mmnotes.txt

mmnotes.txt - Notes
-------------------

These are informal notes on some of the recent proofs and other topics.



(7-Dec-2020) Partial unbundling of ax-7, ax-8, ax-9 (notes by Benoit Jubin)
---------------------------------------------------------------------------

This note discusses the recent partial unbundling of the axiom of
equality ax-7 and the predicate axioms ax-8 and ax-9 in set.mm.

The axiom of equality asserts that equality is a right-Euclidean binary relation
on variables:
  ax-7 |- ( x = y -> ( x = z -> y = z ) )

It can be weakened by adding a DV (disjoint variable) condition on x and y:
  ax7v |- ( x = y -> ( x = z -> y = z ) ) , DV(x,y)

and this scheme can itself be weakened by adding extra DV conditions:
  ax7v1 |- ( x = y -> ( x = z -> y = z ) ) , DV(x,y) , DV(x,z)
  ax7v2 |- ( x = y -> ( x = z -> y = z ) ) , DV(x,y) , DV(y,z)

We prove, in ax7, that either ax7v or the conjunction of ax7v1 and ax7v2
(together with earlier axioms) suffices to recover ax-7.  The proofs are
represented in the following simplified diagram (equid is reflexivity and
equcomiv is unbundled symmetry):

                 --> ax7v1 --> equid --
                /                      \
ax-7 --> ax7v --                        --> equcomiv --> ax7
                \                      /
                 --> ax7v2 ------------

The predicate axioms ax-8 and ax-9 can be similarly weakened, and the proofs are
actually simpler, now that the equality predicate has been proved to be an
equivalence relation on variables.  This is a general result.  If an n-ary
predicate P is added to the langugage, then one has to add the following n
predicate axioms for P:
  ax-P1 |- ( x = y -> ( P(x, z_2, ..., z_n) -> P(y, z_2, ..., z_n) ) )
  ...
  ax-Pn |- ( x = y -> ( P(z_1, ..., z_{n-1}, x) -> P(z_1, ..., z_{n-1}, y) ) )

Any of these axioms can be weakened by adding the DV condition DV(x,y), and it
is also sufficient to replace it by the conjunction of the two schemes:
  ax-Piv1 |- ( x = y ->
         ( P(z_1, ..., x, ..., z_n) -> P(z_1, ..., y, ..., z_n) ) ) , x fresh
  ax-Piv2 |- ( x = y ->
         ( P(z_1, ..., x, ..., z_n) -> P(z_1, ..., y, ..., z_n) ) ) , y fresh

where "fresh" means "disjoint from all other variables".  The proof is similar
to ax8 and ax9 and simply consists in introducing a fresh variable, say t, and
from
  |- ( x = t -> ( P(z_1, ..., x, ..., z_n) -> P(z_1, ..., t, ..., z_n) ) )
  |- ( t = y -> ( P(z_1, ..., t, ..., z_n) -> P(z_1, ..., y, ..., z_n) ) )
  |- ( x = y -> E. t ( x = t /\ t = y ) )
one can prove axPi.

Note that ax-7 can also be seen as the first predicate axiom for the binary
predicate of equality.  This is why it does not appear in Tarski's FOL system,
being a special case of his scheme ( x = y -> ( ph -> ps ) ) where ph is an
atomic formula and ps is obtained from ph by substituting an occurrence of x
for y.  The above paragraphs prove that this scheme can be weakened by adding
the DV condition DV(x,y).


===============================================================================


(21-Dec-2017) Processing of $[ ... $] file inclusions
-----------------------------------------------------

See also the following Google Group posts:

Description and example:
https://groups.google.com/d/msg/metamath/4B85VKSg4j4/8UrpcqR4AwAJ
"Newbie questions":
https://groups.google.com/d/msg/metamath/7uJBdCd9tbc/dwP2jQ3GAgAJ
"Condensed version of set.mm?":
https://groups.google.com/d/msg/metamath/3aSZ5D9FxZk/MfrFfGiaAAAJ
Original proposal:
https://groups.google.com/d/msg/metamath/eI0PE0nPOm0/8O9s1sGlAQAJ

(Updated 31-Dec-2017:  1. 'write source.../no_delete' is now 'write
source.../keep_includes'.   2. Added 'set home_directory' command; see
'help set home_directory'.)
(Updated 1-Jan-2018: Changed 'set home_directory' to 'set root_directory'.)
(Updated 1-Nov-2019: Added Google Group links above.)


1. Enhanced "write source" command
----------------------------------

The "write source" command in metamath.exe will be enhanced with a
"/split" qualifier, which will write included files separately.  The name
of the main (starting) file will be the "write source" argument (as it
is now), and the names of included files will be taken from the original
file inclusions.


2. New markup-type directives related to file inclusions
--------------------------------------------------------

Recall the file inclusion command, "$[ file.mm $]", given in the
Metamath spec.  The spec will be clarified so that, for basic .mm file
verification, this command should be ignored when it occurs inside of a
comment (and it should exist only at the outermost scope, as well).

The metamath.exe program will perform additional actions based on
special markup comments starting "Begin $[", "End $[", and "Skip $[".
These are not part of the Metamath spec and can be ignored by basic
verifiers.  The metamath.exe program allows the .mm file to be written
as a whole or to be split up into modules (with "write source ...
/split"), and this markup controls how the modules will be created.  In
particular, the markup allows us to go back and forth seamlessly between
split .mm files and a single unsplit .mm file.

These markup comments are normally created automatically whenever a .mm
file containing includes is written by "write source" without the
"/split" qualifier.  They can also be inserted by hand to delineate how
the .mm file should be split into modules.  They are converted back to
file inclusions when "write source" is used with the "/split" qualifier.

  "$( Begin $[ file.mm $] $)" - indicates where an included file starts
  "$( End $[ file.mm $] $)" - indicates where an included file ends
  "$( Skip $[ file.mm $] $)" - indicates there was a file inclusion
      at this location in the split files, that wasn't used because
      file.mm was already included earlier.

To summarize:  Split files will have only "$[ file.mm $]" inclusions,
like before.  An unsplit file will have only these three special comments.

(Per the Metamath spec, recall that when a file is included more than
once, only the first inclusion will happen with subsequent ones ignored.
This feature allows us to create subsections of a .mm file that are
themselves stand-alone .mm files.  We need the "Skip" directive to mark
the location of ignored inclusions.)

The "read" command will accept either a single file or split files or
any combination (e.g. when the main file includes a file that originally
also contained includes but was separately written without "/split").
Files can contain any combination of inclusion directives $[ $] and the
3 special comments, except that each "$( Begin $[..." must have a
matching "$( End $[...".


3. Behavior of "read" command
-----------------------------

The "read" command builds an internal buffer corresponding to an unsplit
file.  If "write source" does not have the "/split" qualifier, this
buffer will become the new source file.

When "read" encounters an inclusion command or one of the 3 special
comments, the following actions are taken:

Case 3.1:
---------

"$[ file.mm $]"

  If file.mm has not already been included, this directive will
  be replaced with
  "$( Begin $[ file.mm $] $) <file.mm content> $( End $[ file.mm $] $)"
  If file.mm doesn't exist, an error will be reported.

  If file.mm has already been included, this directive will
  be replaced with "$( Skip $[ file.mm $] $)".

Case 3.2:
---------

"$( Begin $[ file.mm $] $) <file.mm content> $( End $[ file.mm $] $)"

  If file.mm has not already been included, this directive will
  be left alone i.e. will remain
  "$( Begin $[ file.mm $] $) <file.mm content> $( End $[ file.mm $] $)"

  If file.mm has already been included, this directive including the
  <file.mm content> will be replaced with "$( Skip $[ file.mm $] $)".
  Before discarding it, <file.mm content> will be compared to the content
  of file.mm previously included, and if there is a mismatch, a warning
  will be reported.

Case 3.3:
---------

"$( Skip $[ file.mm $] $)"

  If file.mm has not already been included, this directive will
  be replaced with
  "$( Begin $[ file.mm $] $) <file.mm content> $( End $[ file.mm $] $)".
  If file.mm doesn't exist, an error will be reported.

  If file.mm has already been included, this directive will
  be left alone i.e. will remain "$( Skip $[ file.mm $] $)".

Error handling:
---------------

Any comments that don't exactly match the 3 patterns will be silently
ignored i.e. will act as regular comments.  For example,
"$( Skip $[ file.mm $)" and "$( skip $[ file.mm $] $)" will act as
ordinary comments.  In general, it is difficult to draw a line between
what is a comment and what is a markup with a typo, so we take the most
conservative approach of not tolerating any deviation from the patterns.
This shouldn't be a major problem because most of the time the markup
will be generated automatically.

Error messages will be produced for "$( Begin $[.." without a matching
"$( End $[..."  (i.e. with the same file name) and for included files
that are missing.


4. Behavior of "write source" command
-------------------------------------

When "write source" is given without the "/split" qualifier, the
internal buffer (as described above) is written out unchanged.  When
accompanied by the "/split" qualifier, the following actions are taken.

Case 4.1:
---------

"$[ file.mm $]"

  (This directive should never exist in the internal buffer unless
  there is a bug.)

Case 4.2:
---------

"$( Begin $[ file.mm $] $) <file.mm content> $( End $[ file.mm $] $)"

  file.mm will be created containing "<file.mm content>".  The directive
  will be changed to "$[ file.mm $]" in the parent file.

Case 4.3:
---------

"$( Skip $[ file.mm $] $)"

  This directive will be changed to "$[ file.mm $]" in the
  parent file.


5. File creation and deletion
-----------------------------

When "write source" is used with "/split", the main file and all
included files (if they exist) will be overwritten.  As with the
existing "write source", the old versions will be renamed with a "~1"
suffix (and any existing "~1" renamed to "~2" and so on through "~9",
whose existing version will be deleted).

When "write source" is used without "/split", the main file will be
overwritten, and any existing included files will be deleted.  More
precisely, by "deleted" we mean that an existing included file will be
renamed to "~1", any existing "~1" renamed to "~2", etc. until "~9",
which will be actually deleted.  The purpose of doing this is to prevent
accidental edits of included files after the main file is written
without "/split" and thus causing confusing diverging versions.

A new qualifier, "/no_versioning", will be added to "write source" to
turn off the "~n" versioning if it isn't wanted.  (Personally,
versioning has helped me recover from mistakes, and it's easy enough to
"rm *~*" at the end of a work session.)

Another new qualifier, "/keep_includes", will be added to "write source" to
turn off the file deletion when "/split" is not specified.  This can be
useful in odd situations.  For example, suppose main.mm includes abc.mm
and (stand-alone) def.mm, and def.mm also includes abc.mm.  When writing
out def.mm without "/split", by default abc.mm will be deleted, causing
main.mm to fail.  (Another way to recover is to rewrite def.mm with
"/split".  Or recover from abc.mm~1.)


5. Comments inside of includes
------------------------------

A comment inside of a file inclusion, such as
"$[ file.mm $( pred calc $) $]", will be silently deleted when it is
converted to the nonsplit version e.g. "$( Skip $[ file.mm $] $)".
Instead, put the comment before or after the inclusion, such as
"$[ file.mm $] $( pred calc $)".


6. Directories
--------------

Officially, directories aren't supported.  In practice, an included file
in a subdirectory can be specified by its path relative to the current
working directory (the directory from which metamath.exe is launched).
However, it is strongly recommended to use "-" in the file name rather
than directory levels, e.g. set-mbox-nm.mm, and this will be a
requirement for set.mm at this time.

A new command was added to change the working directory assumed by the
program.  See 'help set root_directory'.

Therefore, if included files are present, you shouldn't read set.mm from
another directory with a command such as "read test/set.mm", because
included files will _not_ be assumed to be in test/.  Instead, you
should either launch metamath.exe from the test/ directory, or you
should 'set root_directory test' so that you can type "read set.mm".
(Usually the error messages will let you know right away when your
included files aren't found where expected.)

The reason we don't just extract and use the "test/" prefix of set.mm
automatically is that if we decide to support directories relative to
the root directory, it will be legitimate to "read mbox/mbox-nm.mm",
where mbox/ is a project subdirectory under the root directory.

(End of "(21-Dec-2017) Proposed pocessing of $[ ... $] file inclusions")



(14-May-2017) Dates in set.mm
-----------------------------

Dates below proofs, such as "$( [5-Nov-2016] )$", are now ignored by
metamath.exe (version 0.143, 14-May-2017).  Only the dates in
"(Contributed by...", "(Revised by...", and "(Proof shortened by...)"
are used for the Most Recent Proofs page and elsewhere.

If a "(Contributed by...)" markup tag is not present in a theorem's
comment _and_ the proof is complete, then "save new_proof" or
"save proof" will add "(Contributed by ?who?, dd-mmm-yyyy.)" to the
theorem's comment, where dd-mmm-yyyy is today's date.

You can either change the "?who?" to your name in an editor, or you can
use the new command "set contributor" to specify it before "save
new_proof" or "save proof".  See "help set contributor".

If you are manually pasting proofs into set.mm, say from mmj2, then at
the end of the day you can run "save proof */compressed/fast" to add
missing contributor dates, followed by a global replacement of "?who?"
with your name.

"verify markup" in Version 0.143 includes some additional error checking,
which will cause warnings on older versions of set.mm.  However, it
no longer checks the dates below proofs.

The dates below proofs will be deleted soon in set.mm.  If someone
is using them outside of the metamath program, let me know so I can
postpone the deletion.  The old code to check them can be re-enabled
by uncommenting "#define DATE_BELOW_PROOF" in mmdata.h.

For converting old .mm files to the new "(Contributed by...)" tag, the
program has the following behavior:  the date used is the (earlier) date
below the proof if it exists, otherwise it is today's date.  Thus an old
.mm file can be converted with "read xxx.mm", "save proof
*/compressed/fast", and "write source xxx.mm".  Note that if there are
two dates below the proof, the second one is used, and the first one is
intended for a "(Revised by...)" or "(Proof shortened by...)" tag that
must be inserted by hand.  Searching for "] $) $( [" will identify cases
with two dates that must be handled with manual editing.

Tip:  if you want to revert to the old way of checking (and inserting)
dates below proofs, uncomment the "#define DATE_BELOW_PROOF" in mmdata.h
before compiling.


(11-May-2016) New markup for isolating statements and protecting proofs
-----------------------------------------------------------------------

(Updated 10-Jul-2016:  changed "show restricted" to "show discouraged";
added "set discouragement off"; see below.)

Two optional markup tags have been defined for use inside of
statement description comments:

  "(Proof modification is discouraged.)"
  "(New usage is discouraged.)"

The metamath program has been updated to discourage accidental proof
modification or accidental usage of statements with these tags.

These tags have been added to set.mm in the complex number construction,
axiomatic studies, and obsolete sections, as well as to specific
theorems that normally should be avoided or should not have their proof
changed for various reasons.  I also added them to some mathboxes (AS
and JBN) having theorems or notation that are unconventional.

Most users will never encounter the effect of "discouraged" tags since
they are in areas that are normally not touched or used.

"write source.../rewrap" will prevent the new tags from being broken
across lines.  This is intended to make editing tasks easier.  For
example, if you are doing a major revision to a "discouraged" section
such as the complex number construction, you can change the tags
temporarily (like changing "is discouraged" to "xx discouraged"
throughout the section) then change them back when done.

The following commands recognize "(Proof modification is discouraged.)":

  "prove", "save new_proof"

The following commands recognize "(New usage is discouraged.)":

  "assign", "replace", "improve", "minimize_with"

In the description below, the term "restricted" means a statement's
comment has one or both of the new tags.

Originally, I was going to ask an override question when encountering a
restricted statement, but I decided against that because prompts become
unpredictable, making the user's "flow" awkward and scripts more
difficult to write.  Instead, the user can specify "/override" in the
command to accomplish this.

A warning or error message is issued when there is a potential use or
modification of a restricted statement.  An error message means the
requested action wasn't performed (because the user didn't specify
"/override"), and a warning message means the action was performed but
the user should be aware that the action is "discouraged".  To make the
messages more visible, they have a blank line before and after them, and
they always begin with ">>>" for emphasis.

The behavior of individual commands is as follows.

  "prove" - Without "/override", will give an error message and prevent
  entering the Proof Assistant when a proof is restricted i.e. when the
  statement's comment contains "(Proof modification is discouraged.)".
  With "/override", a warning is issued, but the user may enter the Proof
  Assistant.

  "save new_proof" - Without "/override", will give an error message and
  prevent saving.  With "/override", a warning is issued, but the save is
  allowed.

  "assign", "replace" - Without /override, will give an error message and
  not allow an assignment with a restricted statement i.e. when the
  assigned statement's comment contains "(New usage is discouraged.).
  With /override, will give a warning message but will do the assignment.

  "improve", "minimize_with" - Without /override, will silently skip
  restricted statements during their scans.  With /override, will consider
  all statements and will give a warning if any restricted statements are
  used.

