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mortberg
cubicaltt
Haskell2015–2017
mortberg / cubicaltt

cubicaltt

Experimental implementation of Cubical Type Theory

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4Distinct people
4Mentions
0Explicit recommenders
612GitHub stars
REPOSITORY AT A GLANCE

What to know before exploring

What it is

Experimental implementation of Cubical Type Theory

GitHub

Where it is used

No documented product use has been verified yet.

What people are saying

No recommendation phrase was found in the indexed comments.

No classified warning is attached to this page. Read the discussions before treating silence as approval.

Also in these discussions

No other repository is repeatedly linked in the same discussions.

This week on Hacker News

Nov 7–Nov 14 UTC · Direct repository links and separately labeled related project stories. Indexed through 11/14/2026, 3:56:07 PM UTC.

No direct repository links were found in the indexed HN discussions over the past seven days. Older discussions are available below.

Mentions over time

HN mentions by month4 dated mentions

Observed mentions: 2015-07-11 09:35:14 UTC to 2017-10-31 09:44:10 UTC.

012015-07-01 00:00:00 UTC to 2015-08-01 00:00:00 UTC (exclusive): 1 mentions; 0 submissions, 1 commentsJul 20152015-08-01 00:00:00 UTC to 2015-09-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2015-09-01 00:00:00 UTC to 2015-10-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2015-10-01 00:00:00 UTC to 2015-11-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2015-11-01 00:00:00 UTC to 2015-12-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2015-12-01 00:00:00 UTC to 2016-01-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-01-01 00:00:00 UTC to 2016-02-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-02-01 00:00:00 UTC to 2016-03-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-03-01 00:00:00 UTC to 2016-04-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-04-01 00:00:00 UTC to 2016-05-01 00:00:00 UTC (exclusive): 1 mentions; 1 submissions, 0 comments2016-05-01 00:00:00 UTC to 2016-06-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-06-01 00:00:00 UTC to 2016-07-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-07-01 00:00:00 UTC to 2016-08-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-08-01 00:00:00 UTC to 2016-09-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 commentsAug 20162016-09-01 00:00:00 UTC to 2016-10-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-10-01 00:00:00 UTC to 2016-11-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-11-01 00:00:00 UTC to 2016-12-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2016-12-01 00:00:00 UTC to 2017-01-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-01-01 00:00:00 UTC to 2017-02-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-02-01 00:00:00 UTC to 2017-03-01 00:00:00 UTC (exclusive): 1 mentions; 0 submissions, 1 comments2017-03-01 00:00:00 UTC to 2017-04-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-04-01 00:00:00 UTC to 2017-05-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-05-01 00:00:00 UTC to 2017-06-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-06-01 00:00:00 UTC to 2017-07-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-07-01 00:00:00 UTC to 2017-08-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-08-01 00:00:00 UTC to 2017-09-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-09-01 00:00:00 UTC to 2017-10-01 00:00:00 UTC (exclusive): 0 mentions; 0 submissions, 0 comments2017-10-01 00:00:00 UTC to 2017-11-01 00:00:00 UTC (exclusive): 1 mentions; 0 submissions, 1 commentsOct 2017

Counts come from the repository’s dated source evidence. Index cutoff: 2026-11-02 08:16:59 UTC.

View mention counts and date ranges
UTC ranges include their start and exclude their end. Select a period to filter the source evidence.
PeriodStart (UTC)End (exclusive, UTC)MentionsSubmissionsComments
Jul 20152015-07-01 00:00:00 UTC2015-08-01 00:00:00 UTC101
Aug 20152015-08-01 00:00:00 UTC2015-09-01 00:00:00 UTC000
Sep 20152015-09-01 00:00:00 UTC2015-10-01 00:00:00 UTC000
Oct 20152015-10-01 00:00:00 UTC2015-11-01 00:00:00 UTC000
Nov 20152015-11-01 00:00:00 UTC2015-12-01 00:00:00 UTC000
Dec 20152015-12-01 00:00:00 UTC2016-01-01 00:00:00 UTC000
Jan 20162016-01-01 00:00:00 UTC2016-02-01 00:00:00 UTC000
Feb 20162016-02-01 00:00:00 UTC2016-03-01 00:00:00 UTC000
Mar 20162016-03-01 00:00:00 UTC2016-04-01 00:00:00 UTC000
Apr 20162016-04-01 00:00:00 UTC2016-05-01 00:00:00 UTC110
May 20162016-05-01 00:00:00 UTC2016-06-01 00:00:00 UTC000
Jun 20162016-06-01 00:00:00 UTC2016-07-01 00:00:00 UTC000
Jul 20162016-07-01 00:00:00 UTC2016-08-01 00:00:00 UTC000
Aug 20162016-08-01 00:00:00 UTC2016-09-01 00:00:00 UTC000
Sep 20162016-09-01 00:00:00 UTC2016-10-01 00:00:00 UTC000
Oct 20162016-10-01 00:00:00 UTC2016-11-01 00:00:00 UTC000
Nov 20162016-11-01 00:00:00 UTC2016-12-01 00:00:00 UTC000
Dec 20162016-12-01 00:00:00 UTC2017-01-01 00:00:00 UTC000
Jan 20172017-01-01 00:00:00 UTC2017-02-01 00:00:00 UTC000
Feb 20172017-02-01 00:00:00 UTC2017-03-01 00:00:00 UTC101
Mar 20172017-03-01 00:00:00 UTC2017-04-01 00:00:00 UTC000
Apr 20172017-04-01 00:00:00 UTC2017-05-01 00:00:00 UTC000
May 20172017-05-01 00:00:00 UTC2017-06-01 00:00:00 UTC000
Jun 20172017-06-01 00:00:00 UTC2017-07-01 00:00:00 UTC000
Jul 20172017-07-01 00:00:00 UTC2017-08-01 00:00:00 UTC000
Aug 20172017-08-01 00:00:00 UTC2017-09-01 00:00:00 UTC000
Sep 20172017-09-01 00:00:00 UTC2017-10-01 00:00:00 UTC000
Oct 20172017-10-01 00:00:00 UTC2017-11-01 00:00:00 UTC101

