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Metamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs.^[1] Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others.^[2]

As of December 2020, the set of proved theorems using Metamath is one of the largest bodies of formalized mathematics, containing in particular proofs of 74^[3] of the 100 theorems of the 'Formalizing 100 Theorems' challenge, making it third after HOL Light and Isabelle, but before Coq, Mizar, ProofPower, Lean, Nqthm, ACL2, and Nuprl. There are at least 17 proof verifiers for databases that use the Metamath format.^[4]

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The Metamath language[edit]

The Metamath language is a metalanguage, suitable for developing a wide variety of formal systems. The Metamath language has no specific logic embedded in it. Instead, it can simply be regarded as a way to prove that inference rules (asserted as axioms or proven later) can be applied.The largest database of proved theorems follows conventional ZFC set theory and classic logic, but other databases exist and others can be created.

The Metamath language design is focused on simplicity; the language, employed to state the definitions, axioms, inference rules and theorems is only composed of a handful of keywords, and all the proofs are checked using one simple algorithm based on the substitution of variables (with optional provisos for what variables must remain distinct after a substitution is made).^[6]

Language Basics[edit]

The set of symbols that can be used for constructing formulas is declared using $c(constant symbols)and $v (variable symbols) statements; for example:

The grammar for formulas is specified using a combination of $f (floating (variable-type)hypotheses) and $a (axiomatic assertion) statements; for example:

Axioms and rules of inference are specified with $a statementsalong with ${ and $} for block scoping andoptional $e (essential hypotheses) statements; for example:

Using one construct, $a statements, to capture syntactic rules, axiom schemas, and rules of inference is intended to provide a level of flexibility similar to higher order logical frameworks without a dependency on a complex type system.

Proofs[edit]

Theorems (and derived rules of inference) are written with $p statements;for example:

Note the inclusion of the proof in the $p statement. It abbreviatesthe following detailed proof:

The 'essential' form of the proof elides syntactic details, leaving a more conventional presentation:

Substitution[edit]

All Metamath proof steps use a single substitution rule, which is just the simple replacement of a variable with an expression and not the proper substitution described in works on predicate calculus. Proper substitution, in Metamath databases that support it, is a derived construct instead of one built into the Metamath language itself.

The substitution rule makes no assumption about the logic system in use and only requires that the substitutions of variables are correctly done.

Here is a detailed example of how this algorithm works. Steps 1 and 2 of the theorem 2p2e4 in the Metamath Proof Explorer (set.mm) are depicted left. Let's explain how Metamath uses its substitution algorithm to check that step 2 is the logical consequence of step 1 when you use the theorem opreq2i. Step 2 states that ( 2 + 2 ) = ( 2 + ( 1 + 1 ) ). It is the conclusion of the theorem opreq2i. The theorem opreq2i states that if A = B, then (C F A) = (C F B). This theorem would never appear under this cryptic form in a textbook but its literate formulation is banal: when two quantities are equal, one can replace one by the other in an operation. To check the proof Metamath attempts to unify (C F A) = (C F B) with ( 2 + 2 ) = ( 2 + ( 1 + 1 ) ). There is only one way to do so: unifying C with 2, F with +, A with 2 and B with ( 1 + 1 ). So now Metamath uses the premise of opreq2i. This premise states that A = B. As a consequence of its previous computation, Metamath knows that A should be substituted by 2 and B by ( 1 + 1 ). The premise A = B becomes 2=( 1 + 1 ) and thus step 1 is therefore generated. In its turn step 1 is unified with df-2. df-2 is the definition of the number 2 and states that 2 = ( 1 + 1 ). Here the unification is simply a matter of constants and is straightforward (no problem of variables to substitute). So the verification is finished and these two steps of the proof of 2p2e4 are correct.

When Metamath unifies ( 2 + 2 ) with B it has to check that the syntactical rules are respected. In fact B has the type class thus Metamath has to check that ( 2 + 2 ) is also typed class.

The Metamath proof checker[edit]

The Metamath program is the original program created to manipulate databases written using the Metamath language. It has a text (command line) interface and is written in C. It can read a Metamath database into memory, verify the proofs of a database, modify the database (in particular by adding proofs), and write them back out to storage.

