6. Namespaces and Sections🔗

Names are organized into hierarchical namespaces, which are collections of names. Namespaces are the primary means of organizing APIs in Lean: they provide an ontology of operations, grouping related items. Additionally, while this is not done by giving them names in the namespace, the effects of features such as syntax extensions, instances, and attributes can be attached to a namespace.

Sorting operations into namespaces organizes libraries conceptually, from a global perspective. Any given Lean file will, however, typically not use all names equally. Sections provide a means of ordering a local view of the globally-available collection of names, as well as a way to precisely control the scope of compiler options along with language extensions, instances, and attributes. They also allow parameters shared by many declarations to be declared centrally and propagated as needed using the Lean.Parser.Command.variable : commandDeclares one or more typed variables, or modifies whether already-declared variables are implicit. Introduces variables that can be used in definitions within the same `namespace` or `section` block. When a definition mentions a variable, Lean will add it as an argument of the definition. This is useful in particular when writing many definitions that have parameters in common (see below for an example). Variable declarations have the same flexibility as regular function parameters. In particular they can be [explicit, implicit][binder docs], or [instance implicit][tpil classes] (in which case they can be anonymous). This can be changed, for instance one can turn explicit variable `x` into an implicit one with `variable {x}`. Note that currently, you should avoid changing how variables are bound and declare new variables at the same time; see [issue 2789] for more on this topic. In *theorem bodies* (i.e. proofs), variables are not included based on usage in order to ensure that changes to the proof cannot change the statement of the overall theorem. Instead, variables are only available to the proof if they have been mentioned in the theorem header or in an `include` command or are instance implicit and depend only on such variables. See [*Variables and Sections* from Theorem Proving in Lean][tpil vars] for a more detailed discussion. [tpil vars]: https://lean-lang.org/theorem_proving_in_lean4/dependent_type_theory.html#variables-and-sections (Variables and Sections on Theorem Proving in Lean) [tpil classes]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html (Type classes on Theorem Proving in Lean) [binder docs]: https://leanprover-community.github.io/mathlib4_docs/Lean/Expr.html#Lean.BinderInfo (Documentation for the BinderInfo type) [issue 2789]: https://github.com/leanprover/lean4/issues/2789 (Issue 2789 on github) ## Examples ```lean section variable {α : Type u} -- implicit (a : α) -- explicit [instBEq : BEq α] -- instance implicit, named [Hashable α] -- instance implicit, anonymous def isEqual (b : α) : Bool := a == b #check isEqual -- isEqual.{u} {α : Type u} (a : α) [instBEq : BEq α] (b : α) : Bool variable {a} -- `a` is implicit now def eqComm {b : α} := a == b ↔ b == a #check eqComm -- eqComm.{u} {α : Type u} {a : α} [instBEq : BEq α] {b : α} : Prop end ``` The following shows a typical use of `variable` to factor out definition arguments: ```lean variable (Src : Type) structure Logger where trace : List (Src × String) #check Logger -- Logger (Src : Type) : Type namespace Logger -- switch `Src : Type` to be implicit until the `end Logger` variable {Src} def empty : Logger Src where trace := [] #check empty -- Logger.empty {Src : Type} : Logger Src variable (log : Logger Src) def len := log.trace.length #check len -- Logger.len {Src : Type} (log : Logger Src) : Nat variable (src : Src) [BEq Src] -- at this point all of `log`, `src`, `Src` and the `BEq` instance can all become arguments def filterSrc := log.trace.filterMap fun (src', str') => if src' == src then some str' else none #check filterSrc -- Logger.filterSrc {Src : Type} (log : Logger Src) (src : Src) [inst✝ : BEq Src] : List String def lenSrc := log.filterSrc src |>.length #check lenSrc -- Logger.lenSrc {Src : Type} (log : Logger Src) (src : Src) [inst✝ : BEq Src] : Nat end Logger ``` The following example demonstrates availability of variables in proofs: ```lean variable {α : Type} -- available in the proof as indirectly mentioned through `a` [ToString α] -- available in the proof as `α` is included (a : α) -- available in the proof as mentioned in the header {β : Type} -- not available in the proof [ToString β] -- not available in the proof theorem ex : a = a := rfl ``` After elaboration of the proof, the following warning will be generated to highlight the unused hypothesis: ``` included section variable '[ToString α]' is not used in 'ex', consider excluding it ``` In such cases, the offending variable declaration should be moved down or into a section so that only theorems that do depend on it follow it until the end of the section.variable command.

6.1. Namespaces🔗

Names that contain periods (that aren't inside guillemets) are hierarchical names; the periods separate the components of a name. All but the final component of a name are the namespace, while the final component is the name itself.

Namespaces serve to group related definitions, theorems, types, and other declarations. When a namespace corresponds to a type's name, generalized field notation can be used to access its contents. In addition to organizing names, namespaces also group syntax extensions, attributes, and instances.

Namespaces are orthogonal to modules: a module is a unit of code that is elaborated, compiled, and loaded together, but there is no necessary connection between a module's name and the names that it provides. A module may contain names in any namespace, and the nesting structure of hierarchical modules is unrelated to that of hierarchical namespaces.

There is a root namespace, ordinarily denoted by simply omitting a namespace. It can be explicitly indicated by beginning a name with _root_. This can be necessary in contexts where a name would otherwise be interpreted relative to an ambient namespace (e.g. from a section scope) or local scope.

Explicit Root Namespace

Names in the current namespace take precedence over names in the root namespace. In this example, color in the definition of Forest.statement refers to Forest.color:

def color := "yellow"
namespace Forest
def color := "green"
def statement := s!"Lemons are {color}"
end Forest

"Lemons are green"#eval Forest.statement

"Lemons are green"

Within the Forest namespace, references to color in the root namespace must be qualified with _root_:

namespace Forest
def nextStatement :=
  s!"Ripe lemons are {_root_.color}, not {color}"
end Forest

"Ripe lemons are yellow, not green"#eval Forest.nextStatement

"Ripe lemons are yellow, not green"

6.1.1. Namespaces and Section Scopes

Every section scope has a current namespace, which is determined by the Lean.Parser.Command.namespace : command`namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside the section: * Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the full name `Nat.seventeen`. * Names introduced by `export` declarations are also prefixed by the identifier. * All names starting with `<id>.` become available in the namespace without the prefix. These names are preferred over names introduced by outer namespaces or `open`. * Within a namespace, declarations can be `protected`, which excludes them from the effects of opening the namespace. As with `section`, namespaces can be nested and the scope of a namespace is terminated by a corresponding `end <id>` or the end of the file. `namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.namespace command.The Lean.Parser.Command.namespace : command`namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside the section: * Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the full name `Nat.seventeen`. * Names introduced by `export` declarations are also prefixed by the identifier. * All names starting with `<id>.` become available in the namespace without the prefix. These names are preferred over names introduced by outer namespaces or `open`. * Within a namespace, declarations can be `protected`, which excludes them from the effects of opening the namespace. As with `section`, namespaces can be nested and the scope of a namespace is terminated by a corresponding `end <id>` or the end of the file. `namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.namespace command is described in the section on commands that introduce section scopes. Names that are declared within the section scope are added to the current namespace. If the declared name has more than one component, then its namespace is nested within the current namespace; the body of the declaration's current namespace is the nested namespace. Section scopes also include a set of opened namespaces, which are namespaces whose contents are in scope without additional qualification. Resolving an identifier to a particular name takes the current namespace and opened namespaces into account. However, protected declarations (that is, those with the protected modifier) are not brought into scope when their namespace is opened. The rules for resolving identifiers into names that take the current namespace and opened namespaces into account are described in the section on identifiers as terms.