Here is an example of a session with idALT, which has the comment tag
"(Proof modification is discouraged.)".

    MM> prove idALT

    >>> ?Error: Modification of this statement's proof is discouraged.
    >>> You must use PROVE ... / OVERRIDE to work on it.

    MM> prove idALT/override
    Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to exit.
    You will be working on statement (from "SHOW STATEMENT idALT"):
    101 idALT $p |- ( ph -> ph ) $= ... $.
    Note:  The proof you are starting with is already complete.

    >>> ?Warning: Modification of this statement's proof is discouraged.

    MM-PA> minimize_with id
    Bytes refer to compressed proof size, steps to uncompressed length.
    Proof of "idALT" decreased from 51 to 9 bytes using "id".
    MM-PA> save new_proof/compressed

    >>> ?Error: Attempt to overwrite a proof whose modification is discouraged.
    >>> Use SAVE NEW_PROOF ... / OVERRIDE if you really want to do this.

    MM-PA> save new_proof/compressed/override

    >>> ?Warning: You are overwriting a proof whose modification is discouraged.

    The new proof of "idALT" has been saved internally.
    Remember to use WRITE SOURCE to save changes permanently.
    MM-PA>

Here is an example of a session trying to use re1tbw2 (ax-1 twin) which
has "(New usage is discouraged.)".

    MM> prove r19.12
    Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to exit.
    You will be working on statement (from "SHOW STATEMENT r19.12"):
    $d x y $.  $d y A $.  $d x B $.
    7283 r19.12 $p |- ( E. x e. A A. y e. B ph -> A. y e. B E. x e. A ph ) $= ...
          $.
    Note:  The proof you are starting with is already complete.
    MM-PA> show new_proof /from 70/to 70
     70     ralrimi.2=ax-1    $a |- ( E. x e. A A. y e. B ph -> ( y e. B -> E. x e.
                                                                 A A. y e. B ph ) )
    MM-PA> delete step 70
    A 12-step subproof at step 70 was deleted.  Steps 70:112 are now 59:101.
     59        ralrimi.2=?       $? |- ( E. x e. A A. y e. B ph -> ( y e. B -> E. x
                                                              e. A A. y e. B ph ) )
    MM-PA> assign last re1tbw2

    >>> ?Error: Attempt to assign a statement whose usage is discouraged.
    >>> Use ASSIGN ... / OVERRIDE if you really want to do this.

    MM-PA> assign last re1tbw2/override

    >>> ?Warning: You are assigning a statement whose usage is discouraged.

    MM-PA> undo
    Undid:  ASSIGN LAST re1tbw2 / OVERRIDE
     59        ralrimi.2=?       $? |- ( E. x e. A A. y e. B ph -> ( y e. B -> E. x
                                                              e. A A. y e. B ph ) )
    MM-PA> improve all
    A proof of length 12 was found for step 59.
    Steps 59 and above have been renumbered.
    CONGRATULATIONS!  The proof is complete.  Use SAVE NEW_PROOF to save it.
    Note:  The Proof Assistant does not detect $d violations.  After saving
    the proof, you should verify it with VERIFY PROOF.

Note that "improve all", which scans backwards, skipped over re1tbw2 and
picked up ax-1:

    MM-PA> show new_proof /from 70/to 70
     70     ralrimi.2=ax-1    $a |- ( E. x e. A A. y e. B ph -> ( y e. B -> E. x e.
                                                                 A A. y e. B ph ) )
    MM-PA> undo
    Undid:  IMPROVE ALL
     59        ralrimi.2=?       $? |- ( E. x e. A A. y e. B ph -> ( y e. B -> E. x
                                                              e. A A. y e. B ph ) )

With "/override", it does not skip re1tbw2 but assigns it since it is the
first match encountered (before ax-1 in the backward scan):

    MM-PA> improve all/override

    >>> ?Warning:  Overriding discouraged usage of statement "re1tbw2".

    A proof of length 12 was found for step 59.
    Steps 59 and above have been renumbered.
    CONGRATULATIONS!  The proof is complete.  Use SAVE NEW_PROOF to save it.
    Note:  The Proof Assistant does not detect $d violations.  After saving
    the proof, you should verify it with VERIFY PROOF.
    MM-PA>


If you want to see which statements in a specific section have
restrictions, use "search.../comment" e.g.

    MM> search ax-1~bitr 'is discouraged'/comment
    101 idALT $p "...uted by NM, 5-Aug-1993.) (Proof modification is discouraged.)"
    659 dfbi1gb $p "...ry Bush, 10-Mar-2004.) (Proof modification is discouraged.)"

Program note:  The new markup tags are looked up via the function
getMarkupFlag() in mmdata.c.  Since string searches are slow, the result
of the first search in each statement comment is memoized (saved) so
that subsequent searches can be effectively instant.

Two commands were added primarily for database maintenance:


"show discouraged" will list all of the
statements with "is discouraged" restrictions and their uses in the
database (in case of discouraged usage) or the number of steps (in case
of a proof whose modification is discouraged).  It is verbose and
primarily intended to assist a script to compare a modified database
with an earlier version.  It will not be of interest to most users.

    MM> help show discouraged
    Syntax:  SHOW DISCOURAGED

    This command shows the usage and proof statistics for statements with
    "(Proof modification is discouraged.)" and "(New usage is
    discouraged.)" markup tags in their description comments.  The output
    is intended to be used by scripts that compare a modified .mm file
    to a previous version.

    MM> show discouraged
    ...
    SHOW DISCOURAGED:  Proof modification of "tru2OLD" is discouraged (9 steps).
    SHOW DISCOURAGED:  New usage of "tru2OLD" is discouraged (0 uses).
    SHOW DISCOURAGED:  New usage of "ee22" is discouraged (2 uses).
    SHOW DISCOURAGED:  "ee22" is used by "ee21".
    SHOW DISCOURAGED:  "ee22" is used by "ee33".
    ...

"set discouragement off" will turn off the blocking of commands caused
to "...is discouraged" markup tags.  It does the equivalent of always
specifying "/override" on those commands.  It is intended as a
convenience during maintenance of a "discouraged" area of the database
that the user is very familiar with, such as the construction of complex
numbers.  It is not recommended for most users.

    MM> help set discouragement
    Syntax:  SET DISCOURAGEMENT OFF or SET DISCOURAGEMENT ON

    By default this is set to ON, which means that statements whose
    description comments have the markup tags "(New usage is discouraged.)"
    or "(Proof modification is discouraged.)" will be blocked from usage
    or proof modification.  When this setting is OFF, those actions are no
    longer blocked.  This setting is intended only for the convenience of
    advanced users who are intimately familiar with the database, for use
    when maintaining "discouraged" statements.  SHOW SETTINGS will show you
    the current value.

    MM> set discouragement off
    "(...is discouraged.)" markup tags are no longer honored.

    >>> ?Warning: This setting is intended for advanced users only.  Please turn
    >>> it back ON if you are not intimately familiar with this database.

    MM> set discouragement on
    "(...is discouraged.)" markup tags are now honored.



(10-Mar-2016) metamath program version 0.125
--------------------------------------------

The following changes were made:

1. A new qualifier, '/fast', was added to 'save proof' and 'show proof'.
See the 9-Mar-2016 entry below for an application.

2. Long formulas are no longer wrapped by 'write source.../rewrap' but
should be wrapped by hand to fit in less than 80 columns.  The wrapping
was removed because a human can better judge where to split formulas for
readability.  Comments and indentation are still reformatted as before.

3. Added space between adjacent "}" and "{" in the HTML output.

4. A bug in the /explicit/packed proof format was fixed.  See 'help save
proof' for a list of all formats.

5. To reference a statement by statement number in 'show statement',
'show proof', etc., prefix the number with "#".  For example, 'show
statement #58' will show a1i.  This was added to assist program
debugging but may occasionally be useful for other purposes.  The
complete list of statement lookup formats in shown in 'help search'.





(9-Mar-2016) Procedure to change a variable name in a theorem
--------------------------------------------------------------

The metamath program has been updated in Version 0.125 (10-Mar-2016)
with a new qualifier, '/fast', that merely changes the proof format
without compressing or uncompressing the proof.  This makes format
conversions very fast for making database changes.  The format of the
entire database can be changed from /compressed to /explicit/packed, and
vice-versa, in about a minute each way.

The /explicit/packed format is described here:
https://groups.google.com/d/msg/metamath/xCUNA2ttHew/RXSNzdovBAAJ
You can also look at 'help save proof' in the metamath program.

The basic rules we will be using are:

1. When the proofs are saved in explicit format, you can change $f and
   $e order.

2. When the proofs are saved in compressed format, you can change the
   name of a variable to another if the two variables have adjacent $f's.


======= The conversion procedure: =======

To retrofit the new symbol variables added to set.mm to your theorems,
for example changing "P" to ".+", you can use the following procedure.

First, save all proofs in explict format:

./metamath set.mm
MM> save proof */explicit/packed/fast
MM> write source set.mm
MM> exit

    (Hint: 'save proof *' lists all proofs it is saving.  To suppress this
    output, type "q" at the scrolling question after the first page.  It will
    not really quit; instead, the proof saving will complete silently.)

Next, edit set.mm to place the $f for the "P" adjacent to (either immediately
before or immediately after) the $f for the ".+".  Then resave the proofs
in compressed format:

./metamath set.mm
MM> save proof */compressed/fast
MM> write source set.mm
MM> exit

In your text editor, substitute P for .+ in the theorem you want to
change.  Make sure you include the $p and any $d and $e statements
associated with the theorem.  If the $d and $e statements affect other
theorems in the same block, you will also have to make the P to .+
substitution in those $p's as well.

You are now done with the change.  However, you probably want to restore
the original $f order to make your database compatible with the standard
set.mm.  You can postpone doing this until you have finished making all
of your variable name changes as above.  First, save all proofs in
explict format:

./metamath set.mm
MM> save proof */explicit/packed/fast
MM> write source set.mm
MM> exit

Next, edit set.mm to restore the original $f order. It may be easiest
just to copy and paste the $f section from the standard set.mm.

Finally, you probably want to save all proofs in compressed format
since the file size will be smaller and easier to work with:

./metamath set.mm
MM> save proof */compressed/fast
MM> write source set.mm
MM> exit




(28-Feb-2015) Stefan O'Rear's notes on recent proofs
----------------------------------------------------

# New definitions

df-har: The Hartogs function, which restricts to cardinal successor on initial
ordinals.  Since the latter is somewhat important in higher set theory, I
expect this to get used a bit.  One question about the math symbol: standard
notation for ( har ` x ) is \aleph(x), but I didn't want to overlap the math
symbol used for df-aleph, so I left this as Latin text for now.

df-nzr: Nonzero rings:  A number of properties of linear independence fail for
the zero ring, so I gave a name to all others.

df-wdom: Weak dominance, i.e. dominance considered using onto functions instead
of 1-1.  The starred symbol seems to be relatively standard.  I'm quite pleased
with how ~ wdomd turned out.

df-lindf,df-linds: Definition of a linearly independent family resp. set of
vectors in a module.  Was initially trying to do this with just one definition
but two seems to work much better with corner cases.

# Highlights

hsmex: The class of sets hereditarily smaller than a given set exists; a
formalization of the proof in
http://math.boisestate.edu/~holmes/holmes/hereditary.pdf .  With AC this is
simpler as it follows from the existence of arbitrarily large regular
cardinals.  Intermediate steps use Hartogs numbers and onto functions quite
heavily, so df-har and df-wdom were added to support this; it also uses
iterated unions (ituni*) and order types (otos*), but the former is not a
standard concept and the latter has several definitional issues, so both are
temporary definitions for now.

marypha1,marypha2: P. Hall's marriage theorem, a surprisingly annoying
combinatorial result which is in the dependency chain for the vector space
dimension theorem.

kelac2,dfac21i: Recover the axiom of choice from Tychonoff's theorem (which we
don't have yet).  Required a number of additional results on box-shaped subsets
of cross products, such as boxriin and boxcuts.

In an attempt to unify the empty and nonempty cases, we are now considering
"relative intersections" of the form ( A i^i |^| B ) and ( A i^i |^|_ x e. I B
), assuming that all elements of B are subsets of A, we get the type-theoretic
behavior that an empty intersection is the domain of discourse of a particular
structure, and not the class _V.  Several theorems are added to support this
usage, and the related ( fi ` ( { A } u. B ) ).

lmisfree: There have been a number of recent questions about the correctness of
various definitions of basis.  This hopefully helps to clarify the situation by
proving my earlier conjecture that what we have is exactly what is needed to
witness an isomorphism onto a free module.  Notable intermediate steps here are
islbs (splits our notion of basis into the spanning and independence parts),
islindf4 ("no nontrivial representations of zero"), and lbslcic (only the
cardinality of the index set matters).

Along the way this required separating independent sets and families from the
previous notion of bases; a new "independent sets and families" section
contains the basic properties there.

domunfican: A cancellation law for cardinal arithmetic which came up in
marypha1 but may be of independent interest.

# Possible future directions

Probably a lot of polynomial stuff soon, with maybe a vector space dimension
theorem thrown in for good measure.  A rough priority order, and less final the
farther you go:

1. Fraction ring/field development; rational function fields

2. Recursive decomposition of polynomials

3. AA = ( CCfld IntgRing QQ) and re-re-define _ZZ = ( CCfld IntgRing ZZ ) $.

4. Relating integral elements to finitely generated modules.

5. Hilbert basis theorem

6. Integral closures are rings; aaaddcl/aamulcl.

7. Cowen-Engeler lemma (Schechter UF2); finite choice principles from
ultrafilters

8. M. Hall's marriage theorem

9. The next natural property of independent sets: the exchange theorem and the
finite dimension theorem for vector spaces

10. General dimension theorem from ultrafilters

11. Gauss' lemma on polynomials; ZZ = ( _ZZ i^i QQ )

12. Polynomial rings are UFDs




(17-Feb-2015) Mario Carneiro's notes on recent proofs
-----------------------------------------------------

Notes:

ifbothda:  Common argumentation style for dealing with if, may be good
to know

disjen, disjenex:  When constructing the reals, we needed extra elements
not in CC for +oo and -oo, and for that purpose we used ~P U. CC, ~P ~P
U. CC.  This works great if you only want a few new elements, but for
arbitrarily many elements this approach doesn't work.  So this theorem
generalizes this sort of construction to show that you can build an
arbitrarily large class of sets disjoint from a given base class A.

domss2, domssex2, domssex:  One application of disjen is that you can
turn any injection F :  A -1-1-> B around into G :  B -1-1-onto-> C
where A C_ C, and which is the identity on elements of A. I'm thinking
about taking advantage of this in the field extensions for Cp, since
this way you don't need a canonical injection into the extension but can
actually build the extension around the original field so that it is
literally a subfield of the extension.  It could also be used, if
desired, as a means of building compatible extensions in the
construction of the reals (so that om = NN0).

ghmker:  the kernel of a group homomorphism is a normal subgroup

crngpropd, subrgpropd, lmodpropd, lsspropd, lsppropd, assapropd:
property theorems

tgcnp, tgcn, subbascn:  checking continuity on a subbase or basis for a
topology

ptval:  Product topology.  Important theorems are pttop, ptuni, ptbasfi,
ptpjcn, ptcnp, ptcn.

pt1hmeo, ptunhmeo:  combining these judiciously allows you to show that
indexed product topologies are homeomorphic to iterated binary product
topologies, which for example can be used to prove ptcmpfi, which is
basically Tychonoff for finite index sets.

metdsval, metds0, metdseq0, metdscn:  Properties of the function d(x,A)
which gives the distance from a point to a nonempty set.

lebnum, lebnumii:  The Lebesgue number lemma, a nice result about open
covers in a metric space and a key step in the proof of the covering map
lifting theorem.

df-pi1:  Converted to structure builders, eliminated df-pi1b

pcopt2, pcorev2:  commuted versions of pcopt, pcorev.  Took me a while
to realize that left identity -> right identity and left inverse ->
right inverse does not actually follow from the other proofs, since at
this level we have only a groupoid, not a group.

pi1xfr:  A path induces a group isomorphism on fundamental groups.

df-dv:  Added a second argument to _D for the ambient space.  There is a
drop-in replacement from ( _D ` F ) to ( RR _D F ), and ( CC _D F ) is
the complex derivative of F. The left argument can be any subset of CC,
but since the derivative is not a function if S has isolated points, I
restrict this to S e. { RR , CC } for the primary convenience theorems.

df-lgs:  The Legendre symbol, a tour-de-force of something interesting
coming out of a ridiculous amount of case analysis.  This definition is
actually the Kronecker symbol, which extends the Jacobi symbol which
extends the Legendre symbol to all integers.  The main theorem to be
done in this area is of course the law of quadratic reciprocity, but
currently I'm stuck proving Euler's criterion which is waiting on
polynomials over Z/pZ.  So far the basic theorems prove that ( A /L N )
e. {-1,0,1}, ( A /L N ) =/= 0 iff A,N are coprime, and it is
distributive under multiplication in both arguments.

kur14:  The Kuratowski closure-complement theorem, which I mentioned in
another email.

df-pcon, df-scon:  Path connected spaces and simply connected spaces.  I
hope to use SCon to prove some kind of Cauchy integral theorem, but
we'll see.

txpcon, ptpcon:  products of path-connected spaces are path-connected
(ptpcon is actually an AC-equivalent)

pconpi1:  The fundamental groups of a path-connected space are
isomorphic

sconpi1:  A space is simply connected iff its fundamental group is
trivial

cvxpcon, cvxscon:  a convex subset of CC is simply connected blcvx,
blscon:  a disk in CC is convex and simply connected

df-cvm:  Definition of a covering map.

cvmlift:  The Path Lifting Theorem for covering maps

df-rpm, df-ufd:  define a prime element of a ring and a UFD

psr1val:  Basic theorems for univariate polynomials




(9-Jan-2015) mpbi*an* hypothesis order change
---------------------------------------------

At the suggestion of a couple of people, I changed the hypothesis order
in mpbi*an* (7 theorems) so that the major hypothesis now occurs last
instead of first, in order to make them less annoying to use.  The
theorems changed were:

  mpbiran mpbiran2 mpbir2an mpbi2and mpbir2and mpbir3an mpbir3and

This change affects over 1000 proofs.  The old versions are still there
suffixed with "OLD" and will remain for 1 year.

