.. _extensionsofgallina: Extensions of |Gallina| ======================= |Gallina| is the kernel language of |Coq|. We describe here extensions of |Gallina|’s syntax. .. _record-types: Record types ---------------- The :cmd:`Record` construction is a macro allowing the definition of records as is done in many programming languages. Its syntax is described in the grammar below. In fact, the :cmd:`Record` macro is more general than the usual record types, since it allows also for “manifest” expressions. In this sense, the :cmd:`Record` construction allows defining “signatures”. .. _record_grammar: .. cmd:: {| Record | Structure } @record_definition {* with @record_definition } :name: Record; Structure .. insertprodn record_definition field_body .. prodn:: record_definition ::= {? > } @ident_decl {* @binder } {? : @type } {? @ident } %{ {*; @record_field } %} {? @decl_notations } record_field ::= {* #[ {*, @attr } ] } @name {? @field_body } {? %| @num } {? @decl_notations } field_body ::= {* @binder } @of_type | {* @binder } @of_type := @term | {* @binder } := @term Each :n:`@record_definition` defines a record named by :n:`@ident_decl`. The constructor name is given by :n:`@ident`. If the constructor name is not specified, then the default name :n:`Build_@ident` is used, where :n:`@ident` is the record name. If :n:`@type` is omitted, the default type is :math:`\Type`. The identifiers inside the brackets are the field names. The type of each field :n:`@ident` is :n:`forall {* @binder }, @type`. Notice that the type of an identifier can depend on a previously-given identifier. Thus the order of the fields is important. :n:`@binder` parameters may be applied to the record as a whole or to individual fields. Notations can be attached to fields using the :n:`@decl_notations` annotation. :cmd:`Record` and :cmd:`Structure` are synonyms. 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. More generally, a record may have explicitly defined (a.k.a. manifest) fields. For instance, we might have: :n:`Record @ident {* @binder } : @sort := { @ident__1 : @type__1 ; @ident__2 := @term__2 ; @ident__3 : @type__3 }`. in which case the correctness of :n:`@type__3` may rely on the instance :n:`@term__2` of :n:`@ident__2` and :n:`@term__2` may in turn depend on :n:`@ident__1`. .. example:: The set of rational numbers may be defined as: .. coqtop:: reset all Record Rat : Set := mkRat { sign : bool ; top : nat ; bottom : nat ; Rat_bottom_cond : 0 <> bottom ; Rat_irred_cond : forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1 }. Note here that the fields ``Rat_bottom_cond`` depends on the field ``bottom`` and ``Rat_irred_cond`` depends on both ``top`` and ``bottom``. Let us now see the work done by the ``Record`` macro. First the macro generates a variant type definition with just one constructor: :n:`Variant @ident {* @binder } : @sort := @ident__0 {* @binder }`. To build an object of type :token:`ident`, one should provide the constructor :n:`@ident__0` with the appropriate number of terms filling the fields of the record. .. example:: Let us define the rational :math:`1/2`: .. coqtop:: in Theorem one_two_irred : forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1. Admitted. Definition half := mkRat true 1 2 (O_S 1) one_two_irred. Check half. Alternatively, the following syntax allows creating objects by using named fields, as shown in this grammar. The fields do not have to be in any particular order, nor do they have to be all present if the missing ones can be inferred or prompted for (see :ref:`programs`). .. coqtop:: all Definition half' := {| sign := true; Rat_bottom_cond := O_S 1; Rat_irred_cond := one_two_irred |}. The following settings let you control the display format for types: .. flag:: Printing Records If set, use the record syntax (shown above) as the default display format. You can override the display format for specified types by adding entries to these tables: .. table:: Printing Record @qualid :name: Printing Record Specifies a set of qualids which are displayed as records. Use the :cmd:`Add @table` and :cmd:`Remove @table` commands to update the set of qualids. .. table:: Printing Constructor @qualid :name: Printing Constructor Specifies a set of qualids which are displayed as constructors. Use the :cmd:`Add @table` and :cmd:`Remove @table` commands to update the set of qualids. This syntax can also be used for pattern matching. .. coqtop:: all Eval compute in ( match half with | {| sign := true; top := n |} => n | _ => 0 end). The macro generates also, when it is possible, the projection functions for destructuring an object of type :token:`ident`. These projection functions are given the names of the corresponding fields. If a field is named `_` then no projection is built for it. In our example: .. coqtop:: all Eval compute in top half. Eval compute in bottom half. Eval compute in Rat_bottom_cond half. An alternative syntax for projections based on a dot notation is available: .. coqtop:: all Eval compute in half.(top). .. flag:: Printing Projections This flag activates the dot notation for printing. .. example:: .. coqtop:: all Set Printing Projections. Check top half. .. FIXME: move this to the main grammar in the spec chapter .. _record_projections_grammar: .. insertprodn term_projection term_projection .. prodn:: term_projection ::= @term0 .( @qualid {* @arg } ) | @term0 .( @ @qualid {* @term1 } ) Syntax of Record projections The corresponding grammar rules are given in the preceding grammar. When :token:`qualid` denotes a projection, the syntax :n:`@term0.(@qualid)` is equivalent to :n:`@qualid @term0`, the syntax :n:`@term0.(@qualid {+ @arg })` to :n:`@qualid {+ @arg } @term0`. and the syntax :n:`@term0.(@@qualid {+ @term0 })` to :n:`@@qualid {+ @term0 } @term0`. In each case, :token:`term0` is the object projected and the other arguments are the parameters of the inductive type. .. note:: Records defined with the ``Record`` keyword are not allowed to be recursive (references to the record's name in the type of its field raises an error). To define recursive records, one can use the ``Inductive`` and ``CoInductive`` keywords, resulting in an inductive or co-inductive record. Definition of mutually inductive or co-inductive records are also allowed, as long as all of the types in the block are records. .. note:: Induction schemes are automatically generated for inductive records. Automatic generation of induction schemes for non-recursive records defined with the ``Record`` keyword can be activated with the :flag:`Nonrecursive Elimination Schemes` flag (see :ref:`proofschemes-induction-principles`). .. warn:: @ident cannot be defined. It can happen that the definition of a projection is impossible. This message is followed by an explanation of this impossibility. There may be three reasons: #. The name :token:`ident` already exists in the environment (see :cmd:`Axiom`). #. The body of :token:`ident` uses an incorrect elimination for :token:`ident` (see :cmd:`Fixpoint` and :ref:`Destructors`). #. The type of the projections :token:`ident` depends on previous projections which themselves could not be defined. .. exn:: Records declared with the keyword Record or Structure cannot be recursive. The record name :token:`ident` appears in the type of its fields, but uses the keyword ``Record``. Use the keyword ``Inductive`` or ``CoInductive`` instead. .. exn:: Cannot handle mutually (co)inductive records. Records cannot be defined as part of mutually inductive (or co-inductive) definitions, whether with records only or mixed with standard definitions. During the definition of the one-constructor inductive definition, all the errors of inductive definitions, as described in Section :ref:`gallina-inductive-definitions`, may also occur. .. seealso:: Coercions and records in section :ref:`coercions-classes-as-records` of the chapter devoted to coercions. .. _primitive_projections: Primitive Projections ~~~~~~~~~~~~~~~~~~~~~ .. flag:: Primitive Projections Turns on the use of primitive projections when defining subsequent records (even through the ``Inductive`` and ``CoInductive`` commands). Primitive projections extended the Calculus of Inductive Constructions with a new binary term constructor `r.(p)` representing a primitive projection `p` applied to a record object `r` (i.e., primitive projections are always applied). Even if the record type has parameters, these do not appear in the internal representation of applications of the projection, considerably reducing the sizes of terms when manipulating parameterized records and type checking time. On the user level, primitive projections can be used as a replacement for the usual defined ones, although there are a few notable differences. .. flag:: Printing Primitive Projection Parameters This compatibility flag reconstructs internally omitted parameters at printing time (even though they are absent in the actual AST manipulated by the kernel). Primitive Record Types ++++++++++++++++++++++ When the :flag:`Primitive Projections` flag is on, definitions of record types change meaning. When a type is declared with primitive projections, its :g:`match` construct is disabled (see :ref:`primitive_projections` though). To eliminate the (co-)inductive type, one must use its defined primitive projections. .. The following paragraph is quite redundant with what is above For compatibility, the parameters still appear to the user when printing terms even though they are absent in the actual AST manipulated by the kernel. This can be changed by unsetting the :flag:`Printing Primitive Projection Parameters` flag. There are currently two ways to introduce primitive records types: #. Through the ``Record`` command, in which case the type has to be non-recursive. The defined type enjoys eta-conversion definitionally, that is the generalized form of surjective pairing for records: `r` ``= Build_``\ `R` ``(``\ `r`\ ``.(``\ |p_1|\ ``) …`` `r`\ ``.