\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#3})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]

# Setting properties of a function's arguments

Command Arguments reference arg_specs* , implicits_alt** : args_modifier+,?
  arg_specs`::=`argument_spec`````|`/`````|`&`````|`( argument_spec+ ) % scope​%_ scope*`````|`[ argument_spec+ ] % scope​%_ scope*`````|`{ argument_spec+ } % scope​%_ scope*``argument_spec`::=`!? name % scope​%_ scope*``implicits_alt`::=`name`````|`[ name+ ]`````|`{ name+ }``args_modifier`::=`simpl nomatch`````|`simpl never`````|`default implicits`````|`clear implicits`````|`clear scopes`````|`clear bidirectionality hint`````|`rename`````|`assert`````|`extra scopes`````|`clear scopes and implicits`````|`clear implicits and scopes``

  Specifies properties of the arguments of a function after the function has already been defined. It gives fine-grained control over the elaboration process (i.e. the translation of Gallina language extensions into the core language used by the kernel). The command's effects include:

  - Making arguments implicit. Afterward, implicit arguments must be omitted in any expression that applies `reference`.

  - Declaring that some arguments of a given function should be interpreted in a given notation scope.

  - Affecting when the `simpl` and `cbn` tactics unfold the function. See Effects of Arguments on unfolding.

  - Providing bidirectionality hints. See Bidirectionality hints.

  This command supports the `local` and `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 `local` limits the effect of the command to the current module if it's not in a section. Applying `global` within a section extends the effect outside the current sections and current module in which the command appears.

  > `/`

  >   the function will be unfolded only if it's applied to at least the arguments appearing before the `/`. See Effects of Arguments on unfolding.

  >   Error The / modifier may only occur once.
  > `&`

  >   tells the type checking algorithm to first type check the arguments before the `&` and then to propagate information from that typing context to type check the remaining arguments. See Bidirectionality hints.

  >   Error The & modifier may only occur once.
  > `( argument_spec+ ) %_ scope*`

  >   `(name1 name2 ...)%scope*` is shorthand for `name1%scope* name2%scope* ...`

  > `[ argument_spec+ ] %_ scope*`

  >   declares the enclosed names as implicit, non-maximally inserted. `[name1 name2 ... ]%_scope*` is equivalent to `[name1]%_scope* [name2]%_scope* ...`

  > `{ argument_spec+ } %_ scope*`

  >   declares the enclosed names as implicit, maximally inserted. `{name1 name2 ... }%_scope*` is equivalent to `{name1}%_scope* {name2}%_scope* ...`

  > `!`

  >   the function will be unfolded only if all the arguments marked with `!` evaluate to constructors. See Effects of Arguments on unfolding.

  > `name %_ scope*`

  >   a formal parameter of the function `reference` (i.e. the parameter name used in the function definition). Unless `rename` is specified, the list of `name`s must be a prefix of the formal parameters, including all implicit arguments. `_` can be used to skip over a formal parameter. This construct declares `name` as non-implicit if `clear implicits` is specified or any other `name` in the `Arguments` command is declared implicit. `scope` can be either scope names or their delimiting keys. When multiple scopes are present, notations are interpreted in the leftmost scope containing them. See Binding arguments to scopes.

  >   Deprecated since version 8.19: The `% scope` syntax is deprecated in favor of the currently equivalent `%_ scope`. It will be reused in future versions with the same semantics as in terms.

  >   Error To rename arguments the 'rename' flag must be specified.
  >   Error Flag 'rename' expected to rename name into name.
  >   Error Arguments of section variables such as name may not be renamed.
  > `clear implicits`

  >   makes all implicit arguments into explicit arguments

  >   Error The 'clear implicits' flag must be omitted if implicit annotations are given.
  > `default implicits`

  >   automatically determine the implicit arguments of the object. See Automatic declaration of implicit arguments.

  >   Error The 'default implicits' flag is incompatible with implicit annotations.
  > `rename`

  >   rename implicit arguments for the object. See the example here.

  > `assert`

  >   assert that the object has the expected number of arguments with the expected names. See the example here: Renaming implicit arguments.

  >   Warning This command is just asserting the names of arguments of qualid. If this is what you want, add ': assert' to silence the warning. If you want to clear implicit arguments, add ': clear implicits'. If you want to clear notation scopes, add ': clear scopes'
  > `clear scopes`

  >   clears argument scopes of `reference`

  > `extra scopes`

  >   defines extra argument scopes, to be used in case of coercion to `Funclass` (see Implicit Coercions) or with a computed type.