It has a prove command that enables users to enter a proof, along with mechanisms to search for existing proofs.

The Metamath program can convert statements to HTML or TeX notation;for example, it can output the modus ponens axiom from set.mm as:

Many other programs can process Metamath databases, in particular, there are at least 17 proof verifiers for databases that use the Metamath format.^[7]

Metamath databases[edit]

The Metamath website hosts several databases that store theorems derived from various axiomatic systems. Most databases (.mm files) have an associated interface, called an 'Explorer', which allows one to navigate the statements and proofs interactively on the website, in a user-friendly way. Most databases use a Hilbert system of formal deduction though this is not a requirement.

Metamath Proof Explorer[edit]

The Metamath Proof Explorer (recorded in set.mm) is the main and by far the largest database, with over 23,000 proofs in its main part as of July 2019. It is based on classical first-order logic and ZFC set theory (with the addition of Tarski-Grothendieck set theory when needed, for example in category theory). The database has been maintained for over twenty years (the first proofs in set.mm are dated August 1993). The database contains developments, among other fields, of set theory (ordinals and cardinals, recursion, equivalents of the axiom of choice, the continuum hypothesis...), the construction of the real and complex number systems, order theory, graph theory, abstract algebra, linear algebra, general topology, real and complex analysis, Hilbert spaces, number theory, and elementary geometry. This database was first created by Norman Megill, but as of 2019-10-04 there have been 48 contributors (including Norman Megill).^[8]

The Metamath Proof Explorer references many text books that can be used in conjunction with Metamath.^[9] Thus, people interested in studying mathematics can use Metamath in connection with these books and verify that the proved assertions match the literature.

Intuitionistic Logic Explorer[edit]

This database develops mathematics from a constructive point of view, starting with the axioms of intuitionistic logic and continuing with axiom systems of constructive set theory.

New Foundations Explorer[edit]

This database develops mathematics from Quine's New Foundations set theory.

Higher-Order Logic Explorer[edit]

This database starts with higher-order logic and derives equivalents to axioms of first-order logic and of ZFC set theory.

Databases without explorers[edit]

The Metamath website hosts a few other databases which are not associated with explorers but are nonetheless noteworthy. The database peano.mm written by Robert Solovay formalizes Peano arithmetic. The database nat.mm^[10] formalizes natural deduction. The database miu.mm formalizes the MU puzzle based on the formal system MIU presented in Gödel, Escher, Bach.

Older explorers[edit]

The Metamath website also hosts a few older databases which are not maintained anymore, such as the 'Hilbert Space Explorer', which presents theorems pertaining to Hilbert space theory which have now been merged into the Metamath Proof Explorer, and the 'Quantum Logic Explorer', which develops quantum logic starting with the theory of orthomodular lattices.

Natural Deduction[edit]

Because Metamath has a very generic concept of what a proof is (namely a tree of formulas connected by inference rules) and no specific logic is embedded in the software, Metamath can be used with species of logic as different as Hilbert-style logics or sequents-based logics or even with lambda calculus.

However, Metamath provides no direct support for natural deduction systems. As noted earlier, the database nat.mm formalizes natural deduction. The Metamath Proof Explorer (with its database set.mm) instead use a set of conventions that allow the use of natural deduction approaches within a Hilbert-style logic.

Other works connected to Metamath[edit]

Proof checkers[edit]

Using the design ideas implemented in Metamath, Raph Levien has implemented very small proof checker, mmverify.py, at only 500 lines of Python code.

Ghilbert is a similar though more elaborate language based on mmverify.py.^[11] Levien would like to implement a system where several people could collaborate and his work is emphasizing modularity and connection between small theories.

Using Levien seminal works, many other implementations of the Metamath design principles have been implemented for a broad variety of languages. Juha Arpiainen has implemented his own proof checker in Common Lisp called Bourbaki^[12] and Marnix Klooster has coded a proof checker in Haskell called Hmm.^[13]

Although they all use the overall Metamath approach to formal system checker coding, they also implement new concepts of their own.