(``\ |p_n|\ ``))``. Eta-conversion allows to define dependent elimination for these types as well. #. Through the ``Inductive`` and ``CoInductive`` commands, when the body of the definition is a record declaration of the form ``Build_``\ `R` ``{`` |p_1| ``:`` |t_1|\ ``; … ;`` |p_n| ``:`` |t_n| ``}``. In this case the types can be recursive and eta-conversion is disallowed. These kind of record types differ from their traditional versions in the sense that dependent elimination is not available for them and only non-dependent case analysis can be defined. Reduction +++++++++ The basic reduction rule of a primitive projection is |p_i| ``(Build_``\ `R` |t_1| … |t_n|\ ``)`` :math:`{\rightarrow_{\iota}}` |t_i|. However, to take the :math:`{\delta}` flag into account, projections can be in two states: folded or unfolded. An unfolded primitive projection application obeys the rule above, while the folded version delta-reduces to the unfolded version. This allows to precisely mimic the usual unfolding rules of constants. Projections obey the usual ``simpl`` flags of the ``Arguments`` command in particular. There is currently no way to input unfolded primitive projections at the user-level, and there is no way to display unfolded projections differently from folded ones. Compatibility Projections and :g:`match` ++++++++++++++++++++++++++++++++++++++++ To ease compatibility with ordinary record types, each primitive projection is also defined as a ordinary constant taking parameters and an object of the record type as arguments, and whose body is an application of the unfolded primitive projection of the same name. These constants are used when elaborating partial applications of the projection. One can distinguish them from applications of the primitive projection if the :flag:`Printing Primitive Projection Parameters` flag is off: For a primitive projection application, parameters are printed as underscores while for the compatibility projections they are printed as usual. Additionally, user-written :g:`match` constructs on primitive records are desugared into substitution of the projections, they cannot be printed back as :g:`match` constructs. Variants and extensions of :g:`match` ------------------------------------- .. _mult-match: Multiple and nested pattern matching ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The basic version of :g:`match` allows pattern matching on simple patterns. As an extension, multiple nested patterns or disjunction of patterns are allowed, as in ML-like languages. The extension just acts as a macro that is expanded during parsing into a sequence of match on simple patterns. Especially, a construction defined using the extended match is generally printed under its expanded form (see :flag:`Printing Matching`). .. seealso:: :ref:`extendedpatternmatching`. .. _if-then-else: Pattern-matching on boolean values: the if expression ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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 .. coqtop:: all Definition not (b:bool) := match b with | true => false | false => true end. can be alternatively written .. coqtop:: reset all Definition not (b:bool) := if b then false else true. More generally, for an inductive type with constructors :n:`@ident__1` and :n:`@ident__2`, the following terms are equal: :n:`if @term__0 {? {? as @name } return @term } then @term__1 else @term__2` :n:`match @term__0 {? {? as @name } return @term } with | @ident__1 {* _ } => @term__1 | @ident__2 {* _ } => @term__2 end` .. example:: .. coqtop:: all Check (fun x (H:{x=0}+{x<>0}) => match H with | left _ => true | right _ => false end). Notice that the printing uses the :g:`if` syntax because :g:`sumbool` is declared as such (see :ref:`controlling-match-pp`). .. _irrefutable-patterns: Irrefutable patterns: the destructuring let variants ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Pattern-matching on terms inhabiting inductive type having only one constructor can be alternatively written using :g:`let … in …` constructions. There are two variants of them. First destructuring let syntax ++++++++++++++++++++++++++++++ The expression :n:`let ( {*, @ident__i } ) := @term__0 in @term__1` performs case analysis on :n:`@term__0` whose type must be an inductive type with exactly one constructor. The number of variables :n:`@ident__i` must correspond to the number of arguments of this contrustor. Then, in :n:`@term__1`, these variables are bound to the arguments of the constructor in :n:`@term__0`. For instance, the definition .. coqtop:: reset all Definition fst (A B:Set) (H:A * B) := match H with | pair x y => x end. can be alternatively written .. coqtop:: reset all Definition fst (A B:Set) (p:A * B) := let (x, _) := p in x. Notice that reduction is different from regular :g:`let … in …` construction since it happens only if :n:`@term__0` is in constructor form. Otherwise, the reduction is blocked. The pretty-printing of a definition by matching on a irrefutable pattern can either be done using :g:`match` or the :g:`let` construction (see Section :ref:`controlling-match-pp`). If term inhabits an inductive type with one constructor `C`, we have an equivalence between :: let (ident₁, …, identₙ) [dep_ret_type] := term in term' and :: match term [dep_ret_type] with C ident₁ … identₙ => term' end Second destructuring let syntax +++++++++++++++++++++++++++++++ Another destructuring let syntax is available for inductive types with one constructor by giving an arbitrary pattern instead of just a tuple for all the arguments. For example, the preceding example can be written: .. coqtop:: reset all Definition fst (A B:Set) (p:A*B) := let 'pair x _ := p in x. This is useful to match deeper inside tuples and also to use notations for the pattern, as the syntax :g:`let ’p := t in b` allows arbitrary patterns to do the deconstruction. For example: .. coqtop:: all Definition deep_tuple (A:Set) (x:(A*A)*(A*A)) : A*A*A*A := let '((a,b), (c, d)) := x in (a,b,c,d). Notation " x 'With' p " := (exist _ x p) (at level 20). Definition proj1_sig' (A:Set) (P:A->Prop) (t:{ x:A | P x }) : A := let 'x With p := t in x. When printing definitions which are written using this construct it takes precedence over let printing directives for the datatype under consideration (see Section :ref:`controlling-match-pp`). .. _controlling-match-pp: Controlling pretty-printing of match expressions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The following commands give some control over the pretty-printing of :g:`match` expressions. Printing nested patterns +++++++++++++++++++++++++ .. flag:: Printing Matching The Calculus of Inductive Constructions knows pattern matching only over simple patterns. It is however convenient to re-factorize nested pattern matching into a single pattern matching over a nested pattern. When this flag is on (default), |Coq|’s printer tries to do such limited re-factorization. Turning it off tells |Coq| to print only simple pattern matching problems in the same way as the |Coq| kernel handles them. Factorization of clauses with same right-hand side ++++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Printing Factorizable Match Patterns When several patterns share the same right-hand side, it is additionally possible to share the clauses using disjunctive patterns. Assuming that the printing matching mode is on, this flag (on by default) tells |Coq|'s printer to try to do this kind of factorization. Use of a default clause +++++++++++++++++++++++ .. flag:: Printing Allow Match Default Clause When several patterns share the same right-hand side which do not depend on the arguments of the patterns, yet an extra factorization is possible: the disjunction of patterns can be replaced with a `_` default clause. Assuming that the printing matching mode and the factorization mode are on, this flag (on by default) tells |Coq|'s printer to use a default clause when relevant. Printing of wildcard patterns ++++++++++++++++++++++++++++++ .. flag:: Printing Wildcard Some variables in a pattern may not occur in the right-hand side of the pattern matching clause. When this flag is on (default), the variables having no occurrences in the right-hand side of the pattern matching clause are just printed using the wildcard symbol “_”. Printing of the elimination predicate +++++++++++++++++++++++++++++++++++++ .. flag:: Printing Synth In most of the cases, the type of the result of a matched term is mechanically synthesizable. Especially, if the result type does not depend of the matched term. When this flag is on (default), the result type is not printed when |Coq| knows that it can re- synthesize it. Printing matching on irrefutable patterns ++++++++++++++++++++++++++++++++++++++++++ If an inductive type has just one constructor, pattern matching can be written using the first destructuring let syntax. .. table:: Printing Let @qualid :name: Printing Let Specifies a set of qualids for which pattern matching is displayed using a let expression. Note that this only applies to pattern matching instances entered with :g:`match`. It doesn't affect pattern matching explicitly entered with a destructuring :g:`let`. Use the :cmd:`Add @table` and :cmd:`Remove @table` commands to update this set. Printing matching on booleans +++++++++++++++++++++++++++++ If an inductive type is isomorphic to the boolean type, pattern matching can be written using ``if`` … ``then`` … ``else`` …. This table controls which types are written this way: .. table:: Printing If @qualid :name: Printing If Specifies a set of qualids for which pattern matching is displayed using ``if`` … ``then`` … ``else`` …. Use the :cmd:`Add @table` and :cmd:`Remove @table` commands to update this set. This example emphasizes what the printing settings offer. .. example:: .. coqtop:: all Definition snd (A B:Set) (H:A * B) := match H with | pair x y => y end. Test Printing Let for prod. Print snd. Remove Printing Let prod. Unset Printing Synth. Unset Printing Wildcard. Print snd. .. _advanced-recursive-functions: Advanced recursive functions ---------------------------- The following command is available when the ``FunInd`` library has been loaded via ``Require Import FunInd``: .. cmd:: Function @fix_definition {* with @fix_definition } This command is a generalization of :cmd:`Fixpoint`. It is a wrapper for several ways of defining a function *and* other useful related objects, namely: an induction principle that reflects the recursive structure of the function (see :tacn:`function induction`) and its fixpoint equality. This defines a function similar to those defined by :cmd:`Fixpoint`. As in :cmd:`Fixpoint`, the decreasing argument must be given (unless the function is not recursive), but it might not necessarily be *structurally* decreasing. Use the :n:`@fixannot` clause to name the decreasing argument *and* to describe which kind of decreasing criteria to use to ensure termination of recursive calls. :cmd:`Function` also supports the :n:`with` clause to create mutually recursive definitions, however this feature is limited to structurally recursive functions (i.e. when :n:`@fixannot` is a :n:`struct` clause). See :tacn:`function induction` and :cmd:`Functional Scheme` for how to use the induction principle to reason easily about the function. The form of the :n:`@fixannot` clause determines which definition mechanism :cmd:`Function` uses. (Note that references to :n:`ident` below refer to the name of the function being defined.): * If :n:`@fixannot` is not specified, :cmd:`Function` defines the nonrecursive function :token:`ident` as if it was declared with :cmd:`Definition`. In addition, the following are defined: + :token:`ident`\ ``_rect``, :token:`ident`\ ``_rec`` and :token:`ident`\ ``_ind``, which reflect the pattern matching structure of :token:`term` (see :cmd:`Inductive`); + The inductive :n:`R_@ident` corresponding to the graph of :token:`ident` (silently); + :token:`ident`\ ``_complete`` and :token:`ident`\ ``_correct`` which are inversion information linking the function and its graph. * If :n:`{ struct ... }` is specified, :cmd:`Function` defines the structural recursive function :token:`ident` as if it was declared with :cmd:`Fixpoint`. In addition, the following are defined: + The same objects as above; + The fixpoint equation of :token:`ident`: :n:`@ident`\ ``_equation``. * If :n:`{ measure ... }` or :n:`{ wf ... }` are specified, :cmd:`Function` defines a recursive function by well-founded recursion. The module ``Recdef`` of the standard library must be loaded for this feature. + :n:`{measure @one_term__1 {? @ident } {? @one_term__2 } }`\: where :n:`@ident` is the decreasing argument and :n:`@one_term__1` is a function from the type of :n:`@ident` to :g:`nat` for which the decreasing argument decreases (for the :g:`lt` order on :g:`nat`) for each recursive call of the function. The parameters of the function are bound in :n:`@one_term__1`. + :n:`{wf @one_term @ident }`\: where :n:`@ident` is the decreasing argument and :n:`@one_term` is an ordering relation on the type of :n:`@ident` (i.e. of type `T`\ :math:`_{\sf ident}` → `T`\ :math:`_{\sf ident}` → ``Prop``) for which the decreasing argument decreases for each recursive call of the function. The order must be well-founded. The parameters of the function are bound in :n:`@one_term`. If the clause is ``measure`` or ``wf``, the user is left with some proof obligations that will be used to define the function. These proofs are: proofs that each recursive call is actually decreasing with respect to the given criteria, and (if the criteria is `wf`) a proof that the ordering relation is well-founded. Once proof obligations are discharged, the following objects are defined: + The same objects as with the ``struct`` clause; + The lemma :n:`@ident`\ ``_tcc`` which collects all proof obligations in one property; + The lemmas :n:`@ident`\ ``_terminate`` and :n:`@ident`\ ``_F`` which will be inlined during extraction of :n:`@ident`. The way this recursive function is defined is the subject of several papers by Yves Bertot and Antonia Balaa on the one hand, and Gilles Barthe, Julien Forest, David Pichardie, and Vlad Rusu on the other hand. .. note:: To obtain the right principle, it is better to put rigid parameters of the function as first arguments. For example it is better to define plus like this: .. coqtop:: reset none Require Import FunInd. .. coqtop:: all Function plus (m n : nat) {struct n} : nat := match n with | 0 => m | S p => S (plus m p) end. than like this: .. coqtop:: reset none Require Import FunInd. .. coqtop:: all Function plus (n m : nat) {struct n} : nat := match n with | 0 => m | S p => S (plus p m) end. *Limitations* :token:`term` must be built as a *pure pattern matching tree* (:g:`match … with`) with applications only *at the end* of each branch. :cmd:`Function` does not support partial application of the function being defined. Thus, the following example cannot be accepted due to the presence of partial application of :g:`wrong` in the body of :g:`wrong`: .. coqtop:: none Require List. Import List.ListNotations. .. coqtop:: all fail Function wrong (C:nat) : nat := List.hd 0 (List.map wrong (C::nil)). For now, dependent cases are not treated for non structurally terminating functions. .. exn:: The recursive argument must be specified. :undocumented: .. exn:: No argument name @ident. :undocumented: .. exn:: Cannot use mutual definition with well-founded recursion or measure. :undocumented: .. warn:: Cannot define graph for @ident. The generation of the graph relation (:n:`R_@ident`) used to compute the induction scheme of ident raised a typing error. Only :token:`ident` is defined; the induction scheme will not be generated. This error happens generally when: - the definition uses pattern matching on dependent types, which :cmd:`Function` cannot deal with yet. - the definition is not a *pattern matching tree* as explained above. .. warn:: Cannot define principle(s) for @ident. The generation of the graph relation (:n:`R_@ident`) succeeded but the induction principle could not be built. Only :token:`ident` is defined. Please report. .. warn:: Cannot build functional inversion principle. :tacn:`functional inversion` will not be available for the function. .. seealso:: :ref:`functional-scheme` and :tacn:`function induction` .. _section-mechanism: Section mechanism ----------------- Sections create local contexts which can be shared across multiple definitions. .. example:: Sections are opened by the :cmd:`Section` command, and closed by :cmd:`End`. .. coqtop:: all Section s1. Inside a section, local parameters can be introduced using :cmd:`Variable`, :cmd:`Hypothesis`, or :cmd:`Context` (there are also plural variants for the first two). .. coqtop:: all Variables x y : nat. The command :cmd:`Let` introduces section-wide :ref:`let-in`. These definitions won't persist when the section is closed, and all persistent definitions which depend on `y'` will be prefixed with `let y' := y in`. .. coqtop:: in Let y' := y. Definition x' := S x. Definition x'' := x' + y'. .. coqtop:: all Print x'. Print x''. End s1. Print x'. Print x''. Notice the difference between the value of :g:`x'` and :g:`x''` inside section :g:`s1` and outside. .. cmd:: Section @ident This command is used to open a section named :token:`ident`. Section names do not need to be unique. .. cmd:: End @ident This command closes the section or module named :token:`ident`. See :ref:`Terminating an interactive module or module type definition` for a description of its use with modules. After closing the section, the local declarations (variables and local definitions, see :cmd:`Variable`) are *discharged*, meaning that they stop being visible and that all global objects defined in the section are generalized with respect to the variables and local definitions they each depended on in the section. .. exn:: There is nothing to end. :undocumented: .. exn:: Last block to end has name @ident. :undocumented: .. note:: Most commands, like :cmd:`Hint`, :cmd:`Notation`, option management, … which appear inside a section are canceled when the section is closed. .. cmd:: Let @ident @def_body Let Fixpoint @fix_definition {* with @fix_definition } Let CoFixpoint @cofix_definition {* with @cofix_definition } :name: Let; Let Fixpoint; Let CoFixpoint These commands behave like :cmd:`Definition`, :cmd:`Fixpoint` and :cmd:`CoFixpoint`, except that the declared constant is local to the current section. When the section is closed, all persistent definitions and theorems within it that depend on the constant will be wrapped with a :n:`@term_let` with the same declaration. As for :cmd:`Definition`, :cmd:`Fixpoint` and :cmd:`CoFixpoint`, 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`. .. cmd:: Context {+ @binder } Declare variables in the context of the current section, like :cmd:`Variable`, but also allowing implicit variables, :ref:`implicit-generalization`, and let-binders. .. coqdoc:: Context {A : Type} (a b : A). Context `{EqDec A}. Context (b' := b). .. seealso:: Section :ref:`binders`. Section :ref:`contexts` in chapter :ref:`typeclasses`. Module system ------------- The module system provides a way of packaging related elements together, as well as a means of massive abstraction. .. cmd:: Module {? {| Import | Export } } @ident {* @module_binder } {? @of_module_type } {? := {+<+ @module_expr_inl } } .. insertprodn module_binder module_expr_inl .. prodn:: module_binder ::= ( {? {| Import | Export } } {+ @ident } : @module_type_inl ) module_type_inl ::= ! @module_type | @module_type {? @functor_app_annot } functor_app_annot ::= [ inline at level @num ] | [ no inline ] module_type ::= @qualid | ( @module_type ) | @module_type @module_expr_atom | @module_type with @with_declaration with_declaration ::= Definition @qualid {? @univ_decl } := @term | Module @qualid := @qualid module_expr_atom ::= @qualid | ( {+ @module_expr_atom } ) of_module_type ::= : @module_type_inl | {* <: @module_type_inl } module_expr_inl ::= ! {+ @module_expr_atom } | {+ @module_expr_atom } {? @functor_app_annot } Defines a module named :token:`ident`. See the examples :ref:`here`. The :n:`Import` and :n:`Export` flags specify whether the module should be automatically imported or exported. Specifying :n:`{* @module_binder }` starts a functor with parameters given by the :n:`@module_binder`\s. (A *functor* is a function from modules to modules.) .. todo: would like to find a better term than "interactive", not very descriptive :n:`@of_module_type` specifies the module type. :n:`{+ <: @module_type_inl }` starts a module that satisfies each :n:`@module_type_inl`. :n:`:= {+<+ @module_expr_inl }` specifies the body of a module or functor definition. If it's not specified, then the module is defined *interactively*, meaning that the module is defined as a series of commands terminated with :cmd:`End` instead of in a single :cmd:`Module` command. Interactively defining the :n:`@module_expr_inl`\s in a series of :cmd:`Include` commands is equivalent to giving them all in a single non-interactive :cmd:`Module` command. The ! prefix indicates that any assumption command (such as :cmd:`Axiom`) with an :n:`Inline` clause in the type of the functor arguments will be ignored. .. todo: What is an Inline directive? sb command but still unclear. Maybe referring to the "inline" in functor_app_annot? or assumption_token Inline assum_list? .. cmd:: Module Type @ident {* @module_binder } {* <: @module_type_inl } {? := {+<+ @module_type_inl } } Defines a module type named :n:`@ident`. See the example :ref:`here`. Specifying :n:`{* @module_binder }` starts a functor type with parameters given by the :n:`@module_binder`\s. :n:`:= {+<+ @module_type_inl }` specifies the body of a module or functor type definition. If it's not specified, then the module type is defined *interactively*, meaning that the module type is defined as a series of commands terminated with :cmd:`End` instead of in a single :cmd:`Module Type` command. Interactively defining the :n:`@module_type_inl`\s in a series of :cmd:`Include` commands is equivalent to giving them all in a single non-interactive :cmd:`Module Type` command. .. _terminating_module: **Terminating an interactive module or module type definition** Interactive modules are terminated with the :cmd:`End` command, which is also used to terminate :ref:`Sections`. :n:`End @ident` closes the interactive module or module type :token:`ident`. If the module type was given, the command verifies that the content of the module matches the module type. If the module is not a functor, its components (constants, inductive types, submodules etc.) are now available through the dot notation. .. exn:: No such label @ident. :undocumented: .. exn:: Signature components for label @ident do not match. :undocumented: .. exn:: The field @ident is missing in @qualid. :undocumented: .. |br| raw:: html
