Filters #573
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Filters #573
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plt-amy
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Dec 11, 2025
Co-authored-by: Amélia <me@amelia.how>
ncfavier
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Jan 4, 2026
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| :::{.definition #filter-base} | ||
| An $I$-indexed family $x_i : I \to P$ is a **filter base** of a filter | ||
| $$F \subseteq P$$ if: |
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| $$F \subseteq P$$ if: | |
| $F \subseteq P$ if: |
| ``` | ||
| --> | ||
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| Every principal filter $\uparrow x$ has a filter base consisting of only of element $x$. |
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Maybe link this to and from Order.Instances.Lower?
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| $\int F$, is the poset whose elements are the a $x : P$ such that $F(x)$ holds, where the order | ||
| is given the ordering of $P$. |
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| $\int F$, is the poset whose elements are the a $x : P$ such that $F(x)$ holds, where the order | |
| is given the ordering of $P$. | |
| $\int F$, is the poset whose elements are the $x : P$ such that $F(x)$ holds, where the order | |
| is given by the ordering of $P$. |
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| record Net {od ℓd ℓx} (D : Poset od ℓd) (X : Type ℓx) : Type (od ⊔ ℓd ⊔ ℓx) where | ||
| no-eta-equality | ||
| field | ||
| net : ⌞ D ⌟ → X | ||
| dom-directed : is-upwards-directed D |
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This feels a bit strange: Net D X is empty if D is not upwards-directed. I think it would be cleaner to either have both the indexing poset and the fact that it's upwards directed as parameters or both as fields.
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Description
This PR defines filters in posets, and proves some basic facts about them.
I've also added some basic supporting infrastructure for directed sets. At some point we should port DCPOs over to use this, but this is probably best done in a separate PR.
Checklist
Before submitting a merge request, please check the items below:
support/sort-imports.hs(ornix run --experimental-features nix-command -f . sort-imports).If your change affects many files without adding substantial content, and
you don't want your name to appear on those pages (for example, treewide
refactorings or reformattings), start the commit message and PR title with
chore:.