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Homotopy Type Theory
Archimedean ordered integral domain > history (changes)

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## Definition

< Archimedean integral domain

~~Let ~~

~~$R$~~~~ be an ~~~~ordered integral domain~~~~, and let ~~~~$inj: \mathbb{Z}_{+} \to R$~~~~ be the injection of the positive ~~~~integers~~~~ ~~~~$\mathbb{Z}_{+}$~~~~ into ~~~~$R$~~~~.~~~~
~~### With strict order

~~
~~$R$ is an **Archimedean ordered integral domain** if there is a family of dependent terms

~~
~~$a:R, b:R \vdash \alpha(a, b): ((0 \lt a) \times (0 \lt b)) \to \Vert \sum_{n:\mathbb{Z}_{+}} a \lt inj(n) \cdot b \Vert$

~~
~~### With positivity

~~
~~$R$ is an **Archimedean ordered integral domain** if there is a family of dependent terms

~~
~~$a:R, b:R \vdash \alpha(a, b): (\mathrm{isPositive}(a) \times \mathrm{isPositive}(b)) \to \Vert \sum_{n:\mathbb{Z}_{+}} a \lt inj(n) \cdot b \Vert$

~~
~~## Examples

~~
~~~~
~~## See also

~~
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Last revised on June 16, 2022 at 15:58:09.
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