Mathematics and Computation

A blog about mathematics for computers

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Two new doctors!

Within a month two of my students defended their theses: Dr. Anja Petković Komel just before Christmas, and Dr. Philipp Haselwarter just yesterday. I am very proud of them. Congratulations!

Philipp's thesis An Effective Metatheory for Type Theory has three parts:

  1. A formulation and a study of the notion of finitary type theories and standard type theories. These are closely related to the general type theories that were developed with Peter Lumsdaine, but are tailored for implementation.

  2. A formulation and the study of context-free finitary type theories, which are type theories without explicit contexts. Instead, the variables are annotated with their types. Philipp shows that one can pass between the two versions of type theory.

  3. A novel effectful meta-language Andromeda meta-language (AML) for proof assistants which uses algebraic effects and handlers to allow flexible interaction between a generic proof assistant and the user.

Anja's thesis Meta-analysis of type theories with an application to the design of formal proofs also has three parts:

  1. A formulation and a study of transformations of finitary type theories with an associated category of finitary type theories.

  2. A user-extensible equality checking algorithm for standard type theories which specializes to several existing equality checking algorithms for specific type theories.

  3. A general elaboration theorem in which the transformation of type theories are used to prove that every finitary type theory (not necessarily fully annotated) can be elaborated to a standard type theory (fully annotated one).

In addition, Philipp has done a great amount of work on implementing context-free type theories and the effective meta-language in Andromeda 2, and Anja implemented the generic equality checking algorithm. In the final push to get the theses out the implementation suffered a little bit and is lagging behind. I hope we can bring it up to speed and make it usable. Anja has ideas on how to implement transformations of type theories in a proof assistant.

Of course, I am very happy with the particular results, but I am even happier with the fact that Philipp and Anja made an important step in the development of type theory as a branch of mathematics and computer science: they did not study a particular type theory or a narrow family of them, as has hitherto been the norm, but dependent type theories in general. Their theses contain interesting non-trivial meta-theorems that apply to large classes of type theories, and can no doubt be generalized even further. There is lots of low-hanging fruit out there.

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Jacques Carette asked on Twitter for a refence to the fact that countable choice holds in setoids. I then spent a day formalizing facts about the axiom of choice in setoids in Agda. I noticed something interesting that is worth blogging about.

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Proof assistant communities have grown quite a bit lately. They have active Zulip chats: Lean, Coq, Agda, Isabelle. These are good for discussions, but less so for knowledge accumulation and organization, and are not indexed by the search engines.

I have therefore created a proposal for a new “Proof assistants” StackExchange site. I believe that such a site would complement very well various Zulips dedicated to specific proof assistants. If you favor the idea, please support it by visiting the proposal and

To pass the first stage, we need 60 followers and 40 questions with at least 10 votes to proceed to the next stage.

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Here are the slides of my talk "The dawn of formalized mathematics" from the 8th European Congress of Mathematics, which is taking place online and in Protorož, Slovenia, from June 20 to 26, 2021:

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A while ago, my former student Chuangjie Xu and I computed an integer using a sheaf topos. For that purpose,

  1. we developed our mathematics constructively,
  2. we formalized our mathematics in Martin-Löf type theory, in Agda notation,
  3. we pressed a button, and
  4. after a few seconds we saw the integer we expected in front of us.

Well, it was a few seconds for the computer in steps (3)-(4), but three years for us in steps (1)-(2).

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This is joint work with Marc Bezem, Thierry Coquand, Peter Dybjer.

We use the Burali-Forti argument to show that, in homotopy type theory and univalent foundations, the embedding $$ \mathcal{U} \to \mathcal{U}^+$$ of a universe $\mathcal{U}$ into its successor $\mathcal{U}^+$ is not an equivalence. We also establish this for the types of sets, magmas, monoids and groups. The arguments in this post are also written in Agda.

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It is my pleasure to have the opportunity to speak at the University of Wisconsin Logic seminar. The hosts are graciously keeping the seminar open for everyone. I will speak about a favorite topic of mine:

Synthetic mathematics with an excursion into computability theory

Speaker: Andrej Bauer (University of Ljubljana)
Location: University of Wisconsin Logic seminar
Time: February 8th 2021, 21:30 UTC (3:30pm CST in Wisconsin, 22:30 CET in Ljubljana)
Video link: the Zoom link is available at the seminar page


According to Felix Klein, “synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates”. To put it less eloquently, the synthetic method axiomatizes geometry directly by construing points and lines as primitive notions, whereas the analytic method builds a model, the Euclidean plane, from the real numbers.

Do other branches of mathematics posses the synthetic method, too? For instance, what would “synthetic topology” look like? To build spaces out of sets, as topologists usually do, is the analytic way. The synthetic approach must construe spaces as primitive and axiomatize them directly, without any recourse to sets. It cannot introduce continuity as a desirable property of functions, for that would be like identifying straight lines as the non-bending curves.

