Description:
Here are some of the conventions we use in the
Metamath Proof Explorer (aka "set.mm"), and how they correspond to
typical textbook language (skipping the many cases
where they're identical):
- Notation.
Where possible, the notation attempts to conform to modern
conventions, with variations due to our choice of the axiom system
or to make proofs shorter. However, our notation is strictly
sequential (left-to-right). For example, summation is written in the
form (df-sum 12089) which denotes that index
variable ranges over when evaluating . Thus,
means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 12265). Also, the method of definition,
the axioms for predicate calculus, and the
development of substitution are somewhat different from those found
in standard texts. For example, the expressions for the
axioms were designed for direct derivation of
standard results without excessive use of metatheorems.
(See Theorem 9.7 of [Megill] p. 448 for a rigorous justification.)
The notation is usually explained in more detail when first introduced.
- Axiomatic assertions ($a).
All axiomatic assertions ($a statements)
starting with " " have labels starting
with "ax-" (axioms) or "df-" (definitions). A statement with a
label starting with "ax-" corresponds to what is traditionally
called an axiom. A statement with a label starting with "df-"
introduces new symbols or a new relationship among symbols
that can be eliminated; they always extend the definition of
a wff or class. Metamath blindly treats $a statements as new
given facts but does not try to justify them. The mmj2 program
will justify the definitions as sound, except for 5 (df-bi,
df-cleq, df-clel, df-clab, df-sbc) that require a more complex
metalogical justification by hand.
- Proven axioms.
In some cases we wish to treat an expression as an axiom in
later theorems, even though it can be proved. For example,
we derive the postulates or axioms of complex arithmetic as
theorems of ZFC set theory. For convenience, after deriving
the postulates we re-introduce them as new axioms on
top of set theory. This lets us easily identify which axioms
are needed for a particular complex number proof, without the
obfuscation of the set theory used to derive them. For more, see
http://us.metamath.org/mpegif/mmcomplex.html. When we wish
to use a previously-proven assertion as an axiom, our convention
is that we use the
regular "ax-NAME" label naming convention to define the axiom,
but we precede it with a proof of the same statement with the label
"axNAME" . An example is complex arithmetic axiom ax-1cn 8728,
proven by the preceding theorem ax1cn 8704.
The metamath.exe program will warn if an axiom does not match the
preceding theorem that justifies it if the names match in this way.
- Definitions (df-...).
We encourage definitions to include hypertext links to proven examples.
- Discouraged use and modification.
If something should only be used in limited ways, it is marked with
"(New usage is discouraged.)". This is used, for example, when something
can be constructed in more than one way, and we do not want later
theorems to depend on that specific construction.
This marking is also used if we want later proofs to use proven axioms.
For example, we want later proofs to
use ax-1cn 8728 (not ax1cn 8704) and ax-1ne0 8739 (not ax1ne0 8715), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 8704 and ax1ne0 8715 are marked as "(New usage is discouraged.)".
In some cases a proof should not normally be changed, e.g., when it
demonstrates some specific technique.
These are marked with "(Proof modification is discouraged.)".
- New definitions infrequent.
Typically we are minimalist when introducing new definitions; they are
introduced only when a clear advantage becomes apparent for reducing
the number of symbols, shortening proofs, etc. We generally avoid
the introduction of gratuitous definitions because each one requires
associated theorems and additional elimination steps in proofs.
For example, we use and for inequality expressions, and
use instead of sinh
for the hyperbolic sine.
- Axiom of choice.
The axiom of choice (df-ac 7676) is widely accepted, and ZFC is the most
commonly-accepted fundamental set of axioms for mathematics.
However, there have been and still are some lingering controversies
about the Axiom of Choice. Therefore, where a proof
does not require the axiom of choice, we prefer that proof instead.
E.g., our proof of the Schroeder-Bernstein Theorem (sbth 6914)
does not use the axiom of choice.
In some cases, the weaker axiom of countable choice (ax-cc 7994)
or axiom of dependent choice (ax-dc 8005) can be used instead.
- Variables.
Typically, Greek letters
( = phi, = psi, = chi, etc.),... are used for
propositional (wff) variables;
, , ,... for individual (set) variables;
and , , ,... for class variables.
- Turnstile.