  > `simpl nomatch`

  >   prevents performing a simplification step for `reference` that would expose a match construct in the head position. See Effects of Arguments on unfolding.

  > `simpl never`

  >   prevents performing a simplification step for `reference`. See Effects of Arguments on unfolding.

  > `clear bidirectionality hint`

  >   removes the bidirectionality hint, the `&`

  > `implicits_alt`

  >   use to specify alternative implicit argument declarations for functions that can only be applied to a fixed number of arguments (excluding, for instance, functions whose type is polymorphic). For parsing, the longest list of implicit arguments matching the function application is used to select which implicit arguments are inserted. For printing, the alternative with the most implicit arguments is used; the implict arguments will be omitted if `Printing Implicit` is not set. See the example here.

  Use `About` to view the current implicit arguments setting for a `reference`.

  Or use the `Print Implicit` command to see the implicit arguments of an object (see Displaying implicit arguments).

## Manual declaration of implicit arguments

Example

Inductive list (A : Type) : Type := | nil : list A | cons : A -> list A -> list A.
  list is defined list_rect is defined list_ind is defined list_rec is defined list_sind is defined
Check (cons nat 3 (nil nat)).
  cons nat 3 (nil nat) : list nat
Arguments cons [A] _ _.
Arguments nil {A}.
Check (cons 3 nil).
  cons 3 nil : list nat
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.
  map is defined map is recursively defined (guarded on 4th argument)
Fixpoint length (A : Type) (l : list A) : nat := match l with nil => 0 | cons _ m => S (length A m) end.
  length is defined length is recursively defined (guarded on 2nd argument)
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).
  fun l : list (list nat) => map length l : list (list nat) -> list nat

Example: Multiple alternatives with `implicits_alt`

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).
  fun l : list (list nat) => map length l = map length l : list (list nat) -> Prop

## Automatic declaration of implicit arguments

> The "`default implicits`" `args_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 Controlling strict implicit arguments, Controlling contextual implicit arguments, Controlling reversible-pattern implicit arguments and also Controlling the insertion of implicit arguments not followed by explicit arguments).

Example: Default implicits

Inductive list (A:Set) : Set := | nil : list A | cons : A -> list A -> list A.
  list is defined list_rect is defined list_ind is defined list_rec is defined list_sind is defined
Arguments cons : default implicits.
Print Implicit cons.
  cons : forall [A : Set], A -> list A -> list A Argument A is implicit
Arguments nil : default implicits.
Print Implicit nil.
  nil : forall A : Set, list A
Set Contextual Implicit.
Arguments nil : default implicits.
Print Implicit nil.
  nil : forall {A : Set}, list A Argument A is implicit and maximally inserted

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.

Parameter X : Type.
  X is declared
Definition Relation := X -> X -> Prop.
  Relation is defined
Definition Transitivity (R:Relation) := forall x y:X, R x y -> forall z:X, R y z -> R x z.
  Transitivity is defined
Parameters (R : Relation) (p : Transitivity R).
  R is declared p is declared
Arguments p : default implicits.
Print p.
  *** [ p : Transitivity R ] Expanded type for implicit arguments p : forall [x y : X], R x y -> forall [z : X], R y z -> R x z Arguments p [x y] _ [z] _
Print Implicit p.
  p : forall [x y : X], R x y -> forall [z : X], R y z -> R x z Arguments x, y, z are implicit
Parameters (a b c : X) (r1 : R a b) (r2 : R b c).
  a is declared b is declared c is declared r1 is declared r2 is declared
Check (p r1 r2).
  p r1 r2 : R a c

## Renaming implicit arguments

Example: (continued) Renaming implicit arguments

Arguments p [s t] _ [u] _: rename.
Check (p r1 (u:=c)).
  p r1 (u:=c) : R b c -> R a c
Check (p (s:=a) (t:=b) r1 (u:=c) r2).
  p r1 r2 : R a c
Fail Arguments p [s t] _ [w] _ : assert.
  The command has indeed failed with message: Flag "rename" expected to rename u into w.

## Binding arguments to scopes

> The following command declares that the first two arguments of `plus_fct` are interpreted in the `scope` delimited by the key `F` and the third argument is first interpreted in the scope delimited by the key `R`, then in `F` (when a notation has no interpretation in `R`).

> > Arguments plus_fct (f1 f2)%_F x%_R%_F.