To update your mathbox etc.,

1. Make sure your mathbox is compatible with the the set.mm just prior
to this change, temporarily available here:

  https://us.metamath.org/metamath/set.mm.2015-01-08.bz2

1. In a text editor, suffix all references to mpbi*an* in your proofs
with OLD (e.g. mpbiran to mpbiranOLD).  This will make your proofs
compatible with the current set.mm.

2. Update the current set.mm with your mathbox.

3. For each proof using mpbi*an*OLD, do the following in the metamath
program:

./metamath set.mm
...
MM> prove abc
MM-PA> minimize mpbi*an*/except *OLD/allow_growth
MM-PA> save new_proof/compressed
MM-PA> exit
..,
MM> write source set.mm
MM> exit


(11-Jun-2014) Mario Carneiro's revisions
----------------------------------------

Mario Carneiro finished a major revision of set.mm. Here are his notes:

    syl3anbrc, mpbir3and: simple logic stuff
    fvunsn: eliminate D e. _V hypothesis
    caoprcld, caoprassg, caoprcomg: deduction form
    winafpi: dedth demonstration
    avglt1, avglt2, avgle1, avgle2: average ordering theorem collection
    peano2fzr: recurring lemma
    fzfid: very common use due to deduction-form sum theorems
    seqcl2 thru seq1p: deduction form
    sercaopr2: the 'correct' general form of sercaopr
    seqz: absorbing element in a sequence (note that there are now theorems
    seq1 and seqz, since the old tokens are gone)

    cseq1 thru ser1add, seq1shftid, seqzm1 thru ser0p1i: deleted - all the
    theorems in this section have equivalents in the new seq section,
    although it is often a 4-5 to 1 mapping, since the deduction framework
    makes it easier to have ease of use and full generality at the same time

    df-exp: revised to include negative integer exponents. The theorems for
    nonnegative exponents are exactly the same as before, but for negative
    exponents ( A ^ -u N ) we need the assumption A=/=0 so that we don't
    divide by zero. Relevant new theorems are:
    expneg, expneg2, expnegz, expn1: definition of ( A ^ -u N ) is 1 /  ( A
    ^ N )
    expcl2lem: closure under positive and negative exponents
    rpexpcl: the only existing closure theorem that was changed as a result
    reexpclz, expclz: closure for reals and complexes to integer exponents,
    when the argument is nonzero
    1exp, expne0i, expgt0, expm1, expsub, expordi thru expword2i:
    generalized to negative exponents
    mulexpz, expaddz: the main "hard" theorems in this section, generalizing
    the exponent addition laws to integers. The main reason to keep both the
    new versions and the old (mulexp, expadd) is because the new theorems
    generalize the exponent at the cost of the extra hypothesis A =/= 0.

    discr, discr1: deduction form, generalized to all A e. RR, shortened proof
    rexanuz: the new upper integer form of cvganz, and used as the basis for
    most other manipulations on upper integer sets to replace cau* and cvg*
    theorems
    rexfiuz: finite set generalization of rexanuz
    rexuz3: turn an upper integer quantifier E. j e. ZZ A. k e. (ZZ>= ` j)
    into a restricted upper integer quantifier E. j e. NN A. k e. (ZZ>= ` j)
    rexanuz2: restricted upper integer quantifier form of rexanuz
    r19.29uz: looks sort of like r19.29, but for upper integer quantifiers
    r19.2uz: sort of like r19.2z
    cau3lem: useful on its own for being sufficiently general for both real
    cauchy sequences and cauchy sequences in metric spaces
    cau3: convert a cauchy sequence from two-quantifier form into
    three-quantifier form (this last one is useful because it is compatible
    with rexanuz2)
    cau4: use cau3 to change the base of the cauchy sequence definition
    caubnd2: a cauchy sequence is eventually bounded. This is sufficient for
    most proofs, but I went ahead and proved the original form caubnd as well
    caurei, cauimi, ser1absdifi: deleted since they weren't being used and
    there wasn't a good reason to reformulate them in view of the much more
    general theorems to come
    bcval5: write out the numerator of a binomial coefficient as a sequence
    with arbitrary start and end
    bcn2: N choose 2
    fz1iso: any finite ordered set (in particular, a finite set of integers)
    is isomorphic to a one-based finite sequence
    seqcoll: lemma for soundness of df-sum, stated for general sequences

    clim thru climcn2: deduction form
    addcn2, subcn2, mulcn2: the old approach was a bit backward, using a
    direct proof of continuity to prove that addition etc. is sequentially
    continuous, then using sequential continuity to prove continuity through
    bopcn, whose proof requires CC. To avoid this, now we prove continuity
    directly (writing out the definition since the other continuity theorems
    aren't ready yet) and use this to prove that limits are preserved in
    climadd etc.
    climadd thru serf0: deduction form

    df-sum: a new much more general definition of summation, allowing the
    index set to be either a finite set or a lower bounded subset of the
    integers
    sum2id: assume the argument to a sum is a set
    sumfc: change bound variable
    summo: a 'big' theorem, proving that the new definition is well-defined
    zsum, isum, fsum: the spiritual equivalents to the old dffsum, dfisum,
    showing the definition on upper integer subsets, upper integer sets, and
    nonempty finite sets, respectively.
    sum0: sum of the empty set is zero
    sumz: sum of zero on any summable index set is zero
    fsumf1o: finite sum is unchanged under a bijection
    sumss, fsumss, sumss2: add zeroes to a finite sum to enlarge the index set
    fsumcvg, fsumcvg2, fsumcvg3: a finite sum is convergent (useful for
    treating a finite sum using infinite sum theorems)
    fsumsers: sum over a subset of a finite sequence
    fsumser: sum over a finite sequence (this is the most direct equivalent
    to the old dffsum)
    fsumcl2lem: a set that is closed under addition is closed under nonempty
    finite sums
    fsumcllem: a set containing zero and closed under addition is closed
    under finite sums
    fsumcl, fsumrecl, fsumrpcl: closure under reals, complexes, positive reals
    sumsn, sumsns: sum over a singleton
    fsumm1: break off the last term (this is more general than fsump1,
    because this includes the case where the smaller sum is empty)
    isumclim, isumclim2: relation between infinite series and convergent
    sequences
    sumsplit: generalized for subsets of the integers (but not that useful
    in practice)
    fsum2d, fsumcom2: generalization of fsumcom to sum over non-rectangular
    regions
    fsumxp: sum over a cross product (this theorem has no equivalent under
    the old system)
    abscvgcvg: absolutely convergent implies convergent

    binom: shortened the proof
    divcnv: generalized reccnv for convenience
    arisum: now adds up 1...N instead of 0...N
    expcnv: shortened the proof
    geoser: now sums 0...N-1 since it makes the formula nicer
    cvgrat: shortened the proof
    fsum0diag: shortened the proof (corollary of fsumcom2)
    mertens: generalization of the old proof of efadd

    elcncf2: commuted arguments to elcncf
    cncfco: composition of continuous functions

    ivth, ivth2, ivthle2: shortened the proof (the 'le' version allows the
    value to be equal to one of the endpoints)
    df-tan, df-pi: revised to use shorter dummy-free expressions
    efcllem, efge2le3, efadd, eftlub, eirr, efcn, reeff1o: shortened the proof
    demoivre: swapped with *ALT version and extend to negative exponents
    acdc3lem thru acdcALT: deleted (use axdc* theorems)

    ruc: shortened the proof (yes, this is the second time I've shortened
    ruc. This time I followed the same approach as the original ruc, rather
    than the alternative proof via rpnnen.)

    dvdseq: simplifies some divisibility proofs
    bezout: imported from my mathbox
    dvdsgcd, mulgcd: new proof from bezout
    algrf thru eucalg: deduction form
    df-pc: imported prime count function from my mathbox
    1arith: new proof using the prime count function. This proof has a
    considerably different statement from the original proof, so it is
    perhaps debatable whether this is still the "fundamental theorem of
    arithmetic", but this is the easiest to prove given the tools already
    available, and it is also easier to use in future proofs (although the
    more direct statements pc11 and pcprod are probably more directly
    applicable).

    dscmet, dscopn: the discrete metric generates the discrete topology

    cnmptid thru cnmptcom: This new set of theorems, with names cnmpt*, is
    designed for use in quickly building up continuous functions expressed
    in the mapping notation. The naming convention takes the first number to
    be the number of arguments to the mapping function, and the second
    number is the number of arguments to the function that is applied at the
    top level. If the function that is applied is an atomic operation, a *f
    is suffixed. The "base case" continuous functions are given by cnmptc,
    cnmptid (for constants and the identity), and cnmpt1st, cnmpt2nd for
    two-argument mapping functions. sincn or ipcn provide a good demonstration.

    df-lm: the definition has been changed to represent convergence with
    respect to a topology, rather than a metric space. The old definition
    can be recovered as ( ~~>m ` D ) = ( ~~t ` ( MetOpen ` D ) ), although
    the functions are now required to be partial functions on CC rather than
    just subsets of ( X X. CC ).
    df-cau: the definition has been abbreviated, and the functions are again
    required to be partial functions on CC.
    df-cmet: the definition has been abbreviated
    lmbr, lmbr2, lmbrf: these have the same names as the old versions, but
    now apply to topological convergence; the metric convergence
    equivalents are now called lmmbr, lmmbr2, lmmbrf.
    lmbr thru iscauf: deduction form
    lmmo: was lmuni, now applies to hausdorff spaces instead of metric spaces
    lmcls, lmcld: was previously part of metelcls, but this direction
    doesn't need choice or metric spaces, so it is now separate
    lmcnp, lmcn, lmcn2: continuous functions preserve limits
    metdcn: a metric is a continuous function in its topology
    addcn, subcn, mulcn: modified to use topological product
    fsumcn, fsum2cn: adjusted for compatibility with cnmpt*
    expcn, divccn, sqcn: powers and division by a constant are continuous
    functions (there is still no proof that division is continuous in the
    second argument, but we haven't needed it yet).
    isgrp2d: deduction form of isgrp2i
    ghgrp, ghablo: deduction form of ghgrpi
    ghsubg, ghsubablo: deduction form of ghsubgi

    vacn, smcn, ipcn: generalized one-argument continuity proofs to joint
    continuity (and shortened the proof)
    minvec: shortened the proof

    sincn, coscn: shortened the proof
    pilem*: shortened the proof
    efhalfpi, efipi, sinq12ge0, cosq14ge0: more trig theorems
    sineq0: combined the old sineq0 with sineq0re and sinkpi
    coseq1, efeq1: similar to sineq0
    cosord: generalize cosh111 to [0, pi]
    recosf1o, resinf1o: useful for defining arcsin, arccos
    efif1o, eff1o: shortened the proof
    df-log: abbreviated definition, changed principal domain to (-pi, pi]
    instead of [-pi, pi) - the result is that now log(-1) = i pi rather than
    -i pi, in keeping with standard principal value definition
    relogrn, logrncn: closure for ran log
    logneg, logm1: log of a negative number
    explog, reexplog, relogexp: generalized to negative integer powers

    df-hlim, df-hcau: abbreviated definition
    hvsubass, hvsub32, his35: vector identities
    hlimadd: limit of a sequence of vector sums
    occon3: generalize chsscon2i and prove without CC
    shuni: generalize chocunii to any subspaces with trivial intersection
    pjthmo: the uniqueness part of pjth, which needs no choice
    occl: shortened the proof
    pjth: shortened the proof (from 45 lemmas to 2!) using minvec
    pjpreeq, pjpjpre: a weak version of pjeq that allows some usage of the
    projection operator without assuming the projection theorem
    chscl: the majority of the proof of osum is now here; this proof does
    not need CC even though osum does
    hhcno, hhcnf: relate ConOp and ConFn to the topology Cn predicate
    imaelshi, rnelshi: the image of a subspace under a linear function is a
    subspace
    hmopidmchi, hmopidmpji: the class abstraction { x e. ~H | ( T ` x ) = x
    } is the same as the range of T

    - My mathbox -
    bclbnd: deleted because it is obsolete
    efexple: generalized to N e. ZZ

    - Scott Fenton's mathbox -
    sinccvg: (named for the sinc function: sinc x = sin x / x) shortened the
    proof
    trirecip: shortened the proof
    df-bpoly: abbreviated definition and eliminated the if statement, since
    now you can sum the empty set
    bpolydif: shortened the proof

    - FL's mathbox -
    cmpbvb, fopab2ga, fopab2a, cmpran, riecb, riemtn, fopabco3, dffn5b:
    deleted as duplicates
    df-pro, df-prj: mapping definition
    ispr1, prmapcp2, valpr, isprj1, isprj2: changed P e. Q to P e. V
    prmapcp3: corollary of prmapcp2
    hbcp: imported as hbixp1
    cbicp: shortened the proof
    iserzmulc1b thru seq0p1g: deleted since seqz is gone (there are already
    equivalents for seq)
    df-prod (prod3_ token): inlined into the new df-prod (previously
    df-prod2), since it's not that helpful to have two definitions that are
    so similar
    (I don't really see the point of this definition to begin with - it's
    very similar in scope to what seq does. It's not possible to give it a
    finite sets definition like the new sum_ , since it is defined on
    arbitrary magmas, so it reduces to basically the same as seq, except
    that it is defined on the empty set as well. I thought about moving this
    definition to seq as well, say by having ( seq M ( P , F ) ` ( M - 1 ) )
    = ( Id ` P ), but this makes seqfn a bit asymmetrical and it can't
    really be used to generalize things like seqsplit because you'd want the
    original version anyway, and the sethood requirement is usually a
    distraction anyway.)
    prod0: renamed from valproemset (another indication that the extension
    to the empty set is not that useful is that this theorem is never used)
    prodeq2, prodeq3: exchanged these labels for consistency with ordering
    in the token and with prodeq2d, prodeq3d
    prodeqfv thru fprod2: miscellaneous updates for the modified definition
    clfsebs, fincmpzer, fprodadd, isppm, seqzp2, fprodneg, fprodsub:
    shortened the proof
    df-expsg, df-mmat: these definitions are never used, but I updated them
    to use seq and prod_ in place of seq1 and prod3_
    cmphmp, idhme, cnvhmph, hmphsyma, hmphre, hmeogrp, homcard, eqindhome:
    shortened the proof
    eltpt: imported
    ttcn, ttcn2: use cnmpt* instead
    exopcopn, topgrpsubcn, trhom, ltrhom, cntrset: shortened the proof
    uncon: this is a special case of iuncon - any collection of connected
    subsets that share a point is connected

    - Jeff Madsen's mathbox -
    acdcg, acdc3g, acdc5g: I deleted these as part of the acdc* cleanup, but
    the existing axdc* theorems assume the base set is a set in the
    hypotheses - if this becomes an issue later, these theorems should be
    updated, rather than making *g theorems. As it is, there is no need for
    them yet.
    sdc: shortened the proof
    seq1eq2: seq1 is going away, use seqfveq
    fsumlt, fsumltisum, fsumleisum: generalized to finite sets and imported
    csbrn, trirn, mettrifi: shortened the proof
    mettrifi2: this is basically the same as the new mettrifi
    geomcau, caushft, caures: deduction form, shortened the proof
    metdcn: imported and generalized to joint continuity

    cnmptre, iirevcn, iihalf1cn, iihalf2cn, iimulcn: continuity base cases
    for path homotopy proofs
    elii1, elii2: commonly used lemma in path homotopy proofs
    cncfco: imported
    cnimass: same as elsubsp2
    cnres: same as elsubsp
    cnmpopc: generalized piececn to two-arg functions and adjusted for
    compatibility with cnmpt*

    ishomeo2: this is the same as the new ishomeo
    hmeocn: corollary of hmeobc
    hmeocnv: the same as cnvhmpha
    ctlm thru lmtlm: deleted/imported this whole section because this is
    essentially the same as the new topological limit relation, and all
    theorems here are already represented in main set.mm now
    txcnoprab: the same as cnmpt2t
    txsubsp: imported
    cnresoprab thru cnoprab2c: this set of theorems is very similar to the
    new cnmpt* section (and indeed inspired that section), so they are
    deleted as duplicates
    txmet, txcc: imported
    addcntx thru mulcntx: the same as the new addcn thru mulcn

    bfp: shortened the proof
    df-rrn, rrnval: use mapping notation
    rrnmet, rrncms: shortened the proof

    df-phtpy, df-phtpc: use mapping notation
    phtpyfval thru phtpyco: deduction form, shortened the proof
    isphtpc2: merge with isphtpc
    reparpht: shortened the proof
    df-pco: use mapping notation, also extend the domain to functions that
    may not line up at the endpoints, for simplicity of definition (it
    doesn't affect pi1gp, because that is restricted further to functions
    with the same start and end point anyway)
    pcofval thru pco1: deduction form
    pcocn, pcohtpy, pcopt, pcoass, pcorev: shortened the proof
    pi1fval thru pi1val: deduction form
    pi1gp: shortened the proof



(4-Jan-2014) Wide text editor windows
-------------------------------------

Some people like to work with text editor windows wider than the 79
characters per line convention for set.mm.  The following commands:

  MM> set width 9999
  MM> save proof */compressed
  MM> set width 120
  MM> write source set.mm/rewrap

will put each proof on a single line and will wrap all $a and $p
descriptions at column 120.  If your text editor truncates the display
of lines beyond the end of screen, having each proof on a single line
will reduce scrolling.  (Note that 'save proof */compressed' takes 5-10
minutes depending on CPU speed.)

