9. Structures and Records

We have seen that Lean’s foundational system includes inductive types. We have, moreover, noted that it is a remarkable fact that it is possible to construct a substantial edifice of mathematics based on nothing more than the type universes, Pi types, and inductive types; everything else follows from those. The Lean standard library contains many instances of inductive types (e.g., nat, prod, list), and even the logical connectives are defined using inductive types.

Remember that a non-recursive inductive type that contains only one constructor is called a structure or record. The product type is a structure, as is the dependent product type, that is, the Sigma type. In general, whenever we define a structure S, we usually define projection functions that allow us to “destruct” each instance of S and retrieve the values that are stored in its fields. The functions prod.pr1 and prod.pr2, which return the first and second elements of a pair, are examples of such projections.

When writing programs or formalizing mathematics, it is not uncommon to define structures containing many fields. The structure command, available in Lean, provides infrastructure to support this process. When we define a structure using this command, Lean automatically generates all the projection functions. The structure command also allows us to define new structures based on previously defined ones. Moreover, Lean provides convenient notation for defining instances of a given structure.

9.1. Declaring Structures

The structure command is essentially a “front end” for defining inductive data types. Every structure declaration introduces a namespace with the same name. The general form is as follows:

structure <name> <parameters> <parent-structures> : Sort u :=
  <constructor> :: <fields>

Most parts are optional. Here is an example:

structure point (α : Type) :=
mk :: (x : α) (y : α)

Values of type point are created using point.mk a b, and the fields of a point p are accessed using point.x p and point.y p. The structure command also generates useful recursors and theorems. Here are some of the constructions generated for the declaration above.

#check point              -- a Type
#check point.rec_on       -- the eliminator
#check point.x            -- a projection / field accessor
#check point.y            -- a projection / field accessor

You can obtain the complete list of generated constructions using the command #print prefix.

#print prefix point

Here are some simple theorems and expressions that use the generated constructions. As usual, you can avoid the prefix point by using the command open point.

#reduce point.x (point.mk 10 20)
#reduce point.y (point.mk 10 20)

open point

example (α : Type) (a b : α) : x (mk a b) = a :=
rfl

example (α : Type) (a b : α) : y (mk a b) = b :=
rfl

Given p : point nat, the notation p.x is shorthand for point.x p. This provides a convenient way of accessing the fields of a structure.

def p := point.mk 10 20

#check p.x  -- nat
#reduce p.x  -- 10
#reduce p.y  -- 20

If the constructor is not provided, then a constructor is named mk by default.

structure prod (α : Type) (β : Type) :=
(pr1 : α) (pr2 : β)

#check prod.mk

The dot notation is convenient not just for accessing the projections of a record, but also for applying functions defined in a namespace with the same name. Recall from Section 3.3.1 that if p has type point, the expression p.foo is interpreted as point.foo p, assuming that the first non-implicit argument to foo has type point. The expression p.add q is therefore shorthand for point.add p q in the example below.

structure point (α : Type) :=
mk :: (x : α) (y : α)

namespace point

def add (p q : point ) := mk (p.x + q.x) (p.y + q.y)

end point

def p : point  := point.mk 1 2
def q : point  := point.mk 3 4

#reduce p.add q  -- {x := 4, y := 6}

In the next chapter, you will learn how to define a function like add so that it works generically for elements of point α rather than just point , assuming α has an associated addition operation.

More generally, given an expression p.foo x y z, Lean will insert p at the first non-implicit argument to foo of type point. For example, with the definition of scalar multiplication below, p.smul 3 is interpreted as point.smul 3 p.

structure point (α : Type) :=
mk :: (x : α) (y : α)

def point.smul (n : ) (p : point ) :=
point.mk (n * p.x) (n * p.y)

def p : point  := point.mk 1 2

#reduce p.smul 3  -- {x := 3, y := 6}

It is common to use a similar trick with the list.map function, which takes a list as its second non-implicit argument:

#check @list.map
-- Π {α : Type u_1} {β : Type u_2}, (α → β) → list α → list β

def l : list nat := [1, 2, 3]
def f : nat  nat := λ x, x * x

#eval l.map f  -- [1, 4, 9]

Here l.map f is interpreted as list.map f l.