Editors[edit]

Mel O'Cat designed a system called Mmj2, which provides a graphic user interface for proof entry.^[14] The initial aim of Mel O'Cat was to allow the user to enter the proofs by simply typing the formulas and letting Mmj2 find the appropriate inference rules to connect them. In Metamath on the contrary you may only enter the theorems names. You may not enter the formulas directly. Mmj2 has also the possibility to enter the proof forward or backward (Metamath only allows to enter proof backward). Moreover Mmj2 has a real grammar parser (unlike Metamath). This technical difference brings more comfort to the user. In particular Metamath sometimes hesitates between several formulas it analyzes (most of them being meaningless) and asks the user to choose. In Mmj2 this limitation no longer exists.

There is also a project by William Hale to add a graphical user interface to Metamath called Mmide.^[15] Paul Chapman in its turn is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made.

Milpgame is a proof assistant and a checker (it shows a message only something gone wrong) with a graphic user interface for the Metamath language(set.mm),written by Filip Cernatescu, it is an open source(MIT License) Java application (cross-platform application: Window,Linux,Mac OS). User can enter the demonstration(proof) in two modes : forward and backward relative to the statement to prove. Milpgame checks if a statement is well formed (has a syntactic verifier). It can save unfinished proofs without the use of dummylink theorem. The demonstration is shown as tree, the statements are shown using html definitions (defined in typesetting chapter). Milpgame is distributed as Java .jar(JRE version 6 update 24 written in NetBeans IDE).

See also[edit]

References[edit]

1. ^Megill, Norman; Wheeler, David A. (2019-06-02). Metamath: A Computer Language for Mathematical Proofs (Second ed.). Morrisville, North Carolina, US: Lulul Press. p. 248. ISBN978-0-359-70223-7.
2. ^Megill, Norman. 'What is Metamath?'. Metamath Home Page.
3. ^Metamath 100.
4. ^Megill, Norman. 'Known Metamath proof verifiers'. Retrieved 14 July 2019.CS1 maint: discouraged parameter (link)
5. ^TOC of Theorem List - Metamath Proof Explorer
6. ^Megill,Norman. 'How Proofs Work'. Metamath Proof Explorer Home Page.
7. ^Megill, Norman. 'Known Metamath proof verifiers'. Retrieved 14 July 2019.CS1 maint: discouraged parameter (link)
8. ^Wheeler, David A. 'Metamath set.mm contributions viewed with Gource through 2019-10-04'.
9. ^Megill, Norman. 'Reading suggestions'. Metamath.
10. ^Liné, Frédéric. 'Natural deduction based Metamath system'. Archived from the original on 2012-12-28.CS1 maint: discouraged parameter (link)
11. ^Levien,Raph. 'Ghilbert'.
12. ^Arpiainen, Juha. 'Presentation of Bourbaki'. Archived from the original on 2012-12-28.CS1 maint: discouraged parameter (link)
13. ^Klooster,Marnix. 'Presentation of Hmm'. Archived from the original on 2012-04-02.CS1 maint: discouraged parameter (link)
14. ^O'Cat,Mel. 'Presentation of mmj2'. Archived from the original on December 19, 2013.CS1 maint: discouraged parameter (link)
15. ^Hale, William. 'Presentation of mmide'. Archived from the original on 2012-12-28.CS1 maint: discouraged parameter (link)

External links[edit]

• Metamath: official website.
• What do mathematicians think of Metamath: opinions on Metamath.

2020-12-28 17:33:50 • Filed to: macOS 10.14 Solutions • Proven solutions

To peruse and interact with files more rigorously, it’s imperative for Mac users to know how to access and manipulate the metadata of files. The metadata of a file is a corpus of data originating from the tagging operations of the OS that ensue when you save files to local folder on your Mac. It includes the background information about the file which provide sorting functions with cues to trace out the file’s location.

It’s a cake walk to view metadata in the Mac OS Mojave Finder, and this article lays out the steps in clear outlines. Read on to find out about these steps, and to also find a top-draw answer to the question 'how can I edit file metadata in macOS 10.14?'

