that the type of :n:`@term` - is definitionally equal to :n:`@type`, and registers :n:`@ident` as being of type - :n:`@type`, and bound to value :n:`@term`. - - The form :n:`Definition @ident {* @binder } : @type := @term` is equivalent to - :n:`Definition @ident : forall {* @binder }, @type := fun {* @binder } => @term`. - - .. seealso:: :cmd:`Opaque`, :cmd:`Transparent`, :tacn:`unfold`. - - .. exn:: @ident already exists. - :name: @ident already exists. (Definition) - :undocumented: - - .. exn:: The term @term has type @type while it is expected to have type @type'. - :undocumented: - -.. _gallina-inductive-definitions: - -Inductive types ---------------- - -.. cmd:: Inductive @inductive_definition {* with @inductive_definition } - - .. insertprodn inductive_definition constructor - - .. prodn:: - inductive_definition ::= {? > } @ident_decl {* @binder } {? %| {* @binder } } {? : @type } {? := {? @constructors_or_record } } {? @decl_notations } - constructors_or_record ::= {? %| } {+| @constructor } - | {? @ident } %{ {*; @record_field } %} - constructor ::= @ident {* @binder } {? @of_type } - - This command defines one or more - inductive types and its constructors. Coq generates destructors - depending on the universe that the inductive type belongs to. - - The destructors are named :n:`@ident`\ ``_rect``, :n:`@ident`\ ``_ind``, - :n:`@ident`\ ``_rec`` and :n:`@ident`\ ``_sind``, which - respectively correspond to elimination principles on :g:`Type`, :g:`Prop`, - :g:`Set` and :g:`SProp`. The type of the destructors - expresses structural induction/recursion principles over objects of - type :n:`@ident`. The constant :n:`@ident`\ ``_ind`` is always - generated, whereas :n:`@ident`\ ``_rec`` and :n:`@ident`\ ``_rect`` - may be impossible to derive (for example, when :n:`@ident` is a - proposition). - - This command supports the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`universes(template)`, - :attr:`universes(notemplate)`, :attr:`universes(cumulative)`, - :attr:`universes(noncumulative)` and :attr:`private(matching)` - attributes. - - Mutually inductive types can be defined by including multiple :n:`@inductive_definition`\s. - The :n:`@ident`\s are simultaneously added to the environment before the types of constructors are checked. - Each :n:`@ident` can be used independently thereafter. - See :ref:`mutually_inductive_types`. - - If the entire inductive definition is parameterized with :n:`@binder`\s, the parameters correspond - to a local context in which the entire set of inductive declarations is interpreted. - For this reason, the parameters must be strictly the same for each inductive type. - See :ref:`parametrized-inductive-types`. - - Constructor :n:`@ident`\s can come with :n:`@binder`\s, in which case - the actual type of the constructor is :n:`forall {* @binder }, @type`. - - .. exn:: Non strictly positive occurrence of @ident in @type. - - The types of the constructors have to satisfy a *positivity condition* - (see Section :ref:`positivity`). This condition ensures the soundness of - the inductive definition. The positivity checking can be disabled using - the :flag:`Positivity Checking` flag (see :ref:`controlling-typing-flags`). - - .. exn:: The conclusion of @type is not valid; it must be built from @ident. - - The conclusion of the type of the constructors must be the inductive type - :n:`@ident` being defined (or :n:`@ident` applied to arguments in - the case of annotated inductive types — cf. next section). - -The following subsections show examples of simple inductive types, -simple annotated inductive types, simple parametric inductive types, -mutually inductive types and private (matching) inductive types. - -.. _simple-inductive-types: - -Simple inductive types -~~~~~~~~~~~~~~~~~~~~~~ - -A simple inductive type belongs to a universe that is a simple :n:`@sort`. - -.. example:: - - The set of natural numbers is defined as: - - .. coqtop:: reset all - - Inductive nat : Set := - | O : nat - | S : nat -> nat. - - The type nat is defined as the least :g:`Set` containing :g:`O` and closed by - the :g:`S` constructor. The names :g:`nat`, :g:`O` and :g:`S` are added to the - environment. - - This definition generates four elimination principles: - :g:`nat_rect`, :g:`nat_ind`, :g:`nat_rec` and :g:`nat_sind`. The type of :g:`nat_ind` is: - - .. coqtop:: all - - Check nat_ind. - - This is the well known structural induction principle over natural - numbers, i.e. the second-order form of Peano’s induction principle. It - allows proving universal properties of natural numbers (:g:`forall - n:nat, P n`) by induction on :g:`n`. - - The types of :g:`nat_rect`, :g:`nat_rec` and :g:`nat_sind` are similar, except that they - apply to, respectively, :g:`(P:nat->Type)`, :g:`(P:nat->Set)` and :g:`(P:nat->SProp)`. They correspond to - primitive induction principles (allowing dependent types) respectively - over sorts ```Type``, ``Set`` and ``SProp``. - -In the case where inductive types don't have annotations (the next section -gives an example of annotations), a constructor can be defined -by giving the type of its arguments alone. - -.. example:: - - .. coqtop:: reset none - - Reset nat. - - .. coqtop:: in - - Inductive nat : Set := O | S (_:nat). - -Simple annotated inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -In annotated inductive types, the universe where the inductive type -is defined is no longer a simple :n:`@sort`, but what is called an arity, -which is a type whose conclusion is a :n:`@sort`. - -.. example:: - - As an example of annotated inductive types, let us define the - :g:`even` predicate: - - .. coqtop:: all - - Inductive even : nat -> Prop := - | even_0 : even O - | even_SS : forall n:nat, even n -> even (S (S n)). - - The type :g:`nat->Prop` means that :g:`even` is a unary predicate (inductively - defined) over natural numbers. The type of its two constructors are the - defining clauses of the predicate :g:`even`. The type of :g:`even_ind` is: - - .. coqtop:: all - - Check even_ind. - - From a mathematical point of view, this asserts that the natural numbers satisfying - the predicate :g:`even` are exactly in the smallest set of naturals satisfying the - clauses :g:`even_0` or :g:`even_SS`. This is why, when we want to prove any - predicate :g:`P` over elements of :g:`even`, it is enough to prove it for :g:`O` - and to prove that if any natural number :g:`n` satisfies :g:`P` its double - successor :g:`(S (S n))` satisfies also :g:`P`. This is analogous to the - structural induction principle we got for :g:`nat`. - -.. _parametrized-inductive-types: - -Parameterized inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -In the previous example, each constructor introduces a different -instance of the predicate :g:`even`. In some cases, all the constructors -introduce the same generic instance of the inductive definition, in -which case, instead of an annotation, we use a context of parameters -which are :n:`@binder`\s shared by all the constructors of the definition. - -Parameters differ from inductive type annotations in that the -conclusion of each type of constructor invokes the inductive type with -the same parameter values of its specification. - -.. example:: - - A typical example is the definition of polymorphic lists: - - .. coqtop:: all - - Inductive list (A:Set) : Set := - | nil : list A - | cons : A -> list A -> list A. - - In the type of :g:`nil` and :g:`cons`, we write ":g:`list A`" and not - just ":g:`list`". The constructors :g:`nil` and :g:`cons` have these types: - - .. coqtop:: all - - Check nil. - Check cons. - - Observe that the destructors are also quantified with :g:`(A:Set)`, for example: - - .. coqtop:: all - - Check list_ind. - - Once again, the types of the constructor arguments and of the conclusion can be omitted: - - .. coqtop:: none - - Reset list. - - .. coqtop:: in - - Inductive list (A:Set) : Set := nil | cons (_:A) (_:list A). - -.. note:: - + The constructor type can - recursively invoke the inductive definition on an argument which is not - the parameter itself. - - One can define : - - .. coqtop:: all - - Inductive list2 (A:Set) : Set := - | nil2 : list2 A - | cons2 : A -> list2 (A*A) -> list2 A. - - that can also be written by specifying only the type of the arguments: - - .. coqtop:: all reset - - Inductive list2 (A:Set) : Set := - | nil2 - | cons2 (_:A) (_:list2 (A*A)). - - But the following definition will give an error: - - .. coqtop:: all - - Fail Inductive listw (A:Set) : Set := - | nilw : listw (A*A) - | consw : A -> listw (A*A) -> listw (A*A). - - because the conclusion of the type of constructors should be :g:`listw A` - in both cases. - - + A parameterized inductive definition can be defined using annotations - instead of parameters but it will sometimes give a different (bigger) - sort for the inductive definition and will produce a less convenient - rule for case elimination. - -.. flag:: Uniform Inductive Parameters - - When this flag is set (it is off by default), - inductive definitions are abstracted over their parameters - before type checking constructors, allowing to write: - - .. coqtop:: all - - Set Uniform Inductive Parameters. - Inductive list3 (A:Set) : Set := - | nil3 : list3 - | cons3 : A -> list3 -> list3. - - This behavior is essentially equivalent to starting a new section - and using :cmd:`Context` to give the uniform parameters, like so - (cf. :ref:`section-mechanism`): - - .. coqtop:: all reset - - Section list3. - Context (A:Set). - Inductive list3 : Set := - | nil3 : list3 - | cons3 : A -> list3 -> list3. - End list3. - - For finer control, you can use a ``|`` between the uniform and - the non-uniform parameters: - - .. coqtop:: in reset - - Inductive Acc {A:Type} (R:A->A->Prop) | (x:A) : Prop - := Acc_in : (forall y, R y x -> Acc y) -> Acc x. - - The flag can then be seen as deciding whether the ``|`` is at the - beginning (when the flag is unset) or at the end (when it is set) - of the parameters when not explicitly given. - -.. seealso:: - Section :ref:`inductive-definitions` and the :tacn:`induction` tactic. - -.. _mutually_inductive_types: - -Mutually defined inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. example:: Mutually defined inductive types - - A typical example of mutually inductive data types is trees and - forests. We assume two types :g:`A` and :g:`B` that are given as variables. The types can - be declared like this: - - .. coqtop:: in - - Parameters A B : Set. - - Inductive tree : Set := node : A -> forest -> tree - - with forest : Set := - | leaf : B -> forest - | cons : tree -> forest -> forest. - - This declaration automatically generates eight induction principles. They are not the most - general principles, but they correspond to each inductive part seen as a single inductive definition. - - To illustrate this point on our example, here are the types of :g:`tree_rec` - and :g:`forest_rec`. - - .. coqtop:: all - - Check tree_rec. - - Check forest_rec. - - Assume we want to parameterize our mutual inductive definitions with the - two type variables :g:`A` and :g:`B`, the declaration should be - done as follows: - - .. coqdoc:: - - Inductive tree (A B:Set) : Set := node : A -> forest A B -> tree A B - - with forest (A B:Set) : Set := - | leaf : B -> forest A B - | cons : tree A B -> forest A B -> forest A B. - - Assume we define an inductive definition inside a section - (cf. :ref:`section-mechanism`). When the section is closed, the variables - declared in the section and occurring free in the declaration are added as - parameters to the inductive definition. - -.. seealso:: - A generic command :cmd:`Scheme` is useful to build automatically various - mutual induction principles. - -Private (matching) inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. attr:: private(matching) - - This attribute can be used to forbid the use of the :g:`match` - construct on objects of this inductive type outside of the module - where it is defined. There is also a legacy syntax using the - ``Private`` prefix (cf. :n:`@legacy_attr`). - - The main use case of private (matching) inductive types is to emulate - quotient types / higher-order inductive types in projects such as - the `HoTT library `_. - -.. example:: - - .. coqtop:: all - - Module Foo. - #[ private(matching) ] Inductive my_nat := my_O : my_nat | my_S : my_nat -> my_nat. - Check (fun x : my_nat => match x with my_O => true | my_S _ => false end). - End Foo. - Import Foo. - Fail Check (fun x : my_nat => match x with my_O => true | my_S _ => false end). - -Variants -~~~~~~~~ - -.. cmd:: Variant @variant_definition {* with @variant_definition } - - .. insertprodn variant_definition variant_definition - - .. prodn:: - variant_definition ::= @ident_decl {* @binder } {? %| {* @binder } } {? : @type } := {? %| } {+| @constructor } {? @decl_notations } - - The :cmd:`Variant` command is similar to the :cmd:`Inductive` command, except - that it disallows recursive definition of types (for instance, lists cannot - be defined using :cmd:`Variant`). No induction scheme is generated for - this variant, unless the :flag:`Nonrecursive Elimination Schemes` flag is on. - - This command supports the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`universes(template)`, - :attr:`universes(notemplate)`, :attr:`universes(cumulative)`, - :attr:`universes(noncumulative)` and :attr:`private(matching)` - attributes. - - .. exn:: The @num th argument of @ident must be @ident in @type. - :undocumented: - -.. _coinductive-types: - -Co-inductive types ------------------- - -The objects of an inductive type are well-founded with respect to the -constructors of the type. In other words, such objects contain only a -*finite* number of constructors. Co-inductive types arise from relaxing -this condition, and admitting types whose objects contain an infinity of -constructors. Infinite objects are introduced by a non-ending (but -effective) process of construction, defined in terms of the constructors -of the type. - -.. cmd:: CoInductive @inductive_definition {* with @inductive_definition } - - This command introduces a co-inductive type. - The syntax of the command is the same as the command :cmd:`Inductive`. - No principle of induction is derived from the definition of a co-inductive - type, since such principles only make sense for inductive types. - For co-inductive types, the only elimination principle is case analysis. - - This command supports the :attr:`universes(polymorphic)`, - :attr:`universes(monomorphic)`, :attr:`universes(template)`, - :attr:`universes(notemplate)`, :attr:`universes(cumulative)`, - :attr:`universes(noncumulative)` and :attr:`private(matching)` - attributes. - -.. example:: - - The type of infinite sequences of natural numbers, usually called streams, - is an example of a co-inductive type. - - .. coqtop:: in - - CoInductive Stream : Set := Seq : nat -> Stream -> Stream. - - The usual destructors on streams :g:`hd:Stream->nat` and :g:`tl:Str->Str` - can be defined as follows: - - .. coqtop:: in - - Definition hd (x:Stream) := let (a,s) := x in a. - Definition tl (x:Stream) := let (a,s) := x in s. - -Definitions of co-inductive predicates and blocks of mutually -co-inductive definitions are also allowed. - -.. example:: - - The extensional equality on streams is an example of a co-inductive type: - - .. coqtop:: in - - CoInductive EqSt : Stream -> Stream -> Prop := - eqst : forall s1 s2:Stream, - hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2. - - In order to prove the extensional equality of two streams :g:`s1` and :g:`s2` - we have to construct an infinite proof of equality, that is, an infinite - object of type :g:`(EqSt s1 s2)`. We will see how to introduce infinite - objects in Section :ref:`cofixpoint`. - -Caveat -~~~~~~ - -The ability to define co-inductive types by constructors, hereafter called -*positive co-inductive types*, is known to break subject reduction. The story is -a bit long: this is due to dependent pattern-matching which implies -propositional η-equality, which itself would require full η-conversion for -subject reduction to hold, but full η-conversion is not acceptable as it would -make type checking undecidable. - -Since the introduction of primitive records in Coq 8.5, an alternative -presentation is available, called *negative co-inductive types*. This consists -in defining a co-inductive type as a primitive record type through its -projections. Such a technique is akin to the *co-pattern* style that can be -found in e.g. Agda, and preserves subject reduction. - -The above example can be rewritten in the following way. - -.. coqtop:: none - - Reset Stream. - -.. coqtop:: all - - Set Primitive Projections. - CoInductive Stream : Set := Seq { hd : nat; tl : Stream }. - CoInductive EqSt (s1 s2: Stream) : Prop := eqst { - eqst_hd : hd s1 = hd s2; - eqst_tl : EqSt (tl s1) (tl s2); - }. - -Some properties that hold over positive streams are lost when going to the -negative presentation, typically when they imply equality over streams. -For