What Hacker News says

4 sources
hiker·Oct 31, 2017
Cubical https://github.com/mortberg/cubicaltt is a proof of concept programming language based on constructive (cubical type theory https://arxiv.org/abs/1611.02108) interpretation of HoTT. Agda also added recently support for cubical paths https://agda.readthedocs.io/en/latest/language/cubical.html . I'm also waiting on Lean https://leanprover.github.io to introduce cubical type theory for HoTT. Version 2 of the language had HoTT based on the univalence axiom which was dropped in version 3. With cubical type theory the univalence axiom can be constructively proved and is no longer an axiom.
In Simplicity: A New Language for Blockchains [pdf]
maxiepoo·Feb 5, 2017
Not the OP, but the cool thing about HoTT to a user of proof assistants is that it makes working with "quotients" easier. I put quotients in parentheses because really HoTT is about generalizing the idea of a quotient type. Quotients of sets/algebras are one of the core tools of mathematics and old school type theory doesn't have them so you have to manually keep around equivalence relations and prove over and over again that you respect the relations. In HoTT, there is an extension of inductive types that allows you to, not just have constructors, but also to impose "equalities" so these generalized "quotients" really have first-class status in the language. As far as "exciting developments" in HoTT, the big one right now is Cubical Type Theory [1], which is the first implementation of the new ideas of HoTT where Higher inductive types a
Read the rest
nd the univalence axiom "compute" which means that the proof assistant can do more work for you when you use those features. I just saw a talk about it and from talking to people about it, this means that it won't be too long (< 5 years I predict) before we have this stuff implemented in Agda and/or Coq. Finally, I just want to say to people that are scared off or annoyed by all of the abstract talk about "homotopies" and "cubes", you have to understand that this is very new research and we don't yet know the best ways to use and explain these concepts. I certainly think that people will be able to use this stuff without having to learn anything about homotopy theory, though the intuition will probably help. [1] https://github.com/mortberg/cubicaltt
In Ask HN: What is the most exciting development in your field right now?
Kutta·Jul 11, 2015
Another addendum to the discussion, from an angle more relevant to computer programmers: if we find a computational meaning for the univalence axiom, then it'll be a crazy nuclear weapon of generic programming and code generation. The univalence axiom says that two types are equal if we can convert their values back and forth (without loss of information). For example, one Boolean type may be defined as an enum containing True and False. But we could rename the values to Foo and Bar and it would be still a fine implementation, because it's easy to convert between the two representations. We can imagine more far-fetched representations too, with more complicated conversion functions. But in type theory, if two types are equal, then we can freely substitute one for another in any context! For example, if we have a function "Bool -> Bool", a
Read the rest
nd we know that Bool is equal to Bool2, then we can coerce the function to type "Bool2 -> Bool2". Imagine a complex program that uses a complex piece of data type. If we can provide a correct conversion function to another type (i. e. prove that the type is equivalent to another one), then the univalence axiom can take the conversion function and our program as input, and spits out a new program that uses the new data type, and also preserves all properties of the program. By our current standards of generic programming, this is just insane! Now, the issue is that currently we don't actually have a computational interpretation for the univalence axiom, so the above magic doesn't yet work. But there is meaningful progress towards that, and there are already some experimental implementations that can do some magic, for example this one: https://github.com/mortberg/cubicaltt
In Relation Between Type Theory, Category Theory and Logic

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