How to View Metadata in macOS 10.14 Mojave

There are a number of simple ways to look up file metadata in OS 10.14. The following is a rundown of two easy ways:

1. Using Finder:

When you highlight a file and then press 'Command'+ 'I', a Finder information pane opens to reveal the metadata of the file. This pane gives access to macOS mojave finder metadata of two categories: the first is the 'General' category which displays details such as the file’s format, size, date created and date modified, etc; and the other is the 'More Info' category which gives access to granular specifics of a file like dimension and color profile for image files, and composer, sampling rates, etc for music files.

2. Using The OS Terminal

Despite listing data in two separate categories, the Mac OS Mojave Finder metadata still comes short of a comprehensive collection of a file’s metadata. However, the Terminal’s 'mdls' command line gives access to a complete collection of a file’s metadata. To use the Terminal’s 'mdls' command line to reveal a files metadata, run the Terminal, and then key in 'mdls' followed by a single space. Next, drag and drop the select file from Finder to the open Terminal window, and then hit 'Enter'. After the Terminal successfully executes the command, a list pops up of all the metadata keys and values pertaining to the file. Each key begins with a 'kMDItem' prefix, for instance 'kMDItemPixelWidth' or 'kMDItemContentModificationDate'. The listing features a '=' sign that links each of the keys to their corresponding values.

The Terminal also allows you to access a specific metadata key of a file if you’re not looking for a rundown of the complete metadata. To do this, run the 'mdls' command using the command line scheme: mdls -name KEY FILEPATH

Where 'FILEPATH' represents the name of the select file, and 'KEY' represents the specific metadata key of the select file. There are a plethora of metadata keys that you can find using this command line. Some of the commonly used keys include:

Text

• kMDItemAuthors
• kMDItemAlternateNames
• kMDItemContentCreationDate
• kMDItemContentModificationDate
• kMDItemContentType
• kMDItemContentTypeTree
• kMDItemCopyright
• kMDItemDateAdded
• kMDItemDisplayName
• kMDItemFonts

Audio:

• kMDItemAudioBitRate
• kMDItemAudioChannelCount
• kMDItemAudioSampleRate

Movies:

• kMDItemDurationSeconds
• kMDItemVideoBitRate
• kMDItemTotalBitRate
• kMDItemStreamable

Images:

• kMDItemHasAlphaChannel
• kMDItemIsScreenCapture
• kMDItemScreenCaptureType
• kMDItemResolutionWidthDPI
• kMDItemResolutionHeightDPI
• kMDItemProfileName
• kMDItemPixelWidth
• kMDItemPixelHeight
• kMDItemOrientation

How to Edit Metadata in macOS 10.14 Mojave

The file sorting and searching functions of Mac OS Mojave depend heavily on the metadata of files. Hence, when there are errors and mix ups in the metadata of a file, it becomes quite difficult to trace out the files in the drives. For instance, a music file with a wrong listing of the artist name and genre bears erroneous metadata. It’s commonplace to download or receive files that do not come with accurate metadata. Hence, it’s common for the question 'How can I edit file metadata in macOS 10.14' to cross the mind of a Mac user who’s yet to gain mastery over the metadata of files.

PDFelement provides Mac users with an expedient tool to edit metadata on PDF files. This gives users a wider span of control over the contents and information about PDF files. From structural to administrative and descriptive, PDFelement provides veritable tools for editing nearly every conceivable metadata of any PDF file.

PDFelement’s PDF file management solution offers multi-functional PDF editing tools for editing both the contents and the metadata of the contents of files. You can comment, add sticky notes, highlight, and do even more on the PDF file using PDFelement. You can also convert the PDF file to a wide array of file formats using PDFelement. In addition, PDFelement comes with an impressive array of security features for securing PDF files like the signature and password protection for files.

How to Edit PDF Metadata on macOS 10.14

After downloading and installing PDFelement on your device, you can then proceed to execute the following steps to edit the metadata of a PDF file:

1. Open the PDF file in PDFelement, and then select 'File' and then 'Properties'. This brings you to the 'Document properties' pane where you can get file metadata, including the author, subject, security status, etc.

2. Once the file is open, and you can access the 'Properties' box, you can then proceed to effect changes in various aspects of the file, including the title, author, keywords, etc.

3. To remove an ill-fitting metadata key or value, simply select the undesirable key, and then hit 'Backspace' or 'Delete' key.

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