instance, propositional η-equality is lost when going to the negative -presentation. It is nonetheless logically consistent to recover it through an -axiom. - -.. coqtop:: all - - Axiom Stream_eta : forall s: Stream, s = Seq (hd s) (tl s). - -More generally, as in the case of positive coinductive types, it is consistent -to further identify extensional equality of coinductive types with propositional -equality: - -.. coqtop:: all - - Axiom Stream_ext : forall (s1 s2: Stream), EqSt s1 s2 -> s1 = s2. - -As of Coq 8.9, it is now advised to use negative co-inductive types rather than -their positive counterparts. - -.. seealso:: - :ref:`primitive_projections` for more information about negative - records and primitive projections. - - -Definition of recursive functions ---------------------------------- - -Definition of functions by recursion over inductive objects -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -This section describes the primitive form of definition by recursion over -inductive objects. See the :cmd:`Function` command for more advanced -constructions. - -.. _Fixpoint: - -.. cmd:: Fixpoint @fix_definition {* with @fix_definition } - - .. insertprodn fix_definition fix_definition - - .. prodn:: - fix_definition ::= @ident_decl {* @binder } {? @fixannot } {? : @type } {? := @term } {? @decl_notations } - - This command allows defining functions by pattern matching over inductive - objects using a fixed point construction. The meaning of this declaration is - to define :n:`@ident` as a recursive function with arguments specified by - the :n:`@binder`\s such that :n:`@ident` applied to arguments - corresponding to these :n:`@binder`\s has type :n:`@type`, and is - equivalent to the expression :n:`@term`. The type of :n:`@ident` is - consequently :n:`forall {* @binder }, @type` and its value is equivalent - to :n:`fun {* @binder } => @term`. - - To be accepted, a :cmd:`Fixpoint` definition has to satisfy syntactical - constraints on a special argument called the decreasing argument. They - are needed to ensure that the :cmd:`Fixpoint` definition always terminates. - The point of the :n:`{struct @ident}` annotation (see :n:`@fixannot`) is to - let the user tell the system which argument decreases along the recursive calls. - - The :n:`{struct @ident}` annotation may be left implicit, in which case the - system successively tries arguments from left to right until it finds one - that satisfies the decreasing condition. - - :cmd:`Fixpoint` without the :attr:`program` attribute does not support the - :n:`wf` or :n:`measure` clauses of :n:`@fixannot`. - - The :n:`with` clause allows simultaneously defining several mutual fixpoints. - It is especially useful when defining functions over mutually defined - inductive types. Example: :ref:`Mutual Fixpoints`. - - If :n:`@term` is omitted, :n:`@type` is required and Coq enters proof editing mode. - This can be used to define a term incrementally, in particular by relying on the :tacn:`refine` tactic. - In this case, the proof should be terminated with :cmd:`Defined` in order to define a constant - for which the computational behavior is relevant. See :ref:`proof-editing-mode`. - - .. note:: - - + Some fixpoints may have several arguments that fit as decreasing - arguments, and this choice influences the reduction of the fixpoint. - Hence an explicit annotation must be used if the leftmost decreasing - argument is not the desired one. Writing explicit annotations can also - speed up type checking of large mutual fixpoints. - - + In order to keep the strong normalization property, the fixed point - reduction will only be performed when the argument in position of the - decreasing argument (which type should be in an inductive definition) - starts with a constructor. - - - .. example:: - - One can define the addition function as : - - .. coqtop:: all - - Fixpoint add (n m:nat) {struct n} : nat := - match n with - | O => m - | S p => S (add p m) - end. - - The match operator matches a value (here :g:`n`) with the various - constructors of its (inductive) type. The remaining arguments give the - respective values to be returned, as functions of the parameters of the - corresponding constructor. Thus here when :g:`n` equals :g:`O` we return - :g:`m`, and when :g:`n` equals :g:`(S p)` we return :g:`(S (add p m))`. - - The match operator is formally described in - Section :ref:`match-construction`. - The system recognizes that in the inductive call :g:`(add p m)` the first - argument actually decreases because it is a *pattern variable* coming - from :g:`match n with`. - - .. example:: - - The following definition is not correct and generates an error message: - - .. coqtop:: all - - Fail Fixpoint wrongplus (n m:nat) {struct n} : nat := - match m with - | O => n - | S p => S (wrongplus n p) - end. - - because the declared decreasing argument :g:`n` does not actually - decrease in the recursive call. The function computing the addition over - the second argument should rather be written: - - .. coqtop:: all - - Fixpoint plus (n m:nat) {struct m} : nat := - match m with - | O => n - | S p => S (plus n p) - end. - - .. example:: - - The recursive call may not only be on direct subterms of the recursive - variable :g:`n` but also on a deeper subterm and we can directly write - the function :g:`mod2` which gives the remainder modulo 2 of a natural - number. - - .. coqtop:: all - - Fixpoint mod2 (n:nat) : nat := - match n with - | O => O - | S p => match p with - | O => S O - | S q => mod2 q - end - end. - -.. _example_mutual_fixpoints: - - .. example:: Mutual fixpoints - - The size of trees and forests can be defined the following way: - - .. coqtop:: all - - Fixpoint tree_size (t:tree) : nat := - match t with - | node a f => S (forest_size f) - end - with forest_size (f:forest) : nat := - match f with - | leaf b => 1 - | cons t f' => (tree_size t + forest_size f') - end. - -.. _cofixpoint: - -Definitions of recursive objects in co-inductive types -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. cmd:: CoFixpoint @cofix_definition {* with @cofix_definition } - - .. insertprodn cofix_definition cofix_definition - - .. prodn:: - cofix_definition ::= @ident_decl {* @binder } {? : @type } {? := @term } {? @decl_notations } - - This command introduces a method for constructing an infinite object of a - coinductive type. For example, the stream containing all natural numbers can - be introduced applying the following method to the number :g:`O` (see - Section :ref:`coinductive-types` for the definition of :g:`Stream`, :g:`hd` - and :g:`tl`): - - .. coqtop:: all - - CoFixpoint from (n:nat) : Stream := Seq n (from (S n)). - - Unlike recursive definitions, there is no decreasing argument in a - co-recursive definition. To be admissible, a method of construction must - provide at least one extra constructor of the infinite object for each - iteration. A syntactical guard condition is imposed on co-recursive - definitions in order to ensure this: each recursive call in the - definition must be protected by at least one constructor, and only by - constructors. That is the case in the former definition, where the single - recursive call of :g:`from` is guarded by an application of :g:`Seq`. - On the contrary, the following recursive function does not satisfy the - guard condition: - - .. coqtop:: all - - Fail CoFixpoint filter (p:nat -> bool) (s:Stream) : Stream := - if p (hd s) then Seq (hd s) (filter p (tl s)) else filter p (tl s). - - The elimination of co-recursive definition is done lazily, i.e. the - definition is expanded only when it occurs at the head of an application - which is the argument of a case analysis expression. In any other - context, it is considered as a canonical expression which is completely - evaluated. We can test this using the command :cmd:`Eval`, which computes - the normal forms of a term: - - .. coqtop:: all - - Eval compute in (from 0). - Eval compute in (hd (from 0)). - Eval compute in (tl (from 0)). - - As in the :cmd:`Fixpoint` command, the :n:`with` clause allows simultaneously - defining several mutual cofixpoints. - - If :n:`@term` is omitted, :n:`@type` is required and Coq enters proof editing mode. - This can be used to define a term incrementally, in particular by relying on the :tacn:`refine` tactic. - In this case, the proof should be terminated with :cmd:`Defined` in order to define a constant - for which the computational behavior is relevant. See :ref:`proof-editing-mode`. - -.. _Computations: - -Computations ------------- - -.. insertprodn reduce pattern_occ - -.. prodn:: - reduce ::= Eval @red_expr in - red_expr ::= red - | hnf - | simpl {? @delta_flag } {? @ref_or_pattern_occ } - | cbv {? @strategy_flag } - | cbn {? @strategy_flag } - | lazy {? @strategy_flag } - | compute {? @delta_flag } - | vm_compute {? @ref_or_pattern_occ } - | native_compute {? @ref_or_pattern_occ } - | unfold {+, @unfold_occ } - | fold {+ @one_term } - | pattern {+, @pattern_occ } - | @ident - delta_flag ::= {? - } [ {+ @smart_qualid } ] - strategy_flag ::= {+ @red_flags } - | @delta_flag - red_flags ::= beta - | iota - | match - | fix - | cofix - | zeta - | delta {? @delta_flag } - ref_or_pattern_occ ::= @smart_qualid {? at @occs_nums } - | @one_term {? at @occs_nums } - occs_nums ::= {+ {| @num | @ident } } - | - {| @num | @ident } {* @int_or_var } - int_or_var ::= @int - | @ident - unfold_occ ::= @smart_qualid {? at @occs_nums } - pattern_occ ::= @one_term {? at @occs_nums } - -See :ref:`Conversion-rules`. - -.. todo:: Add text here - -.. _Assertions: - -Assertions and proofs ---------------------- - -An assertion states a proposition (or a type) of which the proof (or an -inhabitant of the type) is interactively built using tactics. The interactive -proof mode is described in Chapter :ref:`proofhandling` and the tactics in -Chapter :ref:`Tactics`. The basic assertion command is: - -.. cmd:: @thm_token @ident_decl {* @binder } : @type {* with @ident_decl {* @binder } : @type } - :name: Theorem; Lemma; Fact; Remark; Corollary; Proposition; Property - - .. insertprodn thm_token thm_token - - .. prodn:: - thm_token ::= Theorem - | Lemma - | Fact - | Remark - | Corollary - | Proposition - | Property - - After the statement is asserted, Coq needs a proof. Once a proof of - :n:`@type` under the assumptions represented by :n:`@binder`\s is given and - validated, the proof is generalized into a proof of :n:`forall {* @binder }, @type` and - the theorem is bound to the name :n:`@ident` in the environment. - - Forms using the :n:`with` clause are useful for theorems that are proved by simultaneous induction - over a mutually inductive assumption, or that assert mutually dependent - statements in some mutual co-inductive type. It is equivalent to - :cmd:`Fixpoint` or :cmd:`CoFixpoint` but using tactics to build the proof of - the statements (or the body of the specification, depending on the point of - view). The inductive or co-inductive types on which the induction or - coinduction has to be done is assumed to be non ambiguous and is guessed by - the system. - - Like in a :cmd:`Fixpoint` or :cmd:`CoFixpoint` definition, the induction hypotheses - have to be used on *structurally smaller* arguments (for a :cmd:`Fixpoint`) or - be *guarded by a constructor* (for a :cmd:`CoFixpoint`). The verification that - recursive proof arguments are correct is done only at the time of registering - the lemma in the environment. To know if the use of induction hypotheses is - correct at some time of the interactive development of a proof, use the - command :cmd:`Guarded`. - - .. exn:: The term @term has type @type which should be Set, Prop or Type. - :undocumented: - - .. exn:: @ident already exists. - :name: @ident already exists. (Theorem) - - The name you provided is already defined. You have then to choose - another name. - - .. exn:: Nested proofs are not allowed unless you turn the Nested Proofs Allowed flag on. - - You are asserting a new statement while already being in proof editing mode. - This feature, called nested proofs, is disabled by default. - To activate it, turn the :flag:`Nested Proofs Allowed` flag on. - -Proofs start with the keyword :cmd:`Proof`. Then Coq enters the proof editing mode -until the proof is completed. In proof editing mode, the user primarily enters -tactics, which are described in chapter :ref:`Tactics`. The user may also enter -commands to manage the proof editing mode. They are described in Chapter -:ref:`proofhandling`. - -When the proof is complete, use the :cmd:`Qed` command so the kernel verifies -the proof and adds it to the environment. - -.. note:: - - #. Several statements can be simultaneously asserted provided the - :flag:`Nested Proofs Allowed` flag was turned on. - - #. Not only other assertions but any vernacular command can be given - while in the process of proving a given assertion. In this case, the - command is understood as if it would have been given before the - statements still to be proved. Nonetheless, this practice is discouraged - and may stop working in future versions. - - #. Proofs ended by :cmd:`Qed` are declared opaque. Their content cannot be - unfolded (see :ref:`performingcomputations`), thus - realizing some form of *proof-irrelevance*. To be able to unfold a - proof, the proof should be ended by :cmd:`Defined`. - - #. :cmd:`Proof` is recommended but can currently be omitted. On the opposite - side, :cmd:`Qed` (or :cmd:`Defined`) is mandatory to validate a proof. - - #. One can also use :cmd:`Admitted` in place of :cmd:`Qed` to turn the - current asserted statement into an axiom and exit the proof editing mode. - -.. _gallina-attributes: - -Attributes ------------ - -.. insertprodn all_attrs legacy_attr - -.. prodn:: - all_attrs ::= {* #[ {*, @attr } ] } {* @legacy_attr } - attr ::= @ident {? @attr_value } - attr_value ::= = @string - | ( {*, @attr } ) - legacy_attr ::= {| Local | Global } - | {| Polymorphic | Monomorphic } - | {| Cumulative | NonCumulative } - | Private - | Program - -Attributes modify the behavior of a command or tactic. -Syntactically, most commands and tactics can be decorated with attributes, but -attributes not supported by the command or tactic will be flagged as errors. - -The order of top-level attributes doesn't affect their meaning. ``#[foo,bar]``, ``#[bar,foo]``, -``#[foo]#[bar]`` and ``#[bar]#[foo]`` are equivalent. - -The legacy attributes (:n:`@legacy_attr`) provide an older, alternate syntax -for certain attributes. They are equivalent to new attributes as follows: - -================ ================================ -Legacy attribute New attribute -================ ================================ -`Local` :attr:`local` -`Global` :attr:`global` -`Polymorphic` :attr:`universes(polymorphic)` -`Monomorphic` :attr:`universes(monomorphic)` -`Cumulative` :attr:`universes(cumulative)` -`NonCumulative` :attr:`universes(noncumulative)` -`Private` :attr:`private(matching)` -`Program` :attr:`program` -================ ================================ - .. attr:: deprecated ( {? since = @string , } {? note = @string } ) :name: deprecated @@ -1709,12 +27,3 @@ Legacy attribute New attribute Proof. now foo. Abort. - -.. warn:: Unsupported attribute - - This warning is an error by default. It is caused by using a - command with some attribute it does not understand. - -.. [1] - Except if the inductive type is empty in which case there is no - equation that can be used to infer the return type. -- cgit v1.2.3 From 57734bc48a98c9dc08b1eebed94363e8c5c8a7b3 Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 13:13:05 +0200 Subject: Create section on writing libraries with only deprecated attributes. --- .../language/gallina-specification-language.rst | 29 ---------------------- 1 file changed, 29 deletions(-) delete mode 100644 doc/sphinx/language/gallina-specification-language.rst (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst deleted file mode 100644 index 91634ea023..0000000000 --- a/doc/sphinx/language/gallina-specification-language.rst +++ /dev/null @@ -1,29 +0,0 @@ -.. attr:: deprecated ( {? since = @string , } {? note = @string } ) - :name: deprecated - - At least one of :n:`since` or :n:`note` must be present. If both are present, - either one may appear first and they must be separated by a comma. - - This attribute is supported by the following commands: :cmd:`Ltac`, - :cmd:`Tactic Notation`, :cmd:`Notation`, :cmd:`Infix`. - - It can trigger the following warnings: - - .. warn:: Tactic @qualid is deprecated since @string__since. @string__note. - Tactic Notation @qualid is deprecated since @string__since. @string__note. - Notation @string is deprecated since @string__since. @string__note. - - :n:`@qualid` or :n:`@string` is the notation, :n:`@string__since` is the version number, - :n:`@string__note` is the note (usually explains the replacement). - - .. example:: - - .. coqtop:: all reset warn - - #[deprecated(since="8.9.0", note="Use idtac instead.")] - Ltac foo := idtac. - - Goal True. - Proof. - now foo. - Abort. -- cgit v1.2.3 From 90285ff50290a49d20d60ffc59725bf87c6acd14 Mon Sep 17 00:00:00 2001 From: Théo Zimmermann Date: Fri, 1 May 2020 13:22:42 +0200 Subject: Move essential vocabulary and syntax conventions to section on basics. --- doc/sphinx/language/cic.rst | 4 +- doc/sphinx/language/core/basic.rst | 354 ++++++++++++++++++--- doc/sphinx/language/core/index.rst | 30 +- doc/sphinx/language/core/records.rst | 7 +- .../language/extensions/implicit-arguments.rst | 44 ++- doc/sphinx/language/gallina-extensions.rst | 10 +- .../language/gallina-specification-language.rst | 151 ++------- 7 files changed, 403 insertions(+), 197 deletions(-) (limited to 'doc/sphinx/language') diff --git a/doc/sphinx/language/cic.rst b/doc/sphinx/language/cic.rst index 09a3897a06..e5af39c8fb 100644 --- a/doc/sphinx/language/cic.rst +++ b/doc/sphinx/language/cic.rst @@ -24,9 +24,9 @@ to a type and takes the form “*for all x of type* :math:`T`, :math:`P`”. The “:math:`x` *of type* :math:`T`” is written “:math:`x:T`”. Informally, “:math:`x:T`” can be thought as “:math:`x` *belongs to* :math:`T`”. -The types of types are *sorts*. Types and sorts are themselves terms +The types of types are called :gdef:`sort`\s. Types and sorts are themselves terms so that terms, types and sorts are all components of a common -syntactic language of terms which is described in Section :ref:`terms` but, +syntactic language of terms which is described in Section :ref:`terms`. But first, we describe sorts. diff --git a/doc/sphinx/language/core/basic.rst b/doc/sphinx/language/core/basic.rst index 03da59e0bf..9473cc5a15 100644 --- a/doc/sphinx/language/core/basic.rst +++ b/doc/sphinx/language/core/basic.rst @@ -1,7 +1,84 @@ +============================= +Basic notions and conventions +============================= + +This section provides some essential notions and conventions for reading +the manual. + +We start by explaining the syntax and lexical conventions used in the +manual. Then, we present the essential vocabulary necessary to read +the rest of the manual. Other terms are defined throughout the manual. +The reader may refer to the :ref:`glossary index ` +for a complete list of defined terms. Finally, we describe the various types of +settings that |Coq| provides. + +Syntax and lexical conventions +------------------------------ + +Syntax conventions +~~~~~~~~~~~~~~~~~~ + +The syntax described in this documentation is equivalent to that +accepted by the |Coq| parser, but the grammar has been edited +to improve readability and presentation. + +In the grammar presented in this manual, the terminal symbols are +black (e.g. :n:`forall`), whereas the nonterminals are green, italic +and hyperlinked (e.g. :n:`@term`). Some syntax is represented +graphically using the following kinds of blocks: + +:n:`{? item }` + An optional item. + +:n:`{+ item }` + A list of one or more items. + +:n:`{* item }` + An optional list of items. + +:n:`{+s item}` + A list of one or more items separated by "s" (e.g. :n:`item__1 s item__2 s item__3`). + +:n:`{*s item}` + An optional list of items separated by "s". + +:n:`{| item__1 | item__2 | ... }` + Alternatives (either :n:`item__1` or :n:`item__2` or ...). + +`Precedence levels +`_ that are +implemented in the |Coq| parser are shown in the documentation by +appending the level to the nonterminal name (as in :n:`@term100` or +:n:`@ltac_expr3`). + +.. note:: + + |Coq| uses an extensible parser. Plugins and the :ref:`notation + system ` can extend the + syntax at run time. Some notations are defined in the prelude, + which is loaded by default. The documented grammar doesn't include + these notations. Precedence levels not used by the base grammar + are omitted from the documentation, even though they could still be + populated by notations or plugins. + + Furthermore, some parsing rules are only activated in certain + contexts (:ref:`interactive proof mode `, + :ref:`custom entries `...). + +.. warning:: + + Given the complexity of these parsing rules, it would be extremely + difficult to create an external program that can properly parse a + |Coq| document. Therefore, tool writers are advised to delegate + parsing to |Coq|, by communicating with it, for instance through + `SerAPI `_. + +.. seealso:: :cmd:`Print Grammar` + .. _lexical-conventions: Lexical conventions -=================== +~~~~~~~~~~~~~~~~~~~ Blanks Space, newline and horizontal tab are considered blanks. @@ -56,24 +133,22 @@ Keywords The following character sequences are reserved keywords that cannot be used as identifiers:: - _ Axiom CoFixpoint Definition Fixpoint Hypothesis IF Parameter Prop - SProp Set Theorem Type Variable as at by cofix discriminated else - end exists exists2 fix for forall fun if in lazymatch let match - multimatch return then using where with + _ Axiom CoFixpoint Definition Fixpoint Hypothesis Parameter Prop + SProp Set Theorem Type Variable as at cofix discriminated else end + fix for forall fun if in let match return then where with - Note that plugins may define additional keywords when they are loaded. + Note that notations and plugins may define additional keywords. Other tokens The set of tokens defined at any given time can vary because the :cmd:`Notation` command can define new tokens. A :cmd:`Require` command may load more notation definitions, while the end of a :cmd:`Section` may remove notations. Some notations - are defined in the basic library (see :ref:`thecoqlibrary`) and are normally + are defined in the standard library (see :ref:`thecoqlibrary`) and are generally loaded automatically at startup time. - Here are the character sequences that Coq directly defines as tokens - without using :cmd:`Notation` (omitting 25 specialized tokens that begin with - ``#int63_``):: + Here are the character sequences that |Coq| directly defines as tokens + without using :cmd:`Notation`:: ! #[ % & ' ( () (bfs) (dfs) ) * ** + , - -> . .( .. ... / : ::= := :> :>> ; < <+ <- <: @@ -87,28 +162,190 @@ Other tokens ``~~`` generate different tokens, whereas if `~~` is not defined, then the two inputs are equivalent. -.. _gallina-attributes: +Essential vocabulary +-------------------- + +This section presents the most essential notions to understand the +rest of the |Coq| manual: :term:`terms ` and :term:`types +` on the one hand, :term:`commands ` and :term:`tactics +` on the other hand. + +.. glossary:: + + term + + Terms are the basic expressions of |Coq|. Terms can represent + mathematical expressions, propositions and proofs, but also + executable programs and program types. + + Here is the top-level syntax of terms. Each of the listed + constructs is presented in a dedicated section. Some of these + constructs (like :n:`@term_forall_or_fun`) are part of the core + language that the kernel of |Coq| understands and are therefore + described in :ref:`this chapter `, while + others (like :n:`@term_if`) are language extensions that are + presented in :ref:`the next chapter `. + + .. insertprodn term qualid_annotated + + .. prodn:: + term ::= @term_forall_or_fun + | @term_let + | @term_if + | @term_fix + | @term_cofix + | @term100 + term100 ::= @term_cast + | @term10 + term10 ::= @term_application + | @one_term + one_term ::= @term_explicit + | @term1 + term1 ::= @term_projection + | @term_scope + | @term0 + term0 ::= @qualid_annotated + | @sort + | @primitive_notations + | @term_evar + | @term_match + | @term_record + | @term_generalizing + | @term_ltac + | ( @term ) + qualid_annotated ::= @qualid {? @univ_annot } + + .. note:: + + Many :term:`commands ` and :term:`tactics ` + use :n:`@one_term` (in the syntax of their arguments) rather + than :n:`@term`. The former need to be enclosed in + parentheses unless they're very simple, such as a single + identifier. This avoids confusing a space-separated list of + terms or identifiers with a :n:`@term_application`. + + type + + To be valid and accepted by the |Coq| kernel, a term needs an + associated type. We express this relationship by “:math:`x` *of + type* :math:`T`”, which we write as “:math:`x:T`”. Informally, + “:math:`x:T`” can be thought as “:math:`x` *belongs to* + :math:`T`”. + + The |Coq| kernel is a type checker: it verifies that a term has + the expected type by applying a set of typing rules (see + :ref:`Typing-rules`). If that's indeed the case, we say that the + term is :gdef:`well-typed`. + + A special feature of the |Coq| language is that types can depend + on terms (we say that the language is `dependently-typed + `_). Because of + this, types and terms share