Current Namespace

Defining an inductive type results in the type's constructors being placed in its namespace, in this case as HotDrink.coffee, HotDrink.tea, and HotDrink.cocoa.

inductive HotDrink where
  | coffee
  | tea
  | cocoa

Outside the namespace, these names must be qualified unless the namespace is opened:

HotDrink.tea : HotDrink#check HotDrink.tea

HotDrink.tea : HotDrink

#check Unknown identifier `tea`tea

Unknown identifier `tea`

section
open HotDrink
HotDrink.tea : HotDrink#check tea
end

HotDrink.tea : HotDrink

If a function is defined directly inside the HotDrink namespace, then the body of the function is elaborated with the current namespace set to HotDrink. The constructors are in scope:

def HotDrink.ofString? : String → Option HotDrink
  | "coffee" => some coffee
  | "tea" => some tea
  | "cocoa" => some cocoa
  | _ => none

Defining another inductive type creates a new namespace:

inductive ColdDrink where
  | water
  | juice

From within the HotDrink namespace, HotDrink.toString can be defined without an explicit prefix. Defining a function in the ColdDrink namespace requires an explicit _root_ qualifier to avoid defining HotDrink.ColdDrink.toString:

namespace HotDrink

def toString : HotDrink → String
  | coffee => "coffee"
  | tea => "tea"
  | cocoa => "cocoa"

def _root_.ColdDrink.toString : ColdDrink → String
  | .water => "water"
  | .juice => "juice"

end HotDrink

The Lean.Parser.Command.open : commandMakes names from other namespaces visible without writing the namespace prefix. Names that are made available with `open` are visible within the current `section` or `namespace` block. This makes referring to (type) definitions and theorems easier, but note that it can also make [scoped instances], notations, and attributes from a different namespace available. The `open` command can be used in a few different ways: * `open Some.Namespace.Path1 Some.Namespace.Path2` makes all non-protected names in `Some.Namespace.Path1` and `Some.Namespace.Path2` available without the prefix, so that `Some.Namespace.Path1.x` and `Some.Namespace.Path2.y` can be referred to by writing only `x` and `y`. * `open Some.Namespace.Path hiding def1 def2` opens all non-protected names in `Some.Namespace.Path` except `def1` and `def2`. * `open Some.Namespace.Path (def1 def2)` only makes `Some.Namespace.Path.def1` and `Some.Namespace.Path.def2` available without the full prefix, so `Some.Namespace.Path.def3` would be unaffected. This works even if `def1` and `def2` are `protected`. * `open Some.Namespace.Path renaming def1 → def1', def2 → def2'` same as `open Some.Namespace.Path (def1 def2)` but `def1`/`def2`'s names are changed to `def1'`/`def2'`. This works even if `def1` and `def2` are `protected`. * `open scoped Some.Namespace.Path1 Some.Namespace.Path2` **only** opens [scoped instances], notations, and attributes from `Namespace1` and `Namespace2`; it does **not** make any other name available. * `open <any of the open shapes above> in` makes the names `open`-ed visible only in the next command or expression. [scoped instance]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html#scoped-instances (Scoped instances in Theorem Proving in Lean) ## Examples ```lean /-- SKI combinators https://en.wikipedia.org/wiki/SKI_combinator_calculus -/ namespace Combinator.Calculus def I (a : α) : α := a def K (a : α) : β → α := fun _ => a def S (x : α → β → γ) (y : α → β) (z : α) : γ := x z (y z) end Combinator.Calculus section -- open everything under `Combinator.Calculus`, *i.e.* `I`, `K` and `S`, -- until the section ends open Combinator.Calculus theorem SKx_eq_K : S K x = I := rfl end -- open everything under `Combinator.Calculus` only for the next command (the next `theorem`, here) open Combinator.Calculus in theorem SKx_eq_K' : S K x = I := rfl section -- open only `S` and `K` under `Combinator.Calculus` open Combinator.Calculus (S K) theorem SKxy_eq_y : S K x y = y := rfl -- `I` is not in scope, we have to use its full path theorem SKxy_eq_Iy : S K x y = Combinator.Calculus.I y := rfl end section open Combinator.Calculus renaming I → identity, K → konstant #check identity #check konstant end section open Combinator.Calculus hiding S #check I #check K end section namespace Demo inductive MyType | val namespace N1 scoped infix:68 " ≋ " => BEq.beq scoped instance : BEq MyType where beq _ _ := true def Alias := MyType end N1 end Demo -- bring `≋` and the instance in scope, but not `Alias` open scoped Demo.N1 #check Demo.MyType.val == Demo.MyType.val #check Demo.MyType.val ≋ Demo.MyType.val -- #check Alias -- unknown identifier 'Alias' end ```open command opens a namespace, making its contents available in the current section scope. There are many variations on opening namespaces, providing flexibility in managing the local scope.

syntaxOpening Namespaces

command ::= ...
    | Makes names from other namespaces visible without writing the namespace prefix.

Names that are made available with `open` are visible within the current `section` or `namespace`
block. This makes referring to (type) definitions and theorems easier, but note that it can also
make [scoped instances], notations, and attributes from a different namespace available.

The `open` command can be used in a few different ways:

* `open Some.Namespace.Path1 Some.Namespace.Path2` makes all non-protected names in
  `Some.Namespace.Path1` and `Some.Namespace.Path2` available without the prefix, so that
  `Some.Namespace.Path1.x` and `Some.Namespace.Path2.y` can be referred to by writing only `x` and
  `y`.

* `open Some.Namespace.Path hiding def1 def2` opens all non-protected names in `Some.Namespace.Path`
  except `def1` and `def2`.

* `open Some.Namespace.Path (def1 def2)` only makes `Some.Namespace.Path.def1` and
  `Some.Namespace.Path.def2` available without the full prefix, so `Some.Namespace.Path.def3` would
  be unaffected.

  This works even if `def1` and `def2` are `protected`.

* `open Some.Namespace.Path renaming def1 → def1', def2 → def2'` same as `open Some.Namespace.Path
  (def1 def2)` but `def1`/`def2`'s names are changed to `def1'`/`def2'`.

  This works even if `def1` and `def2` are `protected`.

* `open scoped Some.Namespace.Path1 Some.Namespace.Path2` **only** opens [scoped instances],
  notations, and attributes from `Namespace1` and `Namespace2`; it does **not** make any other name
  available.

* `open <any of the open shapes above> in` makes the names `open`-ed visible only in the next
  command or expression.

