Dedekind real numbers

Content created by Fredrik Bakke, Egbert Rijke, Louis Wasserman, malarbol, Elif Uskuplu and Vojtěch Štěpančík.

Created on 2023-04-08.
Last modified on 2025-04-25.

module real-numbers.dedekind-real-numbers where

open import elementary-number-theory.inequality-rational-numbers
open import elementary-number-theory.rational-numbers
open import elementary-number-theory.strict-inequality-rational-numbers

open import foundation.action-on-identifications-functions
open import foundation.binary-transport
open import foundation.cartesian-product-types
open import foundation.complements-subtypes
open import foundation.conjunction
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.disjoint-subtypes
open import foundation.disjunction
open import foundation.embeddings
open import foundation.empty-types
open import foundation.equivalences
open import foundation.existential-quantification
open import foundation.function-types
open import foundation.functoriality-cartesian-product-types
open import foundation.functoriality-dependent-pair-types
open import foundation.identity-types
open import foundation.logical-equivalences
open import foundation.negation
open import foundation.powersets
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.sets
open import foundation.subtypes
open import foundation.transport-along-identifications
open import foundation.truncated-types
open import foundation.universal-quantification
open import foundation.universe-levels

open import foundation-core.truncation-levels

open import logic.functoriality-existential-quantification

open import real-numbers.lower-dedekind-real-numbers
open import real-numbers.upper-dedekind-real-numbers

Idea

A Dedekind real number^¶ x consists of a pair of cuts (L , U) of rational numbers, a lower Dedekind cut L and an upper Dedekind cut U.

A lower Dedekind cut L is a subtype of the rational numbers that is

1. Inhabited, meaning it has at least one element.
2. Rounded, meaning that q ∈ L if and only if there exists r, with q < r and r ∈ L.

An upper Dedekind cut is analogous, but must be rounded “in the other direction”: q ∈ U if and only if there is a p < q where p ∈ U.

These cuts present lower and upper bounds on the Dedekind real number, and must satisfy the conditions

1. Disjointness. The cuts lx and uy are disjoint subsets of ℚ.
2. Locatedness. If q < r then q is in the cut L or r is in the cut U.

The collection of all Dedekind real numbers^¶ is denoted ℝ. The Dedekind real numbers will be taken as the standard definition of the real numbers in the agda-unimath library.

Definition

Dedekind cuts

module _
  {l1 l2 : Level} (lx : lower-ℝ l1) (uy : upper-ℝ l2)
  where

  is-disjoint-prop-lower-upper-ℝ : Prop (l1 ⊔ l2)
  is-disjoint-prop-lower-upper-ℝ =
    disjoint-subtype-Prop (cut-lower-ℝ lx) (cut-upper-ℝ uy)

  is-disjoint-lower-upper-ℝ : UU (l1 ⊔ l2)
  is-disjoint-lower-upper-ℝ = type-Prop is-disjoint-prop-lower-upper-ℝ

  is-located-prop-lower-upper-ℝ : Prop (l1 ⊔ l2)
  is-located-prop-lower-upper-ℝ = ∀'
    ( ℚ)
    ( λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (cut-lower-ℝ lx q ∨ cut-upper-ℝ uy r)))

  is-located-lower-upper-ℝ : UU (l1 ⊔ l2)
  is-located-lower-upper-ℝ = type-Prop is-located-prop-lower-upper-ℝ

  is-dedekind-prop-lower-upper-ℝ : Prop (l1 ⊔ l2)
  is-dedekind-prop-lower-upper-ℝ =
    is-disjoint-prop-lower-upper-ℝ ∧ is-located-prop-lower-upper-ℝ

  is-dedekind-lower-upper-ℝ : UU (l1 ⊔ l2)
  is-dedekind-lower-upper-ℝ = type-Prop is-dedekind-prop-lower-upper-ℝ