a common syntax. All types are terms, + but not all terms are types: + + .. insertprodn type type + + .. prodn:: + type ::= @term + + Intuitively, types may be viewed as sets containing terms. We + say that a type is :gdef:`inhabited` if it contains at least one + term (i.e. if we can find a term which is associated with this + type). We call such terms :gdef:`witness`\es. Note that deciding + whether a type is inhabited is `undecidable + `_. + + Formally, types can be used to construct logical foundations for + mathematics alternative to the standard `"set theory" + `_: we call such + logical foundations `"type theories" + `_. |Coq| is based on + the Calculus of Inductive Constructions, which is a particular + instance of type theory. + + sentence + + |Coq| documents are made of a series of sentences that contain + :term:`commands ` or :term:`tactics `, generally + terminated with a period and optionally decorated with + :term:`attributes `. + + .. insertprodn document sentence + + .. prodn:: + document ::= {* @sentence } + sentence ::= {? @attributes } @command . + | {? @attributes } {? @num : } @query_command . + | {? @attributes } {? @toplevel_selector } @ltac_expr {| . | ... } + | @control_command + + :n:`@ltac_expr` syntax supports both simple and compound + :term:`tactics `. For example: ``split`` is a simple + tactic while ``split; auto`` combines two simple tactics. + + command + + A :production:`command` can be used to modify the state of a |Coq| + document, for instance by declaring a new object, or to get + information about the current state. + + By convention, command names begin with uppercase letters. + Commands appear in the HTML documentation in blue or gray boxes + after the label "Command". In the pdf, they appear after the + boldface label "Command:". Commands are listed in the + :ref:`command_index`. Example: + + .. cmd:: Comments {* @string } + + This command prints "Comments ok" and does not change anything + to the state of the document. + + tactic + + Tactics specify how to transform the current proof state as a + step in creating a proof. They are syntactically valid only when + |Coq| is in proof mode, such as after a :cmd:`Theorem` command + and before any subsequent proof-terminating command such as + :cmd:`Qed`. See :ref:`proofhandling` for more on proof mode. + + By convention, tactic names begin with lowercase letters. Tactic + appear in the HTML documentation in blue or gray boxes after the + label "Tactic". In the pdf, they appear after the boldface label + "Tactic:". Tactics are listed in the :ref:`tactic_index`. + +Settings +-------- + +There are several mechanisms for changing the behavior of |Coq|. The +:term:`attribute` mechanism is used to modify the behavior of a single +:term:`sentence`. The :term:`flag`, :term:`option` and :term:`table` +mechanisms are used to modify the behavior of |Coq| more globally in a +document or project. + +.. _attributes: Attributes ------------ +~~~~~~~~~~ + +An :gdef:`attribute` modifies the behavior of a single sentence. +Syntactically, most commands and tactics can be decorated with +attributes (cf. :n:`@sentence`), but attributes not supported by the +command or tactic will trigger :warn:`This command does not support +this attribute`. -.. insertprodn all_attrs legacy_attr +.. insertprodn attributes legacy_attr .. prodn:: - all_attrs ::= {* #[ {*, @attr } ] } {* @legacy_attr } - attr ::= @ident {? @attr_value } + attributes ::= {* #[ {*, @attribute } ] } {* @legacy_attr } + attribute ::= @ident {? @attr_value } attr_value ::= = @string - | ( {*, @attr } ) + | ( {*, @attribute } ) legacy_attr ::= {| Local | Global } | {| Polymorphic | Monomorphic } | {| Cumulative | NonCumulative } | Private | Program -Attributes modify the behavior of a command or tactic. -Syntactically, most commands and tactics can be decorated with attributes, but -attributes not supported by the command or tactic will be flagged as errors. - The order of top-level attributes doesn't affect their meaning. ``#[foo,bar]``, ``#[bar,foo]``, ``#[foo]#[bar]`` and ``#[bar]#[foo]`` are equivalent. @@ -128,22 +365,38 @@ Legacy attribute New attribute `Program` :attr:`program` ================ ================================ -.. warn:: Unsupported attribute +Attributes appear in the HTML documentation in blue or gray boxes +after the label "Attribute". In the pdf, they appear after the +boldface label "Attribute:". Attributes are listed in the +:ref:`attribute_index`. + +.. warn:: This command does not support this attribute: @ident. + :name: This command does not support this attribute + + This warning is configured to behave as an error by default. You + may turn it into a normal warning by using the :opt:`Warnings` option: + + .. coqtop:: none - This warning is an error by default. It is caused by using a - command with some attribute it does not understand. + Set Silent. + + .. coqtop:: all warn + + Set Warnings "unsupported-attributes". + #[ foo ] Comments. .. _flags-options-tables: Flags, Options and Tables ------------------------------ +~~~~~~~~~~~~~~~~~~~~~~~~~ -Coq has many settings to control its behavior. Setting types include flags, options -and tables: +The following types of settings can be used to change the behavior of |Coq| in +subsequent commands and tactics (see :ref:`set_unset_scope_qualifiers` for a +more precise description of the scope of these settings): -* A *flag* has a boolean value, such as :flag:`Asymmetric Patterns`. -* An *option* generally has a numeric or string value, such as :opt:`Firstorder Depth`. -* A *table* contains a set of strings or qualids. +* A :gdef:`flag` has a boolean value, such as :flag:`Universe Polymorphism`. +* An :gdef:`option` generally has a numeric or string value, such as :opt:`Firstorder Depth`. +* A :gdef:`table` contains a set of :token:`string`\s or :token:`qualid`\s. * In addition, some commands provide settings, such as :cmd:`Extraction Language`. .. FIXME Convert "Extraction Language" to an option. @@ -151,6 +404,11 @@ and tables: Flags, options and tables are identified by a series of identifiers, each with an initial capital letter. +Flags, options and tables appear in the HTML documentation in blue or +gray boxes after the labels "Flag", "Option" and "Table". In the pdf, +they appear after a boldface label. They are listed in the +:ref:`options_index`. + .. cmd:: Set @setting_name {? {| @int | @string } } :name: Set @@ -172,10 +430,10 @@ capital letter. This warning message can be raised by :cmd:`Set` and :cmd:`Unset` when :n:`@setting_name` is unknown. It is a warning rather than an error because this helps library authors - produce Coq code that is compatible with several Coq versions. + produce |Coq| code that is compatible with several |Coq| versions. To preserve the same behavior, they may need to set some compatibility flags or options that did not exist in previous - Coq versions. + |Coq| versions. .. cmd:: Unset @setting_name :name: Unset @@ -198,7 +456,7 @@ capital letter. If :n:`@setting_name` is a flag or option, prints its current value. If :n:`@setting_name` is a table: if the `for` clause is specified, reports - whether the table contains each specified value, otherise this is equivalent to + whether the table contains each specified value, otherwise this is equivalent to :cmd:`Print Table`. The `for` clause is not valid for flags and options. .. exn:: There is no flag, option or table with this name: "@setting_name". @@ -230,33 +488,33 @@ capital letter. .. _set_unset_scope_qualifiers: Locality attributes supported by :cmd:`Set` and :cmd:`Unset` -```````````````````````````````````````````````````````````` +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The :cmd:`Set` and :cmd:`Unset` commands support the :attr:`local`, :attr:`global` and :attr:`export` locality attributes: * no attribute: the original setting is *not* restored at the end of the current module or section. -* :attr:`local` (an alternative syntax is to use the ``Local`` - prefix): the setting is applied within the current module or - section. The original value of the setting is restored at the end - of the current module or section. -* :attr:`export` (an alternative syntax is to use the ``Export`` - prefix): similar to :attr:`local`, the original value of the setting - is restored at the end of the current module or section. In - addition, if the value is set in a module, then :cmd:`Import`\-ing - the module sets the option or flag. -* :attr:`global` (an alternative syntax is to use the ``Global`` - prefix): 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 :cmd:`Require`\-ing the file sets the option. +* :attr:`local` (or alternatively, the ``Local`` prefix): the setting + is applied within the current module or section. The original value + of the setting is restored at the end of the current module or + section. +* :attr:`export` (or alternatively, the ``Export`` prefix): similar to + :attr:`local`, the original value of the setting is restored at the + end of the current module or section. In addition, if the value is + set in a module, then :cmd:`Import`\-ing the module sets the option + or flag. +* :attr:`global` (or alternatively, the ``Global`` prefix): 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 + :cmd:`Require`\-ing the file sets the option. Newly opened modules and sections inherit the current settings. .. note:: - The use of the :attr:`global` attribute with the :cmd:`Set` and - :cmd:`Unset` commands is discouraged. If your goal is to define + We discourage using the :attr:`global` attribute with the :cmd:`Set` and + :cmd:`Unset` commands. If your goal is to define project-wide settings, you should rather use the command-line arguments ``-set`` and ``-unset`` for setting flags and options (cf. :ref:`command-line-options`). diff --git a/doc/sphinx/language/core/index.rst b/doc/sphinx/language/core/index.rst index 5ee960d99b..5e83672463 100644 --- a/doc/sphinx/language/core/index.rst +++ b/doc/sphinx/language/core/index.rst @@ -6,23 +6,26 @@ Core language At the heart of the Coq proof assistant is the Coq kernel. While users have access to a language with many convenient features such as -notations, implicit arguments, etc. (that are presented in the -:ref:`next chapter `), such complex terms get translated -down to a core language (the Calculus of Inductive Constructions) that -the kernel understands, and which we present here. Furthermore, while -users can build proofs interactively using tactics (see Chapter +:ref:`notations `, +:ref:`implicit arguments `, etc. (presented in the +:ref:`next chapter `), those features are translated into +the core language (the Calculus of Inductive Constructions) that the +kernel understands, which we present here. Furthermore, while users +can build proofs interactively using tactics (see Chapter :ref:`writing-proofs`), the role of these tactics is to incrementally build a "proof term" which the kernel will verify. More precisely, a -proof term is a term of the Calculus of Inductive Constructions whose -type corresponds to a theorem statement. The kernel is a type checker -which verifies that terms have their expected type. +proof term is a :term:`term` of the Calculus of Inductive +Constructions whose :term:`type` corresponds to a theorem statement. +The kernel is a type checker which verifies that terms have their +expected types. -This separation between the kernel on the one hand and the elaboration -engine and tactics on the other hand follows what is known as the "de -Bruijn criterion" (keeping a small and well delimited trusted code +This separation between the kernel on one hand and the +:ref:`elaboration engine ` and :ref:`tactics +` on the other follows what is known as the :gdef:`de +Bruijn criterion` (keeping a small and well delimited trusted code base within a proof assistant which can be much more complex). This -separation makes it possible to reduce the trust in the whole system -to trusting a smaller, critical component: the kernel. In particular, +separation makes it necessary to trust only a smaller, critical +component (the kernel) instead of the entire system. In particular, users may rely on external plugins that provide advanced and complex tactics without fear of these tactics being buggy, because the kernel will have to check their output. @@ -30,6 +33,7 @@ will have to check their output. .. toctree:: :maxdepth: 1 + basic ../gallina-specification-language ../cic records diff --git a/doc/sphinx/language/core/records.rst b/doc/sphinx/language/core/records.rst index 928378f55e..0080f1d052 100644 --- a/doc/sphinx/language/core/records.rst +++ b/doc/sphinx/language/core/records.rst @@ -15,14 +15,17 @@ expressions. In this sense, the :cmd:`Record` construction allows defining .. cmd:: {| Record | Structure } @record_definition {* with @record_definition } :name: Record; Structure - .. insertprodn record_definition field_body + .. insertprodn record_definition field_def .. prodn:: record_definition ::= {? > } @ident_decl {* @binder } {? : @type } {? @ident } %{ {*; @record_field } %} {? @decl_notations } - record_field ::= {* #[ {*, @attr } ] } @name {? @field_body } {? %| @num } {? @decl_notations } + record_field ::= {* #[ {*, @attribute } ] } @name {? @field_body } {? %| @num } {? @decl_notations } field_body ::= {* @binder } @of_type | {* @binder } @of_type := @term | {* @binder } := @term + term_record ::= %{%| {* @field_def } %|%} + field_def ::= @qualid {* @binder } := @term + Each :n:`@record_definition` defines a record named by :n:`@ident_decl`. The constructor name is given by :n:`@ident`. diff --git a/doc/sphinx/language/extensions/implicit-arguments.rst b/doc/sphinx/language/extensions/implicit-arguments.rst index d93dc00e24..73b1b65097 100644 --- a/doc/sphinx/language/extensions/implicit-arguments.rst +++ b/doc/sphinx/language/extensions/implicit-arguments.rst @@ -351,7 +351,7 @@ application. Use the :n:`(@ident := @term)` form of :token:`arg` to do so, where :token:`ident` is the name of the implicit argument and :token:`term` is its corresponding explicit term. Alternatively, one can deactivate the hiding of implicit arguments for a single function application using the -:n:`@ @qualid {? @univ_annot } {* @term1 }` form of :token:`term10`. +:n:`@@qualid_annotated {+ @term1 }` form of :token:`term_application`. .. example:: Syntax for explicitly giving implicit arguments (continued) @@ -420,6 +420,30 @@ but succeeds in Deactivation of implicit arguments for parsing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +.. insertprodn term_explicit term_explicit + +.. prodn:: + term_explicit ::= @ @qualid_annotated + +This syntax can be used to disable implicit arguments for a single +function. + +.. example:: + + The function `id` has one implicit argument and one explicit + argument. + + .. coqtop:: all reset + + Check (id 0). + Definition id' := @id. + + The function `id'` has no implicit argument. + + .. coqtop:: all + + Check (id' nat 0). + .. flag:: Parsing Explicit Turning this flag on (it is off by default) deactivates the use of implicit arguments. @@ -429,6 +453,19 @@ Deactivation of implicit arguments for parsing to be given as if no arguments were implicit. By symmetry, this also affects printing. +.. example:: + + We can reproduce the example above using the :flag:`Parsing + Explicit` flag: + + .. coqtop:: all reset + + Set Parsing Explicit. + Definition id' := id. + Unset Parsing Explicit. + Check (id 1). + Check (id' nat 1). + .. _canonical-structure-declaration: Canonical structures @@ -606,7 +643,7 @@ Implicit generalization .. index:: `[! ] .. index:: `(! ) -.. insertprodn generalizing_binder typeclass_constraint +.. insertprodn generalizing_binder term_generalizing .. prodn:: generalizing_binder ::= `( {+, @typeclass_constraint } ) @@ -615,7 +652,8 @@ Implicit generalization typeclass_constraint ::= {? ! } @term | %{ @name %} : {? ! } @term | @name : {? ! } @term - + term_generalizing ::= `%{ @term %} + | `( @term ) Implicit generalization is an automatic elaboration of a statement with free variables into a closed statement where these variables are diff --git a/doc/sphinx/language/gallina-extensions.rst b/doc/sphinx/language/gallina-extensions.rst index 51dc169def..5b78280edc 100644 --- a/doc/sphinx/language/gallina-extensions.rst +++ b/doc/sphinx/language/gallina-extensions.rst @@ -30,6 +30,11 @@ under its expanded form (see :flag:`Printing Matching`). Pattern-matching on boolean values: the if expression ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +.. insertprodn term_if term_if + +.. prodn:: + term_if ::= if @term {? {? as @name } return @term100 } then @term else @term + For inductive types with exactly two constructors and for pattern matching expressions that do not depend on the arguments of the constructors, it is possible to use a ``if … then … else`` notation. For instance, the definition @@ -852,7 +857,7 @@ Printing constructions in full .. flag:: Printing All Coercions, implicit arguments, the type of pattern matching, but also - notations (see :ref:`syntaxextensionsandnotationscopes`) can obfuscate the behavior of some + notations (see :ref:`syntax-extensions-and-notation-scopes`) can obfuscate the behavior of some tactics (typically the tactics applying to occurrences of subterms are sensitive to the implicit arguments). Turning this flag on deactivates all high-level printing features such as coercions, @@ -913,7 +918,8 @@ Existential variables .. insertprodn term_evar term_evar .. prodn:: - term_evar ::= ?