[scoped instance]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html#scoped-instances
(Scoped instances in Theorem Proving in Lean)


## Examples

```lean
/-- SKI combinators https://en.wikipedia.org/wiki/SKI_combinator_calculus -/
namespace Combinator.Calculus
  def I (a : α) : α := a
  def K (a : α) : β → α := fun _ => a
  def S (x : α → β → γ) (y : α → β) (z : α) : γ := x z (y z)
end Combinator.Calculus

section
  -- open everything under `Combinator.Calculus`, *i.e.* `I`, `K` and `S`,
  -- until the section ends
  open Combinator.Calculus

  theorem SKx_eq_K : S K x = I := rfl
end

-- open everything under `Combinator.Calculus` only for the next command (the next `theorem`, here)
open Combinator.Calculus in
theorem SKx_eq_K' : S K x = I := rfl

section
  -- open only `S` and `K` under `Combinator.Calculus`
  open Combinator.Calculus (S K)

  theorem SKxy_eq_y : S K x y = y := rfl

  -- `I` is not in scope, we have to use its full path
  theorem SKxy_eq_Iy : S K x y = Combinator.Calculus.I y := rfl
end

section
  open Combinator.Calculus
    renaming
      I → identity,
      K → konstant

  #check identity
  #check konstant
end

section
  open Combinator.Calculus
    hiding S

  #check I
  #check K
end

section
  namespace Demo
    inductive MyType
    | val

    namespace N1
      scoped infix:68 " ≋ " => BEq.beq

      scoped instance : BEq MyType where
        beq _ _ := true

      def Alias := MyType
    end N1
  end Demo

  -- bring `≋` and the instance in scope, but not `Alias`
  open scoped Demo.N1

  #check Demo.MyType.val == Demo.MyType.val
  #check Demo.MyType.val ≋ Demo.MyType.val
  -- #check Alias -- unknown identifier 'Alias'
end
```
open openDecl

open declarationOpening Entire Namespaces

A sequence of one or more identifiers results in each namespace in the sequence being opened:

`openDecl` is the body of an `open` declaration (see `open`) openDecl ::= ...
    | ident ident*

Each namespace in the sequence is considered relative to all currently-open namespaces, yielding a set of namespaces. Every namespace in this set is opened before the next namespace in the sequence is processed.

Opening Nested Namespaces

Namespaces to be opened are considered relative to the currently-open namespaces. If the same component occurs in different namespace paths, a single Lean.Parser.Command.open : commandMakes names from other namespaces visible without writing the namespace prefix. Names that are made available with `open` are visible within the current `section` or `namespace` block. This makes referring to (type) definitions and theorems easier, but note that it can also make [scoped instances], notations, and attributes from a different namespace available. The `open` command can be used in a few different ways: * `open Some.Namespace.Path1 Some.Namespace.Path2` makes all non-protected names in `Some.Namespace.Path1` and `Some.Namespace.Path2` available without the prefix, so that `Some.Namespace.Path1.x` and `Some.Namespace.Path2.y` can be referred to by writing only `x` and `y`. * `open Some.Namespace.Path hiding def1 def2` opens all non-protected names in `Some.Namespace.Path` except `def1` and `def2`. * `open Some.Namespace.Path (def1 def2)` only makes `Some.Namespace.Path.def1` and `Some.Namespace.Path.def2` available without the full prefix, so `Some.Namespace.Path.def3` would be unaffected. This works even if `def1` and `def2` are `protected`. * `open Some.Namespace.Path renaming def1 → def1', def2 → def2'` same as `open Some.Namespace.Path (def1 def2)` but `def1`/`def2`'s names are changed to `def1'`/`def2'`. This works even if `def1` and `def2` are `protected`. * `open scoped Some.Namespace.Path1 Some.Namespace.Path2` **only** opens [scoped instances], notations, and attributes from `Namespace1` and `Namespace2`; it does **not** make any other name available. * `open <any of the open shapes above> in` makes the names `open`-ed visible only in the next command or expression. [scoped instance]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html#scoped-instances (Scoped instances in Theorem Proving in Lean) ## Examples ```lean /-- SKI combinators https://en.wikipedia.org/wiki/SKI_combinator_calculus -/ namespace Combinator.Calculus def I (a : α) : α := a def K (a : α) : β → α := fun _ => a def S (x : α → β → γ) (y : α → β) (z : α) : γ := x z (y z) end Combinator.Calculus section -- open everything under `Combinator.Calculus`, *i.e.* `I`, `K` and `S`, -- until the section ends open Combinator.Calculus theorem SKx_eq_K : S K x = I := rfl end -- open everything under `Combinator.Calculus` only for the next command (the next `theorem`, here) open Combinator.Calculus in theorem SKx_eq_K' : S K x = I := rfl section -- open only `S` and `K` under `Combinator.Calculus` open Combinator.Calculus (S K) theorem SKxy_eq_y : S K x y = y := rfl -- `I` is not in scope, we have to use its full path theorem SKxy_eq_Iy : S K x y = Combinator.Calculus.I y := rfl end section open Combinator.Calculus renaming I → identity, K → konstant #check identity #check konstant end section open Combinator.Calculus hiding S #check I #check K end section namespace Demo inductive MyType | val namespace N1 scoped infix:68 " ≋ " => BEq.beq scoped instance : BEq MyType where beq _ _ := true def Alias := MyType end N1 end Demo -- bring `≋` and the instance in scope, but not `Alias` open scoped Demo.N1 #check Demo.MyType.val == Demo.MyType.val #check Demo.MyType.val ≋ Demo.MyType.val -- #check Alias -- unknown identifier 'Alias' end ```open command can be used to open all of them by iteratively bringing each into scope. This example defines names in a variety of namespaces:

namespace A -- _root_.A
def a1 := 0
namespace B -- _root_.A.B
def a2 := 0
namespace C -- _root_.A.B.C
def a3 := 0
end C
end B
end A
namespace B -- _root_.B
def a4 := 0
namespace C -- _root_.B.C
def a5 := 0
end C
end B
namespace C -- _root_.C
def a6 := 0
end C

The names are:

A.a1
A.B.a2
A.B.C.a3
B.a4
B.C.a5
C.a6

All six names can be brought into scope with a single iterated Lean.Parser.Command.open : commandMakes names from other namespaces visible without writing the namespace prefix. Names that are made available with `open` are visible within the current `section` or `namespace` block. This makes referring to (type) definitions and theorems easier, but note that it can also make [scoped instances], notations, and attributes from a different namespace available. The `open` command can be used in a few different ways: * `open Some.Namespace.Path1 Some.Namespace.Path2` makes all non-protected names in `Some.Namespace.Path1` and `Some.Namespace.Path2` available without the prefix, so that `Some.Namespace.Path1.x` and `Some.Namespace.Path2.y` can be referred to by writing only `x` and `y`. * `open Some.Namespace.Path hiding def1 def2` opens all non-protected names in `Some.Namespace.Path` except `def1` and `def2`. * `open Some.Namespace.Path (def1 def2)` only makes `Some.Namespace.Path.def1` and `Some.Namespace.Path.def2` available without the full prefix, so `Some.Namespace.Path.def3` would be unaffected. This works even if `def1` and `def2` are `protected`. * `open Some.Namespace.Path renaming def1 → def1', def2 → def2'` same as `open Some.Namespace.Path (def1 def2)` but `def1`/`def2`'s names are changed to `def1'`/`def2'`. This works even if `def1` and `def2` are `protected`. * `open scoped Some.Namespace.Path1 Some.Namespace.Path2` **only** opens [scoped instances], notations, and attributes from `Namespace1` and `Namespace2`; it does **not** make any other name available. * `open <any of the open shapes above> in` makes the names `open`-ed visible only in the next command or expression. [scoped instance]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html#scoped-instances (Scoped instances in Theorem Proving in Lean) ## Examples ```lean /-- SKI combinators https://en.wikipedia.org/wiki/SKI_combinator_calculus -/ namespace Combinator.Calculus def I (a : α) : α := a def K (a : α) : β → α := fun _ => a def S (x : α → β → γ) (y : α → β) (z : α) : γ := x z (y z) end Combinator.Calculus section -- open everything under `Combinator.Calculus`, *i.e.* `I`, `K` and `S`, -- until the section ends open Combinator.Calculus theorem SKx_eq_K : S K x = I := rfl end -- open everything under `Combinator.Calculus` only for the next command (the next `theorem`, here) open Combinator.Calculus in theorem SKx_eq_K' : S K x = I := rfl section -- open only `S` and `K` under `Combinator.Calculus` open Combinator.Calculus (S K) theorem SKxy_eq_y : S K x y = y := rfl -- `I` is not in scope, we have to use its full path theorem SKxy_eq_Iy : S K x y = Combinator.Calculus.I y := rfl end section open Combinator.Calculus renaming I → identity, K → konstant #check identity #check konstant end section open Combinator.Calculus hiding S #check I #check K end section namespace Demo inductive MyType | val namespace N1 scoped infix:68 " ≋ " => BEq.beq scoped instance : BEq MyType where beq _ _ := true def Alias := MyType end N1 end Demo -- bring `≋` and the instance in scope, but not `Alias` open scoped Demo.N1 #check Demo.MyType.val == Demo.MyType.val #check Demo.MyType.val ≋ Demo.MyType.val -- #check Alias -- unknown identifier 'Alias' end ```open command:

section
open A B C
example := [a1, a2, a3, a4, a5, a6]
end

If the initial namespace in the command is A.B instead, then neither _root_.A, _root_.B, nor _root_.B.C is opened:

section
open A.B C
example := [Unknown identifier `a1`a1, a2, a3, Unknown identifier `a4`a4, Unknown identifier `a5`a5, a6]
end

Unknown identifier `a1`

Unknown identifier `a4`

Unknown identifier `a5`

Opening A.B makes A.B.C visible as C along with _root_.C, so the subsequent C opens both.

open declarationHiding Names

A hiding declaration specifies a set of names that should not be brought into scope. In contrast to opening an entire namespace, the provided identifier must uniquely designate a namespace to be opened.

`openDecl` is the body of an `open` declaration (see `open`) openDecl ::= ...
    | ident hiding ident ident*

open declarationRenaming

A renaming declaration allows some names from the opened namespace to be renamed; they are accessible under the new name in the current section scope. The provided identifier must uniquely designate a namespace to be opened.

`openDecl` is the body of an `open` declaration (see `open`) openDecl ::= ...
    | ident renaming (ident → ident),*

An ASCII arrow (->) may be used instead of the Unicode arrow (→).

open declarationRestricted Opening

Parentheses indicate that only the names listed in the parentheses should be brought into scope.

`openDecl` is the body of an `open` declaration (see `open`) openDecl ::= ...
    | ident (ident ident*)

The indicated namespace is added to each currently-opened namespace, and each name is considered in each resulting namespace. All of the listed names must be unambiguous; that is, they must exist in exactly one of the considered namespaces.

open declarationScoped Declarations Only

The scoped keyword indicates that all scoped attributes, instances, and syntax from the provided namespaces should be opened, while not making any of the names available.

`openDecl` is the body of an `open` declaration (see `open`) openDecl ::= ...
    | scoped ident ident*

Opening Scoped Declarations

In this example, a scoped notation and a definition are created in the namespace NS:

namespace NS
scoped notation "{!{" e "}!}" => (e, e)
def three := 3
end NS

Outside of the namespace, the notation is not available:

def x := {!{ "pear" }unexpected token '!'; expected '}'!}

<example>:1:21-1:22: unexpected token '!'; expected '}'

An open scoped command makes the notation available:

open scoped NS
def x := {!{ "pear" }!}

However, the name NS.three is not in scope:

def y := Unknown identifier `three`three

Unknown identifier `three`

6.1.2. Exporting Names

Exporting a name makes it available in the current namespace. Unlike a definition, this alias is completely transparent: uses are resolved directly to the original name. Exporting a name to the root namespace makes it available without qualification; the Lean standard library does this for names such as the constructors of Option and key type class methods such as get.

syntaxExporting Names

The export command adds names from other namespaces to the current namespace, as if they had been declared in it. When the current namespace is opened, these exported names are also brought into scope.

command ::= ...
    | Adds names from other namespaces to the current namespace.

The command `export Some.Namespace (name₁ name₂)` makes `name₁` and `name₂`:

- visible in the current namespace without prefix `Some.Namespace`, like `open`, and
- visible from outside the current namespace `N` as `N.name₁` and `N.name₂`.

## Examples

```lean
namespace Morning.Sky
  def star := "venus"
end Morning.Sky

namespace Evening.Sky
  export Morning.Sky (star)
  -- `star` is now in scope
  #check star
end Evening.Sky

-- `star` is visible in `Evening.Sky`
#check Evening.Sky.star
```
export ident (ident*)

Internally, exported names are registered as aliases of their targets. From the perspective of the kernel, only the original name exists; the elaborator resolves aliases as part of resolving identifiers to names.

Exported Names

The declaration of the inductive type Veg.Leafy establishes the constructors Veg.Leafy.spinach and Veg.Leafy.cabbage:

namespace Veg
inductive Leafy where
  | spinach
  | cabbage
export Leafy (spinach)
end Veg
export Veg.Leafy (cabbage)

The first export command makes Veg.Leafy.spinach accessible as Veg.spinach because the current namespace is Veg. The second makes Veg.Leafy.cabbage accessible as cabbage, because the current namespace is the root namespace.

6.2. Section Scopes🔗

Many commands have an effect for the current section scope (sometimes just called "scope" when clear). Every Lean module has a section scope. Nested scopes are created via the Lean.Parser.Command.namespace : command`namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside the section: * Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the full name `Nat.seventeen`. * Names introduced by `export` declarations are also prefixed by the identifier. * All names starting with `<id>.` become available in the namespace without the prefix. These names are preferred over names introduced by outer namespaces or `open`. * Within a namespace, declarations can be `protected`, which excludes them from the effects of opening the namespace. As with `section`, namespaces can be nested and the scope of a namespace is terminated by a corresponding `end <id>` or the end of the file. `namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.namespace and Lean.Parser.Command.section : commandA `section`/`end` pair delimits the scope of `variable`, `include`, `open`, `set_option`, and `local` commands. Sections can be nested. `section <id>` provides a label to the section that has to appear with the matching `end`. In either case, the `end` can be omitted, in which case the section is closed at the end of the file.section commands, as well as the Lean.Parser.Command.in : commandin command combinator.

