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Contents of this page

Other pages
 The Metamath 100 list
 CICM 2016:
Prime Number Theorem (PDF slides)
(PPT)
by Mario Carneiro, with an informal exposition of the related
Dirichlet's theorem [retrieved 4Aug2016].
Models for Metamath (PDF slides)
(PPT)
(PDF paper)
by Mario Carneiro; the paper is also available at
arxiv.org
[retrieved 16Feb2016].
Google Groups discussion [retrieved 4Aug2016].
 CICM 2015:
GCH
implies AC, a Metamath Formalization (YouTube
video) [retrieved 19Jul2015]
(PDF slides)
(PPT)
(PDF paper) [retrieved
19Jul2015] by Mario
Carneiro.
Arithmetic in Metamath
Case Study: Bertrand's Postulate (CICM 2015) (YouTube
video) [retrieved 19Jul2015]
(PDF slides)
(PPT)
(PDF paper) [retrieved
19Jul2015] by Mario Carneiro.
 "Collapsible proof demo"
by Chris Capel (4Jan2011). (JavaScript is required.)
A click on the magnifying glass will
expand, collapse, show substitution, hide substitution depending
on context. Other than an interface that takes getting used to,
this is an interesting proofofconcept demo showing some features
that are possible. Feel free to take over this project to
enhance it. Personally, I would prefer contextindependent
icons that unambiguously indicate their function,
like the [+] for expand and [] for collapse
in Windows Explorer. I would also prefer
to see the proof
fully expanded on loading (and when JavaScript is disabled);
sometimes I just want to see the whole proof at once without
having to open a dozen subtrees.

(Needs volunteer to reformat URLs...)
AsteroidMeta metamath and mmj2 archived wiki
discussions (readonly; partial recovery to Nov. 2008).
Missing pages are
sometimes on archive.org e.g. archive.org
page for metamath (but LaTeX math image files may be missing
after 2013 or so).
 grammarambiguity.txt 
a proof of the unambiguity of set.mm's grammar by Mario Carneiro
(28Apr2015)
 Overview of Metamath
presentation by Norm Megill at Institut Henri Poincaré (2014)
 Open problems and miscellany discussed
at 2003, 2004, and 2005 Argonne workshops
 Shortest known proofs of the
propositional calculus theorems from
Principia Mathematica

Sean B. Palmer has created
some programs
[retrieved 6Oct2017] to verify these
independently and also to search for shorter proofs.
 Editor screenshots and
syntax highlighting for Metamath
 Robert Solovay's Metamath database source file
peano.mm (Peano arithmetic)
 Other Metamath database source files:
miu.mm (Hofstadter's MIUsystem),
bigunifier.mm
(William McCune's unification stress test),
demo0.mm
(toy formal system)
 Current web usage statistics (us.metamath.org mirror only)
summary by month
with links to details;
current raw Webalizer data
 College and university visitors to the
Metamath site in Jan. 2004 (based on old server log)
 Czur ET16 scanner notes
(for NM's convenient reference; probably not interesting to others)
External links

Known Metamath proof verifiers
These are tools that can take the .mm database and verify that
it's correct (or not).
Some, such as Metamathexe program (the
original C program) and mmj2, include functions to
help create proofs.
Note that some programs (like metamathlamp) focus only on
helping create proofs and thus are put in a different list,
while other programs (like metamathknife) are in this list but don't have
significant functions to help create proofs.
The proof verifiers in this list sometimes have a local archived copy.
When an external link is also available, check it to see that you
have the most recent version. If you have or know of a more recent
version or updated external link, let me know so I can update it.
 metamathexe (current version),
previously just called "metamath".
This is the original Metamath proof verifier and proof assistant
(written in C by Norman Megill)
 mmj2 proof assistant and verifier
(written in Java by Mel O'Cat; enhanced and maintained by Mario Carneiro).
Archived AsteroidMeta
wiki page [retrieved 3Aug2016] as of 2011.
 Metamathknife  Metamath system written in Rust, a fork of metamathrs (below).
 mmlean4
[retrieved 7May2021]  a Metamath proof verifier written in Lean, by
Mario Carneiro.

zigmmverify
[retrieved 5May2021]  a Metamath proof verifier written in Zig,
by Marnix Klooster.
 gramm
[retrieved 1Oct2018]  a Metamath
proof verifier written in Antlr4/Java, by Naip Moro.
 Milpgame is a proof assistant
and verifier, written in Java by Filip Cernatescu. For practice
problems, see text linked at mpeuni/mmtheorems.html#problem1.
 mmscala
[retrieved 13Jul2017]  a Metamath proof
verifier written in the Scala
language as part of an ongoing Metamath > MM
import project
 Metamath.jl
[retrieved 12Jun2016]  a Metamath proof verifier written in the Julia
language, by Dan Getz. See also
https://juliaobserver.com/packages/Metamath [retrived 9Jul2017].
 mmamm.m (29Mar2016) 
A verifier written in the Mathematica language by Mario Carneiro, who says
it "is only 74 lines (give or take, lines in Mathematica
are kind of a moving target) and 2885
characters. Of course it is not
very efficient, and it also does not like local $v declarations (which I
think are not in set.mm anymore), but it will catch all
the important errors. It also works with both normal and compressed proofs."