.. note:: #. Interactive modules and module types can be nested. #. Interactive modules and module types can't be defined inside of :ref:`sections`. Sections can be defined inside of interactive modules and module types. #. Hints and notations (:cmd:`Hint` and :cmd:`Notation` commands) can also appear inside interactive modules and module types. Note that with module definitions like: :n:`Module @ident__1 : @module_type := @ident__2.` or :n:`Module @ident__1 : @module_type.` |br| :n:`Include @ident__2.` |br| :n:`End @ident__1.` hints and the like valid for :n:`@ident__1` are the ones defined in :n:`@module_type` rather then those defined in :n:`@ident__2` (or the module body). #. Within an interactive module type definition, the :cmd:`Parameter` command declares a constant instead of definining a new axiom (which it does when not in a module type definition). #. Assumptions such as :cmd:`Axiom` that include the :n:`Inline` clause will be automatically expanded when the functor is applied, except when the function application is prefixed by ``!``. .. cmd:: Include @module_type_inl {* <+ @module_expr_inl } Includes the content of module(s) in the current interactive module. Here :n:`@module_type_inl` can be a module expression or a module type expression. If it is a high-order module or module type expression then the system tries to instantiate :n:`@module_type_inl` with the current interactive module. Including multiple modules is a single :cmd:`Include` is equivalent to including each module in a separate :cmd:`Include` command. .. cmd:: Include Type {+<+ @module_type_inl } .. deprecated:: 8.3 Use :cmd:`Include` instead. .. cmd:: Declare Module {? {| Import | Export } } @ident {* @module_binder } : @module_type_inl Declares a module :token:`ident` of type :token:`module_type_inl`. If :n:`@module_binder`\s are specified, declares a functor with parameters given by the list of :token:`module_binder`\s. .. cmd:: Import {+ @qualid } If :token:`qualid` denotes a valid basic module (i.e. its module type is a signature), makes its components available by their short names. .. example:: .. coqtop:: reset in Module Mod. Definition T:=nat. Check T. End Mod. Check Mod.T. .. coqtop:: all Fail Check T. Import Mod. Check T. Some features defined in modules are activated only when a module is imported. This is for instance the case of notations (see :ref:`Notations`). Declarations made with the :attr:`local` attribute are never imported by the :cmd:`Import` command. Such declarations are only accessible through their fully qualified name. .. example:: .. coqtop:: in Module A. Module B. Local Definition T := nat. End B. End A. Import A. .. coqtop:: all fail Check B.T. .. cmd:: Export {+ @qualid } :name: Export Similar to :cmd:`Import`, except that when the module containing this command is imported, the :n:`{+ @qualid }` are imported as well. .. exn:: @qualid is not a module. :undocumented: .. warn:: Trying to mask the absolute name @qualid! :undocumented: .. cmd:: Print Module @qualid Prints the module type and (optionally) the body of the module :n:`@qualid`. .. cmd:: Print Module Type @qualid Prints the module type corresponding to :n:`@qualid`. .. flag:: Short Module Printing This flag (off by default) disables the printing of the types of fields, leaving only their names, for the commands :cmd:`Print Module` and :cmd:`Print Module Type`. .. _module_examples: Examples ~~~~~~~~ .. example:: Defining a simple module interactively .. coqtop:: in Module M. Definition T := nat. Definition x := 0. .. coqtop:: all Definition y : bool. exact true. .. coqtop:: in Defined. End M. Inside a module one can define constants, prove theorems and do anything else that can be done in the toplevel. Components of a closed module can be accessed using the dot notation: .. coqtop:: all Print M.x. .. _example_def_simple_module_type: .. example:: Defining a simple module type interactively .. coqtop:: in Module Type SIG. Parameter T : Set. Parameter x : T. End SIG. .. _example_filter_module: .. example:: Creating a new module that omits some items from an existing module Since :n:`SIG`, the type of the new module :n:`N`, doesn't define :n:`y` or give the body of :n:`x`, which are not included in :n:`N`. .. coqtop:: all Module N : SIG with Definition T := nat := M. Print N.T. Print N.x. Fail Print N.y. .. reset to remove N (undo in last coqtop block doesn't seem to do that), invisibly redefine M, SIG .. coqtop:: none reset Module M. Definition T := nat. Definition x := 0. Definition y : bool. exact true. Defined. End M. Module Type SIG. Parameter T : Set. Parameter x : T. End SIG. The following definition of :g:`N` using the module type expression :g:`SIG` with :g:`Definition T := nat` is equivalent to the following one: .. todo: what is other definition referred to above? "Module N' : SIG with Definition T := nat. End N`." is not it. .. coqtop:: in Module Type SIG'. Definition T : Set := nat. Parameter x : T. End SIG'. Module N : SIG' := M. If we just want to be sure that our implementation satisfies a given module type without restricting the interface, we can use a transparent constraint .. coqtop:: in Module P <: SIG := M. .. coqtop:: all Print P.y. .. example:: Creating a functor (a module with parameters) .. coqtop:: in Module Two (X Y: SIG). Definition T := (X.T * Y.T)%type. Definition x := (X.x, Y.x). End Two. and apply it to our modules and do some computations: .. coqtop:: in Module Q := Two M N. .. coqtop:: all Eval compute in (fst Q.x + snd Q.x). .. example:: A module type with two sub-modules, sharing some fields .. coqtop:: in Module Type SIG2. Declare Module M1 : SIG. Module M2 <: SIG. Definition T := M1.T. Parameter x : T. End M2. End SIG2. .. coqtop:: in Module Mod <: SIG2. Module M1. Definition T := nat. Definition x := 1. End M1. Module M2 := M. End Mod. Notice that ``M`` is a correct body for the component ``M2`` since its ``T`` component is ``nat`` as specified for ``M1.T``. Libraries and qualified names --------------------------------- .. _names-of-libraries: Names of libraries ~~~~~~~~~~~~~~~~~~ The theories developed in |Coq| are stored in *library files* which are hierarchically classified into *libraries* and *sublibraries*. To express this hierarchy, library names are represented by qualified identifiers qualid, i.e. as list of identifiers separated by dots (see :ref:`gallina-identifiers`). For instance, the library file ``Mult`` of the standard |Coq| library ``Arith`` is named ``Coq.Arith.Mult``. The identifier that starts the name of a library is called a *library root*. All library files of the standard library of |Coq| have the reserved root |Coq| but library filenames based on other roots can be obtained by using |Coq| commands (coqc, coqtop, coqdep, …) options ``-Q`` or ``-R`` (see :ref:`command-line-options`). Also, when an interactive |Coq| session starts, a library of root ``Top`` is started, unless option ``-top`` or ``-notop`` is set (see :ref:`command-line-options`). .. _qualified-names: Qualified names ~~~~~~~~~~~~~~~ Library files are modules which possibly contain submodules which eventually contain constructions (axioms, parameters, definitions, lemmas, theorems, remarks or facts). The *absolute name*, or *full name*, of a construction in some library file is a qualified identifier starting with the logical name of the library file, followed by the sequence of submodules names encapsulating the construction and ended by the proper name of the construction. Typically, the absolute name ``Coq.Init.Logic.eq`` denotes Leibniz’ equality defined in the module Logic in the sublibrary ``Init`` of the standard library of |Coq|. The proper name that ends the name of a construction is the short name (or sometimes base name) of the construction (for instance, the short name of ``Coq.Init.Logic.eq`` is ``eq``). Any partial suffix of the absolute name is a *partially qualified name* (e.g. ``Logic.eq`` is a partially qualified name for ``Coq.Init.Logic.eq``). Especially, the short name of a construction is its shortest partially qualified name. |Coq| does not accept two constructions (definition, theorem, …) with the same absolute name but different constructions can have the same short name (or even same partially qualified names as soon as the full names are different). Notice that the notion of absolute, partially qualified and short names also applies to library filenames. **Visibility** |Coq| maintains a table called the name table which maps partially qualified names of constructions to absolute names. This table is updated by the commands :cmd:`Require`, :cmd:`Import` and :cmd:`Export` and also each time a new declaration is added to the context. An absolute name is called visible from a given short or partially qualified name when this latter name is enough to denote it. This means that the short or partially qualified name is mapped to the absolute name in |Coq| name table. Definitions with the :attr:`local` attribute are only accessible with their fully qualified name (see :ref:`gallina-definitions`). It may happen that a visible name is hidden by the short name or a qualified name of another construction. In this case, the name that has been hidden must be referred to using one more level of qualification. To ensure that a construction always remains accessible, absolute names can never be hidden. .. example:: .. coqtop:: all Check 0. Definition nat := bool. Check 0. Check Datatypes.nat. Locate nat. .. seealso:: Commands :cmd:`Locate` and :cmd:`Locate Library`. .. _libraries-and-filesystem: Libraries and filesystem ~~~~~~~~~~~~~~~~~~~~~~~~ .. note:: The questions described here have been subject to redesign in |Coq| 8.5. Former versions of |Coq| use the same terminology to describe slightly different things. Compiled files (``.vo`` and ``.vio``) store sub-libraries. In order to refer to them inside |Coq|, a translation from file-system names to |Coq| names is needed. In this translation, names in the file system are called *physical* paths while |Coq| names are contrastingly called *logical* names. A logical prefix Lib can be associated with a physical path using the command line option ``-Q`` `path` ``Lib``. All subfolders of path are recursively associated to the logical path ``Lib`` extended with the corresponding suffix coming from the physical path. For instance, the folder ``path/fOO/Bar`` maps to ``Lib.fOO.Bar``. Subdirectories corresponding to invalid |Coq| identifiers are