Repeat the above with 'set width 79' (the default width) to restore the
conventional widths.

Note that comments outside of $a and $p descriptions, such as section
headers, are not affected by 'write source.../rewrap'.  The math
formulas in $a and $p statements are also unchanged, since breaking and
indentation may have been chosen to emphasize the structure of the
formula (exception:  formulas exceeding the 'set width' are wrapped, but
not unwrapped with a larger 'set width').



(2-Dec-2013) Class substitution
-------------------------------

Has the time come to remove the "A e. _V" restriction on uses of df-sbc?

See the comments under df-sbc for the issues.  So far we have been
uncommitted about the behavior at proper classes so as not to conflict
with Quine.  To do this, we have prohibited the direct use of df-sbc and
instead only permitted use of dfsbcq.

However, this has become inconvenient, requiring annoying sethood
justifications for theorems using df-sbc and df-csb that make proofs
longer.

There is no theoretical reason not to define it any way we want for
proper classes, but there are two possible ways that we must choose
from.  We could allow the direct use of df-sbc (always false for proper
classes), which means for example that sbth would still require the
"A e. V" antecedent.  Or we could redefine it as in sbc6g (always true
for proper classes), which means some other kinds of theorems would
require the antecedent.  I'm not sure which is more advantageous, but
I'm inclined to choose df-sbc which seems more natural.  Any opinions
are welcome.



(1-Dec-2013) Definite description binder
----------------------------------------

I changed the symbol for the definite description binder in df-iota and
df-riota to an inverted iota, which was used by Russell and seems more
standard than regular iota.  It is analogous to the inverted A and E for
"for all" and "exists".  I left the tokens "iota" and "iota_" alone for
now, although they should be changed since it's no longer a true iota.
Ideally it should be changed to "i." in analogy with "A." and "E.", but
"i." is already used for the imaginary unit.  "ii" (inverted iota) is
one possibility.  Or maybe "io." since we use the first two letters,
like "ph", for Greek.  Suggestions are welcome.



(8-Oct-2013) Proof repair techniques
------------------------------------

If you discover you have been going down a wrong path while creating a
proof, in some cases there are techniques to help salvage parts that are
already proved so they don't have to be typed in again.  Let me know if
there is a useful technique you use, and I'll post it here.

(1) Jeff Hankins described the following technique he used:

  Here's a cool trick I found useful today.  The theorem dummylink can
  "save" people from incorrect theorem assignments.  Here's what I mean.
  I had used the theorem rcla42ev, but Metamath said that there was a
  disjoint variable violation, so I would have to delete that step.
  However, I had a decent-size proof under the incorrect step and I did
  not want to lose all that hard work.  Here's what I did:  I took the
  last step under rcla42ev and assigned it to a dummylink.  I improved the
  statement under the dummylink to save that part of the proof, then I
  deleted the rcla42ev and cleaned up the disjoint variable stuff.  Once I
  got to the part which I had the proof for already, I was able to improve
  it because the proof was in the dummylink.  After that, I deleted the
  use of dummylink and continued on as normal.  I still had to do all the
  annoying technical rcla4e substitution and bound variable steps, but at
  least I didn't have to do all the exponential and "if N is a natural
  number, N-1 is a nonnegative integer" stuff all over again.

Indeed, dummylink is probably underused as a tool to assist proof
building.  Its description in set.mm contains one suggested method for
how it can be used.  When the proof is complete, 'minimize_with' will
automatically reformat the proof to strip out the dummylink uses.


(2) I often use a quick-and-dirty script called extractproof.cmd that
creates a script to reproduce a proof.   extractproof.cmd contains
the following:

  set width 9999
  open log 1.tmp
  show new_proof
  close log
  set width 79
  tools
  match 1.tmp $ y
  clean 1.tmp ber
  substitute 1.tmp ' ' '%^' 1 ''
  add 1.tmp ! ''
  delete 1.tmp ^ =
  delete 1.tmp " " ""
  !substitute 1.tmp obs "" a ""
  substitute 1.tmp '%' '%assign last ' 1 ''
  add 1.tmp "" " /no_unify"
  reverse 1.tmp
  substitute 1.tmp % \n a ''
  quit

Here is an example of its use.  First, we extract the proof:

  MM> prove a1i
  Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to exit.
  You will be working on statement (from "SHOW STATEMENT a1i"):
  53 a1i.1 $e |- ph $.
  54 a1i $p |- ( ps -> ph ) $= ... $.
  Note:  The proof you are starting with is already complete.
  MM-PA> submit extractproof.cmd/silent

The generated script 1.tmp will contain these lines:

  !9
  assign last ax-mp /no_unify
  !8
  assign last ax-1 /no_unify
  !5
  assign last a1i.1 /no_unify

The step number comment corresponds to the 'show new_proof'
step.  The /no_unify prevents interactive unification while
the script is running.  Now let's test the generated script:

  MM-PA> delete all
  The entire proof was deleted.
  1 a1i=? $? |- ( ps -> ph )
  MM-PA> submit 1.tmp/silent
  MM-PA> unify all/interactive
  No new unifications were made.
  MM-PA> improve all
  A proof of length 1 was found for step 5.
  A proof of length 1 was found for step 4.
  A proof of length 3 was found for step 2.
  A proof of length 1 was found for step 1.
  Steps 1 and above have been renumbered.
  CONGRATULATIONS!  The proof is complete.

For proof repair, I often extract pieces of the generated script to
automatically reprove sections that are correct.  With experience, I've
learned a few tricks to deal with several problems.  For example, if the
proof had an unknown step when the script was generated, the script will
contain 'assign last ?', which must be manually deleted from the generated
script, and all subsequent 'assign last' must be changed to 'assign -1'.



(5-Oct-2013) Improvements in extensible structure utility theorems
------------------------------------------------------------------

I revamped the utility theorems used for extensible structures.  The
large collection of general structure theorems in the "Extensible
structures" section of set.mm have been reduced to just four:  strfvn
(normally not used), ndxarg, ndxid, and strfv.  I added comments to
these to assist understanding their purpose.

Before, it was awkward to work with structures with more than 3
components, since the number of utility theorems was O(N^2) where N is
the number of structure components.  In particular, the old O(N^2)
strNvM theorems have become the single theorem strfv.  To achieve this,
strfv has a hypothesis requiring that the _specific_ structure S be a
function.

Note that, as before, any particular member of say Grp (df-grp) needn't
be a function as long as it has the required values under our df-fv
definition.  (This "opening up" of Grp to possible non-functions
dramatically simplifies many extensible structure theorems and causes no
theoretical problem.)  So, strfv can't be used with an arbitrary member
of Grp.  But practically speaking, we will not use _specific_ structures
that aren't functions, and this limitation (which conforms to the
literature in any case) does not cause a problem.

Extending a structure with a new component (such as groups to rings) is
now O(N), requiring one new theorem per structure component (rngbase,
rngplusg, rngmulr).  In particular, the new theorem fnunsn can be used
to add a component to a previous structure.  Notice its use in building
a ring structure from a group structure in rngfn.

To prove that all structure indices are different, I now successively
show each member is not in the previous set of members rather than
having O(N^2) inequalities.  The largest application of this so far is
in phllem2 in my mathbox, used to prove phlfn with 8 components starting
from lvecfn with 6 components, which in turn comes from rngfn with 3
components.  This shows that an 8-component structure is now practical
to work with.



(8-Sep-2013) New df-seq (sequence generator) and df-sqr (square root)
---------------------------------------------------------------------

Mario Carneiro has updated the sequence generator with the definition,

    df-seq $a |- seq M ( P , F ) = ( rec (
        ( x e. _V , y e. _V |-> <. ( x + 1 ) , ( y P ( F ` ( x + 1 ) ) ) >. ) ,
          <. M , ( F ` M ) >. ) " om ) $.

This rec() is a function on all ordinals 0o, 1o, 2o,... consisting of
the ordered pairs

   { <. 0o , <. M , ( F ` M ) >. >. ,
     <. 1o , <. ( M + 1 ) , ( F ` ( M + 1 ) ) >. >. ,
     <. 2o , <. ( M + 2 ) , ( F ` ( M + 2 ) ) >. >. , ... }

When restricted to the finite ordinals (omega), its range is exactly the
sequence we want:

    { <. M , ( F ` M ) >. >. ,
      <. ( M + 1 ) , ( F ` ( M + 1 ) ) >. >. ,
      <. ( M + 2 ) , ( F ` ( M + 2 ) ) >. >. , ... }

So, we just extract the range and throw away the domain.

This is much simpler than the equivalent triple of previous definitions
for the same thing:

    df-seq1 $a |- seq1 = { <. <. f , g >. , h >. |
                   h = { <. x , y >. | ( x e. NN /\
                     y = ( 2nd ` ( ( rec ( { <. z , w >. |
                         w = <. ( ( 1st ` z ) + 1 ) ,
                           ( ( 2nd ` z ) f ( g ` ( ( 1st ` z ) + 1 ) ) ) >. } ,
                       <. 1 , ( g ` 1 ) >. ) o. `'
       ( rec ( { <. z , w >. | w = ( z + 1 ) } , 1 ) |` om ) ) ` x ) ) ) } } $.
    df-shft $a |- shift = { <. <. f , x >. , g >. | g =
                  { <. y , z >. | ( y e. CC /\ z = ( f ` ( y - x ) ) ) } } $.
    df-seqz $a |- seq = { <. <. x , g >. , h >. |
        h = ( ( ( ( 2nd ` x ) seq1 ( g shift ( 1 - ( 1st ` x ) ) ) )
           shift ( ( 1st ` x ) - 1 ) ) |` { k e. ZZ | ( 1st ` x ) <_ k } ) } $.

The new seq has arguments, unlike the old which was a constant class
symbol.  This allows proper classes for its arguments (in particular P
and F).  The "M" argument is visually separated from the other two since
it acts more like a parameter; it could correspond to a subscript in a
textbook e.g.  "seq_M(P,F)" in LaTeX.

Usually I prefer to have new definitions in the form of new class
constant symbols (thus requiring no new equality, hb*, etc. theorems),
but we made this an exception since Mario wants to use proper classes
for P and M, and also the old seq was rather awkward looking:

  old:  ( <. M , P >. seq F )
  new:  seq M ( P , Q )

The relationship to the old seq0, seq1, and seq (now seqz) are given
by the following theorems:

  seq0fval $p |- ( S seq0 F ) = seq 0 ( S , F )
  seq1fval $p |- ( S seq1 F ) = seq 1 ( S , F )
  seqzfval $p |- ( M e. V -> ( <. M , S >. seqz F ) = seq M ( S , F ) )

Eventually, these 3 can be phased out.

Mario also extended to domain of the square root function to include
all of CC.  The result is still uniquely defined, and corresponds
to the principal value described at
http://en.wikipedia.org/wiki/Square_root#Principal_square_root_of_a_complex_number

    df-sqr $a |- sqr = ( x e. CC |-> ( iota_ y e. CC ( ( y ^ 2 ) = x /\
      0 <_ ( Re ` y ) /\ ( _i x. y ) e/ RR+ ) ) ) $.

Many existing theorems were affected by these changes; see the list
at the top of set.mm for 8-Sep-2013.



(3-Jun-2013) Adding or deleting antecedents in a theorem
--------------------------------------------------------

Sometimes I might not initially know that a certain antecedent is
required for a proof, but discover it only when I'm deep into the proof.
Other times, I may have redundant antecedents that are not needed by the
proof.  And, sometimes I just want to rearrange the antecedents for
better appearance or easier use later on.  In all these cases, it is
annoying and time-consuming to have to re-enter the proof from scratch
to account for a modified conjunction of antecedents.

To make the task of editing antecedents easier, I use a submit script
called "unlink-ant.cmd" which is listed below at the end of this note.

This method has a limitation:  it assumes all antecedent linkages are
done via simp*, which will occur when we chain to referenced theorems
via syl*anc as described in the mmnotes.txt entry below of 19-Mar-2012.
In other situations, additional manual step deletion may be required, or
it might be possible to enhance the script.

As an example, we will use divdivdiv, which was described in the
19-Mar-2012 entry below.  You can follow these steps to test the script.

  MM> read set.mm
  MM> prove divdivdiv
  ...

If a proof is incomplete, make sure it is in the state left by 'unify
all/interactive' then 'improve all'.

From inside the Proof Assistant,
run the script to delete all steps linking antecedents:

  MM-PA> submit unlink-ant.cmd/silent

At this point, the proof will have the antecedent linkages stripped.
Replace the original proof in set.mm with this stripped-down proof,
either in an editor or with 'save new_proof' then 'write source'.

  MM-PA> show new_proof/normal
  Proof of "divdivdiv":
  ---------Clip out the proof below this line to put it in the source file:
      ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? divcl syl111anc
      ? ? ? ? ? ? ? ? ? ? divcl syl111anc ? ? mulcom syl11anc ? ? ? ? ? ? ? ? ? ?
      ? ? ? ? divmuldiv syl22anc eqtrd opreq2d ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
      ? ? ? ? ? ? ? ? ? ? ? ? divmuldiv syl22anc ? ? ? ? ? ? ? ? ? ? ? ? ? mulcom
      syl11anc opreq1d ? ? ? ? ? ? ? ? ? ? ? ? mulcl syl11anc ? ? ? ? ? ? ? ? ? ?
      ? ? mulne0 syl22anc ? divid syl11anc 3eqtrd opreq1d ? ? ? ? ? ? ? ? ? ? ? ?
      ? ? ? divcl syl111anc ? ? ? ? ? ? ? ? ? ? divcl syl111anc ? ? ? ? ? ? ? ? ?
      ? divcl syl111anc ? ? ? mulass syl111anc ? ? ? ? ? ? ? ? ? ? ? ? ? divcl
      syl111anc ? mulid2 syl 3eqtr3d eqtr3d ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? divcl
      syl111anc ? ? ? ? ? ? ? ? ? ? ? ? ? mulcl syl11anc ? ? ? ? ? ? ? ? mulcl
      syl11anc ? ? ? ? ? ? ? mulne0 ad2ant2lr ? ? divcl syl111anc ? ? ? ? ? ? ? ?
      ? ? divcl syl111anc ? ? ? ? ? divne0 adantl ? ? ? divmul syl112anc mpbird
      $.
  ---------The proof of "divdivdiv" to clip out ends above this line.
  MM-PA>

You can now edit set.mm to rearrange, add, or delete antecedents.  When
done, go back into MM-PA:

  MM-PA> unify all/interactive
  MM-PA> improve all
  ...
  CONGRATULATIONS!  The proof is complete.

Note that the file 3.tmp produced by the script will contain a list of
all antecedents that are needed.  This is useful for determining whether
the theorem has unused ones.  (The list may be slightly redundant, as
the ones with /\ below show.)