smm
[retrieved 6Sep2015] 
the smm, smm2, smm3 Metamath proof verifiers
(written in JavaScript by Stefan O'Rear) with a longterm
goal of providing another proof assistant. By using multiple threads,
smm3 verifies set.mm in 0.7s on a 2core, 2way SMT Intel i5
1.6GHz CPU as of June 18, 2016. See this
Google Groups post [retrieved 21Jun2016].
 MM Tool [retrieved 6Sep2015] 
a Metamath proof verifier and editor that runs in a browser
(written in JavaScript by Ivan Kuckir). He also has written an
Introduction
to Metamath that summarizes the basics of the Metamath language,
which you may find useful.
 Igor [retrieved
24Sep2015]  A proof assistant for Metamath, specialized for set.mm
(written in a custom language by Drahflow; in progress)
 mmverify.py (27Jan2013) 
the mmverify Metamath proof verifier
(written in 350 lines of Python by Raph Levien)

hmm.zip (3Nov2006)  the hmm
Metamath proof verifier (written in 400 lines of Haskell by Marnix
Klooster). External link: hmm [retrieved
3Aug2016]
 verify.lua (21Oct2006) 
a Metamath proof verifier
(written in 380 lines of Lua by Martin Kiselkov; needs 40 min
to verify set.mm, but provides an interesting example for learning Lua).
External link: verify.lua
[retrieved 3Aug2016]
 Verifier.cs (29Oct2010)  a
Metamath proof verifier (written in 550 lines of C# by Chris Capel).
External link: verifier.cs
[retrieved 3Aug2016]
 checkmm.cpp (9Dec2010) 
the checkmm Metamath proof verifier
(written in C++ by Eric Schmidt)
 smetamathrs 
Metamath system engine [retrieved 3Aug2016] (written in Rust by Stefan
O'Rear).
 metamathada
(16Jul2017)  a simple Metamath proof checker [retrieved 15Aug2016]
(written in Ada by Tony Häger). It should probably be considered to be
under development; see this this Google
Group message for its limitations. A zip download is also attached
to that message.
 Metamatix is a verified implementation of a Metamath proof checker
written using Coq.
 Sean B. Palmer has a
web page
[retrieved 6Oct2017] that runs the metamath (metamathexe) C program
in JavaScript. He writes,
"When you load that page it emulates an entire Linux stack running on
an OpenRISC CPU simulator written by Sebastian Macke. I added Metamath
to its filesystem, compiled it, and set it up so that it runs in
screen. It takes about 30 seconds to boot to Metamath on this page in
Firefox and Chrome for me... To verify all proofs took my browser about
half an hour. But at least this may serve as a curiosity."
Other tools for Metamath
There are many other kinds of tools for Metamath.
The metamathexe and
metamathknife
programs, previously mentioned, include a number of support functions
that you may find useful.
Here are some other tools for Metamath:
 Metamath app  Bill Hale's app for Apple
desktop and iPad computers allows you to browse Metamath theorems and
their proofs. Do a search for "Metamath" to find it on Apple's app
store. It is free and has no ads or inapp purchases. For a demo of
this app, you can view the two YouTube videos
[retrieved 27Feb2018]. While its main display resembles the Metamath
web pages you are familiar with, a key feature is a "claim" screen to
view the breakdown of a single step of a proof (see the second video).
An online connection is not required since it stores set.mm locally.
(Bill Hale is also the author of mmide.)

Holophrasm
[retrieved 22Aug2016]  a Metamathspecialized automated theorem prover
written in Python by Daniel Whalen. The paper describes using deep
learning to prove 14% of its test propositions from set.mm.
Other links: paper
[retrieved 22Aug2016], working
release [retrieved 22Aug2016].
 metamathtest
[retrieved 15Aug2016], by David A. Wheeler, contains test cases
(positive and negative) for Metamath verifiers and automatically runs
them against a collection of verifiers. Last
execution run [retrieved 15Aug2016].

EMetamath
[retrieved 11Apr2018]  Metamath plugin for Eclipse IDE, with mmj2
inside (written
in Java by Thierry Arnoux).
Google Groups discussion [retrieved 10Jul2016].