skipped, and, by convention, subdirectories named ``CVS`` or ``_darcs`` are skipped too. Thanks to this mechanism, ``.vo`` files are made available through the logical name of the folder they are in, extended with their own basename. For example, the name associated to the file ``path/fOO/Bar/File.vo`` is ``Lib.fOO.Bar.File``. The same caveat applies for invalid identifiers. When compiling a source file, the ``.vo`` file stores its logical name, so that an error is issued if it is loaded with the wrong loadpath afterwards. Some folders have a special status and are automatically put in the path. |Coq| commands associate automatically a logical path to files in the repository trees rooted at the directory from where the command is launched, ``coqlib/user-contrib/``, the directories listed in the ``$COQPATH``, ``${XDG_DATA_HOME}/coq/`` and ``${XDG_DATA_DIRS}/coq/`` environment variables (see `XDG base directory specification `_) with the same physical-to-logical translation and with an empty logical prefix. The command line option ``-R`` is a variant of ``-Q`` which has the strictly same behavior regarding loadpaths, but which also makes the corresponding ``.vo`` files available through their short names in a way similar to the :cmd:`Import` command. For instance, ``-R path Lib`` associates to the file ``/path/fOO/Bar/File.vo`` the logical name ``Lib.fOO.Bar.File``, but allows this file to be accessed through the short names ``fOO.Bar.File,Bar.File`` and ``File``. If several files with identical base name are present in different subdirectories of a recursive loadpath, which of these files is found first may be system- dependent and explicit qualification is recommended. The ``From`` argument of the ``Require`` command can be used to bypass the implicit shortening by providing an absolute root to the required file (see :ref:`compiled-files`). There also exists another independent loadpath mechanism attached to OCaml object files (``.cmo`` or ``.cmxs``) rather than |Coq| object files as described above. The OCaml loadpath is managed using the option ``-I`` `path` (in the OCaml world, there is neither a notion of logical name prefix nor a way to access files in subdirectories of path). See the command :cmd:`Declare ML Module` in :ref:`compiled-files` to understand the need of the OCaml loadpath. See :ref:`command-line-options` for a more general view over the |Coq| command line options. .. _ImplicitArguments: Implicit arguments ------------------ An implicit argument of a function is an argument which can be inferred from contextual knowledge. There are different kinds of implicit arguments that can be considered implicit in different ways. There are also various commands to control the setting or the inference of implicit arguments. The different kinds of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Implicit arguments inferable from the knowledge of other arguments of a function ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ The first kind of implicit arguments covers the arguments that are inferable from the knowledge of the type of other arguments of the function, or of the type of the surrounding context of the application. Especially, such implicit arguments correspond to parameters dependent in the type of the function. Typical implicit arguments are the type arguments in polymorphic functions. There are several kinds of such implicit arguments. **Strict Implicit Arguments** An implicit argument can be either strict or non strict. An implicit argument is said to be *strict* if, whatever the other arguments of the function are, it is still inferable from the type of some other argument. Technically, an implicit argument is strict if it corresponds to a parameter which is not applied to a variable which itself is another parameter of the function (since this parameter may erase its arguments), not in the body of a match, and not itself applied or matched against patterns (since the original form of the argument can be lost by reduction). For instance, the first argument of :: cons: forall A:Set, A -> list A -> list A in module ``List.v`` is strict because :g:`list` is an inductive type and :g:`A` will always be inferable from the type :g:`list A` of the third argument of :g:`cons`. Also, the first argument of :g:`cons` is strict with respect to the second one, since the first argument is exactly the type of the second argument. On the contrary, the second argument of a term of type :: forall P:nat->Prop, forall n:nat, P n -> ex nat P is implicit but not strict, since it can only be inferred from the type :g:`P n` of the third argument and if :g:`P` is, e.g., :g:`fun _ => True`, it reduces to an expression where ``n`` does not occur any longer. The first argument :g:`P` is implicit but not strict either because it can only be inferred from :g:`P n` and :g:`P` is not canonically inferable from an arbitrary :g:`n` and the normal form of :g:`P n`. Consider, e.g., that :g:`n` is :math:`0` and the third argument has type :g:`True`, then any :g:`P` of the form :: fun n => match n with 0 => True | _ => anything end would be a solution of the inference problem. **Contextual Implicit Arguments** An implicit argument can be *contextual* or not. An implicit argument is said *contextual* if it can be inferred only from the knowledge of the type of the context of the current expression. For instance, the only argument of:: nil : forall A:Set, list A` is contextual. Similarly, both arguments of a term of type:: forall P:nat->Prop, forall n:nat, P n \/ n = 0 are contextual (moreover, :g:`n` is strict and :g:`P` is not). **Reversible-Pattern Implicit Arguments** There is another class of implicit arguments that can be reinferred unambiguously if all the types of the remaining arguments are known. This is the class of implicit arguments occurring in the type of another argument in position of reversible pattern, which means it is at the head of an application but applied only to uninstantiated distinct variables. Such an implicit argument is called *reversible- pattern implicit argument*. A typical example is the argument :g:`P` of nat_rec in :: nat_rec : forall P : nat -> Set, P 0 -> (forall n : nat, P n -> P (S n)) -> forall x : nat, P x (:g:`P` is reinferable by abstracting over :g:`n` in the type :g:`P n`). See :ref:`controlling-rev-pattern-implicit-args` for the automatic declaration of reversible-pattern implicit arguments. Implicit arguments inferable by resolution ++++++++++++++++++++++++++++++++++++++++++ This corresponds to a class of non-dependent implicit arguments that are solved based on the structure of their type only. Maximal or non maximal insertion of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case a function is partially applied, and the next argument to be applied is an implicit argument, two disciplines are applicable. In the first case, the function is considered to have no arguments furtherly: one says that the implicit argument is not maximally inserted. In the second case, the function is considered to be implicitly applied to the implicit arguments it is waiting for: one says that the implicit argument is maximally inserted. Each implicit argument can be declared to be inserted maximally or non maximally. In Coq, maximally-inserted implicit arguments are written between curly braces "{ }" and non-maximally-inserted implicit arguments are written in square brackets "[ ]". .. seealso:: :flag:`Maximal Implicit Insertion` Trailing Implicit Arguments +++++++++++++++++++++++++++ An implicit argument is considered trailing when all following arguments are declared implicit. Trailing implicit arguments cannot be declared non maximally inserted, otherwise they would never be inserted. .. exn:: Argument @name is a trailing implicit, so it can't be declared non maximal. Please use %{ %} instead of [ ]. For instance: .. coqtop:: all fail Fail Definition double [n] := n + n. Casual use of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In a given expression, if it is clear that some argument of a function can be inferred from the type of the other arguments, the user can force the given argument to be guessed by replacing it by “_”. If possible, the correct argument will be automatically generated. .. exn:: Cannot infer a term for this placeholder. :name: Cannot infer a term for this placeholder. (Casual use of implicit arguments) |Coq| was not able to deduce an instantiation of a “_”. .. _declare-implicit-args: Declaration of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In case one wants that some arguments of a given object (constant, inductive types, constructors, assumptions, local or not) are always inferred by |Coq|, one may declare once and for all which are the expected implicit arguments of this object. There are two ways to do this, *a priori* and *a posteriori*. Implicit Argument Binders +++++++++++++++++++++++++ .. insertprodn implicit_binders implicit_binders .. prodn:: implicit_binders ::= %{ {+ @name } {? : @type } %} | [ {+ @name } {? : @type } ] In the first setting, one wants to explicitly give the implicit arguments of a declared object as part of its definition. To do this, one has to surround the bindings of implicit arguments by curly braces or square braces: .. coqtop:: all Definition id {A : Type} (x : A) : A := x. This automatically declares the argument A of id as a maximally inserted implicit argument. One can then do as-if the argument was absent in every situation but still be able to specify it if needed: .. coqtop:: all Definition compose {A B C} (g : B -> C) (f : A -> B) := fun x => g (f x). Goal forall A, compose id id = id (A:=A). For non maximally inserted implicit arguments, use square brackets: .. coqtop:: all Fixpoint map [A B : Type] (f : A -> B) (l : list A) : list B := match l with | nil => nil | cons a t => cons (f a) (map f t) end. Print Implicit map. The syntax is supported in all top-level definitions: :cmd:`Definition`, :cmd:`Fixpoint`, :cmd:`Lemma` and so on. For (co-)inductive datatype declarations, the semantics are the following: an inductive parameter declared as an implicit argument need not be repeated in the inductive definition and will become implicit for the inductive type and the constructors. For example: .. coqtop:: all Inductive list {A : Type} : Type := | nil : list | cons : A -> list -> list. Print list. One can always specify the parameter if it is not uniform using the usual implicit arguments disambiguation syntax. The syntax is also supported in internal binders. For instance, in the following kinds of expressions, the type of each declaration present in :token:`binders` can be bracketed to mark the declaration as implicit: :n:`fun (@ident:forall {* @binder }, @type) => @term`, :n:`forall (@ident:forall {* @binder }, @type), @type`, :n:`let @ident {* @binder } := @term in @term`, :n:`fix @ident {* @binder } := @term in @term` and :n:`cofix @ident {* @binder } := @term in @term`. Here is an example: .. coqtop:: all Axiom Ax : forall (f:forall {A} (a:A), A * A), let g {A} (x y:A) := (x,y) in f 0 = g 0 0. .. warn:: Ignoring implicit binder declaration in unexpected position This is triggered when setting an argument implicit in an expression which does not correspond to the type of an assumption or to the body of a definition. Here is an example: .. coqtop:: all warn Definition f := forall {y}, y = 0. .. warn:: Making shadowed name of implicit argument accessible by position This is triggered when two variables of same name are set implicit in the same block of binders, in which case the first occurrence is considered to be unnamed. Here is an example: .. coqtop:: all warn Check let g {x:nat} (H:x=x) {x} (H:x=x) := x in 0. Declaring Implicit Arguments ++++++++++++++++++++++++++++ .. cmd:: Arguments @smart_qualid {* @argument_spec_block } {* , {* @more_implicits_block } } {? : {+, @arguments_modifier } } :name: Arguments .. insertprodn smart_qualid arguments_modifier .. prodn:: smart_qualid ::= @qualid | @by_notation by_notation ::= @string {? % @ident } argument_spec_block ::= @argument_spec | / | & | ( {+ @argument_spec } ) {? % @ident } | [ {+ @argument_spec } ] {? % @ident } | %{ {+ @argument_spec } %} {? % @ident } argument_spec ::= {? ! } @name {? % @ident } more_implicits_block ::= @name | [ {+ @name } ] | %{ {+ @name } %} arguments_modifier ::= simpl nomatch | simpl never | default implicits | clear bidirectionality hint | clear implicits | clear scopes | clear scopes and implicits | clear implicits and scopes | rename | assert | extra scopes This command sets implicit arguments *a posteriori*, where the list of :n:`@name`\s is a prefix of the list of arguments of :n:`@smart_qualid`. Arguments in square brackets are declared as implicit and arguments in curly brackets are declared as maximally inserted. After the command is issued, implicit arguments can and must be omitted in any expression that applies :token:`qualid`. This command supports the :attr:`local` and :attr:`global` attributes. Default behavior is to limit the effect to the current section but also to extend their effect outside the current module or library file. Applying :attr:`local` limits the effect of the command to the current module if it's not in a section. Applying :attr:`global` within a section extends the effect outside the current sections and current module if the command occurs. A command containing :n:`@argument_spec_block & @argument_spec_block` provides :ref:`bidirectionality_hints`. Use the :n:`@more_implicits_block` to specify multiple implicit arguments declarations for names of constants, inductive types, constructors and lemmas that can only be applied to a fixed number of arguments (excluding, for instance, constants whose type is polymorphic). The longest applicable list of implicit arguments will be used to select which implicit arguments are inserted. For printing, the omitted arguments are the ones of the longest list of implicit arguments of the sequence. See the example :ref:`here`. The :n:`@arguments_modifier` values have various effects: * :n:`clear implicits` - clears implicit arguments * :n:`default implicits` - automatically determine the implicit arguments of the object. See :ref:`auto_decl_implicit_args`. * :n:`rename` - rename implicit arguments for the object * :n:`assert` - assert that the object has the expected number of arguments with the expected names. See the example here: :ref:`renaming_implicit_arguments`. .. exn:: The / modifier may only occur once. :undocumented: .. exn:: The & modifier may only occur once. :undocumented: .. example:: .. coqtop:: reset all Inductive list (A : Type) : Type := | nil : list A | cons : A -> list A -> list A. Check (cons nat 3 (nil nat)). Arguments cons [A] _ _. Arguments nil {A}. Check (cons 3 nil). Fixpoint map (A B : Type) (f : A -> B) (l : list A) : list B := match l with nil => nil | cons a t => cons (f a) (map A B f t) end. Fixpoint length (A : Type) (l : list A) : nat := match l with nil => 0 | cons _ m => S (length A m) end. Arguments map [A B] f l. Arguments length {A} l. (* A has to be maximally inserted *) Check (fun l:list (list nat) => map length l). .. _example_more_implicits: .. example:: Multiple implicit arguments with :n:`@more_implicits_block` .. coqtop:: all Arguments map [A B] f l, [A] B f l, A B f l. Check (fun l => map length l = map (list nat) nat length l). .. note:: Use the :cmd:`Print Implicit` command to see the implicit arguments of an object (see :ref:`displaying-implicit-args`). .. _auto_decl_implicit_args: Automatic declaration of implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The :n:`default implicits @arguments_modifier` clause tells |Coq| to automatically determine the implicit arguments of the object. Auto-detection is governed by flags specifying whether strict, contextual, or reversible-pattern implicit arguments must be considered or not (see :ref:`controlling-strict-implicit-args`, :ref:`controlling-contextual-implicit-args`, :ref:`controlling-rev-pattern-implicit-args` and also :ref:`controlling-insertion-implicit-args`). .. example:: Default implicits .. coqtop:: reset all Inductive list (A:Set) : Set := | nil : list A | cons : A -> list A -> list A. Arguments cons : default implicits. Print Implicit cons. Arguments nil : default implicits. Print Implicit nil. Set Contextual Implicit. Arguments nil : default implicits. Print Implicit nil. The computation of implicit arguments takes account of the unfolding of constants. For instance, the variable ``p`` below has type ``(Transitivity R)`` which is reducible to ``forall x,y:U, R x y -> forall z:U, R y z -> R x z``. As the variables ``x``, ``y`` and ``z`` appear strictly in the body of the type, they are implicit. .. coqtop:: all Parameter X : Type. Definition Relation := X -> X -> Prop. Definition Transitivity (R:Relation) := forall x y:X, R x y -> forall z:X, R y z -> R x z. Parameters (R : Relation) (p : Transitivity R). Arguments p : default implicits. Print p. Print Implicit p. Parameters (a b c : X) (r1 : R a b) (r2 : R b c). Check (p r1 r2). Mode for automatic declaration of implicit arguments ++++++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Implicit Arguments This flag (off by default) allows to systematically declare implicit the arguments detectable as such. Auto-detection of implicit arguments is governed by flags controlling whether strict and contextual implicit arguments have to be considered or not. .. _controlling-strict-implicit-args: Controlling strict implicit arguments +++++++++++++++++++++++++++++++++++++ .. flag:: Strict Implicit When the mode for automatic declaration of implicit arguments is on, the default is to automatically set implicit only the strict implicit arguments plus, for historical reasons, a small subset of the non-strict implicit arguments. To relax this constraint and to set implicit all non strict implicit arguments by default, you can turn this flag off. .. flag:: Strongly Strict Implicit Use this flag (off by default) to capture exactly the strict implicit arguments and no more than the strict implicit arguments. .. _controlling-contextual-implicit-args: Controlling contextual implicit arguments +++++++++++++++++++++++++++++++++++++++++ .. flag:: Contextual Implicit By default, |Coq| does not automatically set implicit the contextual implicit arguments. You can turn this flag on to tell |Coq| to also infer contextual implicit argument. .. _controlling-rev-pattern-implicit-args: Controlling reversible-pattern implicit arguments +++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Reversible Pattern Implicit By default, |Coq| does not automatically set implicit the reversible-pattern implicit arguments. You can turn this flag on to tell |Coq| to also infer reversible-pattern implicit argument. .. _controlling-insertion-implicit-args: Controlling the insertion of implicit arguments not followed by explicit arguments ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ .. flag:: Maximal Implicit Insertion Assuming the implicit argument mode is on, this flag (off by default) declares implicit arguments to be automatically inserted when a function is partially applied and the next argument of the function is an implicit one. Combining manual declaration and automatic declaration ++++++++++++++++++++++++++++++++++++++++++++++++++++++ When some arguments are manually specified implicit with binders in a definition and the automatic declaration mode in on, the manual implicit arguments are added to the automatically declared ones. In that case, and when the flag :flag:`Maximal Implicit Insertion` is set to off, some trailing implicit arguments can be inferred to be non maximally inserted. In this case, they are converted to maximally inserted ones. .. example:: .. coqtop:: all Set Implicit Arguments. Axiom eq0_le0 : forall (n : nat) (x : n = 0), n <= 0. Print Implicit eq0_le0. Axiom eq0_le0' : forall (n : nat) {x : n = 0}, n <= 0. Print Implicit eq0_le0'. .. _explicit-applications: Explicit applications ~~~~~~~~~~~~~~~~~~~~~ In presence of non-strict or contextual arguments, or in presence of partial applications, the synthesis of implicit arguments may fail, so one may have to explicitly give certain implicit arguments of an 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`. .. example:: Syntax for explicitly giving implicit arguments (continued) .. coqtop:: all Check (p r1 (z:=c)). Check (p (x:=a) (y:=b) r1 (z:=c) r2). .. _renaming_implicit_arguments: Renaming implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. example:: (continued) Renaming implicit arguments .. coqtop:: all Arguments p [s t] _ [u] _: rename. Check (p r1 (u:=c)). Check (p (s:=a) (t:=b) r1 (u:=c) r2). Fail Arguments p [s t] _ [w] _ : assert. .. _displaying-implicit-args: Displaying implicit arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. cmd:: Print Implicit @smart_qualid Displays the implicit arguments associated with an object, identifying which arguments are applied maximally or not. Displaying implicit arguments when pretty-printing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. flag:: Printing Implicit By default, the basic pretty-printing rules hide the inferrable implicit arguments of an application. Turn this flag on to force printing all implicit arguments. .. flag:: Printing Implicit Defensive By default, the basic pretty-printing rules display implicit arguments that are not detected as strict implicit arguments. This “defensive” mode can quickly make the display cumbersome so this can be deactivated by turning this flag off. .. seealso:: :flag:`Printing All`. Interaction with subtyping ~~~~~~~~~~~~~~~~~~~~~~~~~~ When an