  MM-PA> more 3.tmp
  ( B e. CC /\ B =/= 0 )
  ( C e. CC /\ C =/= 0 )
  ( D e. CC /\ D =/= 0 )
  A e. CC
  B =/= 0
  B e. CC
  C =/= 0
  C e. CC
  D =/= 0
  D e. CC


The file unlink-ant.cmd is listed below.

! unlink-ant.cmd - delete all steps linking antecedents to proof
! Run this at the MM-PA prompt with 'submit delant.cmd/silent' on a
! complete or incomplete proof.  Make sure the starting proof is in
! the state left by 'unify all/interactive' then 'improve all'.  Output:
!   1. The list of required antecedents is contained in 3.tmp.
!   2. The proof shown by 'show new_proof/normal' will have antecedent
!      linkages removed, so that the antecedent conjunction can be
!      edited (antecedents added, deleted, or reorganized).  Later,
!      'unify all/interactive' then 'improve all' will reconnect them.
!   3. The original set.mm is saved in 2.tmp in case something goes wrong.
! Temporary files used: 1.tmp, 2.tmp, 3.tmp, 4.tmp.
! We assume all antecedent linkages are done via simp*, which are normally
! chained to a referenced theorem via syl*anc (see mmnotes.txt entry
! of 19-Mar-2012).  In other situations, additional manual step deletion
! may be required.
save new_proof/compressed
write source 2.tmp
set width 9999
open log 1.tmp
show new_proof
close log
set width 79
tools
copy 1.tmp 3.tmp
! prevent simpld, simprd deletion
substitute 1.tmp '=simpld' '=ximpld' a ''
substitute 1.tmp '=simprd' '=ximprd' a ''
! extract the steps to be deleted into 1.tmp
match 1.tmp '=simp' ''
clean 1.tmp ber
delete 1.tmp ' '  ''
add 1.tmp 'delete step ' ''
reverse 1.tmp
! get list of all required antecedents into 3.tmp
copy 3.tmp 4.tmp
match 3.tmp '=simp' ''
match 4.tmp '=? ' ''
copy 3.tmp,4.tmp 3.tmp
delete 3.tmp '' ' ->'
unduplicate 3.tmp
clean 3.tmp ber
add 3.tmp '' '$'
substitute 3.tmp ' )$' '' 1 ''
quit
submit 1.tmp
delete floating_hypotheses
!(end of unlink-ant.cmd)

-------------------------------------------------------------------------------


(22-May-2013) New metamath program features
-------------------------------------------

1. A /FORBID qualifier was added to MINIMIZE_WITH.  Stronger than
/EXCEPT, it will also exclude any statements that _depend_ on the
statements in the list (based on the algorithm for SHOW TRACE_BACK).
For example,

  MM-PA> MINIMIZE_WITH * /FORBID ax-inf*,ax-ac

will prevent statements depending on ax-inf, ax-inf2, and ax-ac from
being used.

2. A /MATCH qualifier was added to SHOW_TRACEBACK.  For example,

  MM> SHOW TRACE_BACK cp /AXIOMS/MATCH ax-ac,ax-inf*

will list only "ax-inf2" instead of a long list of axioms and
definitions to sort through.

These changes are in Version 0.07.91 20-May-2013.



(18-May-2013) Separate axioms for complex numbers
-------------------------------------------------

The real and complex numbers are now derived from a set of axioms
separate from their construction.  This isolates them better from the
construction.  It also provides more meaningful information in the
'SHOW TRACE_BACK /ESSENTIAL /AXIOMS' command.

The construction theorem that derives the axiom is called the same name
except that the prefix "ax-" is changed to "ax".  For example, the axiom
for closure of addition is ax-addcl, and the theorem that derives it is
axaddcl.



(27-Feb-2013) *OLD cleanup by Scott Fenton
------------------------------------------

Scott Fenton did a large cleanup of set.mm by eliminating many
*OLD references.  The following proofs were changed:

  oancom ficardom alephon omsublim domtriomlem axdc3lem2 axcclem cfom
  lemul1i lemul2i lemul1a ltmul12a mulgt1 ltmulgt11 gt0div ge0div ltdiv2
  lediv2 lediv12a ledivp1 ledivp1i flhalf nnwo infmssuzcl expord2 expmwordi
  exple1 sqlecan sqeqori crreczi facwordi faclbnd faclbnd6 facavg
  fsumabs2mul 0.999... cvgratlem1ALT cvgratlem1 erelem3 efaddlem11
  efaddlem15 efaddlem16 efaddlem20 efaddlem22 eftlex ef1tllem ef01tllem1
  eflti efcnlem2 sin01bndlem2 cos01bndlem2 cos2bnd sin02gt0 sin4lt0
  alephsuc3 bcthlem1 minveclem27 cospi cos2pi sinq12gt0t hvsubcan hvsubcan2
  bcs2 norm1exi chocunii projlem18 pjthlem10 pjthlem12 omlsilem pjoc1i
  shscli shsvs shsvsi shsleji shsidmi spanuni h1de2bi h1de2ctlem spansni
  spansnmul spansnss spanunsni hoscl hodcl osumlem2 sumspansn spansncvi
  pjaddii pjmulii pjss2i pjssmii pjocini hoaddcomi hodsi hoaddassi
  hocadddiri hocsubdiri nmopub2tALT nmfnleub2 hoddii lnophsi hmops
  nmcopexlem3 nmcopexlem5 nmcfnexlem3 nmcfnexlem5 cnlnadjlem2 cnlnadjlem7
  nmopadjlem adjadd nmoptrii nmopcoadji leopadd leopmuli leopnmid
  hmopidmchlem pjsdii pjddii pjscji pjtoi strlem1 sumdmdii cdjreui cdj1i
  cdj3lem1 nndivsub epos intnat atcvrne atcvrj2b cvrat4 2llnm3 2llnm4
  cdlema2 pmapjat1 2polcon4b paddun lhpocnle lhpmat idltrn ltrnmw trl0
  ltrnnidn cdleme2



(25-Feb-2013) Sethood antecedents
---------------------------------

I changed many antecedents to use e.g.  "A e. V" instead of "A e. B" to
indicate that A must be a set.  For example, uniexg was changed from
"( A e. B -> U. A e. _V )" to "( A e. V -> U. A e. _V )".  I think the
variable V is more suggestive of the _V that it will often be replaced
with, better indicating the purpose of the antecedent.

I changed only theorems whose $f order would not change, so there is no
impact on any proofs.  Eventually, we could change the others such as
funopfvg, but it would mean changing all the proofs referencing them, so
it probably won't be done soon.  But I think "A e. V" is a good
convention to follow for future theorems.



(21-Feb-2013) Changes to syl*
-----------------------------

I changed the order of the hypotheses of 76 syl* theorems for a better
logical "flow", per a suggestion from Mario Carneiro.  This was a big
change, affecting about 6400 proofs (about 1/3 of set.mm).  If you
have made changes to your mathbox that aren't in set.mm, I can update
your mathbox for you if you send it along with the set.mm it works with.

You can also update it yourself as follows.

Step 1.  Copy your mathbox into a file called mathbox.mm.

Step 2.  Copy and paste the following lines into the MM> prompt
(or put them in a SUBMIT script if you wish):

    tools
    copy mathbox.mm tmp.mm
    add tmp.mm '' ' '
    substitute tmp.mm ' sylanOLD ' ' sylanOLDOLD ' all ''
    substitute tmp.mm ' sylan ' ' sylanOLD ' all ''
    substitute tmp.mm ' sylanb ' ' sylanbOLD ' all ''
    substitute tmp.mm ' sylanbr ' ' sylanbrOLD ' all ''
    substitute tmp.mm ' sylan2OLD ' ' sylan2OLDOLD ' all ''
    substitute tmp.mm ' sylan2 ' ' sylan2OLD ' all ''
    substitute tmp.mm ' sylan2b ' ' sylan2bOLD ' all ''
    substitute tmp.mm ' sylan2br ' ' sylan2brOLD ' all ''
    substitute tmp.mm ' syl2an ' ' syl2anOLD ' all ''
    substitute tmp.mm ' syl2anb ' ' syl2anbOLD ' all ''
    substitute tmp.mm ' syl2anbr ' ' syl2anbrOLD ' all ''
    substitute tmp.mm ' syland ' ' sylandOLD ' all ''
    substitute tmp.mm ' sylan2d ' ' sylan2dOLD ' all ''
    substitute tmp.mm ' syl2and ' ' syl2andOLD ' all ''
    substitute tmp.mm ' sylanl1 ' ' sylanl1OLD ' all ''
    substitute tmp.mm ' sylanl2 ' ' sylanl2OLD ' all ''
    substitute tmp.mm ' sylanr1 ' ' sylanr1OLD ' all ''
    substitute tmp.mm ' sylanr2 ' ' sylanr2OLD ' all ''
    substitute tmp.mm ' sylani ' ' sylaniOLD ' all ''
    substitute tmp.mm ' sylan2i ' ' sylan2iOLD ' all ''
    substitute tmp.mm ' syl2ani ' ' syl2aniOLD ' all ''
    substitute tmp.mm ' sylancl ' ' sylanclOLD ' all ''
    substitute tmp.mm ' sylancr ' ' sylancrOLD ' all ''
    substitute tmp.mm ' sylanbrc ' ' sylanbrcOLD ' all ''
    substitute tmp.mm ' sylancb ' ' sylancbOLD ' all ''
    substitute tmp.mm ' sylancbr ' ' sylancbrOLD ' all ''
    substitute tmp.mm ' syl3an1OLD ' ' syl3an1OLDOLD ' all ''
    substitute tmp.mm ' syl3an1 ' ' syl3an1OLD ' all ''
    substitute tmp.mm ' syl3an2 ' ' syl3an2OLD ' all ''
    substitute tmp.mm ' syl3an3 ' ' syl3an3OLD ' all ''
    substitute tmp.mm ' syl3an1b ' ' syl3an1bOLD ' all ''
    substitute tmp.mm ' syl3an2b ' ' syl3an2bOLD ' all ''
    substitute tmp.mm ' syl3an3b ' ' syl3an3bOLD ' all ''
    substitute tmp.mm ' syl3an1br ' ' syl3an1brOLD ' all ''
    substitute tmp.mm ' syl3an2br ' ' syl3an2brOLD ' all ''
    substitute tmp.mm ' syl3an3br ' ' syl3an3brOLD ' all ''
    substitute tmp.mm ' syl3an ' ' syl3anOLD ' all ''
    substitute tmp.mm ' syl3anb ' ' syl3anbOLD ' all ''
    substitute tmp.mm ' syl3anbr ' ' syl3anbrOLD ' all ''
    substitute tmp.mm ' syld3an3 ' ' syld3an3OLD ' all ''
    substitute tmp.mm ' syld3an1 ' ' syld3an1OLD ' all ''
    substitute tmp.mm ' syld3an2 ' ' syld3an2OLD ' all ''
    substitute tmp.mm ' syl3anl1 ' ' syl3anl1OLD ' all ''
    substitute tmp.mm ' syl3anl2 ' ' syl3anl2OLD ' all ''
    substitute tmp.mm ' syl3anl3 ' ' syl3anl3OLD ' all ''
    substitute tmp.mm ' syl3anl ' ' syl3anlOLD ' all ''
    substitute tmp.mm ' syl3anr1 ' ' syl3anr1OLD ' all ''
    substitute tmp.mm ' syl3anr2 ' ' syl3anr2OLD ' all ''
    substitute tmp.mm ' syl3anr3 ' ' syl3anr3OLD ' all ''
    substitute tmp.mm ' syl5OLD ' ' syl5OLDOLD ' all ''
    substitute tmp.mm ' syl5com ' ' syl5comOLD ' all ''
    substitute tmp.mm ' syl5 ' ' syl5OLD ' all ''
    substitute tmp.mm ' syl5d ' ' syl5dOLD ' all ''
    substitute tmp.mm ' syl5ib ' ' syl5ibOLD ' all ''
    substitute tmp.mm ' syl5ibr ' ' syl5ibrOLD ' all ''
    substitute tmp.mm ' syl5bi ' ' syl5biOLD ' all ''
    substitute tmp.mm ' syl5cbi ' ' syl5cbiOLD ' all ''
    substitute tmp.mm ' syl5bir ' ' syl5birOLD ' all ''
    substitute tmp.mm ' syl5cbir ' ' syl5cbirOLD ' all ''
    substitute tmp.mm ' syl5bb ' ' syl5bbOLD ' all ''
    substitute tmp.mm ' syl5rbb ' ' syl5rbbOLD ' all ''
    substitute tmp.mm ' syl5bbr ' ' syl5bbrOLD ' all ''
    substitute tmp.mm ' syl5rbbr ' ' syl5rbbrOLD ' all ''
    substitute tmp.mm ' syl5eq ' ' syl5eqOLD ' all ''
    substitute tmp.mm ' syl5req ' ' syl5reqOLD ' all ''
    substitute tmp.mm ' syl5eqr ' ' syl5eqrOLD ' all ''
    substitute tmp.mm ' syl5reqr ' ' syl5reqrOLD ' all ''
    substitute tmp.mm ' syl5eqel ' ' syl5eqelOLD ' all ''
    substitute tmp.mm ' syl5eqelr ' ' syl5eqelrOLD ' all ''
    substitute tmp.mm ' syl5eleq ' ' syl5eleqOLD ' all ''
    substitute tmp.mm ' syl5eleqr ' ' syl5eleqrOLD ' all ''
    substitute tmp.mm ' syl5eqner ' ' syl5eqnerOLD ' all ''
    substitute tmp.mm ' syl5ss ' ' syl5ssOLD ' all ''
    substitute tmp.mm ' syl5ssr ' ' syl5ssrOLD ' all ''
    substitute tmp.mm ' syl5eqbr ' ' syl5eqbrOLD ' all ''
    substitute tmp.mm ' syl5eqbrr ' ' syl5eqbrrOLD ' all ''
    substitute tmp.mm ' syl5breq ' ' syl5breqOLD ' all ''
    substitute tmp.mm ' syl5breqr ' ' syl5breqrOLD ' all ''
    substitute tmp.mm ' syl7OLD ' ' syl7OLDOLD ' all ''
    substitute tmp.mm ' syl7 ' ' syl7OLD ' all ''
    substitute tmp.mm ' syl7ib ' ' syl7ibOLD ' all ''
    substitute tmp.mm ' syl6ss ' ' syl6sseq ' all ''
    substitute tmp.mm ' syl6ssr ' ' syl6sseqr ' all ''
    clean tmp.mm e
    copy tmp.mm mathbox.mm
    exit

Step 3.  Place mathbox.mm into the most recent set.mm.
'Verify proof *' should show no errors related to these changes.
You may want to delete the tmp.mm* files to clean up your directory.

Step 4 (optional):  Replace the syl*OLD references in your new set.mm
with the new versions.  You can find these with 'show usage syl*OLD'.
For example, suppose your theorem 'xyz' uses sylanOLD.  In the Proof
Assistant, type in the following commands (or run them from a script):

    MM> prove xyz
    MM-PA> minimize_with syl*an*,syl5*,syl7*/except *OLD/allow_growth
    MM-PA> save new_proof/compressed
    MM-PA> exit

If you don't replace the syl*OLDs, then I will do that when you next
submit your mathbox using my scripts; it is an easy change for me.  I
will keep the syl*OLDs in set.mm for about a year.

In the new syl5* names, "ib" and "bi" are swapped to reflect the
swapping of implication and biconditional in the new hypothesis order.
See the list at the top of set.mm for a few other renamings.



(4-Nov-2012) Structures have been renamed
-----------------------------------------

I made the changes of the 30-Oct-2012 proposal below.  All changes can
be seen at the top of set.mm in the 4-Nov-12 entries.

I also changed conflicting labels so that the "NEW" suffix could be
removed from the new structure theorems.  There are about 50 labels
with "NEW" removed, and another 50 or so older labels were renamed
to prevent conflict.  The old label renames generally follow this
scheme, where the added "o" stands for the "Op" added to "GrpOp" etc.:

  *grp* becomes *grpo*
  *abl* becomes *ablo*
  *ring* becomes *ringo*
  *divrng* becomes *divrngo*

Note that only conflicting ones were renamed.  Thus, for example,
isgrp2i for GrpOp was _not_ renamed to "isgrpo2i" because it didn't
conflict with any "NEW".  I didn't rename them in order to keep the
size of the change list smaller.

If people prefer that I rename all the GrpOp labels from *grp* to *grpo*
for better consistency, that is no big deal for me.  Let me know.


(30-Oct-2012) Proposed renaming of structures
              -------------------------------

For a few structures in the literature, the pair that includes a base
set is called a "space", such as topological space or metric space.  In
this case, the main name (without "space") is one of the pair's
elements:  the topology on a (base) set, the metric of a metric space.
This is reflected in set.mm's current naming:  Top vs.  TopSp, Met vs.
MetSp.

