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Formalizing the Ring of Witt Vectors

less than 1 minute read

Published:

Johan Commelin and I have released a preprint of our new paper, Formalizing the Ring of Witt Vectors! (Update, Nov 25: the paper has been accepted at CPP ‘21 and recognized as a Distinguished Paper.) We describe the construction of the p-typical Witt vectors and their ring structure in Lean, and show that the ring of Witt vectors over the integers modulo p is isomorphic to the ring of p-adic integers.

Metaprogramming in Lean tutorial

less than 1 minute read

Published:

Johan Commelin and Patrick Massot are organizing the workshop Lean for the Curious Mathematician, taking place virtually July 13-17, 2020. The workshop aims to give working mathematicians a hands-on introduction to formal proof.

Logic and Modeling course information

less than 1 minute read

Published:

In Spring 2020, I taught the course Logic and Modeling at the VU. The course was held entirely online due to the COVID-19 pandemic.

The Lean mathematical library

less than 1 minute read

Published:

Along with a number of other contributors, I have written a paper about the Lean mathematical library. The paper describes a large-scale, open project to which 68 people (and counting) have contributed. Rather than attribute the paper to some subset of this group, we decided to release it as a community effort. (Update, Nov 28: our paper has been accepted at CPP!)

Lean Together 2019

Published:

From the 7th to the 11th of January, 2019, we hosted a workshop Lean Together in Amsterdam. This workshop was meant to serve a few purposes. First, it was a meeting for Lean developers and users who have been working remotely on the system and mathematics libraries. Second, it was a kickoff party for Jasmin Blanchette’s Lean Forward NWO project. A key part of this project is “mathematical outreach”; we want to put proof assistants in the hands of mathematicians, and through collaboration with computer scientists, advance the world of formal mathematics.

A formal proof of Hensel’s lemma over the p-adic integers

less than 1 minute read

Published:

Here you will find the supplementary materials to my paper A formal proof of Hensel’s lemma over the p-adic integers. This paper was published at CPP 2019. For more information, contact Robert Y. Lewis.

The Formal Development

The project described in the linked paper has been incorporated into the Lean mathlib repository. The current state of the p-adic number development can be found in the directory /data/padics/. See also the pull requests here and here.

Since mathlib is regularly updated, we preserve a branch here that shows its state when this paper was written.

A bi-directional extensible interface between Lean and Mathematica

Published:

Minchao Wu and I have been working on a project that connects the computer algebra system Mathematica to the Lean theorem prover. Here you will find the supplementary materials corresponding to our paper draft.

An early version of this work was presented at PxTP. For more information, contact Robert Y. Lewis and Minchao Wu.

Versioning

These tools and examples are compatible with various versions of Lean. In any of the GitHub repositories, look for the lean-x.y.z branch for a version of the project compatible with Lean x.y.z. The core tool will automatically be updated to new versions of Lean as long as compatibility is maintained. Unfortunately this automatic upgrading is not possible for the example repositories due to licensing issues, but we will do it manually; please contact the authors if you need a fresh update on short notice, or try running leanproject up yourself.

The link is known to work with Mathematica versions 11 and 12, and perhaps with earlier versions.

The lean-3.17.1 branches, and any below, are frozen as of August 19, 2020. All claims in our paper draft have been tested to be true of this version, and we have not intentionally falsified any in later versions.

The link requires you to have Python installed with python3 in the system path.

We strongly recommend installing Lean with the elan version manager and leanproject supporting tool, following the “regular install” instructions.

The core tool for using Mathematica from Lean is available on GitHub. This project can be added as a dependency to any Lean project using the leanpkg package manager. Note that, if you install the project as a dependency using leanpkg, you do not need to download it separately. It will be downloaded automatically to the _target/deps/mathematica directory of your project.

To use the link in this direction, you must start the Mathematica server. Do this either by

• running math --noprompt -run '<<"server2.m" in the _target/deps/mathematica/src directory of your project, or
• opening the Mathematica frontend and evaluating <<"/path/to/_target/deps/mathematica/src/server2.m" (with the correct path).

The code for using Lean from Mathematica is also available on GitHub. This project uses the above as a dependency. See directions below for using it.

Examples of calling Mathematica from Lean

A project containing examples of the Mathematica-from-Lean link in action is available here.