> When interpreting a term, if some of the arguments of `reference` are built from a notation, then this notation is interpreted in the scope stack extended by the scopes bound (if any) to this argument. The effect of these scopes is limited to the argument itself. It does not propagate to subterms but the subterms that, after interpretation of the notation, turn to be themselves arguments of a reference are interpreted according to the argument scopes bound to this reference.

Note

In notations, the subterms matching the identifiers of the notations are interpreted in the scope in which the identifiers occurred at the time of the declaration of the notation. Here is an example:

Parameter g : bool -> bool.
  g is declared
Declare Scope mybool_scope.
Notation "@@" := true (only parsing) : bool_scope.
Notation "@@" := false (only parsing): mybool_scope.
Bind Scope bool_scope with bool.
Notation "# x #" := (g x) (at level 40).
Check # @@ #.
  # true # : bool
Arguments g _%_mybool_scope.
Check # @@ #.
  # true # : bool
Delimit Scope mybool_scope with mybool.
Check # @@%mybool #.
  # false # : bool

## Effects of `Arguments` on unfolding

- `simpl never` indicates that a constant should not be unfolded by `cbn` or `simpl` when in head position. Note that in the case of `simpl`, the modifier does not apply to reduction of the main argument of a `match`, `fix`, primitive projection, or of an unfoldable constant hiding a `match`, `fix` or primitive projection.

  Example

  Arguments Nat.sub n m : simpl never.

  After that command an expression like `(Nat.sub (S x) y)` is left untouched by the tactics `cbn` and `simpl`.

  Otherwise, an expression like `(Nat.sub (S x) 0) + 1` reduces to `S (x + 1)` for `simpl` because `Nat.sub` is the main argument of `+` in this case.

- A constant can be marked to be unfolded only if it's applied to at least the arguments appearing before the `/` in a `Arguments` command.

  Example

  Definition fcomp A B C f (g : A -> B) (x : A) : C := f (g x).
    fcomp is defined
  Arguments fcomp {A B C} f g x /.
  Notation "f \o g" := (fcomp f g) (at level 50).

  After that command the expression `(f \o g)` is left untouched by `simpl` while `((f \o g) t)` is reduced to `(f (g t))`. The same mechanism can be used to make a constant volatile, i.e. always unfolded.

  Example

  Definition volatile := fun x : nat => x.
    volatile is defined
  Arguments volatile / x.

- A constant can be marked to be unfolded only if an entire set of arguments evaluates to a constructor. The `!` symbol can be used to mark such arguments.

  Example

  Arguments minus !n !m.

  After that command, the expression `(minus (S x) y)` is left untouched by `simpl`, while `(minus (S x) (S y))` is reduced to `(minus x y)`.

- `simpl nomatch` indicates that a constant should not be unfolded if it would expose a `match` construct in the head position. This affects the `cbn`, `simpl` and `hnf` tactics.

  Example

  Arguments minus n m : simpl nomatch.

  In this case, `(minus (S (S x)) (S y))` is simplified to `(minus (S x) y)` even if an extra simplification is possible.

  In detail: the tactic `simpl` first applies βι-reduction. Then, it expands transparent constants and tries to reduce further using βι-reduction. But, when no ι rule is applied after unfolding then δ-reductions are not applied. For instance trying to use `simpl` on `(plus n O) = n` changes nothing.

## 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 `Arguments` command containing `arg_specs1 & arg_specs2` provides bidirectionality hints. It tells the typechecking algorithm, when type checking applications of `qualid`, to first type check the arguments in `arg_specs1` and then propagate information from the typing context to type check the remaining arguments (in `arg_specs2`).

Example: Bidirectionality hints

In a context where a coercion was declared from `bool` to `nat`:

Definition b2n (b : bool) := if b then 1 else 0.
  b2n is defined
Coercion b2n : bool >-> nat.
  b2n is now a coercion

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:

Fail Check (ex_intro _ true _ : exists n : nat, n > 0).
  The command has indeed failed with message: The term "ex_intro ?P true ?y" has type "exists y, ?P y" while it is expected to have type "exists n : nat, n > 0" (cannot unify "bool" and "nat").

However, a suitable bidirectionality hint makes the example work:

Arguments ex_intro _ _ & _ _.
Check (ex_intro _ true _ : exists n : nat, n > 0).
  ex_intro (fun n : nat => n > 0) true ?g : exists n : nat, n > 0 : exists n : nat, n > 0 where ?g : [ |- (fun n : nat => n > 0) true]

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.

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