[ @ident ] + term_evar ::= _ + | ?[ @ident ] | ?[ ?@ident ] | ?@ident {? @%{ {+; @ident := @term } %} } diff --git a/doc/sphinx/language/gallina-specification-language.rst b/doc/sphinx/language/gallina-specification-language.rst index e43fa84e67..353bed1b3d 100644 --- a/doc/sphinx/language/gallina-specification-language.rst +++ b/doc/sphinx/language/gallina-specification-language.rst @@ -7,108 +7,13 @@ This chapter describes Gallina, the specification language of Coq. It allows developing mathematical theories and to prove specifications of programs. The theories are built from axioms, hypotheses, parameters, lemmas, theorems and -definitions of constants, functions, predicates and sets. The syntax of logical -objects involved in theories is described in Section :ref:`term`. The -language of commands, called *The Vernacular* is described in Section -:ref:`vernacular`. - -In Coq, logical objects are typed to ensure their logical correctness. The -rules implemented by the typing algorithm are described in Chapter :ref:`calculusofinductiveconstructions`. - - -.. About the grammars in the manual - ================================ - - Grammars are presented in Backus-Naur form (BNF). Terminal symbols are - set in black ``typewriter font``. In addition, there are special notations for - regular expressions. - - An expression enclosed in square brackets ``[…]`` means at most one - occurrence of this expression (this corresponds to an optional - component). - - The notation “``entry sep … sep entry``” stands for a non empty sequence - of expressions parsed by entry and separated by the literal “``sep``” [1]_. - - Similarly, the notation “``entry … entry``” stands for a non empty - sequence of expressions parsed by the “``entry``” entry, without any - separator between. - - At the end, the notation “``[entry sep … sep entry]``” stands for a - possibly empty sequence of expressions parsed by the “``entry``” entry, - separated by the literal “``sep``”. +definitions of constants, functions, predicates and sets. .. _term: Terms ===== -Syntax of terms ---------------- - -The following grammars describe the basic syntax of the terms of the -*Calculus of Inductive Constructions* (also called Cic). The formal -presentation of Cic is given in Chapter :ref:`calculusofinductiveconstructions`. Extensions of this syntax -are given in Chapter :ref:`extensionsofgallina`. How to customize the syntax -is described in Chapter :ref:`syntaxextensionsandnotationscopes`. - -.. insertprodn term field_def - -.. prodn:: - term ::= forall @open_binders , @term - | fun @open_binders => @term - | @term_let - | if @term {? {? as @name } return @term100 } then @term else @term - | @term_fix - | @term_cofix - | @term100 - term100 ::= @term_cast - | @term10 - term10 ::= @term1 {+ @arg } - | @ @qualid {? @univ_annot } {* @term1 } - | @term1 - arg ::= ( @ident := @term ) - | @term1 - one_term ::= @term1 - | @ @qualid {? @univ_annot } - term1 ::= @term_projection - | @term0 % @scope_key - | @term0 - term0 ::= @qualid {? @univ_annot } - | @sort - | @numeral - | @string - | _ - | @term_evar - | @term_match - | ( @term ) - | %{%| {* @field_def } %|%} - | `%{ @term %} - | `( @term ) - | ltac : ( @ltac_expr ) - field_def ::= @qualid {* @binder } := @term - -.. note:: - - Many commands and tactics use :n:`@one_term` rather than :n:`@term`. - The former need to be enclosed in parentheses unless they're very - simple, such as a single identifier. This avoids confusing a space-separated - list of terms with a :n:`@term1` applied to a list of arguments. - -.. _types: - -Types ------ - -.. prodn:: - type ::= @term - -:n:`@type`\s are a subset of :n:`@term`\s; not every :n:`@term` is a :n:`@type`. -Every term has an associated type, which -can be determined by applying the :ref:`typing-rules`. Distinct terms -may share the same type, for example 0 and 1 are both of type `nat`, the -natural numbers. - .. _gallina-identifiers: Qualified identifiers and simple identifiers @@ -134,9 +39,15 @@ Field identifiers, written :n:`@field_ident`, are identifiers prefixed by Numerals and strings -------------------- +.. insertprodn primitive_notations primitive_notations + +.. prodn:: + primitive_notations ::= @numeral + | @string + Numerals and strings have no predefined semantics in the calculus. They are merely notations that can be bound to objects through the notation mechanism -(see Chapter :ref:`syntaxextensionsandnotationscopes` for details). +(see Chapter :ref:`syntax-extensions-and-notation-scopes` for details). Initially, numerals are bound to Peano’s representation of natural numbers (see :ref:`datatypes`). @@ -263,6 +174,12 @@ Section :ref:`let-in`). Products: forall ---------------- +.. insertprodn term_forall_or_fun term_forall_or_fun + +.. prodn:: + term_forall_or_fun ::= forall @open_binders , @term + | fun @open_binders => @term + The expression :n:`forall @ident : @type, @term` denotes the *product* of the variable :n:`@ident` of type :n:`@type`, over the term :n:`@term`. As for abstractions, :g:`forall` is followed by a binder list, and products @@ -284,6 +201,14 @@ the propositional implication and function types. Applications ------------ +.. insertprodn term_application arg + +.. prodn:: + term_application ::= @term1 {+ @arg } + | @ @qualid_annotated {+ @term1 } + arg ::= ( @ident := @term ) + | @term1 + :n:`@term__fun @term` denotes applying the function :n:`@term__fun` to :token:`term`. :n:`@term__fun {+ @term__i }` denotes applying @@ -545,34 +470,6 @@ co-recursion. It is the local counterpart of the :cmd:`CoFixpoint` command. When The Vernacular ============== -.. insertprodn vernacular sentence - -.. prodn:: - vernacular ::= {* @sentence } - sentence ::= {? @all_attrs } @command . - | {? @all_attrs } {? @num : } @query_command . - | {? @all_attrs } {? @toplevel_selector } @ltac_expr {| . | ... } - | @control_command - -The top-level input to |Coq| is a series of :n:`@sentence`\s, -which are :production:`tactic`\s or :production:`command`\s, -generally terminated with a period -and optionally decorated with :ref:`gallina-attributes`. :n:`@ltac_expr` syntax supports both simple -and compound tactics. For example: ``split`` is a simple tactic while ``split; auto`` combines two -simple tactics. - -Tactics specify how to transform the current proof state as a step in creating a proof. They -are syntactically valid only when |Coq| is in proof mode, such as after a :cmd:`Theorem` command -and before any subsequent proof-terminating command such as :cmd:`Qed`. See :ref:`proofhandling` for more -on proof mode. - -By convention, command names begin with uppercase letters, while -tactic names begin with lowercase letters. Commands appear in the -HTML documentation in blue boxes after the label "Command". In the pdf, they appear -after the boldface label "Command:". Commands are listed in the :ref:`command_index`. - -Similarly, tactics appear after the label "Tactic". Tactics are listed in the :ref:`tactic_index`. - .. _gallina-assumptions: Assumptions @@ -608,7 +505,7 @@ has type :n:`@type`. of an object of this type) is accepted as a postulate. :cmd:`Axiom`, :cmd:`Conjecture`, :cmd:`Parameter` and their plural forms - are equivalent. They can take the :attr:`local` attribute (see :ref:`gallina-attributes`), + are equivalent. They can take the :attr:`local` :term:`attribute`, which makes the defined :n:`@ident`\s accessible by :cmd:`Import` and its variants only through their fully qualified names. @@ -675,7 +572,7 @@ Section :ref:`typing-rules`. | {* @binder } : @type These commands bind :n:`@term` to the name :n:`@ident` in the environment, - provided that :n:`@term` is well-typed. They can take the :attr:`local` attribute (see :ref:`gallina-attributes`), + provided that :n:`@term` is well-typed. They can take the :attr:`local` :term:`attribute`, which makes the defined :n:`@ident` accessible by :cmd:`Import` and its variants only through their fully qualified names. If :n:`@reduce` is present then :n:`@ident` is bound to the result of the specified -- cgit v1.2.3