The Dedekind real numbers

ℝ : (l : Level) → UU (lsuc l)
ℝ l = Σ (lower-ℝ l) (λ lx → Σ (upper-ℝ l) (is-dedekind-lower-upper-ℝ lx))

real-lower-upper-ℝ :
  {l : Level} →
  (lx : lower-ℝ l) (uy : upper-ℝ l) →
  is-disjoint-lower-upper-ℝ lx uy →
  is-located-lower-upper-ℝ lx uy →
  ℝ l
real-lower-upper-ℝ lx uy H K =
  lx , uy , H , K

module _
  {l : Level} (x : ℝ l)
  where

  lower-real-ℝ : lower-ℝ l
  lower-real-ℝ = pr1 x

  upper-real-ℝ : upper-ℝ l
  upper-real-ℝ = pr1 (pr2 x)

  lower-cut-ℝ : subtype l ℚ
  lower-cut-ℝ = cut-lower-ℝ lower-real-ℝ

  upper-cut-ℝ : subtype l ℚ
  upper-cut-ℝ = cut-upper-ℝ upper-real-ℝ

  is-in-lower-cut-ℝ : ℚ → UU l
  is-in-lower-cut-ℝ = is-in-subtype lower-cut-ℝ

  is-in-upper-cut-ℝ : ℚ → UU l
  is-in-upper-cut-ℝ = is-in-subtype upper-cut-ℝ

  is-dedekind-ℝ : is-dedekind-lower-upper-ℝ lower-real-ℝ upper-real-ℝ
  is-dedekind-ℝ = pr2 (pr2 x)

  is-inhabited-lower-cut-ℝ : exists ℚ lower-cut-ℝ
  is-inhabited-lower-cut-ℝ = is-inhabited-cut-lower-ℝ lower-real-ℝ

  is-inhabited-upper-cut-ℝ : exists ℚ upper-cut-ℝ
  is-inhabited-upper-cut-ℝ = is-inhabited-cut-upper-ℝ upper-real-ℝ

  is-rounded-lower-cut-ℝ :
    (q : ℚ) →
    is-in-lower-cut-ℝ q ↔
    exists ℚ (λ r → (le-ℚ-Prop q r) ∧ (lower-cut-ℝ r))
  is-rounded-lower-cut-ℝ = is-rounded-cut-lower-ℝ lower-real-ℝ

  is-rounded-upper-cut-ℝ :
    (r : ℚ) →
    is-in-upper-cut-ℝ r ↔
    exists ℚ (λ q → (le-ℚ-Prop q r) ∧ (upper-cut-ℝ q))
  is-rounded-upper-cut-ℝ = is-rounded-cut-upper-ℝ upper-real-ℝ

  is-disjoint-cut-ℝ : disjoint-subtype lower-cut-ℝ upper-cut-ℝ
  is-disjoint-cut-ℝ = pr1 (pr2 (pr2 x))

  is-located-lower-upper-cut-ℝ :
    (q r : ℚ) → le-ℚ q r →
    type-disjunction-Prop (lower-cut-ℝ q) (upper-cut-ℝ r)
  is-located-lower-upper-cut-ℝ = pr2 (pr2 (pr2 x))

  cut-ℝ : subtype l ℚ
  cut-ℝ q =
    coproduct-Prop
      ( lower-cut-ℝ q)
      ( upper-cut-ℝ q)
      ( ev-pair ( is-disjoint-cut-ℝ q))

  is-in-cut-ℝ : ℚ → UU l
  is-in-cut-ℝ = is-in-subtype cut-ℝ

Properties

The Dedekind real numbers form a set

abstract
  is-set-ℝ : (l : Level) → is-set (ℝ l)
  is-set-ℝ l =
    is-set-Σ
      ( is-set-lower-ℝ l)
      ( λ x →
        is-set-Σ
          ( is-set-upper-ℝ l)
          ( λ y →
            ( is-set-is-prop
              ( is-prop-type-Prop
                ( is-dedekind-prop-lower-upper-ℝ x y)))))

ℝ-Set : (l : Level) → Set (lsuc l)
ℝ-Set l = ℝ l , is-set-ℝ l

Properties of lower/upper Dedekind cuts

Lower and upper Dedekind cuts are closed under the standard ordering on the rationals

module _
  {l : Level} (x : ℝ l) (p q : ℚ)
  where

  le-lower-cut-ℝ :
    le-ℚ p q →
    is-in-lower-cut-ℝ x q →
    is-in-lower-cut-ℝ x p
  le-lower-cut-ℝ = is-in-cut-le-ℚ-lower-ℝ (lower-real-ℝ x) p q

  leq-lower-cut-ℝ :
    leq-ℚ p q →
    is-in-lower-cut-ℝ x q →
    is-in-lower-cut-ℝ x p
  leq-lower-cut-ℝ = is-in-cut-leq-ℚ-lower-ℝ (lower-real-ℝ x) p q

  le-upper-cut-ℝ :
    le-ℚ p q →
    is-in-upper-cut-ℝ x p →
    is-in-upper-cut-ℝ x q
  le-upper-cut-ℝ = is-in-cut-le-ℚ-upper-ℝ (upper-real-ℝ x) p q

  leq-upper-cut-ℝ :
    leq-ℚ p q →
    is-in-upper-cut-ℝ x p →
    is-in-upper-cut-ℝ x q
  leq-upper-cut-ℝ = is-in-cut-leq-ℚ-upper-ℝ (upper-real-ℝ x) p q