The following data are tracked in section scopes:

The Current Namespace: The current namespace is the namespace into which new declarations will be defined. Additionally, name resolution includes all prefixes of the current namespace in the scope for global names.
Opened Namespaces: When a namespace is opened, its names become available without an explicit prefix in the current scope. Additionally, scoped attributes and scoped syntax extensions in namespaces that have been opened are active in the current section scope.
Options: Compiler options are reverted to their original values at the end of the scope in which they were modified.
Section Variables: Section variables are names (or instance implicit parameters) that are automatically added as parameters to definitions. They are also added as universally-quantified assumptions to theorems when they occur in the theorem's statement.

6.2.1. Controlling Section Scopes🔗

The Lean.Parser.Command.section : commandA `section`/`end` pair delimits the scope of `variable`, `include`, `open`, `set_option`, and `local` commands. Sections can be nested. `section <id>` provides a label to the section that has to appear with the matching `end`. In either case, the `end` can be omitted, in which case the section is closed at the end of the file.section command creates a new section scope, but does not modify the current namespace, opened namespaces, or section variables. Changes made to the section scope are reverted when the section ends. Sections may optionally be named; the Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end command that closes a named section must use the same name. If section names have multiple components (that is, if they contain .-separated names), then multiple nested sections are introduced. Section names have no other effect, and are a readability aid.

syntaxSections

command ::= ...
    | A `section`/`end` pair delimits the scope of `variable`, `include`, `open`, `set_option`, and `local`
commands. Sections can be nested. `section <id>` provides a label to the section that has to appear
with the matching `end`. In either case, the `end` can be omitted, in which case the section is
closed at the end of the file.
section ident?

Named Section

The name english is defined in the Greetings namespace.

def Greetings.english := "Hello"

Outside its namespace, it cannot be evaluated.

#eval Unknown identifier `english`english

Unknown identifier `english`

Opening a section allows modifications to the global scope to be contained. This section is named Greetings.

section Greetings

Even though the section name matches the definition's namespace, the name is not in scope because section names are purely for readability and ease of refactoring.

#eval Unknown identifier `english`english

Unknown identifier `english`

Opening the namespace Greetings brings Greetings.english as english:

open Greetings

"Hello"#eval english

"Hello"

The section's name must be used to close it.

Missing name after `end`: Expected the current scope name `Greetings`

Hint: To end the current scope `Greetings`, specify its name:
  end ̲G̲r̲e̲e̲t̲i̲n̲g̲s̲end

Missing name after `end`: Expected the current scope name `Greetings`

Hint: To end the current scope `Greetings`, specify its name:
  end ̲G̲r̲e̲e̲t̲i̲n̲g̲s̲

end Greetings

When the section is closed, the effects of the Lean.Parser.Command.open : commandMakes names from other namespaces visible without writing the namespace prefix. Names that are made available with `open` are visible within the current `section` or `namespace` block. This makes referring to (type) definitions and theorems easier, but note that it can also make [scoped instances], notations, and attributes from a different namespace available. The `open` command can be used in a few different ways: * `open Some.Namespace.Path1 Some.Namespace.Path2` makes all non-protected names in `Some.Namespace.Path1` and `Some.Namespace.Path2` available without the prefix, so that `Some.Namespace.Path1.x` and `Some.Namespace.Path2.y` can be referred to by writing only `x` and `y`. * `open Some.Namespace.Path hiding def1 def2` opens all non-protected names in `Some.Namespace.Path` except `def1` and `def2`. * `open Some.Namespace.Path (def1 def2)` only makes `Some.Namespace.Path.def1` and `Some.Namespace.Path.def2` available without the full prefix, so `Some.Namespace.Path.def3` would be unaffected. This works even if `def1` and `def2` are `protected`. * `open Some.Namespace.Path renaming def1 → def1', def2 → def2'` same as `open Some.Namespace.Path (def1 def2)` but `def1`/`def2`'s names are changed to `def1'`/`def2'`. This works even if `def1` and `def2` are `protected`. * `open scoped Some.Namespace.Path1 Some.Namespace.Path2` **only** opens [scoped instances], notations, and attributes from `Namespace1` and `Namespace2`; it does **not** make any other name available. * `open <any of the open shapes above> in` makes the names `open`-ed visible only in the next command or expression. [scoped instance]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html#scoped-instances (Scoped instances in Theorem Proving in Lean) ## Examples ```lean /-- SKI combinators https://en.wikipedia.org/wiki/SKI_combinator_calculus -/ namespace Combinator.Calculus def I (a : α) : α := a def K (a : α) : β → α := fun _ => a def S (x : α → β → γ) (y : α → β) (z : α) : γ := x z (y z) end Combinator.Calculus section -- open everything under `Combinator.Calculus`, *i.e.* `I`, `K` and `S`, -- until the section ends open Combinator.Calculus theorem SKx_eq_K : S K x = I := rfl end -- open everything under `Combinator.Calculus` only for the next command (the next `theorem`, here) open Combinator.Calculus in theorem SKx_eq_K' : S K x = I := rfl section -- open only `S` and `K` under `Combinator.Calculus` open Combinator.Calculus (S K) theorem SKxy_eq_y : S K x y = y := rfl -- `I` is not in scope, we have to use its full path theorem SKxy_eq_Iy : S K x y = Combinator.Calculus.I y := rfl end section open Combinator.Calculus renaming I → identity, K → konstant #check identity #check konstant end section open Combinator.Calculus hiding S #check I #check K end section namespace Demo inductive MyType | val namespace N1 scoped infix:68 " ≋ " => BEq.beq scoped instance : BEq MyType where beq _ _ := true def Alias := MyType end N1 end Demo -- bring `≋` and the instance in scope, but not `Alias` open scoped Demo.N1 #check Demo.MyType.val == Demo.MyType.val #check Demo.MyType.val ≋ Demo.MyType.val -- #check Alias -- unknown identifier 'Alias' end ```open command are reverted.

#eval Unknown identifier `english`english

Unknown identifier `english`

The Lean.Parser.Command.namespace : command`namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside the section: * Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the full name `Nat.seventeen`. * Names introduced by `export` declarations are also prefixed by the identifier. * All names starting with `<id>.` become available in the namespace without the prefix. These names are preferred over names introduced by outer namespaces or `open`. * Within a namespace, declarations can be `protected`, which excludes them from the effects of opening the namespace. As with `section`, namespaces can be nested and the scope of a namespace is terminated by a corresponding `end <id>` or the end of the file. `namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.namespace command creates a new section scope. Within this section scope, the current namespace is the name provided in the command, interpreted relative to the current namespace in the surrounding section scope. Like sections, changes made to the section scope are reverted when the namespace's scope ends.

