module Data.Power where

# Power Setsπ

The **power set** of a type
$X$
is the collection of all maps from
$X$
into the universe of propositional
types. Since the universe of all
$n$-types
is a
$(n+1)$-type
(by `n-Type-is-hlevel`

), and function
types have the same h-level as their codomain (by `fun-is-hlevel`

), the power set of a type
$X$
is always a set. We denote the power set of
$X$
by
$\mathbb{P}(X)$.

β : Type β β Type β β X = X β Ξ© β-is-set : is-set (β X) β-is-set = hlevel 2

The **membership** relation is defined by applying the
predicate and projecting the underlying type of the proposition: We say
that
$x$
is an element of
$P$
if
$P(x)$
is inhabited.

_ : β {x : X} {P : β X} β x β P β‘ β£ P x β£ _ = refl

The **subset** relation is defined as is done
traditionally: If
$x \in X$
implies
$x \in Y$,
for
$X, Y : \mathbb{P}(T)$,
then
$X \subseteq Y$.

_β_ : β X β β X β Type _ X β Y = β x β x β X β x β Y

By function and propositional extensionality, two subsets of $X$ are equal when they contain the same elements, i.e., they assign identical propositions to each inhabitant of $X$.

β-ext : {A B : β X} β A β B β B β A β A β‘ B β-ext {A = A} {B = B} AβB BβA = funext Ξ» x β Ξ©-ua (AβB x) (BβA x)

## Lattice Structureπ

The type
$\mathbb{P}(X)$
has a lattice structure, with the order given by subset
inclusion. We call the meets **intersections** and the
joins **unions**.

maximal : β X maximal _ = el β€ hlevel! minimal : β X minimal _ = el (Lift _ β₯) hlevel! _β©_ : β X β β X β β X (A β© B) x = el (x β A Γ x β B) hlevel!

Note that in the definition of union, we must truncate the coproduct, since there is nothing which guarantees that A and B are disjoint subsets.

_βͺ_ : β X β β X β β X (A βͺ B) x = elΞ© (x β A β x β B) infixr 22 _β©_ infixr 21 _βͺ_