Totally bounded sets are propositionally decidable #1625
+239
−29
Conversation
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
fredrik-bakke
added
logic
univalent-combinatorics
real-numbers
metric-spaces
and removed
logic
labels
src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
Outdated
Show resolved
Hide resolved
src/metric-spaces/inhabited-totally-bounded-subspaces-metric-spaces.lagda.md
Outdated
Show resolved
Hide resolved
src/univalent-combinatorics/inhabited-finitely-enumerable-types.lagda.md
Outdated
Show resolved
Hide resolved
src/univalent-combinatorics/inhabited-finitely-enumerable-types.lagda.md
Outdated
Show resolved
Hide resolved
fredrik-bakke
approved these changes
Oct 24, 2025
Co-authored-by: Fredrik Bakke <[email protected]>
Co-authored-by: Fredrik Bakke <[email protected]>
…s.lagda.md Co-authored-by: Fredrik Bakke <[email protected]>
Co-authored-by: Fredrik Bakke <[email protected]>
…paces.lagda.md Co-authored-by: Fredrik Bakke <[email protected]>
…ded' into decidable-inhabited-totally-bounded
| ( approximation-net-Metric-Space X ε S) | ||
| ``` | ||
|
|
||
| ### Given any net for a metric space `X`, it is propositionally decidable whether `X` is inhabited |
There was a problem hiding this comment.
Being inhabited is already a proposition
Suggested change
| ### Given any net for a metric space `X`, it is propositionally decidable whether `X` is inhabited | |
| ### Given any net for a metric space `X`, it is decidable whether `X` is inhabited |
There was a problem hiding this comment.
I'm trying to reference https://unimath.github.io/agda-unimath/logic.propositionally-decidable-types.html?highlight=Propositionally#propositionally-decidable-types specifically, which seems to be setting up a different term?
There was a problem hiding this comment.
Maybe that should be "X is propositionally decidable"?
Comment on lines
+160
to
+166
| ( finitely-enumerable-subset-net-Metric-Space | ||
| ( metric-space-subset-ℝ S) | ||
| ( δ) | ||
| ( M δ) , | ||
| backward-implication | ||
| ( is-coinhabited-net-Metric-Space (metric-space-subset-ℝ S) δ (M δ)) | ||
| ( |S|)) |
There was a problem hiding this comment.
This was better before
There was a problem hiding this comment.
even though coinhabitedness is a stronger result, either direction is still useful, as demonstrated here, and can/should still be recorded.
Comment on lines
+210
to
+237
| ### The standard finite types are decidable | ||
|
|
||
| ```agda | ||
| is-decidable-Fin : (n : ℕ) → is-decidable (Fin n) | ||
| is-decidable-Fin zero-ℕ = inr (λ ()) | ||
| is-decidable-Fin (succ-ℕ n) = inl (neg-one-Fin n) | ||
|
|
||
| is-inhabited-or-empty-Fin : (n : ℕ) → is-inhabited-or-empty (Fin n) | ||
| is-inhabited-or-empty-Fin n = | ||
| is-inhabited-or-empty-is-decidable (is-decidable-Fin n) | ||
| ``` | ||
|
|
||
| ### The finite types are propositionally decidable | ||
|
|
||
| ```agda | ||
| module _ | ||
| {l : Level} (X : Finite-Type l) | ||
| where | ||
|
|
||
| is-inhabited-or-empty-type-Finite-Type : | ||
| is-inhabited-or-empty (type-Finite-Type X) | ||
| is-inhabited-or-empty-type-Finite-Type = | ||
| rec-trunc-Prop | ||
| ( is-inhabited-or-empty-Prop (type-Finite-Type X)) | ||
| ( λ (n , Fin-n≃X) → | ||
| is-inhabited-or-empty-equiv' Fin-n≃X (is-inhabited-or-empty-Fin n)) | ||
| ( is-finite-type-Finite-Type X) | ||
| ``` |
There was a problem hiding this comment.
These results are about not-necessarily-inhabited finite types, so they should go in standard-finite-types and finite-types respectively
Comment on lines
+114
to
+117
| ( λ (N , Fin-n↠X) → | ||
| is-inhabited-or-empty-surjection | ||
| ( Fin-n↠X) | ||
| ( is-inhabited-or-empty-Fin N)) |
There was a problem hiding this comment.
Suggested change
| ( λ (N , Fin-n↠X) → | |
| is-inhabited-or-empty-surjection | |
| ( Fin-n↠X) | |
| ( is-inhabited-or-empty-Fin N)) | |
| ( λ (n , Fin-n↠X) → | |
| is-inhabited-or-empty-surjection | |
| ( Fin-n↠X) | |
| ( is-inhabited-or-empty-Fin n)) |
There was a problem hiding this comment.
since you use a lower case n in Fin-n↠X
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.
...and so are lots of other simple types for which that principle wasn't written down yet?