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module Types.product where

open import Lang.dataStructures using (
  Bool; true; false;
;; singleton;; List;
  one; two; three; four; five; six; seven; eight; nine; ten; zero; succ;
  _::_; [])

open import Agda.Primitive using (Level; __; lsuc; lzero)

Product types

A cartesian product of two types A and B can be defined as a pair of objects (a, b), where aA, and bB.

data _××_ (A B : Set) : Set where
  _,,_  : A  B  A ×× B

infixr 4 _××_

Cartesian products can be constructed as:

oneTwo = one ,, two

oneTwoThree = one ,, (two ,, three)

Construction

While the above definition of cartesian products is intuitive, it’s algebraic definition captures the most abstract aspect of a product:

A cartesian product, in set theoretic language, for two sets A and B is defined as:

A × B = { (a , b) | a ∈ A ~and~ b ∈ B }

In type theory, we look at another way of defining product types, just by looking at them as objects in themselves: For an object X, we call X a product type if:

  1. There exists two functions, lets call them proj₁ and proj₂ such that they can be used to “extract” the contents of the product X:
proj₁ : {L R : Set}  (L × R)  L
(l × r) = l

proj₂ : {L R : Set}  (L × R)  R
(l × r) = r
  1. If there is any another object A, such that the functions proj₁ₐ and proj₂ₐ satisfied the above condition for A, then there exists a function, fₐ₀ such that:
fₐ₀ : A  X

Note: The above is pseudo-code, the proof is below.

The second condition is the condition for the unique-ness of the product, i.e. for all objects having projections to left and right, there is one through which all projections go through. This object through which we can route all other similar objects is called the “product”. We call this one object the “Product” This is better visualized in this diagram:

Figure 1: Product

Dependent Pair Types or Σ-types

A Dependent type is a type whose definition depends on a value. A dependent pair type is a product type whose second type depends on the first. They are of the form (x : A, B(x)). The notation in type theory looks like this for binary dependent pairs:

\[ \sum_{x : A} B(x) \]

with ternary dependent pair types being represented as:

\[ \sum_{x : A} \sum_{y : B(x)} C(y) \]

and so on.

The record type is a special syntax for representing dependent or Σ (sigma) types. They provide some syntactic sugars like constructors:

record Σ {a b} (A : Set a) (B : A  Set b) : Set (a ⊔ b) where
  constructor _,_
  field
    fst : A
    snd : B fst

open Σ public

infixr 4 _,_

For constructing and studying algebraic structures, a pattern generally followed is: 1. Choose a number of sets of objects. 2. Choose a number of binary operations. 3. Select a bunch properties of the binary operations that they have to adhere to. 4. Combine objects, operations and their properties to form higher mathematical objects.

A combination of a bunch of objects, operations and their properties can be represented as a product type, and is where records are extensively used. Example: if, say Prop₁ and Prop₂ are two properties, an object that satisfies both is a record with both properties as fields:

data prop1 : Set where
data prop2 : Set where

record Satisfies (x : prop1)(y : prop2) : Set where
  field
    p1 : prop1
    p2 : prop2

We extensively use records where we use this pattern.

Utils

Product constructor

_×_ :  {a b} (A : Set a) (B : Set b)  Set (a ⊔ b)
A × B = Σ A  x  B)

Application of a product

Apply a pair of functions to a pair of objects.

<_,_> :  {a b c} {A : Set a} {B : A  Set b} {C :  {x}  B x  Set c}
        (f : (x : A)  B x)
         ((x : A)  C (f x))
         ((x : A)  Σ (B x) C)
< f , g > x = (f x , g x)

Map

Mapping a pair of functions f and g over a product:

map :  {a b p q} {A : Set a} {B : Set b} {P : A  Set p} {Q : B  Set q}
         (f : A  B)
         (∀ {x}  P x  Q (f x))
         Σ A P
         Σ B Q
map f g (x , y) = (f x , g y)

Swap

swap :  {a b} {A : Set a} {B : Set b}  A × B  B × A
swap (x , y) = (y , x)

Co-product types

A “disjoint union” 𝕌 of X and Y has the property that every element of 𝕌 either maps to an element of X or Y but not both. Co-products, also called as “sum” types can be thought of as a disjoint union of two objects.

Mathematically, an object X ⋃ Y is a co-product of objects X and Y if,

  1. There exists two functions inj₁ and inj₂ such that:
inj₁ : {A B : Set}  A  (AB)
a = (a ∪ b)

inj₂ : {A B : Set}  B  (AB)
b = (a ∪ b)
  1. If there is any another object A, such that the functions inj₁ₐ and inj₂ₐ satisfied the above condition, then there exists a function, fₐ₀ such that:
fₐ₀ : X  A
data __ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
  inj₁ : (x : A)  A ∪ B
  inj₂ : (y : B)  A ∪ B

Co-product types are similar to product types, except with reversed arrows (they are “dual” to products):

Figure 3: Coproducts

Maybe

Just like the cartesian product is the representative type of a product, the Maybe type fills that role for the co-product. This happens to be a very popular datatype in functional programming languages like haskell Maybe, scala Option etc and is part of a widely used pattern for error handling. The Maybe type is a disjoint union of something (a type) or nothing (or an error type). These types can be used to encapsulate behavior of functions that either return a value or an error.

data Maybe {a} (A : Set a) : Set a where
  just    : (x : A)  Maybe A
  nothing : Maybe A

API

Eliminator

maybe :  {a b} {A : Set a} {B : Maybe A  Set b}
         ((x : A)  B (just x))
         B nothing
         (x : Maybe A)
         B x
maybe j n (just x) = j x
maybe j n nothing  = n

Map

A Maybe is also a structure which can be map-ed over:

smap :  {a b} {A : Set a} {B : Set b}  (A  B)  Maybe A  Maybe B
smap f (just x) = just (f x)
smap f nothing  = nothing

Zip

zip :  {a b c} {A : Set a} {B : Set b} {C : Set c}
         Maybe A
         Maybe B
         Maybe (A × B)
zip (just a) (just b) = just (a , b)
zip _ _ = nothing

Dependent Function Types / Π-types