structures_and_records.md

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, dependent arrow 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.

Recall 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 (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.fst and prod.snd, 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.

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> where
      <constructor> :: <fields>

Most parts are optional. Here is an example:

structure Point (α : Type u) where
  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 (but p.x and p.y also work, see below). The structure command also generates useful recursors and theorems. Here are some of the constructions generated for the declaration above.

# structure Point (α : Type u) where
#  mk :: (x : α) (y : α)
#check Point       -- a Type
#check @Point.rec  -- the eliminator
#check @Point.mk   -- the constructor
#check @Point.x    -- a projection
#check @Point.y    -- a projection

If the constructor name is not provided, then a constructor is named mk by default. You can also avoid the parentheses around field names if you add a line break between each field.

structure Point (α : Type u) where
  x : α
  y : α

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.

# structure Point (α : Type u) where
#  x : α
#  y : α
#eval Point.x (Point.mk 10 20)
#eval Point.y (Point.mk 10 20)

open Point

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

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

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

# structure Point (α : Type u) where
#  x : α
#  y : α
def p := Point.mk 10 20

#check p.x  -- Nat
#eval p.x   -- 10
#eval p.y   -- 20

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 the Conjunction section 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 u) where
  x : α
  y : α
  deriving Repr

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

def p : Point Nat := Point.mk 1 2
def q : Point Nat := Point.mk 3 4

#eval 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 Nat, assuming α has an associated addition operation.

More generally, given an expression p.foo x y z where p : Point, Lean will insert p at the first argument to Point.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 u) where
#  x : α
#  y : α
#  deriving Repr
def Point.smul (n : Nat) (p : Point Nat) :=
  Point.mk (n * p.x) (n * p.y)

def p : Point Nat := Point.mk 1 2

#eval 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

def xs : List Nat := [1, 2, 3]
def f : Nat → Nat := fun x => x * x

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

Here xs.map f is interpreted as List.map f xs.

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.

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

The suffix : structure-type 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 u) where
  x : α
  y : α

#check { x := 10, y := 20 : Point Nat }  -- Point ℕ
#check { y := 20, x := 10 : Point _ }
#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 flags an error indicating the corresponding placeholder could not be synthesized.

structure MyStruct where
    {α : Type u}
    {β : Type v}
    a : α
    b : β

#check { a := 10, b := true : MyStruct }

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 previously defined structure object s by adding the annotation s with before the field assignments. If more than one record object is provided, then they are visited in order until Lean finds one that 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 u) where
  x : α
  y : α
  deriving Repr

def p : Point Nat :=
  { x := 1, y := 2 }

#eval { p with y := 3 }  -- { x := 1, y := 3 }
#eval { p with x := 4 }  -- { x := 4, y := 2 }

structure Point3 (α : Type u) where
  x : α
  y : α
  z : α

def q : Point3 Nat :=
  { x := 5, y := 5, z := 5 }

def r : Point3 Nat :=
  { p, q with x := 6 }

example : r.x = 6 := rfl
example : r.y = 2 := rfl
example : r.z = 5 := rfl

Inheritance

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

structure Point (α : Type u) where
  x : α
  y : α

inductive Color where
  | red | green | blue

structure ColorPoint (α : Type u) extends Point α where
  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 u) where
  x : α
  y : α
  z : α

structure RGBValue where
  red : Nat
  green : Nat
  blue : Nat

structure RedGreenPoint (α : Type u) extends Point α, RGBValue where
  no_blue : blue = 0

def p : Point Nat :=
  { x := 10, y := 10, z := 20 }

def rgp : RedGreenPoint Nat :=
  { p with red := 200, green := 40, blue := 0, no_blue := rfl }

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