To close a namespace, the Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end command requires a suffix of the current namespace, which is removed. All section scopes introduced by the Lean.Parser.Command.namespace : command`namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside the section: * Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the full name `Nat.seventeen`. * Names introduced by `export` declarations are also prefixed by the identifier. * All names starting with `<id>.` become available in the namespace without the prefix. These names are preferred over names introduced by outer namespaces or `open`. * Within a namespace, declarations can be `protected`, which excludes them from the effects of opening the namespace. As with `section`, namespaces can be nested and the scope of a namespace is terminated by a corresponding `end <id>` or the end of the file. `namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.namespace command that introduced part of that suffix are closed.

syntaxNamespace Declarations

The namespace command modifies the current namespace by appending the provided identifier.

creates a section scope that lasts either until an Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end command or the end of the file.

command ::= ...
    | `namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside
the section:
* Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the
  full name `Nat.seventeen`.
* Names introduced by `export` declarations are also prefixed by the identifier.
* All names starting with `<id>.` become available in the namespace without the prefix. These names
  are preferred over names introduced by outer namespaces or `open`.
* Within a namespace, declarations can be `protected`, which excludes them from the effects of
  opening the namespace.

As with `section`, namespaces can be nested and the scope of a namespace is terminated by a
corresponding `end <id>` or the end of the file.

`namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.
namespace ident

syntaxSection and Namespace Terminators

Without an identifier, Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end closes the most recently opened section, which must be anonymous.

command ::= ...
    | `end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed
with `end <id>`. The `end` command is optional at the end of a file.
end

With an identifier, it closes the most recently opened section section or namespace. If it is a section, the identifier must be a suffix of the concatenated names of the sections opened since the most recent Lean.Parser.Command.namespace : command`namespace <id>` opens a section with label `<id>` that influences naming and name resolution inside the section: * Declarations names are prefixed: `def seventeen : ℕ := 17` inside a namespace `Nat` is given the full name `Nat.seventeen`. * Names introduced by `export` declarations are also prefixed by the identifier. * All names starting with `<id>.` become available in the namespace without the prefix. These names are preferred over names introduced by outer namespaces or `open`. * Within a namespace, declarations can be `protected`, which excludes them from the effects of opening the namespace. As with `section`, namespaces can be nested and the scope of a namespace is terminated by a corresponding `end <id>` or the end of the file. `namespace` also acts like `section` in delimiting the scope of `variable`, `open`, and other scoped commands.namespace command. If it is a namespace, then the identifier must be a suffix of the current namespace's extensions since the most recent Lean.Parser.Command.section : commandA `section`/`end` pair delimits the scope of `variable`, `include`, `open`, `set_option`, and `local` commands. Sections can be nested. `section <id>` provides a label to the section that has to appear with the matching `end`. In either case, the `end` can be omitted, in which case the section is closed at the end of the file.section that is still open; afterwards, the current namespace will have had this suffix removed.

command ::= ...
    | `end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed
with `end <id>`. The `end` command is optional at the end of a file.
end ident

The Lean.Parser.Command.mutual : commandend that closes a Lean.Parser.Command.mutual : commandmutual block is part of the syntax of Lean.Parser.Command.mutual : commandmutual, rather than the Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end command.

Nesting Namespaces and Sections

Namespaces and sections may be nested. A single Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end command may close one or more namespaces or one or more sections, but not a mix of the two.

After setting the current namespace to A.B.C with two separate commands, B.C may be removed with a single Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end:

namespace A.B
namespace C
end B.C

At this point, the current namespace is A.

Next, an anonymous section and the namespace D.E are opened:

section
namespace D.E

At this point, the current namespace is A.D.E. An Lean.Parser.Command.end : command`end` closes a `section` or `namespace` scope. If the scope is named `<id>`, it has to be closed with `end <id>`. The `end` command is optional at the end of a file.end command cannot close all three due to the intervening section:

Invalid name after `end`: Expected `D.E`, but found `A.D.E`end A.D.E

Invalid name after `end`: Expected `D.E`, but found `A.D.E`

Instead, namespaces and sections must be ended separately.

end D.E
end
end A

Rather than opening a section for a single command, the Lean.Parser.Command.in : commandin combinator can be used to create single-command section scope. The Lean.Parser.Command.in : commandin combinator is right-associative, allowing multiple scope modifications to be stacked.

syntaxLocal Section Scopes

The in command combinator introduces a section scope for a single command.

command ::= ...
    | command in
      command

Using Lean.Parser.Command.in : commandin for Local Scopes

The contents of a namespace can be made available for a single command using Lean.Parser.Command.in : commandin.

def Dessert.cupcake := "delicious"

open Dessert in
"delicious"#eval cupcake

After the single command, the effects of Lean.Parser.Command.open : commandMakes names from other namespaces visible without writing the namespace prefix. Names that are made available with `open` are visible within the current `section` or `namespace` block. This makes referring to (type) definitions and theorems easier, but note that it can also make [scoped instances], notations, and attributes from a different namespace available. The `open` command can be used in a few different ways: * `open Some.Namespace.Path1 Some.Namespace.Path2` makes all non-protected names in `Some.Namespace.Path1` and `Some.Namespace.Path2` available without the prefix, so that `Some.Namespace.Path1.x` and `Some.Namespace.Path2.y` can be referred to by writing only `x` and `y`. * `open Some.Namespace.Path hiding def1 def2` opens all non-protected names in `Some.Namespace.Path` except `def1` and `def2`. * `open Some.Namespace.Path (def1 def2)` only makes `Some.Namespace.Path.def1` and `Some.Namespace.Path.def2` available without the full prefix, so `Some.Namespace.Path.def3` would be unaffected. This works even if `def1` and `def2` are `protected`. * `open Some.Namespace.Path renaming def1 → def1', def2 → def2'` same as `open Some.Namespace.Path (def1 def2)` but `def1`/`def2`'s names are changed to `def1'`/`def2'`. This works even if `def1` and `def2` are `protected`. * `open scoped Some.Namespace.Path1 Some.Namespace.Path2` **only** opens [scoped instances], notations, and attributes from `Namespace1` and `Namespace2`; it does **not** make any other name available. * `open <any of the open shapes above> in` makes the names `open`-ed visible only in the next command or expression. [scoped instance]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html#scoped-instances (Scoped instances in Theorem Proving in Lean) ## Examples ```lean /-- SKI combinators https://en.wikipedia.org/wiki/SKI_combinator_calculus -/ namespace Combinator.Calculus def I (a : α) : α := a def K (a : α) : β → α := fun _ => a def S (x : α → β → γ) (y : α → β) (z : α) : γ := x z (y z) end Combinator.Calculus section -- open everything under `Combinator.Calculus`, *i.e.* `I`, `K` and `S`, -- until the section ends open Combinator.Calculus theorem SKx_eq_K : S K x = I := rfl end -- open everything under `Combinator.Calculus` only for the next command (the next `theorem`, here) open Combinator.Calculus in theorem SKx_eq_K' : S K x = I := rfl section -- open only `S` and `K` under `Combinator.Calculus` open Combinator.Calculus (S K) theorem SKxy_eq_y : S K x y = y := rfl -- `I` is not in scope, we have to use its full path theorem SKxy_eq_Iy : S K x y = Combinator.Calculus.I y := rfl end section open Combinator.Calculus renaming I → identity, K → konstant #check identity #check konstant end section open Combinator.Calculus hiding S #check I #check K end section namespace Demo inductive MyType | val namespace N1 scoped infix:68 " ≋ " => BEq.beq scoped instance : BEq MyType where beq _ _ := true def Alias := MyType end N1 end Demo -- bring `≋` and the instance in scope, but not `Alias` open scoped Demo.N1 #check Demo.MyType.val == Demo.MyType.val #check Demo.MyType.val ≋ Demo.MyType.val -- #check Alias -- unknown identifier 'Alias' end ```open are reverted.

