Complements of subtypes

Content created by Fredrik Bakke, Jonathan Prieto-Cubides and Egbert Rijke.

Created on 2022-09-12.
Last modified on 2023-10-20.

module foundation.complements-subtypes where
Imports 
open import foundation.decidable-propositions
open import foundation.decidable-subtypes
open import foundation.full-subtypes
open import foundation.negation
open import foundation.propositional-truncations
open import foundation.unions-subtypes
open import foundation.universe-levels

open import foundation-core.function-types
open import foundation-core.subtypes

Idea

The complement of a subtype P of A consists of the elements that are not in P.

Definition

Complements of subtypes

complement-subtype :
  {l1 l2 : Level} {A : UU l1} → subtype l2 A → subtype l2 A
complement-subtype P x = neg-Prop (P x)

Complements of decidable subtypes

complement-decidable-subtype :
  {l1 l2 : Level} {A : UU l1} → decidable-subtype l2 A → decidable-subtype l2 A
complement-decidable-subtype P x = neg-Decidable-Prop (P x)

Properties

The union of a subtype P with its complement is the full subtype if and only if P is a decidable subtype

module _
  {l1 l2 : Level} {A : UU l1}
  where

  is-full-union-subtype-complement-subtype :
    (P : subtype l2 A) → is-decidable-subtype P →
    is-full-subtype (union-subtype P (complement-subtype P))
  is-full-union-subtype-complement-subtype P d x =
    unit-trunc-Prop (d x)

  is-decidable-subtype-is-full-union-subtype-complement-subtype :
    (P : subtype l2 A) →
    is-full-subtype (union-subtype P (complement-subtype P)) →
    is-decidable-subtype P
  is-decidable-subtype-is-full-union-subtype-complement-subtype P H x =
    apply-universal-property-trunc-Prop
      ( H x)
      ( is-decidable-Prop (P x))
      ( id)

  is-full-union-subtype-complement-decidable-subtype :
    (P : decidable-subtype l2 A) →
    is-full-decidable-subtype
      ( union-decidable-subtype P (complement-decidable-subtype P))
  is-full-union-subtype-complement-decidable-subtype P =
    is-full-union-subtype-complement-subtype
      ( subtype-decidable-subtype P)
      ( is-decidable-decidable-subtype P)

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