In Idris/Haskell, one can prove properties of data by annotating the types and using GADT constructors, such as with Vect, however, this requires hardcoding the property into the type (e.g. a Vect has to be a separate type from a List). Is it possible to have types with an open set of properties (such as list carrying both a length and running average), for example by overloading constructors or using something in the vein of Effect?
Related Questions in HASKELL
- Overlapping UICollectionView in storyboard
- Cannot pod spec lint because of undeclared type errors
- Is the transactionReceipt data present in dataWithContentsOfURL?
- UIWebView Screen Fitting Issue
- ZXingObjC encoding issues
- iOS: None of the valid provisioning profiles allowed the specific entitlements
- How to hide "Now playing url" in control center
- CloudKit: Preventing Duplicate Records
- Slow performance on ipad erasing image
- Swift code with multiple NSDateFormatter - optimization
Related Questions in PROOF
- Overlapping UICollectionView in storyboard
- Cannot pod spec lint because of undeclared type errors
- Is the transactionReceipt data present in dataWithContentsOfURL?
- UIWebView Screen Fitting Issue
- ZXingObjC encoding issues
- iOS: None of the valid provisioning profiles allowed the specific entitlements
- How to hide "Now playing url" in control center
- CloudKit: Preventing Duplicate Records
- Slow performance on ipad erasing image
- Swift code with multiple NSDateFormatter - optimization
Related Questions in CATEGORY-THEORY
- Overlapping UICollectionView in storyboard
- Cannot pod spec lint because of undeclared type errors
- Is the transactionReceipt data present in dataWithContentsOfURL?
- UIWebView Screen Fitting Issue
- ZXingObjC encoding issues
- iOS: None of the valid provisioning profiles allowed the specific entitlements
- How to hide "Now playing url" in control center
- CloudKit: Preventing Duplicate Records
- Slow performance on ipad erasing image
- Swift code with multiple NSDateFormatter - optimization
Related Questions in CORRECTNESS
- Overlapping UICollectionView in storyboard
- Cannot pod spec lint because of undeclared type errors
- Is the transactionReceipt data present in dataWithContentsOfURL?
- UIWebView Screen Fitting Issue
- ZXingObjC encoding issues
- iOS: None of the valid provisioning profiles allowed the specific entitlements
- How to hide "Now playing url" in control center
- CloudKit: Preventing Duplicate Records
- Slow performance on ipad erasing image
- Swift code with multiple NSDateFormatter - optimization
Related Questions in IDRIS
- Overlapping UICollectionView in storyboard
- Cannot pod spec lint because of undeclared type errors
- Is the transactionReceipt data present in dataWithContentsOfURL?
- UIWebView Screen Fitting Issue
- ZXingObjC encoding issues
- iOS: None of the valid provisioning profiles allowed the specific entitlements
- How to hide "Now playing url" in control center
- CloudKit: Preventing Duplicate Records
- Slow performance on ipad erasing image
- Swift code with multiple NSDateFormatter - optimization
Popular Questions
- How do I undo the most recent local commits in Git?
- How can I remove a specific item from an array in JavaScript?
- How do I delete a Git branch locally and remotely?
- Find all files containing a specific text (string) on Linux?
- How do I revert a Git repository to a previous commit?
- How do I create an HTML button that acts like a link?
- How do I check out a remote Git branch?
- How do I force "git pull" to overwrite local files?
- How do I list all files of a directory?
- How to check whether a string contains a substring in JavaScript?
- How do I redirect to another webpage?
- How can I iterate over rows in a Pandas DataFrame?
- How do I convert a String to an int in Java?
- Does Python have a string 'contains' substring method?
- How do I check if a string contains a specific word?
Popular Tags
Trending Questions
- UIImageView Frame Doesn't Reflect Constraints
- Is it possible to use adb commands to click on a view by finding its ID?
- How to create a new web character symbol recognizable by html/javascript?
- Why isn't my CSS3 animation smooth in Google Chrome (but very smooth on other browsers)?
- Heap Gives Page Fault
- Connect ffmpeg to Visual Studio 2008
- Both Object- and ValueAnimator jumps when Duration is set above API LvL 24
- How to avoid default initialization of objects in std::vector?
- second argument of the command line arguments in a format other than char** argv or char* argv[]
- How to improve efficiency of algorithm which generates next lexicographic permutation?
- Navigating to the another actvity app getting crash in android
- How to read the particular message format in android and store in sqlite database?
- Resetting inventory status after order is cancelled
- Efficiently compute powers of X in SSE/AVX
- Insert into an external database using ajax and php : POST 500 (Internal Server Error)
I believe that McBride has answered that question (for Type Theory) in his ornament paper (pdf). The concept you are looking for is the one of an algebraic ornament (emphasis mine):
Now, let's write some code. I've put the whole thing in a gist because I'm going to interleave comments in here. Also, I'm using Agda but it should be easy to tranlate to Idris.
We start by defining what it means to be an algebra delivering
B
s acting on a list ofA
s. We need a base case (a value of typeB
) as well as way to combine the head of the list with the induction hypothesis.Given this definition, we can define a type of lists of
A
s indexed byB
s whose value is precisely the result of the computation corresponding to a givenListAlg A B
. In thenil
case, the result is the base case provided to us by the algebra (proj₁ alg
) whilst in thecons
case, we simply combine the induction hypothesis with the new head using the second projection:Ok, let's import some libraries and see a couple of examples now:
The algebra computing the length of a list is given by
0
as the base case andconst suc
as the way to combine anA
and the length of the tail to build the length of the current list. Hence:If the elements are natural numbers then they can be summed. The algebra corresponding to that has
0
as the base case and_+_
as the way to combine anℕ
together with the sum of the elements contained in the tail. Hence:Crazy thought: If we have two algebras working on the same elements, we can combine them! This way we'll track 2 invariants rather than one!
An now the examples:
If we are tracking the length, then we can define Vectors:
And have, e.g., this vector of length 4:
If we are tracking the sum of the elements, then we can define a notion of distribution: a statistical distribution is a list of probabilities whose sum is 100:
And we can define a uniform one for instance:
Finally, by combining the length and sum algebras, we can implement the notion of sized distribution.
And give this nice uniform distribution for a 4 elements set: