Open
Conversation
fredrik-bakke
approved these changes
Feb 5, 2026
| [subtypes](foundation-core.subtypes.md) `A` and `A'` of `X` that are | ||
| [apart](measure-theory.apart-subtypes.md). | ||
|
|
||
| This definition follows {{#cite Zeuner22}}.. |
Collaborator
There was a problem hiding this comment.
Suggested change
| This definition follows {{#cite Zeuner22}}.. | |
| This definition follows {{#cite Zeuner22}}. |
Comment on lines
+25
to
+30
| A | ||
| {{#concept "complemented subtype" Disambiguation="of a type with apartness" Agda=complemented-subtype-Type-With-Apartness}} | ||
| of a type `X` equipped with an | ||
| [apartness relation](foundation.apartness-relations.md) is an ordered pair of | ||
| [subtypes](foundation-core.subtypes.md) `A` and `A'` of `X` that are | ||
| [apart](measure-theory.apart-subtypes.md). |
Collaborator
There was a problem hiding this comment.
I find this choice of terminology a bit confusing, since it is certainly not a subtype with a complement, but a subtype equipped with a subtype of its complement. Maybe you can say something more about how these objects should be interpreted and what justifies the name?
Collaborator
There was a problem hiding this comment.
It should also be noted that this concept is not to be confused with "decidable subtypes", i.e., subtypes
| `(A, A')` and `(B, B')` of a type equipped with an | ||
| [apartness relation](foundation.apartness-relations.md), we define the | ||
| [large poset](order-theory.large-posets.md) induced by the relation | ||
| `A ≤ B ≔ A ⊆ B ∧ B' ⊆ A'`. |
Collaborator
There was a problem hiding this comment.
Suggested change
| `A ≤ B ≔ A ⊆ B ∧ B' ⊆ A'`. | |
| `(A ≤ B) ≔ ((A ⊆ B) ∧ (B' ⊆ A'))`. |
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
Starting to build infrastructure based around https://arxiv.org/abs/2207.04000 to make a viable measure theory.