"", meaning "It is provable that," is the first token
of all assertions
and hypotheses that aren't syntax constructions. This is a standard
convention in logic. For us, it also prevents any ambiguity with
statements that are syntax constructions, such as "wff ".
- Substitution.
" " should be read "the wff that results from the
proper substitution of for in wff ." See df-sb 1884
and the related df-sbc 2936 and df-csb 3024.
- Is-a set.
" " should be read
"Class is a set (i.e. exists)."
This is a convenient convention based on
Definition 2.9 of [Quine] p. 19. See df-v 2742 and isset 2744.
- Converse.
"" should be read "converse of (relation) "
and is the same as the more standard notation R^{-1}
(the standard notation is ambiguous). See df-cnv 4642.
This can be used to define a subset, e.g., df-tan 12280 notates
"the set of values whose cosine is a nonzero complex number" as
.
- Function application.
"()" should be read "the value
of function at " and has the same meaning as the more
familiar but ambiguous notation F(x). For example,
(see cos0 12357). The left apostrophe notation
originated with Peano and was adopted in Definition *30.01 of
[WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
Definition 6.11 of [TakeutiZaring] p. 26. See df-fv 4654.
In the ASCII (input) representation there are spaces around the grave
accent; there is a single accent when it is used directly,
and it is doubled within comments.
- Infix and parentheses.
When a function that takes two classes and produces a class
is applied as part of an infix expression, the expression is always
surrounded by parentheses (see df-ov 5760).
For example, the in ; see 2p2e4 9774.
Function application is itself an example of this.
Similarly, predicate expressions
in infix form that take two or three wffs and produce a wff
are also always surrounded by parentheses, such as
, , , and
(see wi 6, df-or 361, df-an 362, and df-bi 179 respectively).
In contrast, a binary relation (which compares two classes and
produces a wff) applied in an infix expression is not
surrounded by parentheses.
This includes set membership (see wel 1622),
equality (see df-cleq 2249),
subset (see df-ss 3108), and
less-than (see df-lt 8683). For the general definition
of a binary relation in the form , see df-br 3964.
For example, ( see 0lt1 9229) does not use parentheses.
- Unary minus.
The symbol is used to indicate a unary minus, e.g., .
It is specially defined because it is so commonly used.
See cneg 8971.
- Function definition.
Functions are typically defined by first defining the constant symbol
(using $c) and declaring that its symbol is a class with the
label cNAME (e.g., ccos 12273).
The function is then defined labelled df-NAME; definitions
are typically given using the maps-to notation (e.g., df-cos 12279).
Typically there are other proofs such as its
closure labelled NAMEcl (e.g., coscl 12334), its
function application form labelled NAMEval (e.g., cosval 12330),
and at least one simple value (e.g., cos0 12357).
- Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
; (df-fac 11220 and fac4 11227).
- Unambiguous symbols.
A given symbol has a single unambiguous meaning in general.
Thus, where the literature might use the same symbol with different
meanings, here we use different (variant) symbols for different
meanings. These variant symbols often have suffixes, subscripts,
or underlines to distinguish them. For example, here
"" always means the value zero (df-0 8677), while
"" is the group identity element (df-0g 13331),
"" is the poset zero (df-p0 14072),
"" is the zero polynomial (df-0p 18952),
"" is the zero vector in a normed complex vector space
(df-0v 21079), and
"" is a class variable for use as a connective symbol
(this is used, for example, in p0val 14074).
There are other class variables used as connective symbols
where traditional notation would use ambiguous symbols, including
"", "", "", and "".
These symbols are very similar to traditional notation, but because
they are different symbols they eliminate ambiguity.
- Natural numbers.
There are different definitions of "natural" numbers in the literature.
We use (df-n 9680) for the integer numbers starting
from 1, and (df-n0 9898) for the set of nonnegative integers
starting at zero.
- Decimal numbers.
Numbers larger than ten are often expressed in base 10
using the decimal constructor df-dec 10057, e.g.,
;;; (see 4001prm 13070 for a proof that 4001 is prime).
- Theorem forms.
We often call a theorem a "deduction" whenever the conclusion and
all hypotheses are each prefixed
with the same antecedent . Deductions are often the
preferred form for theorems because they
allow us to easily use the theorem in places where
(in traditional textbook formalizations) the standard
Deduction Theorem would be used. See, for example, a1d 24.