implicit argument can be inferred from the type of more than one of the other arguments, then only the type of the first of these arguments is taken into account, and not an upper type of all of them. As a consequence, the inference of the implicit argument of “=” fails in .. coqtop:: all Fail Check nat = Prop. but succeeds in .. coqtop:: all Check Prop = nat. Deactivation of implicit arguments for parsing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. flag:: Parsing Explicit Turning this flag on (it is off by default) deactivates the use of implicit arguments. In this case, all arguments of constants, inductive types, constructors, etc, including the arguments declared as implicit, have to be given as if no arguments were implicit. By symmetry, this also affects printing. .. _canonical-structure-declaration: Canonical structures ~~~~~~~~~~~~~~~~~~~~ A canonical structure is an instance of a record/structure type that can be used to solve unification problems involving a projection applied to an unknown structure instance (an implicit argument) and a value. The complete documentation of canonical structures can be found in :ref:`canonicalstructures`; here only a simple example is given. .. cmd:: Canonical {? Structure } @smart_qualid Canonical {? Structure } @ident_decl @def_body :name: Canonical Structure; _ The first form of this command declares an existing :n:`@smart_qualid` as a canonical instance of a structure (a record). The second form defines a new constant as if the :cmd:`Definition` command had been used, then declares it as a canonical instance as if the first form had been used on the defined object. This command supports the :attr:`local` attribute. When used, the structure is canonical only within the :cmd:`Section` containing it. Assume that :token:`qualid` denotes an object ``(Build_struct`` |c_1| … |c_n| ``)`` in the structure :g:`struct` of which the fields are |x_1|, …, |x_n|. Then, each time an equation of the form ``(``\ |x_i| ``_)`` |eq_beta_delta_iota_zeta| |c_i| has to be solved during the type checking process, :token:`qualid` is used as a solution. Otherwise said, :token:`qualid` is canonically used to extend the field |c_i| into a complete structure built on |c_i|. Canonical structures are particularly useful when mixed with coercions and strict implicit arguments. .. example:: Here is an example. .. coqtop:: all Require Import Relations. Require Import EqNat. Set Implicit Arguments. Unset Strict Implicit. Structure Setoid : Type := {Carrier :> Set; Equal : relation Carrier; Prf_equiv : equivalence Carrier Equal}. Definition is_law (A B:Setoid) (f:A -> B) := forall x y:A, Equal x y -> Equal (f x) (f y). Axiom eq_nat_equiv : equivalence nat eq_nat. Definition nat_setoid : Setoid := Build_Setoid eq_nat_equiv. Canonical nat_setoid. Thanks to :g:`nat_setoid` declared as canonical, the implicit arguments :g:`A` and :g:`B` can be synthesized in the next statement. .. coqtop:: all abort Lemma is_law_S : is_law S. .. note:: If a same field occurs in several canonical structures, then only the structure declared first as canonical is considered. .. attr:: canonical(false) To prevent a field from being involved in the inference of canonical instances, its declaration can be annotated with the :attr:`canonical(false)` attribute (cf. the syntax of :n:`@record_field`). .. example:: For instance, when declaring the :g:`Setoid` structure above, the :g:`Prf_equiv` field declaration could be written as follows. .. coqdoc:: #[canonical(false)] Prf_equiv : equivalence Carrier Equal See :ref:`canonicalstructures` for a more realistic example. .. attr:: canonical This attribute can decorate a :cmd:`Definition` or :cmd:`Let` command. It is equivalent to having a :cmd:`Canonical Structure` declaration just after the command. .. cmd:: Print Canonical Projections {* @smart_qualid } This displays the list of global names that are components of some canonical structure. For each of them, the canonical structure of which it is a projection is indicated. If constants are given as its arguments, only the unification rules that involve or are synthesized from simultaneously all given constants will be shown. .. example:: For instance, the above example gives the following output: .. coqtop:: all Print Canonical Projections. .. coqtop:: all Print Canonical Projections nat. .. note:: The last line in the first example would not show up if the corresponding projection (namely :g:`Prf_equiv`) were annotated as not canonical, as described above. Implicit types of variables ~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is possible to bind variable names to a given type (e.g. in a development using arithmetic, it may be convenient to bind the names :g:`n` or :g:`m` to the type :g:`nat` of natural numbers). .. cmd:: Implicit {| Type | Types } @reserv_list :name: Implicit Type; Implicit Types .. insertprodn reserv_list simple_reserv .. prodn:: reserv_list ::= {+ ( @simple_reserv ) } | @simple_reserv simple_reserv ::= {+ @ident } : @type Sets the type of bound variables starting with :token:`ident` (either :token:`ident` itself or :token:`ident` followed by one or more single quotes, underscore or digits) to :token:`type` (unless the bound variable is already declared with an explicit type, in which case, that type will be used). .. example:: .. coqtop:: all Require Import List. Implicit Types m n : nat. Lemma cons_inj_nat : forall m n l, n :: l = m :: l -> n = m. Proof. intros m n. Abort. Lemma cons_inj_bool : forall (m n:bool) l, n :: l = m :: l -> n = m. Abort. .. flag:: Printing Use Implicit Types By default, the type of bound variables is not printed when the variable name is associated to an implicit type which matches the actual type of the variable. This feature can be deactivated by turning this flag off. .. _implicit-generalization: Implicit generalization ~~~~~~~~~~~~~~~~~~~~~~~ .. index:: `{ } .. index:: `[ ] .. index:: `( ) .. index:: `{! } .. index:: `[! ] .. index:: `(! ) .. insertprodn generalizing_binder typeclass_constraint .. prodn:: generalizing_binder ::= `( {+, @typeclass_constraint } ) | `%{ {+, @typeclass_constraint } %} | `[ {+, @typeclass_constraint } ] typeclass_constraint ::= {? ! } @term | %{ @name %} : {? ! } @term | @name : {? ! } @term Implicit generalization is an automatic elaboration of a statement with free variables into a closed statement where these variables are quantified explicitly. Use the :cmd:`Generalizable` command to designate which variables should be generalized. It is activated for a binder by prefixing a \`, and for terms by surrounding it with \`{ }, or \`[ ] or \`( ). Terms surrounded by \`{ } introduce their free variables as maximally inserted implicit arguments, terms surrounded by \`[ ] introduce them as non maximally inserted implicit arguments and terms surrounded by \`( ) introduce them as explicit arguments. Generalizing binders always introduce their free variables as maximally inserted implicit arguments. The binder itself introduces its argument as usual. In the following statement, ``A`` and ``y`` are automatically generalized, ``A`` is implicit and ``x``, ``y`` and the anonymous equality argument are explicit. .. coqtop:: all reset Generalizable All Variables. Definition sym `(x:A) : `(x = y -> y = x) := fun _ p => eq_sym p. Print sym. Dually to normal binders, the name is optional but the type is required: .. coqtop:: all Check (forall `{x = y :> A}, y = x). When generalizing a binder whose type is a typeclass, its own class arguments are omitted from the syntax and are generalized using automatic names, without instance search. Other arguments are also generalized unless provided. This produces a fully general statement. this behaviour may be disabled by prefixing the type with a ``!`` or by forcing the typeclass name to be an explicit application using ``@`` (however the later ignores implicit argument information). .. coqtop:: all Class Op (A:Type) := op : A -> A -> A. Class Commutative (A:Type) `(Op A) := commutative : forall x y, op x y = op y x. Instance nat_op : Op nat := plus. Set Printing Implicit. Check (forall `{Commutative }, True). Check (forall `{Commutative nat}, True). Fail Check (forall `{Commutative nat _}, True). Fail Check (forall `{!Commutative nat}, True). Arguments Commutative _ {_}. Check (forall `{!Commutative nat}, True). Check (forall `{@Commutative nat plus}, True). Multiple binders can be merged using ``,`` as a separator: .. coqtop:: all Check (forall `{Commutative A, Hnat : !Commutative nat}, True). .. cmd:: Generalizable {| {| Variable | Variables } {+ @ident } | All Variables | No Variables } Controls the set of generalizable identifiers. By default, no variables are generalizable. This command supports the :attr:`global` attribute. The :n:`{| Variable | Variables } {+ @ident }` form allows generalization of only the given :n:`@ident`\s. Using this command multiple times adds to the allowed identifiers. The other forms clear the list of :n:`@ident`\s. The :n:`All Variables` form generalizes all free variables in the context that appear under a generalization delimiter. This may result in confusing errors in case of typos. In such cases, the context will probably contain some unexpected generalized variables. The :n:`No Variables` form disables implicit generalization entirely. This is the default behavior (before any :cmd:`Generalizable` command has been entered). .. _Coercions: Coercions --------- Coercions can be used to implicitly inject terms from one *class* in which they reside into another one. A *class* is either a sort (denoted by the keyword ``Sortclass``), a product type (denoted by the keyword ``Funclass``), or a type constructor (denoted by its name), e.g. an inductive type or any constant with a type of the form :n:`forall {+ @binder }, @sort`. Then the user is able to apply an object that is not a function, but can be coerced to a function, and more generally to consider that a term of type ``A`` is of type ``B`` provided that there is a declared coercion between ``A`` and ``B``. More details and examples, and a description of the commands related to coercions are provided in :ref:`implicitcoercions`. .. _printing_constructions_full: Printing constructions in full ------------------------------ .. flag:: Printing All Coercions, implicit arguments, the type of pattern matching, but also notations (see :ref:`syntaxextensionsandinterpretationscopes`) 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, implicit arguments, returned type of pattern matching, notations and various syntactic sugar for pattern matching or record projections. Otherwise said, :flag:`Printing All` includes the effects of the flags :flag:`Printing Implicit`, :flag:`Printing Coercions`, :flag:`Printing Synth`, :flag:`Printing Projections`, and :flag:`Printing Notations`. To reactivate the high-level printing features, use the command ``Unset Printing All``. .. _printing-universes: Printing universes ------------------ .. flag:: Printing Universes Turn this flag on to activate the display of the actual level of each occurrence of :g:`Type`. See :ref:`Sorts` for details. This wizard flag, in combination with :flag:`Printing All` can help to diagnose failures to unify terms apparently identical but internally different in the Calculus of Inductive Constructions. .. cmd:: Print {? Sorted } Universes {? Subgraph ( {* @qualid } ) } {? @string } :name: Print Universes This command can be used to print the constraints on the internal level of the occurrences of :math:`\Type` (see :ref:`Sorts`). The :n:`Subgraph` clause limits the printed graph to the requested names (adjusting constraints to preserve the implied transitive constraints between kept universes). The :n:`Sorted` clause makes each universe equivalent to a numbered label reflecting its level (with a linear ordering) in the universe hierarchy. :n:`@string` is an optional output filename. If :n:`@string` ends in ``.dot`` or ``.gv``, the constraints are printed in the DOT language, and can be processed by Graphviz tools. The format is unspecified if `string` doesn’t end in ``.dot`` or ``.gv``. .. _existential-variables: Existential variables --------------------- .. insertprodn term_evar term_evar .. prodn:: term_evar ::= ?