For most other structures, the name usually refers to the entire
structure.  In modern literature, a group is almost alway an ordered
pair of a base set and an operation.

Prior to set theory, such as Cayley (1854), a group was a base set
"accompanied" by an operation in a way not formally defined.  Ordered
pairs had not been formally defined at this point in time.  Although a
group member was a member of the base set, I doubt that the base set in
isolation, independent of any operation, would have been called a group.

For brevity, modern authors sometimes also use "group" to mean the "base
set of the group", using context to disambiguate.  For example, a group
member is a member of the base set, since that is the only meaning that
makes sense.  And most of them explain this informal dual usage.  But
the formal meaning is still understood to be the ordered pair, which is
how they (in modern literature) usually define "group".

On the other hand, I don't recall ever seeing the isolated operation
itself referred to as a group.

I propose the following renaming to suggest, at least to some extent,
the literature usage.

  OLD       NEW

  Open      MetOpen     open sets of a metric space
  Grp       GrpOp       an operation for a group
  GrpNEW    Grp         a group
  SubGrp    SubGrpOp    the operation for a subgroup of a group
  Ring      RingOps     the pair of operations for a ring
  DivRing   DivRingOps  the pair of operations for a division ring
  Poset     PosetRel    an ordering relation for a poset
  PosetNEW  Poset       a poset
  Lat       LatRel      an ordering relation for a lattice
  LatNEW    Lat         a lattice
  Dir       DirRel      an ordering relation for a directed set
  Toset     TosetRel    an ordering relation for a totally ordered set
  TopSet    TopOpen     the topology extractor for a TopSp

For structures that exist only in mathboxes, I won't do any renaming,
but I may rename them if they are moved to the main set.mm.

I may also rename CVec, NrmCVec, CPreHil, CBan, CHil to CVecOps, etc.,
although I need to figure out what to with them.  In a way they are too
specialized (for CC rather than general fields).  I might move them to
my mathbox, since I don't think anyone is using them.



(17-Sep-2012) Differences between REPLACE and ASSIGN
              --------------------------------------

ASSIGN can only accept step numbers that haven't been assigned yet
(i.e. those in SHOW NEW_PROOF/UNKNOWN).  REPLACE can potentially
accept any step number.

The assignment of $e and $f hypotheses of the theorem being proved must
be done with ASSIGN.  REPLACE does not (currently) allow them.

REPLACE fails if it does not find a complete subproof for the
step being replaced.  ASSIGN fails only if the conclusion of the
assigned statement cannot be unified with the unknown step.

REPLACE takes longer to run than ASSIGN.  In fact, the algorithm
used by REPLACE is the same as IMPROVE <step> /3 /DEPTH 1 /SUBPROOF
but using only the specified statement rather than scanning the entire
database.

Unlike IMPROVE, REPLACE can be used with steps having "shared" working
variables ($nn).  A "shared" working variable means that it is used
outside of any existing subproof of the replaced step.  In this case,
REPLACE attempts a guess at an assignment for the working variable, and
usually seems to be right.  However, it is possible that this
"aggressive" assignment can be wrong, and REPLACE will issue the
warning,

    Assignments to shared working variables ($nn) are guesses.  If
    incorrect, to undo DELETE STEP <step>, INITIALIZE, UNIFY, then assign
    them manually with LET and try REPLACE again.

I have also found that I can usually recover with just DELETE FLOATING,
INITIALIZE, UNIFY.  This is what I try first so that I can salvage what
REPLACE found.



(15-Sep-2012) Enhancements to label and step number arguments
              -----------------------------------------------

Version 0.07.81 (14-Sep-2012) of the metamath program has two minor
enhancements.  I added the first one primarily to make the REPLACE and
ASSIGN syntax uniform.  I have found it useful to try e.g.  "REPLACE
LAST abc" first, then if it fails use "ASSIGN LAST abc".  When REPLACE
is successful, it eliminates the tedium of manually assigning the
hypotheses.  In the future, we could have REPLACE automatically call
ASSIGN when it fails, but I want to get some experience with it first.

Enhancement #1:  Statement numbers can now be specified with +nn, which
means the nn'th unassigned step from the first in SHOW
NEW_PROOF/UNKNOWN.  This complements the previously existing -nn, which
means the nn'th step from the last in SHOW NEW_PROOF/UNKNOWN.  This
change affects ASSIGN, REPLACE, IMPROVE, and LET STEP and makes their
step their step specification argument uniform.

For example, consider:

    MM-PA> show new_proof/unknown
    6   mpd.1=? $? |- ( ph -> ( ph -> ph ) )
    7   mpd.2=? $? |- ( ph -> ( ( ph -> ph ) -> ph ) )

ASSIGN LAST or ASSIGN -0 refers to step 7.
ASSIGN -1 refers to step 6.
ASSIGN FIRST or ASSIGN +0 refers to step 6.
ASSIGN +1 refers to step 7.
ASSIGN 6 refers to step 6.
ASSIGN +6 is illegal, because there aren't at least 7 unknown steps.
ASSIGN 1 is illegal, because step 1 isn't unknown.

Enhancement #2:  The unique label argument needed for PROVE, ASSIGN, and
REPLACE now allows wildcards, provided that there is a unique match.
This can sometimes save typing.  For example, "into*3" matches only the
theorem "intopcoaconlem3".  So instead of typing the long theorem name,
you can use "into*3" as a shortcut:

    MM> prove into*3
    Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to
    exit.
    You will be working on statement (from "SHOW STATEMENT intopcoaconlem3"):

If you use too few characters for a unique match, the program will tell you:

    MM> prove int*3
    ?This command requires a unique label, but there are 3 (allowed)
    matches for "int*3".  The first 2 are "intmin3" and "inttop3".  Use
    SHOW LABELS "int*3" to see all matches.

Of course, you can check in advance to see if it's unique by typing
SHOW LABELS into*3.

It is also now easier to assign $e hypotheses.  For example,

    MM-PA> ASSIGN LAST *.a

will assign the $e hypothesis ending with ".a", because there are no
$a or $p statements in set.mm ending with ".a".  (Any $e hypothesis
not belonging to the statement being proved are ignored by the
wildcard scan.)



(12-Sep-2012) New IMPROVE qualifiers and improved REPLACE command
              ---------------------------------------------------

New IMPROVE qualifiers
----------------------

Version 0.07.80 (4-Sep-2012) of the metamath program has the additional
qualifiers /2, /3, and /SUBPROOFS for the IMPROVE command.  When using
this version, and especially the new qualifiers, please save your work
since the runtime can be unpredictable (possibly hours).  There is no
way to cancel except to abort the program.  And, because of the new
code, there is the possibility of a bug I didn't find in testing.  (If
you find a bug, please let me know, of course.)

From HELP IMPROVE:

        / 1 - Use the traditional search algorithm used in earlier versions
            of this program.  It is the default.  It tries to match cut-free
            statements only (those having not having variables in their
            hypotheses that are not in the conclusion).  It is the fastest
            method when it can find a proof.
        / 2 - Try to match statements with cuts.  It also tries to match
            steps containing working ($nn) variables when they don't share
            working variables with the rest of the proof.  It runs slower
            than / 1.
        / 3 - Attempt to find (cut-free) proofs of $e hypotheses that result
            from a trial match, unlike / 2, which only attempts (cut-free)
            proofs of $f hypotheses.  It runs much slower than / 1, and you
            may prefer to use it with specific statements.  For example, if
            step 456 is unknown, you may want to use IMPROVE 456 / 3 rather
            than IMPROVE ALL / 3.  Note that / 3 respects the / DEPTH
            qualifier, although at the expense of additional run time.
        / SUBPROOFS - Look at each subproof that isn't completely known, and
            try to see if it can be proved independently.  This qualifier is
            meaningful only for IMPROVE ALL / 2 or IMPROVE ALL / 3.  It may
            take a very long time to run, especially with / 3.

    Note that / 2 includes the search of / 1, and / 3 includes / 2.
    Specifying / 1 / 2 / 3 has the same effect as specifying just / 3, so
    there is no need to specify more than one.  Finally, since / 1 is the
    default, you never need to use it; it is included for completeness (or
    in case the default is changed in the future).

Here is an example you can duplicate if you want:

    MM> prove eqtr3i
    Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to exit.
    You will be working on statement (from "SHOW STATEMENT eqtr3i"):
    5507 eqtr3i.1 $e |- A = B $.
    5508 eqtr3i.2 $e |- A = C $.
    5509 eqtr3i $p |- B = C $= ... $.
    Note:  The proof you are starting with is already complete.
    MM-PA> delete all
    The entire proof was deleted.
    1 eqtr3i=? $? |- B = C
    MM-PA> improve all
    No new subproofs were found.
    MM-PA> improve all /2
    Pass 1:  Trying to match cut-free statements...
    Pass 2:  Trying to match all statements...
    No new subproofs were found.
    MM-PA> improve all /3
    Pass 1:  Trying to match cut-free statements...
    Pass 2:  Trying to match all statements, with cut-free hypothesis matches...
    No new subproofs were found.
    MM-PA> improve all /3 /depth 1
    Pass 1:  Trying to match cut-free statements...
    Pass 2:  Trying to match all statements, with cut-free hypothesis matches...
    A proof of length 9 was found for step 1.
    Steps 1 and above have been renumbered.
    CONGRATULATIONS!  The proof is complete.  Use SAVE NEW_PROOF to save it.
    Note:  The Proof Assistant does not detect $d violations.  After saving
    the proof, you should verify it with VERIFY PROOF.
    MM-PA> show new_proof
    6     eqcomi.1=eqtr3i.1 $e |- A = B
    7   eqtri.1=eqcomi    $p |- B = A
    8   eqtri.2=eqtr3i.2  $e |- A = C
    9 eqtr3i=eqtri      $p |- B = C
    MM-PA>

Explanation:

IMPROVE ALL /1 (default i.e. old algorithm) didn't even consider eqtri
because it is not cut free i.e. it has variable "A" in the hypotheses
that isn't in the conclusion.

IMPROVE ALL /2 did not prove it because there was no match to hypothesis
eqtri.1 "|- B = A" in the hypotheses (or other parts of the proof, if
there were any) of eqtr3i.

IMPROVE ALL /3 alone didn't prove it because there was no statement with
0 $e hypotheses that matched eqtri.1 "|- B = A".  It did not consider
eqcomi, which has a $e hypothesis.

IMPROVE ALL /3 /DEPTH 1 considered eqcomi, which has 1 $e hypothesis,
and the hypothesis eqcomi.1 matched eqtr3i.1, proving the theorem.

Each of these qualifiers takes longer to run than the previous.  In
addition, it is better to try depth 0 (default) first, then /depth 1,
then /depth 2,... because (in addition to much greater runtime) each
increasing depth may result in a less efficient proof.  For example,
'improve /3 /depth 2' finds a proof, but it will have a redundant use of
idi (try it).

The /SUBPROOF qualifier
-----------------------

The /SUBPROOF qualifier is occasionally useful if you have a proof that
is a tangled mess with many unknown steps, and you want to see if
something simpler will prove parts of it.  Originally I had /SUBPROOF as
the default for /2 and /3, but later I made it a separate qualifier
because it can take a huge amount of runtime, especially with /3.

The /SUBPROOF qualifier found a proof for the following (somewhat
contrived) example that you can duplicate if you want.

    MM> prove dalem22
    Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to exit.
    You will be working on statement (from "SHOW STATEMENT dalem22"):
    70869 dalem.ph $e |- ( ph <-> ( ( ( K e. HL /\ C e. A ) /\ ( P e. A /\ Q e.
          A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e.
          O ) /\ ( ( -. C L ( P J Q ) /\ -. C L ( Q J R ) /\ -. C L ( R J P ) )
          /\ ( -. C L ( S J T ) /\ -. C L ( T J U ) /\ -. C L ( U J S ) ) /\ (
          C L ( P J S ) /\ C L ( Q J T ) /\ C L ( R J U ) ) ) ) ) $.
    70870 dalem.l $e |- L = ( le ` K ) $.
    70871 dalem.j $e |- J = ( join ` K ) $.
    70872 dalem.a $e |- A = ( Atoms ` K ) $.
    70873 dalem.ps $e |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c L Y /\ ( d =/=
          c /\ -. d L Y /\ C L ( c J d ) ) ) ) $.
    70905 dalem22.o $e |- O = ( LPlanes ` K ) $.
    70906 dalem22.y $e |- Y = ( ( P J Q ) J R ) $.
    70907 dalem22.z $e |- Z = ( ( S J T ) J U ) $.
    70908 dalem22 $p |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c J d ) J ( P J S ) ) e.
          O ) $= ... $.
    Note:  The proof you are starting with is already complete.
    MM-PA> delete step 70
    A 34-step subproof at step 70 was deleted.  Steps 70:276 are now 37:243.
     37   mpbid.min=?           $? |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c J d ) (
                                                  meet ` K ) ( P J S ) ) e. A )
    MM-PA> assign last dalem21
    To undo the assignment, DELETE STEP 65 and if needed INITIALIZE, UNIFY.
     56     dalem.ph=?            $? |- ( ph <-> ( ( ( $12 e. HL /\ $13 e. A )
    /\ ( P e. A /\ $14 e. A /\ $15 e. A ) /\ ( S e. A /\ $16 e. A /\ $17 e. A )
    ) /\ ( Y e. $18 /\ Z e. $18 ) /\ ( ( -. $13 $19 ( P J $14 ) /\ -. $13 $19 (
      $14 J $15 ) /\ -. $13 $19 ( $15 J P ) ) /\ ( -. $13 $19 ( S J $16 ) /\ -.
    $13 $19 ( $16 J $17 ) /\ -. $13 $19 ( $17 J S ) ) /\ ( $13 $19 ( P J S ) /\
                         $13 $19 ( $14 J $16 ) /\ $13 $19 ( $15 J $17 ) ) ) ) )
     57     dalem.l=?             $? |- $19 = ( le ` $12 )
     58     dalem.j=?             $? |- J = ( join ` $12 )
     59     dalem.a=?             $? |- A = ( Atoms ` $12 )
     60     dalem.ps=?            $? |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c
                    $19 Y /\ ( d =/= c /\ -. d $19 Y /\ $13 $19 ( c J d ) ) ) )
     61     dalem21.m=?           $? |- ( meet ` K ) = ( meet ` $12 )
     62     dalem21.o=?           $? |- $18 = ( LPlanes ` $12 )
     63     dalem21.y=?           $? |- Y = ( ( P J $14 ) J $15 )
     64     dalem21.z=?           $? |- Z = ( ( S J $16 ) J $17 )
    MM-PA> improve all /2 /subproof
    Pass 1:  Trying to match cut-free statements...
    A proof of length 1 was found for step 55.
    A proof of length 1 was found for step 54.
    A proof of length 1 was found for step 53.
    A proof of length 1 was found for step 52.
    A proof of length 3 was found for step 50.
    A proof of length 1 was found for step 47.
    A proof of length 1 was found for step 44.
    A proof of length 1 was found for step 41.
    A proof of length 1 was found for step 39.
    A proof of length 1 was found for step 38.
    A proof of length 1 was found for step 37.
    Pass 2:  Trying to match all statements...
    Pass 3:  Trying to replace incomplete subproofs...
    A proof of length 34 was found for step 67.
    Steps 37 and above have been renumbered.
    CONGRATULATIONS!  The proof is complete.  Use SAVE NEW_PROOF to save it.
    Note:  The Proof Assistant does not detect $d violations.  After saving
    the proof, you should verify it with VERIFY PROOF.
    MM-PA>

In this case, just before "IMPROVE ALL /2/SUBPROOF", step 37 (which
became step 65 after the last assignment) had the known content "$p |- (
( ph /\ Y = Z /\ ps ) -> ( ( c J d ) ( meet ` K ) ( P J S ) ) e. A )".
However, the subproof ending at dalem21 contained a hopeless mess of $nn
work variables.

The /subproof algorithm did the following.  It scanned the proof and saw
that step 65 had an incomplete subproof.  So it then tried to prove step
65 (using the /2 algorithm) completely independently of the existing
incomplete subproof and found a new proof.  It then deleted the existing
subproof and replaced it with the one it found..  It didn't matter what
the existing subproof had:  it could have been total "garbage" with
nothing at all to do with the final proof that was found.

I don't yet have a good feel for when /subproof will help or not.
However, sometimes it can take a very long time to run, so you should
"save new_proof" then "write source" before you try it, in case you have
to abort it.