The easiest way to try this out is to download the project using leanproject:

leanproject get robertylewis/mathematica_examples
cd mathematica_examples/_target/deps/mathematica/src
math --noprompt -run '<<"server2.m"'


This will download the core link, the examples, Lean’s mathematical library mathlib (on which the examples depend), and precompiled binaries for mathlib. If you use an alternate installation method, you may have to compile mathlib yourself, which can take a long time.

You can then see the link in action in any file in mathematica_examples/src. Note: if you use the VSCode editor, you must use the “Open Folder” feature to open the mathematica_examples directory. Opening individual files will not work. An easy way to do this from the command line, starting in the same directory you started in above, is

code mathematica_examples


Examples of calling Lean from Mathematica

These examples and the code to make them work are on GitHub.

As above, it is easiest to install using leanproject:

leanproject get minchaowu/mm_lean


The file LeanProof.nb in the mm_lean directory contains examples of the Lean-from-Mathematica link. You will need to evaluate the first few configuration lines to start a Lean server; after that, the examples below can be tried in any order.

Possible issues

The Mathematica-from-Lean link expects a python3 executable in the path. Create an alias if you do not have this.

The link communicates over port 10000. If this is taken, change the port numbers in client2.py and server2.m.

There are known issues with Mathematica’s socket server functionality on certain versions of Linux. This may cause the server to use excess CPU.

The full, unabbreviated form of the expression x^2-2*x+1 in Lean is

app (app (app (app (const (name.mk_string add (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_has_add (name.mk_string distrib (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_distrib (name.mk_string ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ring (name.mk_string division_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_division_ring (name.mk_string field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_field (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil)))))))) (app (app (app (app (const (name.mk_string sub (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string add_group_has_sub (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_add_group (name.mk_string add_comm_group (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_add_comm_group (name.mk_string ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ring (name.mk_string division_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_division_ring (name.mk_string field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_field (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil))))))))) (app (app (app (app (const (name.mk_string mul (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_has_mul (name.mk_string mul_zero_class (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_mul_zero_class (name.mk_string semiring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_semiring (name.mk_string ordered_semiring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ordered_semiring (name.mk_string ordered_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ordered_ring (name.mk_string linear_ordered_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_linear_ordered_ring (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil))))))))) (local_const (name.mk_numeral 1 (name.mk_numeral 18 (name.anonymous))) (name.mk_string x (name.anonymous)) (binder_info.default) (const (name.mk_string real (name.anonymous)) (level_list.nil)))) (local_const (name.mk_numeral 1 (name.mk_numeral 18 (name.anonymous))) (name.mk_string x (name.anonymous)) (binder_info.default) (const (name.mk_string real (name.anonymous)) (level_list.nil))))) (app (app (app (app (const (name.mk_string mul (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_has_mul (name.mk_string mul_zero_class (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_mul_zero_class (name.mk_string semiring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_semiring (name.mk_string ordered_semiring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ordered_semiring (name.mk_string ordered_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ordered_ring (name.mk_string linear_ordered_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_linear_ordered_ring (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil))))))))) (app (app (app (const (name.mk_string bit0 (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_has_add (name.mk_string distrib (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_distrib (name.mk_string ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_ring (name.mk_string division_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_division_ring (name.mk_string field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_field (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil)))))))) (app (app (const (name.mk_string one (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_has_one (name.mk_string zero_ne_one_class (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_zero_ne_one_class (name.mk_string division_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_division_ring (name.mk_string field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_field (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil))))))))) (local_const (name.mk_numeral 1 (name.mk_numeral 18 (name.anonymous))) (name.mk_string x (name.anonymous)) (binder_info.default) (const (name.mk_string real (name.anonymous)) (level_list.nil)))))) (app (app (const (name.mk_string one (name.anonymous)) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_has_one (name.mk_string zero_ne_one_class (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_zero_ne_one_class (name.mk_string division_ring (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_division_ring (name.mk_string field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (app (app (const (name.mk_string to_field (name.mk_string linear_ordered_field (name.anonymous))) (level_list.cons (level.zero) (level_list.nil))) (const (name.mk_string real (name.anonymous)) (level_list.nil))) (const (name.mk_string h (name.anonymous)) (level_list.nil)))))))


The Mathematica version of this same expression is:

"LeanApp[LeanApp[LeanApp[LeanApp[LeanConst[LeanNameMkString["add", LeanNameAnonymous], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_has_add", LeanNameMkString["distrib", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_distrib", LeanNameMkString["ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ring", LeanNameMkString["division_ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_division_ring", LeanNameMkString["field", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_field", LeanNameMkString["linear_ordered_field", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanConst[LeanNameMkString["h", LeanNameAnonymous], LeanLevelListNil]]]]]]], LeanApp[LeanApp[LeanApp[LeanApp[LeanConst[LeanNameMkString["sub", LeanNameAnonymous], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["add_group_has_sub", LeanNameAnonymous], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_add_group", LeanNameMkString["add_comm_group", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_add_comm_group", LeanNameMkString["ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ring", LeanNameMkString["division_ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_division_ring", LeanNameMkString["field", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_field", LeanNameMkString["linear_ordered_field", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanConst[LeanNameMkString["h", LeanNameAnonymous], LeanLevelListNil]]]]]]]], LeanApp[LeanApp[LeanApp[LeanApp[LeanConst[LeanNameMkString["mul", LeanNameAnonymous], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_has_mul", LeanNameMkString["mul_zero_class", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_mul_zero_class", LeanNameMkString["semiring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_semiring", LeanNameMkString["ordered_semiring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ordered_semiring", LeanNameMkString["ordered_ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ordered_ring", LeanNameMkString["linear_ordered_ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_linear_ordered_ring", LeanNameMkString["linear_ordered_field", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanConst[LeanNameMkString["h", LeanNameAnonymous], LeanLevelListNil]]]]]]]], LeanLocal[LeanNameMkNum[1843, LeanNameMkNum[17, LeanNameAnonymous]], LeanNameMkString["x", LeanNameAnonymous], BinderInfoDefault, LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]]], LeanLocal[LeanNameMkNum[1843, LeanNameMkNum[17, LeanNameAnonymous]], LeanNameMkString["x", LeanNameAnonymous], BinderInfoDefault, LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]]]], LeanApp[LeanApp[LeanApp[LeanApp[LeanConst[LeanNameMkString["mul", LeanNameAnonymous], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_has_mul", LeanNameMkString["mul_zero_class", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_mul_zero_class", LeanNameMkString["semiring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_semiring", LeanNameMkString["ordered_semiring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ordered_semiring", LeanNameMkString["ordered_ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ordered_ring", LeanNameMkString["linear_ordered_ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_linear_ordered_ring", LeanNameMkString["linear_ordered_field", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanConst[LeanNameMkString["h", LeanNameAnonymous], LeanLevelListNil]]]]]]]], LeanApp[LeanApp[LeanApp[LeanConst[LeanNameMkString["bit0", LeanNameAnonymous], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_has_add", LeanNameMkString["distrib", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_distrib", LeanNameMkString["ring", LeanNameAnonymous]], LeanLevelListCons[LeanZeroLevel, LeanLevelListNil]], LeanConst[LeanNameMkString["real", LeanNameAnonymous], LeanLevelListNil]], LeanApp[LeanApp[LeanConst[LeanNameMkString["to_ring", 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The Nature of Mathematical Reasoning

less than 1 minute read

Published:

In 2015 I tought a course at Carnegie Mellon titled The Nature of Mathematical Reasoning. This course, aimed at non-mathematicians, took a philosophical look at the historical and modern practice of mathematics. The course materials can be found here.

I haven’t transferred the course materials to my new website design yet. This page is to preserve a link from the main site to the materials.

Paper Title Number 2

Published in Journal 1, 2010

This paper is about the number 2. The number 3 is left for future work.

Recommended citation: Your Name, You. (2010). "Paper Title Number 2." Journal 1. 1(2). http://academicpages.github.io/files/paper2.pdf

Paper Title Number 3

Published in Journal 1, 2015

This paper is about the number 3. The number 4 is left for future work.

Recommended citation: Your Name, You. (2015). "Paper Title Number 3." Journal 1. 1(3). http://academicpages.github.io/files/paper3.pdf

Formalizing the Ring of Witt Vectors

Published in Certified Programs and Proofs, 2021

This paper is about the number 1. The number 2 is left for future work.

Formalizing the Ring of Witt Vectors

Published in Certified Programs and Proofs, 2021

This paper is about the number 1. The number 2 is left for future work.

Recommended citation: Your Name, You. (2009). "Paper Title Number 1." Journal 1. 1(1). http://robertylewis.com/files/witt-vectors.pdf

Teaching experience 1

Undergraduate course, University 1, Department, 2014

This is a description of a teaching experience. You can use markdown like any other post.

Teaching experience 2

Workshop, University 1, Department, 2015

This is a description of a teaching experience. You can use markdown like any other post.