Elements of the lower cut are lower bounds of the upper cut

module _
  {l : Level} (x : ℝ l) (p q : ℚ)
  where

  le-lower-upper-cut-ℝ :
    is-in-lower-cut-ℝ x p →
    is-in-upper-cut-ℝ x q →
    le-ℚ p q
  le-lower-upper-cut-ℝ H H' =
    rec-coproduct
      ( id)
      ( λ I →
        ex-falso
          ( is-disjoint-cut-ℝ x p
              ( H , leq-upper-cut-ℝ x q p I H')))
      ( decide-le-leq-ℚ p q)

Characterisation of each cut by the other

The lower cut is the subtype of rationals bounded above by some element of the complement of the upper cut

module _
  {l : Level} (x : ℝ l)
  where

  is-lower-complement-upper-cut-ℝ-Prop : (p q : ℚ) → Prop l
  is-lower-complement-upper-cut-ℝ-Prop p q =
    ( le-ℚ-Prop p q) ∧ (¬' (upper-cut-ℝ x q))

  lower-complement-upper-cut-ℝ : subtype l ℚ
  lower-complement-upper-cut-ℝ p =
    ∃ ℚ (is-lower-complement-upper-cut-ℝ-Prop p)

module _
  {l : Level} (x : ℝ l)
  where

  subset-lower-cut-lower-complement-upper-cut-ℝ :
    lower-complement-upper-cut-ℝ x ⊆ lower-cut-ℝ x
  subset-lower-cut-lower-complement-upper-cut-ℝ p =
    elim-exists
      ( lower-cut-ℝ x p)
      ( λ q I →
        map-right-unit-law-disjunction-is-empty-Prop
          ( lower-cut-ℝ x p)
          ( upper-cut-ℝ x q)
          ( pr2 I)
          ( is-located-lower-upper-cut-ℝ x p q ( pr1 I)))

  subset-lower-complement-upper-cut-lower-cut-ℝ :
    lower-cut-ℝ x ⊆ lower-complement-upper-cut-ℝ x
  subset-lower-complement-upper-cut-lower-cut-ℝ p H =
    map-tot-exists
      ( λ q → map-product id (λ L U → is-disjoint-cut-ℝ x q (L , U)))
      ( pr1 (is-rounded-lower-cut-ℝ x p) H)

  eq-lower-cut-lower-complement-upper-cut-ℝ :
    lower-complement-upper-cut-ℝ x ＝ lower-cut-ℝ x
  eq-lower-cut-lower-complement-upper-cut-ℝ =
    antisymmetric-leq-subtype
      ( lower-complement-upper-cut-ℝ x)
      ( lower-cut-ℝ x)
      ( subset-lower-cut-lower-complement-upper-cut-ℝ)
      ( subset-lower-complement-upper-cut-lower-cut-ℝ)

The upper cut is the subtype of rationals bounded below by some element of the complement of the lower cut

module _
  {l : Level} (x : ℝ l)
  where

  is-upper-complement-lower-cut-ℝ-Prop : (q p : ℚ) → Prop l
  is-upper-complement-lower-cut-ℝ-Prop q p =
    (le-ℚ-Prop p q) ∧ (¬' (lower-cut-ℝ x p))

  upper-complement-lower-cut-ℝ : subtype l ℚ
  upper-complement-lower-cut-ℝ q =
    ∃ ℚ (is-upper-complement-lower-cut-ℝ-Prop q)

module _
  {l : Level} (x : ℝ l)
  where

  subset-upper-cut-upper-complement-lower-cut-ℝ :
    upper-complement-lower-cut-ℝ x ⊆ upper-cut-ℝ x
  subset-upper-cut-upper-complement-lower-cut-ℝ q =
    elim-exists
      ( upper-cut-ℝ x q)
      ( λ p I →
        map-left-unit-law-disjunction-is-empty-Prop
          ( lower-cut-ℝ x p)
          ( upper-cut-ℝ x q)
          ( pr2 I)
          ( is-located-lower-upper-cut-ℝ x p q ( pr1 I)))