Improved REPLACE command
------------------------

Recall the first part of the previous example:

    MM> prove dalem22
    Entering the Proof Assistant.  HELP PROOF_ASSISTANT for help, EXIT to exit.
    You will be working on statement (from "SHOW STATEMENT dalem22"):
    70869 dalem.ph $e |- ( ph <-> ( ( ( K e. HL /\ C e. A ) /\ ( P e. A /\ Q e.
          A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e.
          O ) /\ ( ( -. C L ( P J Q ) /\ -. C L ( Q J R ) /\ -. C L ( R J P ) )
          /\ ( -. C L ( S J T ) /\ -. C L ( T J U ) /\ -. C L ( U J S ) ) /\ (
          C L ( P J S ) /\ C L ( Q J T ) /\ C L ( R J U ) ) ) ) ) $.
    70870 dalem.l $e |- L = ( le ` K ) $.
    70871 dalem.j $e |- J = ( join ` K ) $.
    70872 dalem.a $e |- A = ( Atoms ` K ) $.
    70873 dalem.ps $e |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c L Y /\ ( d =/=
          c /\ -. d L Y /\ C L ( c J d ) ) ) ) $.
    70905 dalem22.o $e |- O = ( LPlanes ` K ) $.
    70906 dalem22.y $e |- Y = ( ( P J Q ) J R ) $.
    70907 dalem22.z $e |- Z = ( ( S J T ) J U ) $.
    70908 dalem22 $p |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c J d ) J ( P J S ) ) e.
          O ) $= ... $.
    Note:  The proof you are starting with is already complete.
    MM-PA> delete step 70
    A 34-step subproof at step 70 was deleted.  Steps 70:276 are now 37:243.
     37   mpbid.min=?           $? |- ( ( ph /\ Y = Z /\ ps ) -> ( ( c J d ) (
                                                  meet ` K ) ( P J S ) ) e. A )

Suppose we know in advance that dalem21 is the correct statement to
assign to step 37.  If we type "ASSIGN LAST dalem21", we get the 9
hypotheses, as shown in the previous section, that have to be assigned
by hand; even worse, there are a lot of $nn work variables in those
hypotheses that we have to figure out.

The REPLACE command has been enhanced so that it can "replace" any step
at all, even unknown steps that have not been assigned yet, provided
that it can find a match to all of the hypotheses of the REPLACE
label argument.

So above, instead of "ASSIGN LAST dalem21", we can type
"REPLACE LAST dalem21".  We will see:

    MM-PA> replace last dalem21
    CONGRATULATIONS!  The proof is complete.  Use SAVE NEW_PROOF to save it.
    Note:  The Proof Assistant does not detect $d violations.  After saving
    the proof, you should verify it with VERIFY PROOF.

Thus we have eliminated the tedium of having to assign by hand the 9
hypotheses resulting from ASSIGN.  If REPLACE cannot find matches for
all the hypotheses, it will say so, and in that case you will have to go
back to ASSIGN to complete the proof.  But in a lot of cases, it may be
worth a try.

Note that you can also REPLACE a step that has been assigned, even with
the same label it has been assigned to.  For example, if we typed
"ASSIGN LAST dalem21", then step 37 would become step 65, and we can
then try "REPLACE 65 dalem21".



(24-Mar-2012) MINIMIZE_WITH usage suggestions
              -------------------------------

Recently, the metamath program was changed so the the scan order in the
MINIMIZE_WITH command is now bottom to top instead of top to bottom.  I
chose this because empirically, slightly shorter proofs often result
because more "advanced" theorems get used first.  For large proofs, it
can be useful to run the original order using the new /REVERSE
qualifier, to see which order results in the smallest proof.

The script that I almost always use is the following:

  MINIMIZE_WITH * /BRIEF /EXCEPT 3*tr*,3syl,*OLD,ee*
  MINIMIZE_WITH 3*tr* /BRIEF /EXCEPT *OLD
  SHOW NEW_PROOF /COMPRESSED

Running the 3*tr*'s last sometimes results in a more optimal use of
3bitr*, 3eqtr*, etc.  The ee* exclusion forces standard theorems to
be used in place of identical ones in Alan Sare's utility set.

You may need to use /NO_DISTINCT with some theorems to avoid
distinct variable conflicts.  (At some future point hopefully
this won't be required.)

I've noticed that minimizing with 3syl often increases rather than
decreases the size of the compressed proof.  If I suspect 3syl could
be useful, I save the proof, then MINIMIZE_WITH 3syl; if the compressed
proof becomes shorter, I use it, otherwise I discard it.

On larger proofs, I save the source before minimization into
separate file (SAVE NEW_PROOF then WRITE SOURCE xxx), then run
the following script on the saved file:

  MINIMIZE_WITH * /BRIEF /EXCEPT 3*tr*,3syl,*OLD,ee* /REVERSE
  MINIMIZE_WITH 3*tr* /BRIEF /EXCEPT *OLD /REVERSE
  SHOW NEW_PROOF /COMPRESSED

I then pick whichever compressed proof is the shortest.

Finally, recall that mathboxes are excluded unless you use the
/INCLUDE_MATHBOXES (/I) qualifier.  Right now, your own mathbox is also
excluded, unfortuately; I hope to fix this in a future release.



(19-Mar-2012) A new approach to antecedent manipulation
              -----------------------------------------

In the 18-Aug-2011 entry below, I described the problem of having to
"waste" proof steps to manipulate antecedents in order to match the
antecedents of referenced theorems.  My initial proposal, which was to
adopt "canonical" parenthesizations, had a number of problems described
in the 31-Aug-2011 entry below, and I gave up on that approach.

Here I will described a new approach I have been using in my mathbox for
a while and seems to work reasonably well.  I think it is much easier to
use than the sequence of antecedent commuting (such as ancom1s),
re-parenthesizations (such as anassrs), and adding antecendents (such as
ad2ant2r) that has "traditionally" been used in set.mm and which can be
tedious and frustrating to do optimally.  Moreover, it often results in
a shorter compressed proof.  The uncompressed proof tends to be longer,
but usually this isn't a concern since we ordinarily store compressed
proofs.

Note that the new method is not a requirement for new proofs but is
simply a suggested method available to the user.

At the end of this entry I compare an example proof (divdivdiv)
using the old and new methods.


New utility theorems
--------------------

For use by the new method, I added to set.mm 117 simp* theorems
(simplification to a single conjunct, simpll1 through simp333) and 28
syl*anc theorems (syllogism with contraction, syl11anc through
syl333anc).  The simp* theorems will handle all possible simplifications
of double and triple conjunctions nested up to 2 levels deep.  The
syl*anc theorems will handle all possible target antecedents with double
and triple conjunctions nested up to 1 level deep.

Even though this is a lot of theorems, I think that over time they will
result in a set.mm size reduction via shorter proofs that will result.

They are consistently named.  For example, in the simp* example

  simp31l $p |- ( ( ta /\ et /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=

the

  "3" means the 3rd member of the outermost triple conjunction,
  "1" means the 1st member of the next-nested triple conjunction, and
  "l" means the left member of the next-nested double conjunction.

In general, "1", "2", "3" refer to triple conjunctions and "l", "r" to
double conjunctions.

The following example shows the syl*anc naming convention:

  syl231anc.1 $e |- ( ph -> ps ) $.
  syl231anc.2 $e |- ( ph -> ch ) $.
  syl231anc.3 $e |- ( ph -> th ) $.
  syl231anc.4 $e |- ( ph -> ta ) $.
  syl231anc.5 $e |- ( ph -> et ) $.
  syl231anc.6 $e |- ( ph -> ze ) $.
  syl231anc.7 $e |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ze ) -> si ) $.
  syl231anc $p |- ( ph -> si ) $= ... $.

the "231" (a string of length 3) means there are 3 outer conjuncts
composed of a double ("2"), a triple ("3"), and a "unary" ("1")
conjunction respectively.

In a proof, the simp* and syl*anc theorems would be used in the following way:
        ...
   sylxxxanc.1=simpxxx       |- (main antecedent) -> (piece of ref'd antecdnt)
   sylxxxanc.2=simpxxx       |- (main antecedent) -> (piece of ref'd antecdnt)
   sylxxxanc.3=simpxxx       |- (main antecedent) -> (piece of ref'd antecdnt)
   sylxxxanc.4=(ref'd thm)   |- (ref'd antecent) -> (ref'd theorem)
 (target)=sylxxxanc        |- (main antecedent) -> (ref'd theorem)
        ...


The new procedure
-----------------

You can use the following procedure is used to create this kind of proof,
without having to remember the names of the simp* and syl*anc theorems.
(See divdivdiv below for an example.)

  1. Use syl to connect the main antecedent to the reference theorem.

  2. Assign the referenced theorem to syl.2.

  3. Apply 3jca and jca as many times as necessary to break up the
     expression now assigned to syl.1 into individual conjuncts on the
     right-hand side.  (Hint: if in doubt whether the outermost conjunction
     is double or triple, try 3jca first.)

  4. Use IMPROVE ALL to assign the simp* theorems to the jca and 3jca
     hypotheses automatically.

When the proof is done, use 'MINIMIZE_WITH syl*anc / BRIEF'.  The
syl*anc theorems will automatically replace the syl and shorten the
proof.

Sometimes the jca and 3jca breakups won't be proved automatically
with MINIMIZE_WITH, such as when the closure of a compound operation
is needed.  In that case, repeat the above 4 steps recursively:
apply syl, assign the closure theorem to syl.2, and break up
syl.1 with 3jca and jca.


Guidelines for new theorems
---------------------------

In order for new theorems to be compatible with uses of syl*anc in
theorems referencing them, the important thing (for up to 9 antecedent
conjuncts) is that

  1. In the theorem's statement, the conjunct nesting level in the
     antecedent should be no greater than one.

For example, ( ( ph /\ ps ) /\ ( ch /\ th ) ) is acceptable,
whereas ( ( ( ph /\ ps ) /\ ch ) /\ th ) is not.  (For more than 9
conjuncts, which is rare, I don't have a guideline.)

The simp* theorems provide for all possibilities up to 2 nesting levels.
Thus additional antecedent conjuncts can be present as needed inside of
the proof (as would be the case, for example, before applying
pm2.61dan).  The main guideline for antecedents inside of proofs is that

  2. In a proof step, the conjunct nesting level in the antecedent
     should be no greater than two.

The maximum number of antecedent conjuncts inside of a proof to which
simp* can apply is 27, achieved with triple conjunctions nested 2
levels deep.


Obsolete theorems
-----------------

I plan to make the following changes within a week or so.

The following 4 existing theorems will be replaced by new ones with a
different hypothesis order.  (The new ones are in set.mm already.)

  Obsolete    Will be replaced by    |    Obsolete    Will be replaced by
  --------    -------------------    |    --------    -------------------
                                     |
  sylanc      syl11anc               |    syl3anc     syl111anc
  syl2anc     syl22anc               |    syl3an2c    syl13anc

The following 49 existing theorems will be renamed for naming consistency.

  Old name    New name will be       |    Old name    New name will be
  --------    ----------------       |    --------    ----------------
                                     |
  sylan31c    syl21anc               |    3simp1i     simp1i
  sylan32c    syl12anc               |    3simp2i     simp2i
  pm3.26im    simplim                |    3simp3i     simp3i
  pm3.27im    simprim                |    3simp1d     simp1d
  pm3.26      simpl                  |    3simp2d     simp2d
  pm3.26i     simpli                 |    3simp3d     simp3d
  pm3.26d     simplld                |    3simp1bi    simp1bi
  pm3.26bi    simplbi                |    3simp2bi    simp2bi
  pm3.27      simpr                  |    3simp3bi    simp3bi
  pm3.27i     simpri                 |    3simp1l     simp1l
  pm3.27d     simprd                 |    3simp1r     simp1r
  pm3.27bi    simprbi                |    3simp2l     simp2l
  pm3.26bda   simprbda               |    3simp2r     simp2r
  pm3.27bda   simplbda               |    3simp3l     simp3l
  pm3.26bi2   simplbi2               |    3simp3r     simp3r
  pm3.26bi2VD simplbi2VD             |    3simp11     simp11
  3simp1      simp1                  |    3simp12     simp12
  3simp2      simp2                  |    3simp13     simp13
  3simp3      simp3                  |    3simp21     simp21
  3simpl1     simpl1                 |    3simp22     simp22
  3simpl2     simpl2                 |    3simp23     simp23
  3simpl3     simpl3                 |    3simp31     simp31
  3simpr1     simpr1                 |    3simp32     simp32
  3simpr2     simpr2                 |    3simp33     simp33
  3simpr3     simpr3                 |


Example - proof of theorem divdivdiv
------------------------------------

Compare:
https://us.metamath.org/mpegif/divdivdiv.html - new
https://us.metamath.org/mpegif/divdivdivOLD.html - old

As an example, I re-proved the existing divdivdiv (which I remember as
being somewhat tedious to achieve a short proof for using the traditional
method).  You can compare them as divdivdiv vs. divdivdivOLD.  The
uncompressed proof of divdivdiv is about 25% larger than that of
divdivdivOLD, but the compressed proof is slightly smaller and there are
fewer steps on the web page.

As an example in this proof, one of the steps needed was:

   555         eqtrd.2=?            $? |- ( ( ( A e. CC /\ ( B e. CC /\ B
  =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) -> (
                  ( A / B ) x. ( D / C ) ) = ( ( A x. D ) / ( B x. C ) ) )

The consequent matches divmuldiv.  To connect the antecedent, I
performed the following steps:

  assign last syl
  assign last divmuldiv

obtaining:

   558           syl.1=?              $? |- ( ( ( A e. CC /\ ( B e. CC /\
  B =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) ->
    ( ( A e. CC /\ D e. CC ) /\ ( ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C
                                                             =/= 0 ) ) ) )

Next I broke up the consequent with jca's:

  assign last jca
  improve all
  assign last jca
  improve all
    ...

until no conjunctions remained in the consequent.

Finally, I did "MINIMIZE_WITH * /BRIEF" resulting in:

   648           sylXanc.1=simpll     $p |- ( ( ( A e. CC /\ ( B e. CC /\
  B =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) ->
                                                                 A e. CC )
   673           sylXanc.2=simprrl    $p |- ( ( ( A e. CC /\ ( B e. CC /\
  B =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) ->
                                                                 D e. CC )
   699           sylXanc.3=simplr     $p |- ( ( ( A e. CC /\ ( B e. CC /\
  B =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) ->
                                                  ( B e. CC /\ B =/= 0 ) )
   725           sylXanc.4=simprl     $p |- ( ( ( A e. CC /\ ( B e. CC /\
  B =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) ->
                                                  ( C e. CC /\ C =/= 0 ) )
   730           syl22anc.5=divmuldiv $p |- ( ( ( A e. CC /\ D e. CC ) /\
  Press <return> for more, Q <ret> quit, S <ret> scroll, B <ret> back up...
  ( ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) ) ) -> ( ( A / B ) x.
                               ( D / C ) ) = ( ( A x. D ) / ( B x. C ) ) )
   731         eqtrd.2=syl22anc     $p |- ( ( ( A e. CC /\ ( B e. CC /\ B
  =/= 0 ) ) /\ ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) ) -> (
                  ( A / B ) x. ( D / C ) ) = ( ( A x. D ) / ( B x. C ) ) )

Note that syl, jca, and 3jca were the only theorems I had to know the
names of.  I didn't need to remember the names of simprrl, syl22anc,
etc. since they will happen automatically.


(End of 19-Mar-2012 A new approach to antecedent manipulation)

-------------------------------------------------------------------------
-------------------------------------------------------------------------



(15-Sep-2011) Partial functions and restricted iota
              -------------------------------------

This is a proposal to add definition df-riota below.  As usual, any
comments are welcome.

The current iota definition is

  df-iota $a |- ( iota x ph ) = U. { y | { x | ph } = { y } }

Consider a poset (partially ordered set) with a base set B and a
relation R.  The supremum (least upper bound) of a subset S (of the base
set B) is the unique member of B (if there is one) such that

    A. y e. S y R x /\ A. z e. B ( A. y e. S y R z -> x R z )

The LUB is a partial function on the subsets of B:  for some subsets it
may "exist" (in the textbook sense of this verb), and for others it may
not.  There are several ways to deal with partial functions.  One useful
way is to define "the value exists" as meaning "the value is a member of
B" i.e.  "E. x e. B ...", since the domain of discourse is the base set
B. This is analogous to the set-theoretical meaning of "exists" meaning
"is a member of the domain of discourse _V."

Now, it turns out that the set ~P U. B is not a member of B, which can
be proved without invoking the Axiom of Regularity (see theorem
pwuninel).  We can define an "undefined value" function:

  ( Undef ` B ) = ~P U. B    so that   ( Undef ` B ) e/ B

This device can be used to work with partial functions generally, the
case of posets being one example.