#eval Unknown identifier `cupcake`cupcake

Unknown identifier `cupcake`

6.2.2. Section Variables🔗

Section variables are parameters that are automatically added to declarations that mention them. This occurs whether or not the option autoImplicit is true. Section variables may be implicit, strict implicit, or explicit; instance implicit section variables are treated specially.

When the name of a section variable is encountered in a non-theorem declaration, it is added as a parameter. Any instance implicit section variables that mention the variable are also added. If any of the variables that were added depend on other variables, then those variables are added as well; this process is iterated until no more dependencies remain. All section variables are added in the order in which they are declared, before all other parameters. Section variables are added only when they occur in the statement of a theorem. Otherwise, modifying the proof of a theorem could change its statement if the proof term made use of a section variable.

Variables are declared using the Lean.Parser.Command.variable : commandDeclares one or more typed variables, or modifies whether already-declared variables are implicit. Introduces variables that can be used in definitions within the same `namespace` or `section` block. When a definition mentions a variable, Lean will add it as an argument of the definition. This is useful in particular when writing many definitions that have parameters in common (see below for an example). Variable declarations have the same flexibility as regular function parameters. In particular they can be [explicit, implicit][binder docs], or [instance implicit][tpil classes] (in which case they can be anonymous). This can be changed, for instance one can turn explicit variable `x` into an implicit one with `variable {x}`. Note that currently, you should avoid changing how variables are bound and declare new variables at the same time; see [issue 2789] for more on this topic. In *theorem bodies* (i.e. proofs), variables are not included based on usage in order to ensure that changes to the proof cannot change the statement of the overall theorem. Instead, variables are only available to the proof if they have been mentioned in the theorem header or in an `include` command or are instance implicit and depend only on such variables. See [*Variables and Sections* from Theorem Proving in Lean][tpil vars] for a more detailed discussion. [tpil vars]: https://lean-lang.org/theorem_proving_in_lean4/dependent_type_theory.html#variables-and-sections (Variables and Sections on Theorem Proving in Lean) [tpil classes]: https://lean-lang.org/theorem_proving_in_lean4/type_classes.html (Type classes on Theorem Proving in Lean) [binder docs]: https://leanprover-community.github.io/mathlib4_docs/Lean/Expr.html#Lean.BinderInfo (Documentation for the BinderInfo type) [issue 2789]: https://github.com/leanprover/lean4/issues/2789 (Issue 2789 on github) ## Examples ```lean section variable {α : Type u} -- implicit (a : α) -- explicit [instBEq : BEq α] -- instance implicit, named [Hashable α] -- instance implicit, anonymous def isEqual (b : α) : Bool := a == b #check isEqual -- isEqual.{u} {α : Type u} (a : α) [instBEq : BEq α] (b : α) : Bool variable {a} -- `a` is implicit now def eqComm {b : α} := a == b ↔ b == a #check eqComm -- eqComm.{u} {α : Type u} {a : α} [instBEq : BEq α] {b : α} : Prop end ``` The following shows a typical use of `variable` to factor out definition arguments: ```lean variable (Src : Type) structure Logger where trace : List (Src × String) #check Logger -- Logger (Src : Type) : Type namespace Logger -- switch `Src : Type` to be implicit until the `end Logger` variable {Src} def empty : Logger Src where trace := [] #check empty -- Logger.empty {Src : Type} : Logger Src variable (log : Logger Src) def len := log.trace.length #check len -- Logger.len {Src : Type} (log : Logger Src) : Nat variable (src : Src) [BEq Src] -- at this point all of `log`, `src`, `Src` and the `BEq` instance can all become arguments def filterSrc := log.trace.filterMap fun (src', str') => if src' == src then some str' else none #check filterSrc -- Logger.filterSrc {Src : Type} (log : Logger Src) (src : Src) [inst✝ : BEq Src] : List String def lenSrc := log.filterSrc src |>.length #check lenSrc -- Logger.lenSrc {Src : Type} (log : Logger Src) (src : Src) [inst✝ : BEq Src] : Nat end Logger ``` The following example demonstrates availability of variables in proofs: ```lean variable {α : Type} -- available in the proof as indirectly mentioned through `a` [ToString α] -- available in the proof as `α` is included (a : α) -- available in the proof as mentioned in the header {β : Type} -- not available in the proof [ToString β] -- not available in the proof theorem ex : a = a := rfl ``` After elaboration of the proof, the following warning will be generated to highlight the unused hypothesis: ``` included section variable '[ToString α]' is not used in 'ex', consider excluding it ``` In such cases, the offending variable declaration should be moved down or into a section so that only theorems that do depend on it follow it until the end of the section.variable command.

syntaxVariable Declarations

command ::= ...
    | Declares one or more typed variables, or modifies whether already-declared variables are
  implicit.

Introduces variables that can be used in definitions within the same `namespace` or `section` block.
When a definition mentions a variable, Lean will add it as an argument of the definition. This is
useful in particular when writing many definitions that have parameters in common (see below for an
example).

Variable declarations have the same flexibility as regular function parameters. In particular they
can be [explicit, implicit][binder docs], or [instance implicit][tpil classes] (in which case they
can be anonymous). This can be changed, for instance one can turn explicit variable `x` into an
implicit one with `variable {x}`. Note that currently, you should avoid changing how variables are
bound and declare new variables at the same time; see [issue 2789] for more on this topic.

In *theorem bodies* (i.e. proofs), variables are not included based on usage in order to ensure that
changes to the proof cannot change the statement of the overall theorem. Instead, variables are only
available to the proof if they have been mentioned in the theorem header or in an `include` command
or are instance implicit and depend only on such variables.

See [*Variables and Sections* from Theorem Proving in Lean][tpil vars] for a more detailed
discussion.