  subset-upper-complement-lower-cut-upper-cut-ℝ :
    upper-cut-ℝ x ⊆ upper-complement-lower-cut-ℝ x
  subset-upper-complement-lower-cut-upper-cut-ℝ q H =
    map-tot-exists
      ( λ p → map-product id (λ U L → is-disjoint-cut-ℝ x p (L , U)))
      ( pr1 (is-rounded-upper-cut-ℝ x q) H)

  eq-upper-cut-upper-complement-lower-cut-ℝ :
    upper-complement-lower-cut-ℝ x ＝ upper-cut-ℝ x
  eq-upper-cut-upper-complement-lower-cut-ℝ =
    antisymmetric-leq-subtype
      ( upper-complement-lower-cut-ℝ x)
      ( upper-cut-ℝ x)
      ( subset-upper-cut-upper-complement-lower-cut-ℝ)
      ( subset-upper-complement-lower-cut-upper-cut-ℝ)

The lower/upper cut of a real determines the other

module _
  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
  where

  subset-lower-cut-upper-cut-ℝ :
    upper-cut-ℝ y ⊆ upper-cut-ℝ x → lower-cut-ℝ x ⊆ lower-cut-ℝ y
  subset-lower-cut-upper-cut-ℝ H =
    binary-tr
      ( _⊆_)
      ( eq-lower-cut-lower-complement-upper-cut-ℝ x)
      ( eq-lower-cut-lower-complement-upper-cut-ℝ y)
      ( λ p → map-tot-exists (λ q → tot (λ _ K → K ∘ H q)))

  subset-upper-cut-lower-cut-ℝ :
    lower-cut-ℝ x ⊆ lower-cut-ℝ y → upper-cut-ℝ y ⊆ upper-cut-ℝ x
  subset-upper-cut-lower-cut-ℝ H =
    binary-tr
      ( _⊆_)
      ( eq-upper-cut-upper-complement-lower-cut-ℝ y)
      ( eq-upper-cut-upper-complement-lower-cut-ℝ x)
      ( λ q → map-tot-exists (λ p → tot (λ _ K → K ∘ H p)))

module _
  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
  where

  sim-lower-cut-sim-upper-cut-ℝ :
    sim-subtype (upper-cut-ℝ x) (upper-cut-ℝ y) →
    sim-subtype (lower-cut-ℝ x) (lower-cut-ℝ y)
  pr1 (sim-lower-cut-sim-upper-cut-ℝ (_ , uy⊆ux)) =
    subset-lower-cut-upper-cut-ℝ x y uy⊆ux
  pr2 (sim-lower-cut-sim-upper-cut-ℝ (ux⊆uy , _)) =
    subset-lower-cut-upper-cut-ℝ y x ux⊆uy

  sim-upper-cut-sim-lower-cut-ℝ :
    sim-subtype (lower-cut-ℝ x) (lower-cut-ℝ y) →
    sim-subtype (upper-cut-ℝ x) (upper-cut-ℝ y)
  pr1 (sim-upper-cut-sim-lower-cut-ℝ (_ , ly⊆lx)) =
    subset-upper-cut-lower-cut-ℝ y x ly⊆lx
  pr2 (sim-upper-cut-sim-lower-cut-ℝ (lx⊆ly , _)) =
    subset-upper-cut-lower-cut-ℝ x y lx⊆ly

module _
  {l : Level} (x y : ℝ l)
  where

  eq-lower-cut-eq-upper-cut-ℝ :
    upper-cut-ℝ x ＝ upper-cut-ℝ y → lower-cut-ℝ x ＝ lower-cut-ℝ y
  eq-lower-cut-eq-upper-cut-ℝ H =
    eq-sim-subtype
      ( lower-cut-ℝ x)
      ( lower-cut-ℝ y)
      ( sim-lower-cut-sim-upper-cut-ℝ
        ( x)
        ( y)
        ( tr
          ( sim-subtype (upper-cut-ℝ x))
          ( H)
          ( refl-sim-subtype (upper-cut-ℝ x))))

  eq-lower-real-eq-upper-real-ℝ :
    upper-real-ℝ x ＝ upper-real-ℝ y → lower-real-ℝ x ＝ lower-real-ℝ y
  eq-lower-real-eq-upper-real-ℝ ux=uy =
    eq-eq-cut-lower-ℝ
      ( lower-real-ℝ x)
      ( lower-real-ℝ y)
      ( eq-lower-cut-eq-upper-cut-ℝ (ap cut-upper-ℝ ux=uy))