Metamath
plugin for Eclipse based on Xtext
[retrieved 3Aug2016]  Metamath plugin for Eclipse IDE (written
in Xtext by Marnix Klooster).
Google Groups discussion [retrieved 22Oct2015].
 completeusersproof
(14Sep2016, updated 12Apr2018)  Alan Sare's completeusersproof
software that enhances mmj2 for certain kinds of proofs. Documentation
can be found in the __README.TXT file in the zip file (also reproduced
here).
(Sadly, Alan Sare passed away on Mar. 23, 2019. The linked .zip file contains
the last version that he provided.)

mmide.zip (15Aug2006)  mmide
provides a graphical user interface for displaying the output of the
Metamath program commands (written in Python by William Hale). External
link: mmide [retrieved 3Aug2016].
Metamathrelated programs

XPuzzle [retrieved 1Sep2020]  a
puzzle with math formulas derived from the Metamath system (an Android
App by Filip Cernatescu). At the bottom of the web page is a link to
the Google Play Store. See also this Google Group
post.
 gh_verify [retrieved
3Aug2016]  a verifier for the Ghilbert language (written in Python by
Raph Levien)
 shullivan0.05.zip
(14Jan2007)  a verifier for the Ghilbert language (written in C by
Dan Krejsa; loads and verifies the translated 2008 set.mm in 500 ms).
External link:
http://home.alamedanet.net/~dan.krejsa/shullivan/shullivan.html [broken
3Aug2016].
 bourbaki.zip (20Dec2006)  a
proof checker for Bourbaki, a custom Lisplike language related to
Metamath (written in Common Lisp by Juha Arpiainen). External link: Bourbaki proof checker [retrieved 3Aug2016].
 Raph Levien has developed a related language called Ghilbert
that strives to improve upon Metamath by guaranteeing the soundness of
definitions and providing features useful for collaborative work.
 jhilbert8.zip (24Jun2009) 
JHilbert (written in Java by Alexander Klauer), a proof verifier for
collaborative theorem proving "in the spirit of Ghilbert" and the engine
behind Wikiproofs [retrieved
3Aug2016] External links: JHilbert [retrieved 3Aug2016],
JHilbert [retrieved 3Aug2016].
 mdl0.8.772.tar.gz
(10Apr2013)  Russell Math verifier (written in C++ by D. Yu Vlasov).
Built upon Metamath as a high level superstructure with an automatic
proving facility, described in a paper [retrieved 3Aug2016]
(in Russian) and reviewed here [retrieved 3Aug2016]. External link: rusellmath.org [retrieved 3Aug2016]
SourceForge page: Mathematics
Development Language [retrieved 3Aug2016].
Mathematica program to generate
arithmetic proof steps
(11Jul2015) Mario Carneiro has developed a
reference implementation of a Mathematica program, arithmetic.nb, to fill in missing
arithmetic steps in an mmj2 proof worksheet. An example of its use was
for the proof of log2ub, which proves
that log(2) < 253/365. The starting worksheet log2uborig.mmp has 150 steps, 12
of them incomplete (the most complicated unproven assertion being
53057*365 < 253*(3^7)*5*7). The Mathematica program processes this
file in about 0.25 seconds to produce a completed proof worksheet (log2ub.mmp) with 716 steps.
Reading this worksheet into mmj2 generates the final compressed proof of
8167 bytes, which can be decreased to the current 6942 bytes by a
separate proof optimization algorithm (the 'minimize' command in the
metamath program).
Study of complex number axiomatization
In June 2012,
Eric Schmidt discovered that two of our
Axioms for Complex Numbers,
namely
axaddcom (now addcom) and
ax0id (now addid1), were redundant.
His results are described in
schmidtcnaxioms.pdf
(LaTeX source
schmidtcnaxioms.tex),
which also includes independence results for the remaining
axioms.
In addition, ax1id (now mulid1)
for complex numbers can be
weakened to ax1rid for reals.
In Jan. 2013,
Scott Fenton formalized Eric's work, resulting in
23 instead of 25 axioms for real and
complex numbers in set.mm. The
Axioms for Complex Numbers
page has been updated with these results.
An interesting part
of the proof, showing how commutativity of addition follows from
other laws, is in
addcomi. Gérard Lang pointed
out that this
holds for rings generally, which is now shown by theorem
ringcom.
Natural deduction in Metamath
In July 2014,
Mario Carneiro presented a talk, "Natural Deduction in the
Metamath Proof Language," at the
6PCM conference [retrieved 12Jul2015]. It describes the
natural deduction emulation method (prefixing hypotheses and theorems
with
"
") that we
now commonly use in set.mm to achieve significant savings in proof sizes.
See
natded.pdf for slides and
natded.mp3 for audio.
Older pages
Hilbert Space Explorer 
Extends ZFC set theory into Hilbert space,
which is the foundation for quantum mechanics. Includes over
1,000 complete formal proofs.



Quantum Logic Explorer 
Starts from the orthomodular lattice properties proved in the
Hilbert Space Explorer and takes you into
quantum logic with around 1,000 proofs.



Metamath Solitaire  A Java
applet
that demonstrates simple proofs.
Builtin axiom systems
include ZFC; modal, intuitionistic, and quantum logics; and Tarski's
plane geometry.



Metamath Music Page  Strictly
for fun. You can listen
to what mathematical proofs "sound" like!



Guest links
The links in this section are provided as a courtesy to
correspondents who have requested them. They may be commercial in
nature and may or may not be related to Metamath. The Metamath project
does not necessarily endorse these links and receives no compensation
for posting them. They may be removed by us at any time for any reason.
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