It is indeed possible to build the synthetic worlds of topology, smooth analysis, measure theory, and computability. In each of them, the basic structure – topological, smooth, measurable, computable – is implicit by virtue of permeating everything, even logic itself. The synthetic worlds demand an economy of thought that the unaccustomed mind finds frustrating at first, but eventually rewards it with new elegance and conceptual clarity. The synthetic method is still fruitfully related to the analytic method by interpretation of the former in models provided by the latter.

We demonstrate the approach by taking a closer look at synthetic computability, whose central axiom states that there are countably many countable subsets of the natural numbers. The axiom is validated and explained by its interpretation in the effective topos, where it corresponds to the familiar fact that the computably enumerable sets may be computably enumerated. Classic theorems of computability may be proved in a straightforward manner, without reference to any notion of computation. We end by showing that in synthetic computability Turing reducibility is expressed in terms of sequential continuity of maps between directed-complete partial orders.

The slides and the video recording of the talk are now available.

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After a short pause, our next talk in the series will be given by Jamie Vicary, who will present a proof assistant in which the proofs are drawn!

Introducing A proof assistant for geometrical higher category theory

Time: Thursday, November 26, 2020 from 15:00 to 16:00 (Central European Time, UTC+1)
Location: online at Zoom ID 989 0478 8985
Speaker: Jamie Vicary (University of Cambridge)
Proof assistant:


Weak higher categories can be difficult to work with algebraically, with the weak structure potentially leading to considerable bureaucracy. Conjecturally, every weak infty-category is equivalent to a "semistrict" one, in which unitors and associators are trivial; such a setting might reduce the burden of constructing large proofs. In this talk, I will present the proof assistant, which allows direct construction of composites in a finitely-generated semistrict (infty,infty)-category. The terms of the proof assistant have a geometrical interpretation as string diagrams, and interaction with the proof assistant is entirely geometrical, by clicking and dragging with the mouse, completely unlike more traditional computer algebra systems. I will give an outline of the underlying theoretical foundations, and demonstrate use of the proof assistant to construct some nontrivial homotopies, rendered in 2d and 3d. I will close with some speculations about the possible interaction of such a system with more traditional type-theoretical approaches. (Joint work with Lukas Heidemann, Nick Hu and David Reutter.)


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The preprint version of the paper A general definition of dependent type theories has finally appeared on the arXiv! Over three years ago Peter Lumsdaine invited me to work on the topic, which I gladly accepted, and dragged my student Philipp Haselwarter into it. We set out to give an answer to the queation:

What is type theory, precisely?

At least for me the motivation to work on such a thankless topic came from Vladimir Voevodsky, who would ask the question to type-theoretic audiences. Some took him to be a troll and others a newcomer who just had to learn more type theory. I was among the latter, but eventually the question got through to me – I could point to any number of specific examples of type theories, but not a comprehensive and mathematically precise definition of the general concept.

It is too easy to dismiss the question by claiming that type theory is an open-ended concept which therefore cannot be completely captured by any mathematical definition. Of course it is open-ended, but it does not follow at all that we should not even attempt to define it. If geometers were equally fatalistic about the open-ended notion of space we would never have had modern geometry, topology, sheaves – heck, half of 20th century mathematics would not be there!

Of course, we are neither the first nor the last to give a definition of type theory, and that is how things should be. We claim no priority or supremacy over other definitions and views of type theory. Our approach could perhaps be described as "concrete" and "proof-theoretic":

  1. We wanted to stay close to traditional syntax.
  2. We gave a complete and precise definition.
  3. We aimed for a level of generality that allows useful meta-theory of a wide range of type theories.

One can argue each of the above points, and we have done so among ourselves many times. Nevertheless, I feel that we have accomplished something worthwhile – but the ultimate judges will be our readers, or lack of them. You are kindly invited to take a look at the paper.

Download PDF:

I should not forget to mention that Peter, with modest help from Philipp and me, formalized almost the entire paper in Coq! See the repository general-type-theories at GitHub.

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I am happy to announce that we are restarting the "Every proof assistants" series of talks with Anders Mörtberg who will talk about Cubical Agda. Note that we are moving the seminar time to a more reasonable hour, at least as far as the working people in Europe are concerned.

Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types

Time: Thursday, September 17, 2020 from 15:00 to 16:00 (Central European Summer Time, UTC+2)
Location: online at Zoom ID 989 0478 8985
Speaker: Anders Mörtberg (Stockholm University)
Proof assistant: Cubical Agda

Abstract: The dependently typed programming language Agda has recently been extended with a cubical mode which provides extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically to proof assistants based on dependent type theory which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types. In the talk I will discuss how Agda was extended to a full-blown proof assistant with native support for univalence and a general schema of higher inductive types. I will also show a variety of examples of how to use Cubical Agda in practice to reason about mathematics and computer science.

The talk video recording is available.

We have more talks in store, but we will space them out a bit to give slots to our local seminar.

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