  eq-upper-cut-eq-lower-cut-ℝ :
    lower-cut-ℝ x ＝ lower-cut-ℝ y → upper-cut-ℝ x ＝ upper-cut-ℝ y
  eq-upper-cut-eq-lower-cut-ℝ H =
    antisymmetric-leq-subtype
      ( upper-cut-ℝ x)
      ( upper-cut-ℝ y)
      ( subset-upper-cut-lower-cut-ℝ y x
        ( pr2 ∘ has-same-elements-eq-subtype
          ( lower-cut-ℝ x)
          ( lower-cut-ℝ y)
          ( H)))
      ( subset-upper-cut-lower-cut-ℝ x y
        ( pr1 ∘ has-same-elements-eq-subtype
          ( lower-cut-ℝ x)
          ( lower-cut-ℝ y)
          ( H)))

  eq-upper-real-eq-lower-real-ℝ :
    lower-real-ℝ x ＝ lower-real-ℝ y → upper-real-ℝ x ＝ upper-real-ℝ y
  eq-upper-real-eq-lower-real-ℝ lx=ly =
    eq-eq-cut-upper-ℝ
      ( upper-real-ℝ x)
      ( upper-real-ℝ y)
      ( eq-upper-cut-eq-lower-cut-ℝ (ap cut-lower-ℝ lx=ly))

The map from a real number to its lower real is an embedding

module _
  {l : Level}
  (lx : lower-ℝ l)
  where

  is-dedekind-cut-lower-ℝ-Prop : Prop (lsuc l)
  pr1 is-dedekind-cut-lower-ℝ-Prop =
    Σ (upper-ℝ l) (is-dedekind-lower-upper-ℝ lx)
  pr2 is-dedekind-cut-lower-ℝ-Prop =
    is-prop-all-elements-equal
    ( λ uy uy' →
      eq-type-subtype
        ( is-dedekind-prop-lower-upper-ℝ lx)
        ( eq-eq-cut-upper-ℝ
          ( pr1 uy)
          ( pr1 uy')
          ( eq-upper-cut-eq-lower-cut-ℝ (lx , uy) (lx , uy') refl)))

is-emb-lower-real : {l : Level} → is-emb (lower-real-ℝ {l})
is-emb-lower-real = is-emb-inclusion-subtype is-dedekind-cut-lower-ℝ-Prop

Two real numbers with the same lower/upper real are equal

module _
  {l : Level} (x y : ℝ l)
  where

  eq-eq-lower-real-ℝ : lower-real-ℝ x ＝ lower-real-ℝ y → x ＝ y
  eq-eq-lower-real-ℝ = eq-type-subtype is-dedekind-cut-lower-ℝ-Prop

  eq-eq-upper-real-ℝ : upper-real-ℝ x ＝ upper-real-ℝ y → x ＝ y
  eq-eq-upper-real-ℝ = eq-eq-lower-real-ℝ ∘ (eq-lower-real-eq-upper-real-ℝ x y)

  eq-eq-lower-cut-ℝ : lower-cut-ℝ x ＝ lower-cut-ℝ y → x ＝ y
  eq-eq-lower-cut-ℝ lcx=lcy =
    eq-eq-lower-real-ℝ
      ( eq-eq-cut-lower-ℝ (lower-real-ℝ x) (lower-real-ℝ y) lcx=lcy)

  eq-eq-upper-cut-ℝ : upper-cut-ℝ x ＝ upper-cut-ℝ y → x ＝ y
  eq-eq-upper-cut-ℝ = eq-eq-lower-cut-ℝ ∘ (eq-lower-cut-eq-upper-cut-ℝ x y)

References

This page follows parts of Section 11.2 in [UF13].

[UF13]

The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, 2013. URL: https://homotopytypetheory.org/book/, arXiv:1308.0729.

External links

• Real number at Wikidata
• Set of real numbers at Wikidata
• DedekindReals.Type at TypeTopology
• Dedekind cut at $n$Lab
• Dedekind cut at Wikipedia
• Construction of the real numbers by Dedekind cuts at Wikipedia
• Dedekind cut at Britannica

Recent changes

• 2025-04-25. Fredrik Bakke. chore: Fix some links (#1411).
• 2025-02-27. Louis Wasserman. Similarity reasoning for real numbers (#1338).
• 2025-02-25. Louis Wasserman. Disjoint subtypes (#1344).
• 2025-02-16. Louis Wasserman. Break the Dedekind reals into lower and upper Dedekind reals (#1314).
• 2025-02-08. Fredrik Bakke. Functorial action of existential quantifications, and applications in real-numbers (#1298).