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lowasser committed Feb 15, 2025
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Expand Up @@ -49,8 +49,8 @@ is
if for any [positive](elementary-number-theory.positive-rational-numbers.md)
[rational number](elementary-number-theory.rational-numbers.md) `ε : ℚ⁺`, there
exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that `0 < q - p < ε`.
Intuitively, when `L` and `U` are the lower and upper cuts of a real number `x`, then `p` and `q`
are rational approximations of `x` to within `ε`.
Intuitively, when `L` and `U` are the lower and upper cuts of a real number `x`,
then `p` and `q` are rational approximations of `x` to within `ε`.

This follows parts of Section 11 in {{#cite BauerTaylor2009}}.

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Expand Up @@ -40,10 +40,11 @@ open import real-numbers.upper-dedekind-real-numbers

## Idea

The {{#concept "negation" Disambiguation="of a one-sided Dedekind real number" Agda=neg-lower-ℝ Agda=neg-upper-ℝ}} of a
[lower Dedekind real number](real-numbers.lower-dedekind-real-numbers.md) is an
[upper Dedekind real number](real-numbers.upper-dedekind-real-numbers.md) whose
cut contains negations of elements in the lower cut, and vice versa.
The
{{#concept "negation" Disambiguation="of a one-sided Dedekind real number" Agda=neg-lower-ℝ Agda=neg-upper-ℝ}}
of a [lower Dedekind real number](real-numbers.lower-dedekind-real-numbers.md)
is an [upper Dedekind real number](real-numbers.upper-dedekind-real-numbers.md)
whose cut contains negations of elements in the lower cut, and vice versa.

## Definitions

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