If you have a structure definition that depends on a type, you can make it polymorphic over universe levels using a previously declared universe variable, declaring a universe variable on the fly, or using an underscore:

universe u

structure point (α : Type u) :=
mk :: (x : α) (y : α)

structure {v} point2 (α : Type v) :=
mk :: (x : α) (y : α)

structure point3 (α : Type _) :=
mk :: (x : α) (y : α)

#check @point
#check @point2
#check @point3

The three variations have the same net effect. The annotations in the next example force the parameters α and β to be types from the same universe, and set the return type to also be in the same universe.

structure {u} prod (α : Type u) (β : Type u) :
  Type (max 1 u) :=
(pr1 : α) (pr2 : β)

set_option pp.universes true
#check prod.mk

The set_option command above instructs Lean to display the universe levels. Here we have used max 1 l as the resultant universe level to ensure the universe level is never 0 even when the parameter α and β are propositions. Recall that in Lean, Type 0 is Prop, which is impredicative and proof irrelevant.

We can use the anonymous constructor notation to build structure values whenever the expected type is known.

structure {u} prod (α : Type u) (β : Type u) :
  Type (max 1 u) :=
(pr1 : α) (pr2 : β)

example : prod nat nat :=
1, 2

#check (⟨1, 2 : prod nat nat)

9.2. Objects

We have been using constructors to create elements of a structure type. For structures containing many fields, this is often inconvenient, because we have to remember the order in which the fields were defined. Lean therefore provides the following alternative notations for defining elements of a structure type.

{ structure-name . (<field-name> := <expr>)* }
or
{ (<field-name> := <expr>)* }

The prefix structure-name . can be omitted whenever the name of the structure can be inferred from the expected type. For example, we use this notation to define “points.” The order that the fields are specified does not matter, so all the expressions below define the same point.

structure point (α : Type) :=
mk :: (x : α) (y : α)

#check { point . x := 10, y := 20 }  -- point ℕ
#check { point . y := 20, x := 10 }
#check ({x := 10, y := 20} : point nat)

example : point nat :=
{ y := 20, x := 10 }

If the value of a field is not specified, Lean tries to infer it. If the unspecified fields cannot be inferred, Lean signs an error indicating the corresponding placeholder could not be synthesized.

structure my_struct :=
mk :: {α : Type} {β : Type} (a : α) (b : β)

#check { my_struct . a := 10, b := true }

Record update is another common operation which amounts to creating a new record object by modifying the value of one or more fields in an old one. Lean allows you to specify that unassigned fields in the specification of a record should be taken from a previous defined record object r by adding the annotation ..r after the field assignments. If more than one record object is provided, then they are visited in order until Lean finds one the contains the unspecified field. Lean raises an error if any of the field names remain unspecified after all the objects are visited.

structure point (α : Type) :=
mk :: (x : α) (y : α)

def p : point nat :=
{x := 1, y := 2}

#reduce {y := 3, ..p}  -- {x := 1, y := 3}
#reduce {x := 4, ..p}  -- {x := 4, y := 2}

structure point3 (α : Type) :=
mk :: (x : α) (y : α) (z : α)

def q : point3 nat :=
{x := 5, y := 5, z := 5}

def r : point3 nat := {x := 6, ..p, ..q}

#print r  -- {x := 6, y := p.y, z := q.z}
#reduce r  -- {x := 6, y := 2, z := 5}

9.3. Inheritance

We can extend existing structures by adding new fields. This feature allow us to simulate a form of inheritance.

structure point (α : Type) :=
mk :: (x : α) (y : α)

inductive color
| red | green | blue

structure color_point (α : Type) extends point α :=
mk :: (c : color)

In the next example, we define a structure using multiple inheritance, and then define an object using objects of the parent structures.

structure point (α : Type) :=
(x : α) (y : α) (z : α)

structure rgb_val :=
(red : nat) (green : nat) (blue : nat)

structure red_green_point (α : Type) extends point α, rgb_val :=
(no_blue : blue = 0)

def p   : point nat := {x := 10, y := 10, z := 20}
def rgp : red_green_point nat :=
{red := 200, green := 40, blue := 0, no_blue := rfl, ..p}

example : rgp.x   = 10 := rfl
example : rgp.red = 200 := rfl