A deduction hypothesis can have an indirect antecedent via definitions,
e.g., see lhop 19290. Deductions have a label suffix of "d"
if there are other forms of the same theorem. By contrast,
we tend to call the simpler version with no common antecedent an
"inference" and suffix its label with "i"; compare theorem a1i 12.
Finally, a "tautology" would be the form with no hypotheses,
and its label would have no suffix. For example, see pm2.43 49,
pm2.43i 45, and pm2.43d 46.
- Label names.
Every statement has a unique identifying label; here are few
additional label name conventions.
Hypotheses have the name of final axiom or theorem, followed by
".", followed by a unique id.
We sometimes suffix with "v" the label of a theorem
eliminating a hypothesis such as in 19.21 1771 via the use of
distinct variable conditions combined with nfv 1629. Conversely, we
sometimes suffix with "f" the label of a theorem introducing such a
hypothesis to eliminate the need for the distinct variable condition;
e.g. euf 2123 derived from df-eu 2121. The "f" stands for "not free in"
which is less restrictive than "does not occur in."
We sometimes suffix with "s" the label of an inference
that manipulates an
antecedent, leaving the consequent unchanged. The "s" means that the
inference eliminates the need for a syllogism (syl 17) -type inference
in a proof. When an inference is converted to a
theorem by eliminating an "is a set" hypothesis, we sometimes
suffix the theorem form with "g" (for "more general") as in
uniex 4453 vs. uniexg 4454.
Label names are relatively short while suggesting their purpose, e.g.,
2p2e4 9774 is the label for because it is
"2 plus 2 equals 4".
When creating a new theorem or axiom, try to reuse abbreviations
used elsewhere. A comment should explain the first use of an
abbreviation.
- Deduction theorem.
The Deduction Theorem is a metalogical theorem that
provides an algorithm for constructing a proof of a theorem from
the proof of its corresponding deduction.
In ordinary mathematics, no one actually carries out the algorithm,
because (in its most basic form) it involves an exponential
explosion of the number of proof steps as more hypotheses are
eliminated. Instead, in ordinary mathematics the Deduction Theorem
is invoked simply to claim that something can be done in principle,
without actually doing it. For more details, see
http://us.metamath.org/mpegif/mmdeduction.html.
The Deduction Theorem is a metalogical theorem that cannot be
applied directly in metamath, and the explosion of steps would
be a problem anyway, so alternatives are used. One alternative
we use sometimes is the "weak deduction theorem" dedth 3547,
which works in certain cases in set theory. We also
sometimes use dedhb 2886.
However, the primary mechanism we use today for
emulating the deduction theorem is to write proofs in the deduction
theorem form (aka "deduction style") described earlier; the
prefixed mimics the context in a deduction proof system.
In practice this mechanism works very well.
This approach is described in the
deduction form
and natural deduction page; a list of translations
for common natural deduction rules is given in natded 4.
- Recursion. We define recursive functions
using various "recursion constructors". These allow us to define, with
compact direct definitions, functions that are usually defined in
textbooks with indirect self-referencing recursive definitions.
This produces compact definition and much simpler proofs, and
greatly reduces the risk of creating unsound definitions.
Examples of recursion constructors include recs in df-recs 6321,
in df-rdg 6356, seq_{𝜔} in df-seqom 6393, and
in df-seq 10978.
These have characteristic function and initial value .
(_{g} in df-gsum 13332 isn't really designed for arbitrary recursion,
but you could do it with the right magma.)
The logically primary one is df-recs 6321, but for the "average user"
the most useful one is probably df-seq 10978- provided that
a countable sequence is sufficient for the recursion.
- Extensible structures.
Mathematics includes many structures such as
ring, group, poset, etc. We define an "extensible structure"
which is then used to define group, ring, poset, etc.
This allows theorems from more general structures (groups)
to be reused for more specialized structures (rings) without
having to reprove them. See df-struct 13077.
- Organizing proofs.
Humans have trouble understanding long proofs.
It is often preferable to break longer proofs into
smaller parts (just as with traditional proofs). in Metamath
this is done by creating separate proofs of the separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 6904 is the first lemma for sbth 6914. Also, consider proving
reusable results separately, so that others will be able to easily
reuse that part of your work.
- Hypertext links.