[tpil vars]:
https://lean-lang.org/theorem_proving_in_lean4/dependent_type_theory.html#variables-and-sections
(Variables and Sections on Theorem Proving in Lean) [tpil classes]:
https://lean-lang.org/theorem_proving_in_lean4/type_classes.html (Type classes on Theorem Proving in
Lean) [binder docs]:
https://leanprover-community.github.io/mathlib4_docs/Lean/Expr.html#Lean.BinderInfo (Documentation
for the BinderInfo type) [issue 2789]: https://github.com/leanprover/lean4/issues/2789 (Issue 2789
on github)

## Examples

```lean
section
  variable
    {α : Type u}      -- implicit
    (a : α)           -- explicit
    [instBEq : BEq α] -- instance implicit, named
    [Hashable α]      -- instance implicit, anonymous

  def isEqual (b : α) : Bool :=
    a == b

  #check isEqual
  -- isEqual.{u} {α : Type u} (a : α) [instBEq : BEq α] (b : α) : Bool

  variable
    {a} -- `a` is implicit now

  def eqComm {b : α} := a == b ↔ b == a

  #check eqComm
  -- eqComm.{u} {α : Type u} {a : α} [instBEq : BEq α] {b : α} : Prop
end
```

The following shows a typical use of `variable` to factor out definition arguments:

```lean
variable (Src : Type)

structure Logger where
  trace : List (Src × String)
#check Logger
-- Logger (Src : Type) : Type

namespace Logger
  -- switch `Src : Type` to be implicit until the `end Logger`
  variable {Src}

  def empty : Logger Src where
    trace := []
  #check empty
  -- Logger.empty {Src : Type} : Logger Src

  variable (log : Logger Src)

  def len :=
    log.trace.length
  #check len
  -- Logger.len {Src : Type} (log : Logger Src) : Nat

  variable (src : Src) [BEq Src]

  -- at this point all of `log`, `src`, `Src` and the `BEq` instance can all become arguments

  def filterSrc :=
    log.trace.filterMap
      fun (src', str') => if src' == src then some str' else none
  #check filterSrc
  -- Logger.filterSrc {Src : Type} (log : Logger Src) (src : Src) [inst✝ : BEq Src] : List String

  def lenSrc :=
    log.filterSrc src |>.length
  #check lenSrc
  -- Logger.lenSrc {Src : Type} (log : Logger Src) (src : Src) [inst✝ : BEq Src] : Nat
end Logger
```

The following example demonstrates availability of variables in proofs:
```lean
variable
  {α : Type}    -- available in the proof as indirectly mentioned through `a`
  [ToString α]  -- available in the proof as `α` is included
  (a : α)       -- available in the proof as mentioned in the header
  {β : Type}    -- not available in the proof
  [ToString β]  -- not available in the proof

theorem ex : a = a := rfl
```
After elaboration of the proof, the following warning will be generated to highlight the unused
hypothesis:
```
included section variable '[ToString α]' is not used in 'ex', consider excluding it
```
In such cases, the offending variable declaration should be moved down or into a section so that
only theorems that do depend on it follow it until the end of the section.
variable bracketedBinder bracketedBinder*

The bracketed binders allowed after variable match the syntax used in definition headers.

Section Variables

In this section, automatic implicit parameters are disabled, but a number of section variables are defined.

section
set_option autoImplicit false
universe u
variable {α : Type u} (xs : List α) [Zero α] [Add α]

Because automatic implicit parameters are disabled, the following definition fails:

def addAll (lst : List Unknown identifier `β`β) : Unknown identifier `β`β :=
  lst.foldr (init := 0) (·+ ·)

Unknown identifier `β`

On the other hand, not even xs needs to be written directly in the definition:

def addAll :=
  xs.foldr (init := 0) (·+ ·)

To add a section variable to a theorem even if it is not explicitly mentioned in the statement, mark the variable with the Lean.Parser.Command.include : command`include eeny meeny` instructs Lean to include the section `variable`s `eeny` and `meeny` in all theorems in the remainder of the current section, differing from the default behavior of conditionally including variables based on use in the theorem header. Other commands are not affected. `include` is usually followed by `in theorem ...` to limit the inclusion to the subsequent declaration.include command. All variables marked for inclusion are added to all theorems. The Lean.Parser.Command.omit : command`omit` instructs Lean to not include a variable previously `include`d. Apart from variable names, it can also refer to typeclass instance variables by type using the syntax `omit [TypeOfInst]`, in which case all instance variables that unify with the given type are omitted. `omit` should usually only be used in conjunction with `in` in order to keep the section structure simple.omit command removes the inclusion mark from a variable; it's typically a good idea to use it with Lean.Parser.Command.in : commandin.

Included and Omitted Section Variables

This section's variables include a predicate as well as everything needed to prove that it holds universally, along with a useless extra assumption.

section
variable {p : Nat → Prop}
variable (pZero : p 0) (pStep : ∀ n, p n → p (n + 1))
variable (pFifteen : p 15)

However, only p is added to this theorem's assumptions, so it cannot be proved.

theorem p_all : ∀ n, p n := unsolved goals
zerop:Nat → Prop⊢ p 0

succp:Nat → Propn✝:Nata✝:p n✝⊢ p (n✝ + 1)byp:Nat → Prop⊢ ∀ (n : Nat), p n
  intro np:Nat → Propn:Nat⊢ p n
  induction nzerop:Nat → Prop⊢ p 0succp:Nat → Propn✝:Nata✝:p n✝⊢ p (n✝ + 1)

The Lean.Parser.Command.include : command`include eeny meeny` instructs Lean to include the section `variable`s `eeny` and `meeny` in all theorems in the remainder of the current section, differing from the default behavior of conditionally including variables based on use in the theorem header. Other commands are not affected. `include` is usually followed by `in theorem ...` to limit the inclusion to the subsequent declaration.include command causes the additional assumptions to be added unconditionally:

include pZero pStep pFifteen

automatically included section variable(s) unused in theorem `p_all`:
  pFifteen
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit pFifteen in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`theorem p_all : ∀ n, p n := byp✝:Nat → ProppFifteen:p✝ 15p:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)⊢ ∀ (n : Nat), p n
  intro np✝:Nat → ProppFifteen:p✝ 15p:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)n:Nat⊢ p n
  induction nzerop✝:Nat → ProppFifteen:p✝ 15p:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)⊢ p 0succp✝:Nat → ProppFifteen:p✝ 15p:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)n✝:Nata✝:p n✝⊢ p (n✝ + 1) <;>zerop✝:Nat → ProppFifteen:p✝ 15p:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)⊢ p 0succp✝:Nat → ProppFifteen:p✝ 15p:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)n✝:Nata✝:p n✝⊢ p (n✝ + 1) simp [*]All goals completed! 🐙

Because the spurious assumption pFifteen was inserted, Lean issues a warning:

automatically included section variable(s) unused in theorem `p_all`:
  pFifteen
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit pFifteen in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`

This can be avoided by using Lean.Parser.Command.omit : command`omit` instructs Lean to not include a variable previously `include`d. Apart from variable names, it can also refer to typeclass instance variables by type using the syntax `omit [TypeOfInst]`, in which case all instance variables that unify with the given type are omitted. `omit` should usually only be used in conjunction with `in` in order to keep the section structure simple.omitto remove pFifteen:

include pZero pStep pFifteen

omit pFifteen in
theorem p_all : ∀ n, p n := byp:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)⊢ ∀ (n : Nat), p n
  intro np:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)n:Nat⊢ p n
  induction nzerop:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)⊢ p 0succp:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)n✝:Nata✝:p n✝⊢ p (n✝ + 1) <;>zerop:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)⊢ p 0succp:Nat → ProppZero:p 0pStep:∀ (n : Nat), p n → p (n + 1)n✝:Nata✝:p n✝⊢ p (n✝ + 1) simp [*]All goals completed! 🐙

end