[ @ident ] | ?[ ?@ident ] | ?@ident {? @%{ {+; @ident := @term } %} } |Coq| terms can include existential variables which represents unknown subterms to eventually be replaced by actual subterms. Existential variables are generated in place of unsolvable implicit arguments or “_” placeholders when using commands such as ``Check`` (see Section :ref:`requests-to-the-environment`) or when using tactics such as :tacn:`refine`, as well as in place of unsolvable instances when using tactics such that :tacn:`eapply`. An existential variable is defined in a context, which is the context of variables of the placeholder which generated the existential variable, and a type, which is the expected type of the placeholder. As a consequence of typing constraints, existential variables can be duplicated in such a way that they possibly appear in different contexts than their defining context. Thus, any occurrence of a given existential variable comes with an instance of its original context. In the simple case, when an existential variable denotes the placeholder which generated it, or is used in the same context as the one in which it was generated, the context is not displayed and the existential variable is represented by “?” followed by an identifier. .. coqtop:: all Parameter identity : forall (X:Set), X -> X. Check identity _ _. Check identity _ (fun x => _). In the general case, when an existential variable :n:`?@ident` appears outside of its context of definition, its instance, written under the form :n:`{ {*; @ident := @term} }` is appending to its name, indicating how the variables of its defining context are instantiated. The variables of the context of the existential variables which are instantiated by themselves are not written, unless the :flag:`Printing Existential Instances` flag is on (see Section :ref:`explicit-display-existentials`), and this is why an existential variable used in the same context as its context of definition is written with no instance. .. coqtop:: all Check (fun x y => _) 0 1. Set Printing Existential Instances. Check (fun x y => _) 0 1. Existential variables can be named by the user upon creation using the syntax :n:`?[@ident]`. This is useful when the existential variable needs to be explicitly handled later in the script (e.g. with a named-goal selector, see :ref:`goal-selectors`). .. _explicit-display-existentials: Explicit displaying of existential instances for pretty-printing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. flag:: Printing Existential Instances This flag (off by default) activates the full display of how the context of an existential variable is instantiated at each of the occurrences of the existential variable. .. _tactics-in-terms: Solving existential variables using tactics ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Instead of letting the unification engine try to solve an existential variable by itself, one can also provide an explicit hole together with a tactic to solve it. Using the syntax ``ltac:(``\ `tacexpr`\ ``)``, the user can put a tactic anywhere a term is expected. The order of resolution is not specified and is implementation-dependent. The inner tactic may use any variable defined in its scope, including repeated alternations between variables introduced by term binding as well as those introduced by tactic binding. The expression `tacexpr` can be any tactic expression as described in :ref:`ltac`. .. coqtop:: all Definition foo (x : nat) : nat := ltac:(exact x). This construction is useful when one wants to define complicated terms using highly automated tactics without resorting to writing the proof-term by means of the interactive proof engine. .. _primitive-integers: Primitive Integers ------------------ The language of terms features 63-bit machine integers as values. The type of such a value is *axiomatized*; it is declared through the following sentence (excerpt from the :g:`Int63` module): .. coqdoc:: Primitive int := #int63_type. This type is equipped with a few operators, that must be similarly declared. For instance, equality of two primitive integers can be decided using the :g:`Int63.eqb` function, declared and specified as follows: .. coqdoc:: Primitive eqb := #int63_eq. Notation "m '==' n" := (eqb m n) (at level 70, no associativity) : int63_scope. Axiom eqb_correct : forall i j, (i == j)%int63 = true -> i = j. The complete set of such operators can be obtained looking at the :g:`Int63` module. These primitive declarations are regular axioms. As such, they must be trusted and are listed by the :g:`Print Assumptions` command, as in the following example. .. coqtop:: in reset From Coq Require Import Int63. Lemma one_minus_one_is_zero : (1 - 1 = 0)%int63. Proof. apply eqb_correct; vm_compute; reflexivity. Qed. .. coqtop:: all Print Assumptions one_minus_one_is_zero. The reduction machines (:tacn:`vm_compute`, :tacn:`native_compute`) implement dedicated, efficient, rules to reduce the applications of these primitive operations. The extraction of these primitives can be customized similarly to the extraction of regular axioms (see :ref:`extraction`). Nonetheless, the :g:`ExtrOCamlInt63` module can be used when extracting to OCaml: it maps the Coq primitives to types and functions of a :g:`Uint63` module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq. Literal values (at type :g:`Int63.int`) are extracted to literal OCaml values wrapped into the :g:`Uint63.of_int` (resp. :g:`Uint63.of_int64`) constructor on 64-bit (resp. 32-bit) platforms. Currently, this cannot be customized (see the function :g:`Uint63.compile` from the kernel). .. _primitive-floats: Primitive Floats ---------------- The language of terms features Binary64 floating-point numbers as values. The type of such a value is *axiomatized*; it is declared through the following sentence (excerpt from the :g:`PrimFloat` module): .. coqdoc:: Primitive float := #float64_type. This type is equipped with a few operators, that must be similarly declared. For instance, the product of two primitive floats can be computed using the :g:`PrimFloat.mul` function, declared and specified as follows: .. coqdoc:: Primitive mul := #float64_mul. Notation "x * y" := (mul x y) : float_scope. Axiom mul_spec : forall x y, Prim2SF (x * y)%float = SF64mul (Prim2SF x) (Prim2SF y). where :g:`Prim2SF` is defined in the :g:`FloatOps` module. The set of such operators is described in section :ref:`floats_library`. These primitive declarations are regular axioms. As such, they must be trusted, and are listed by the :g:`Print Assumptions` command. The reduction machines (:tacn:`vm_compute`, :tacn:`native_compute`) implement dedicated, efficient rules to reduce the applications of these primitive operations, using the floating-point processor operators that are assumed to comply with the IEEE 754 standard for floating-point arithmetic. The extraction of these primitives can be customized similarly to the extraction of regular axioms (see :ref:`extraction`). Nonetheless, the :g:`ExtrOCamlFloats` module can be used when extracting to OCaml: it maps the Coq primitives to types and functions of a :g:`Float64` module. Said OCaml module is not produced by extraction. Instead, it has to be provided by the user (if they want to compile or execute the extracted code). For instance, an implementation of this module can be taken from the kernel of Coq. Literal values (of type :g:`Float64.t`) are extracted to literal OCaml values (of type :g:`float`) written in hexadecimal notation and wrapped into the :g:`Float64.of_float` constructor, e.g.: :g:`Float64.of_float (0x1p+0)`. .. _bidirectionality_hints: Bidirectionality hints ---------------------- When type-checking an application, Coq normally does not use information from the context to infer the types of the arguments. It only checks after the fact that the type inferred for the application is coherent with the expected type. Bidirectionality hints make it possible to specify that after type-checking the first arguments of an application, typing information should be propagated from the context to help inferring the types of the remaining arguments. An :cmd:`Arguments` command containing :n:`@argument_spec_block__1 & @argument_spec_block__2` provides :ref:`bidirectionality_hints`. It tells the typechecking algorithm, when type-checking applications of :n:`@qualid`, to first type-check the arguments in :n:`@argument_spec_block__1` and then propagate information from the typing context to type-check the remaining arguments (in :n:`@argument_spec_block__2`). .. example:: Bidirectionality hints In a context where a coercion was declared from ``bool`` to ``nat``: .. coqtop:: in reset Definition b2n (b : bool) := if b then 1 else 0. Coercion b2n : bool >-> nat. Coq cannot automatically coerce existential statements over ``bool`` to statements over ``nat``, because the need for inserting a coercion is known only from the expected type of a subterm: .. coqtop:: all Fail Check (ex_intro _ true _ : exists n : nat, n > 0). However, a suitable bidirectionality hint makes the example work: .. coqtop:: all Arguments ex_intro _ _ & _ _. Check (ex_intro _ true _ : exists n : nat, n > 0). Coq will attempt to produce a term which uses the arguments you provided, but in some cases involving Program mode the arguments after the bidirectionality starts may be replaced by convertible but syntactically different terms.