We strongly encourage comments to have many links to related material,
with accompanying text that explains the relationship.
These can help readers understand the context.
Links to other statements, or to HTTP/HTTPS URLs,
can be inserted in ASCII source text by prepending a
space-separated tilde.
When metamath.exe is used to generate HTML it automatically inserts
hypertext links for syntax used (e.g., every symbol used),
every axiom and definition depended on, the justification
for each step in a proof, and to both the
next and previous assertion.
- Bibliography references.
Please include a bibliographic reference to any external material used.
A name in square brackets in a comment indicates a
bibliographic reference. The full reference must be of the form
KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER -
note that this is a very specific form that requires a page number.
There should be no comma between the author reference and the
"p." (a constant indicator).
Whitespace, comma, period, or semicolon should follow NUMBER.
An example is Theorem 3.1 of [Monk1] p. 22,
The KEYWORD, which is not case-sensitive,
must be one of the following: Axiom, Chapter, Compare, Condition,
Corollary, Definition, Equation, Example, Exercise, Figure, Item,
Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem,
Property, Proposition, Remark, Rule, Scheme, Section, or Theorem.
The IDENTIFIER is optional, as in for example
"Remark in [Monk1] p. 22".
The NOISEWORDS(S) are zero or more from the list: from, in, of, on.
The AUTHOR(S) must be present in the file identified with the
htmlbibliography assignment (e.g. mmset.html) as a named anchor
(NAME=). If there is more than one document by the same author(s),
add a numeric suffix (as shown here).
The NUMBER is a page number, and may be any alphanumeric string such as
an integer or Roman numeral.
Note that we require page numbers in comments for individual
$a or $p statements. We allow names in square brackets without
page numbers (a reference to an entire document) in
heading comments.
- Input format.
The input is in ASCII with two-space indents.
Tab characters are not allowed.
Use embedded math comments or HTML entities for non-ASCII characters
(e.g., "é" for "é").
- Information on syntax, axioms, and definitions.
For a hyperlinked list of syntax, axioms, and definitions, see
http://us.metamath.org/mpegif/mmdefinitions.html.
If you have questions about a specific symbol or axiom, it is best
to go directly to its definition to learn more about it.
The generated HTML for each theorem and axiom includes hypertext
links to each symbol's definition.
Here are some conventions that address distinctness or freeness of a
variable:
- is read " is not free in (wff) ";
see df-nf 1540.
Similarly, is read is not free in (class) ,
see df-nfc 2381.
- "$d x y $." should be read "Assume x and y are distinct
variables."
- "$d x $." should be read "Assume x does not occur in phi $."
Sometimes a theorem is proved using
(df-nf 1540) in place of
"$d $." when a more general result is desired;
ax-17 1628 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2123 from df-eu 2121.
- "$d x A $." should be read "Assume x is not a variable occurring in
class A."
- "$d x A $. $d x ps $. $e |- $."
is an idiom
often used instead of explicit substitution, meaning "Assume psi results
from the proper substitution of A for x in phi."
- " " occurs early in some cases, and
should be read "If x and y are distinct
variables, then..." This antecedent provides us with a technical
device (called a "distinctor" in Section 7 of [Megill] p. 444)
to avoid the need for the
$d statement early in our development of predicate calculus, permitting
unrestricted substitutions as conceptually simple as those in
propositional calculus. However, the $d eventually becomes a
requirement, and after that this device is rarely used.
Here is more information about our processes for checking and
contributing to this work:
- Multiple verifiers.
This entire file is verified by multiple independently-implemented
verifiers when it is checked in, giving us extremely high
confidence that all proofs follow from the assumptions.
The checkers also check for various other problems such as
overly long lines.
- Rewrapped line length.
The input file routinely has its text wrapped using
metamath 'read set.mm' 'set width 79' 'write source set.mm /rewrap' .
- Discouraged information.
A separate file named "discouraged" lists all
discouraged statements and uses of them, and this file is checked.
If you change the use of discouraged things, you will need to change
this file.
This makes it obvious when there is a change to anything discouraged
(triggering further review).
- Proposing specific changes.
Please propose specific changes as pull requests (PRs) against the
"develop" branch of set.mm, at:
https://github.com/metamath/set.mm/tree/develop
- Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/forum/#!forum/metamath.
(Contributed by DAW, 27-Dec-2016.) |