The problem with using the standard iota for defining the LUB is that
the iota returns the empty set (/) when it is not meaningful, and (/)
could be a member of B.  In order to get a guaranteed non-member of B
when the LUB doesn't "exist", we can't (easily) use the standard iota
but need the following awkward and non-intuitive definition of LUB:

  ( lub ` S ) = { t | E. u ( u = { x e. B | ( A. y e. S y R x
                              /\ A. z e. B ( A. y e. S y R z -> x R z ) ) }
                      /\ t = if ( E. v u = { v } , U. u , ( Undef ` B ) ) ) }

(The purpose of the "{ t | E. u ... }" is to avoid having to repeat
"A. y e. S y R x /\ A. z e. B ( A. y e. S y R z -> x R z )" twice,
which would be required if we used df-iota.)

We can introduce a "restricted iota" as follows:

  df-riota $a |- ( iota x e. A ph ) = if ( E! x e. A ph ,
                    ( iota x ( x e. A /\ ph ) ) , ( Undef ` A ) )

Using df-riota, the LUB becomes

  ( lub ` S ) = ( iota x e. B
         ( A. y e. S y R x /\ A. z e. B ( A. y e. S y R z -> x R z ) ) )

Thus df-riota above, even though somewhat awkward, seems to provide the
most useful tool.  Some properties we would have are:

  iotariota $p |- ( iota x ph ) = ( iota x e. _V ph )

  riotaiota $p |- ( E! x e. A ph -> ( iota x e. A ph ) = ( iota x ( x e. A
      /\ ph ) ) )

Most importantly, the "closure" of restricted iota is equivalent to
its "existence" in the textbook sense:

  riotaclb.1 $e |- A e. _V $.
  riotaclb $p |- ( E! x e. A ph <-> ( iota x e. A ph ) e. A )

thus making it very useful for working with partial functions.


(End of 15-Sep-2011 restricted iota proposal)
---------------------------------------------



(6-Sep-2011) cnaddablNEW vs. cnaddabl2NEW (continuation of 5-Sep-2011)
             ---------------------------------------------------------

Yesterday, I wrote:  "For some things like proving cnaddablxNEW (the new
version of cnaddabl), it seems we have to reference the actual
finite-sequence structure."  Today I added a function to construct an
explicit member of a structure class given its components, called
StrBldr (df-strbldr), and I proved cnaddablxNEW with a
"scaffold-independent" notation, mainly to demonstrate a way to do it.

An advantage of using StrBldr is that we can later change the
definition of df-struct (e.g. to use a different ordered n-tuple
definition, such as a sequence on ordinals) without changing theorems.

A disadvantage of using StrBldr is longer expressions (at least for this
example).  It also hides the actual "thing" that the complex addition
group "is" in an abstract (and non-standard) way.

cnaddablxNEW is an example of the "scaffold-dependent" method:

cnaddablx.1NEW $e |- G = { <. 1 , CC >. , <. 2 , + >. } $.
cnaddablxNEW $p |- G e. AbelNEW $= ... $.

Note that the explicit structure { <. 1 , CC >. , <. 2 , + >. }, using
only elementary set-theory symbols, is shown.  In other words, the
"scaffold" is a 2-member finite sequence on NN, which we show
explicitly.  This would change if we adopted a different n-tuple
definition.

cnaddablNEW is an example of the "scaffold-independent" method:

cnaddabl.1NEW $e |- G = StrBldr ( 2 , g , ( ( base ` g ) = CC /\ ( +g ` g ) = + ) ) $.
cnaddablNEW $p |- G e. AbelNEW $= ... $.

In cnaddablNEW, there is no reference to structure holding the group
components (base and operation).  If later we changed the definition of
Struct to use a different n-tuple definition, the theorem cnaddabl2NEW
would not change.

I don't know which might be better for long-term use.  In any case,
these theorems (in my mathbox) are experimental, and I don't plan to put
them into the main part of set.mm in the near future.  As usual, any
comments are welcome.


(5-Sep-2011) Structures for groups, rings, etc.
             ----------------------------------

In general, the way we extend structures in the present set.mm is
awkward.  For example, an inner product space is not a vector space in
the strict formal sense, but we must undergo a mapping to apply vector
space theorems to inner product spaces.

For small structures like groups, the "trick" of specifying the group
completely by its operation makes the statement of some theorems
shorter.  We extract the base set of the group from the range of this
operation.  But as structures get more complex (more specialized), we
have no uniform way to extend them.

With a view towards more flexibility in the future, I propose a new
format for structures using what I call "extensible structure builders".
The structures will be finite sequences, i.e. functions on ( 1 ...  N )
where N e. NN.  For example, a group consists of any finite sequence
with at least 2 members, a base and the group operation.  A ring is any
(Abelian) group specialized with a a multiplication operation as the 3rd
member.  Thus any ring is also a group, and any theorem about groups
automatically applies to rings.  This kind of extensible specialization
becomes important with things like the transition from lattices to
ortholattices, vector spaces to inner product spaces, etc.  In other
words, "every Boolean algebra is a lattice" becomes a literal, rigorous
statement, not just an informal way of expressing a homomorphism.

I propose to use finite sequences instead of ordered n-tuples because
the theory of functions is well-developed, whereas ordered n-tuples
become awkward with more than 3 or 4 members, involving e.g.  ( 1st ` (
2nd ` ( 1st...))) to extract a member.  I made them sequences on NN
rather than om (omega) because we have a richer set of theorems for NN.
But in principle om could also be used, and it would be closer to the
ZFC axioms, and the Axiom of Infinity would not be needed to prove say
simple theorems of group theory.

In my mathbox, I have shown a proposed definition of the extensible
structure builder Struct along with some theorems about groups and rings
"translated" to use it.  For the most part, the translations were
straightforward and mechanical, as can be seen by comparing theorems
with "NEW" after them with to their originals already in set.mm.

In df-struct, we define Struct(N,f,ph), where N is an integer and ph is
a wff containing f as a free variable.  In Struct(N,f,ph), f is a
bound variable.

  df-struct $a |- Struct ( N , f , ph ) = { f | ( E. m e. NN ( N <_ m
                     /\ f Fn ( 1 ... m ) ) /\ ph ) } $.

In other words, a structure with say 2 members (such as a group) is any
sequence with _at least_ 2 members whose first 2 members satisfy the
requirements specified by wff ph.  This means we can extend (specialize)
a group to become a ring without destroying its property of being a
group.

When possible, it desirable not to reference members of a structure
directly via the sequence values (which is highly dependent on the
definition of Struct(N,f,ph)).  Instead, we can define "extractors" of
the components.  This way, if in the future we decide on a different
"scaffold" (such as a sequence on finite ordinals instead of NN) most
theorems will remain unchanged.  For example, for groups and rings, we
define

  df-base  $a |- base = ( f e. _V |-> ( f ` 1 ) ) $.
  df-plusg $a |-   +g = ( f e. _V |-> ( f ` 2 ) ) $.
  df-mulr  $a |-   .r = ( f e. _V |-> ( f ` 3 ) ) $.

Then we can reference "(base ` G)" rather than "(G ` 1)".

I put df-struct in my mathbox.  To illustrate it, I added the
corresponding new definitions df-grpNEW, df-ablNEW, and df-ringNEW
(which define classes GrpNEW, AbelNEW, and RingNEW) along with several
dozen theorems taken from the main set.mm, with NEW appended to their
names.  We have the nice inclusions

     GrpNEW C_ AbelNEW C_ RingNEW,

making it easy to reuse theorems for more general structures with less
general structures.  These inclusions can be continued through division
rings, vectors spaces, and inner product spaces.

You may want to compare the "scaffold-independent" isgrpiNEW to the old
isgrpi.  There is also a "scaffold-dependent" version isgrpixNEW (the
"x" after "isgrpi" means "explicit" i.e. scaffold-dependent).  For some
things like proving cnaddablxNEW (the new version of cnaddabl), it seems
we have to reference the actual finite-sequence structure.  I have some
ideas for avoiding that, but so far they seem to make things more
complex rather than simpler, and I didn't put them in.  (Later - I did
put them in; see df-strbldr and related theorems and note of
6-Sep-2011.)

Also note the analogous zaddablNEW showing the integers are a group:
we do not have to restrict the "+" operation to ZZ.

Comments are welcome.



(31-Aug-2011) Canonical conjunctions - followup
              ---------------------------------

It looks like there is no strong consensus on any of the proposed
methods, so for the time being we'll continue to use whatever ad hoc
parenthesization seems to fit the need at hand.  Perhaps this is best
anyway.




(18-Aug-2011) Canonical conjunctions
              ----------------------

Background
----------

A significant portion of many proofs consists of manipulating
antecendents to rearrange the parenthesization of conjuncts, change the
order of conjuncts, etc.  One of the problems has to do with the many
ways that a conjunction can be parenthesized.  In the case of df-an
(binary conjuncts), the number of parenthesizations for 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, and 12 conjuncts are 1, 2, 5, 14, 42, 132, 429, 1430,
4862, 16796, and 58786 respectively (Catalan numbers).  For mixed df-an
and df-3an it is of course even larger.  It is impractical to have
utility theorems for all possible transformations from one
parenthesization to another; e.g. we would need 42 * 41 = 1722 utility
theorems, analogous to anasss, to handle all possible transformations of
6-conjunct antecedents with df-an parenthesizations.  So we compromise
and use a sequence of manipulations with a much smaller set of
transformations in order to achieve the desired transformation.

To reduce the number of such antecedent manipulation steps as well as
the number of special-purpose utility theorems, I propose that for all
new theorems, we adopt a canonical parenthesization for all conjunctions
(whether in antecedents or not).  Of course there would be certain early
exceptions such as anass that are intrinsically about the
parenthesization.  Over time, existing theorems and their proofs can
also be retrofitted to shorten them.  I believe this can significantly
reduce the size of some proofs as well as make them easier to follow,
since the reader doesn't have to mentally skip over increasingly nested
steps doing nothing but boring antecedent transformations.


Summary of proposals
--------------------

I will describe three possible methods.  (A fifth method from a
Metamath Google Groups posting by FL on 9-Aug-2011, is a hybrid of
Method #1 and Method #2 and not presented here.)

Method #1:  Same nesting level
Advantages:  Arguably most "natural"; fewest changes to retrofit set.mm
Disadvantages:  Neither minimum parentheses nor minimum nesting

Method #2:  Recursive grouping
Advantages:  Minimal parentheses; simple algorithm
Disadvantages:  No minimum nesting

Method #3:  Symmetry
Advantages:  Minimal parentheses; minimal nesting under constraint of
    symmetry
Disadvantages:  Apparently not "natural" (rarely used in current
    set.mm); no simple algorithm to derive n+1st case from nth case;
    nesting is not theoretical minimum.

Method #4:  Attempt at minimal nesting
Advantages:  Appears to have minimum parentheses and minimum nesting.
Disadvantages:  Not intuitive; somewhat complex algorithm.


Method #1:  Same nesting level
------------------------------

I collected all antecedents using 2an and/or 3an, with the results
summarized in Table 1 below.  The empirically most frequent patterns are
arguably the "most natural".  For 2 through 6 conjuncts, these are:

  2210 (ph/\ps)
  1005 (ph/\ps/\ch)
   239 ((ph/\ps)/\(ch/\th))
    58 ((ph/\ps/\ch)/\(th/\ta))
    24 ((ph/\ps/\ch)/\(th/\ta/\et))

(For 7 through 12 conjuncts there isn't enough data for a clear pattern
to emerge.)  The pattern here is that all conjuncts are at the same
nesting level, at the expense of both minimum nesting and minimum
parentheses.  The rules would be:

1. Using 3an as much as possible, put all conjuncts at the same nesting
level.

2. At a given level, 2an's go to the right of 3an's.

For 2 through 12 conjuncts, these rules give:

(ph/\ps)
(ph/\ps/\ch)
((ph/\ps)/\(ch/\th))
((ph/\ps/\ch)/\(th/\ta))
((ph/\ps/\ch)/\(th/\ta/\et))
((ph/\ps/\ch)/\(th/\ta)/\(et/\ze))
((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si))
((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))
(((ph/\ps/\ch)/\(th/\ta/\et))/\((ze/\si)/\(rh/\la)))
(((ph/\ps/\ch)/\(th/\ta/\et))/\((ze/\si/\rh)/\(la/\ka)))
(((ph/\ps/\ch)/\(th/\ta/\et))/\((ze/\si/\rh)/\(la/\ka/\mu)))

Table 1:  Frequency of parenthesization patterns in set.mm

#cases parenthesization
------ ----------------
  2210 (ph/\ps)
  1005 (ph/\ps/\ch)
    62 (ph/\(ps/\ch))
    74 ((ph/\ps)/\ch)
     8 ((ph/\ps)/\ch/\th)
    13 (ph/\(ps/\ch)/\th)
    50 (ph/\ps/\(ch/\th))
   101 ((ph/\ps/\ch)/\th)
   128 (ph/\(ps/\ch/\th))
     6 (((ph/\ps)/\ch)/\th)
     6 (ph/\((ps/\ch)/\th))
     7 (ph/\(ps/\(ch/\th)))
    10 ((ph/\(ps/\ch))/\th)
   239 ((ph/\ps)/\(ch/\th))
     1 (ph/\ps/\(ch/\th)/\ta)
     4 (ph/\ps/\(ch/\th/\ta))
     5 ((ph/\ps/\ch)/\th/\ta)
     1 (((ph/\ps)/\ch/\th)/\ta)
     1 ((ph/\ps)/\ch/\(th/\ta))
     2 ((ph/\ps/\(ch/\th))/\ta)
     2 (ph/\((ps/\ch/\th)/\ta))
     4 ((ph/\(ps/\ch/\th))/\ta)
     6 ((ph/\ps)/\(ch/\th)/\ta)
    19 (ph/\(ps/\ch)/\(th/\ta))
    21 ((ph/\ps)/\(ch/\th/\ta))
    58 ((ph/\ps/\ch)/\(th/\ta))
     1 (((ph/\ps)/\ch)/\(th/\ta))
     1 ((ph/\(ps/\(ch/\th)))/\ta)
     1 ((ph/\ps)/\((ch/\th)/\ta))
     1 (ph/\(((ps/\ch)/\th)/\ta))
     1 (ph/\((ps/\(ch/\th))/\ta))
     2 ((ph/\(ps/\ch))/\(th/\ta))
     4 (ph/\((ps/\ch)/\(th/\ta)))
     5 (((ph/\ps)/\(ch/\th))/\ta)
     1 (ph/\(ps/\ch/\th)/\(ta/\et))
     2 (((ph/\ps/\ch)/\th/\ta)/\et)
    24 ((ph/\ps/\ch)/\(th/\ta/\et))
     1 (((ph/\ps/\ch)/\th)/\(ta/\et))
    13 ((ph/\ps)/\(ch/\th)/\(ta/\et))
     1 (((ph/\ps/\ch)/\th/\ta)/\et/\ze)
     1 ((((ph/\ps)/\ch)/\(th/\ta))/\et)
     1 (((ph/\ps)/\ch)/\(th/\(ta/\et)))
     1 ((ph/\(ps/\ch))/\((th/\ta)/\et))
     1 (ph/\(ps/\((ch/\th)/\(ta/\et))))
     2 ((ph/\(ps/\ch))/\(th/\(ta/\et)))
     2 ((ph/\ps)/\((ch/\(th/\ta))/\et))
     3 (((ph/\ps)/\(ch/\th))/\(ta/\et))
     3 (((ph/\ps)/\ch)/\((th/\ta)/\et))
     7 ((ph/\ps)/\((ch/\th)/\(ta/\et)))
     1 (ph/\((ps/\ch/\th)/\(ta/\et)/\ze))
     3 (((ph/\ps)/\(ch/\th)/\(ta/\et))/\ze)
     4 ((ph/\ps)/\(ch/\th/\ta)/\(et/\ze/\si))
     2 (((ph/\ps)/\(ch/\th))/\((ta/\et)/\ze))
     2 ((ph/\(ps/\ch))/\((th/\ta)/\(et/\ze)))
     2 (ph/\((ps/\ch)/\((th/\ta)/\(et/\ze))))
     1 (ph/\((ps/\ch/\th)/\(ta/\et)/\(ze/\si)))
     1 (ph/\(ps/\ch/\th)/\((ta/\et)/\(ze/\si)))
     1 ((ph/\ps/\ch)/\th/\(ta/\et/\(ze/\si/\rh)))
     1 (((ph/\ps)/\(ch/\th)/\(ta/\et))/\(ze/\si))
     2 ((((ph/\ps)/\(ch/\th))/\(ta/\et))/\(ze/\si))
     4 ((ph/\ps)/\((ch/\((th/\ta)/\(et/\ze)))/\si))
     8 (((ph/\ps)/\(ch/\th))/\((ta/\et)/\(ze/\si)))
     1 ((((ph/\ps)/\(ch/\th))/\((ta/\et)/\(ze/\si)))/\((rh/\la)/\ka))
     1 ((((ph/\ps)/\ch)/\((th/\ta)/\et))/\(((ze/\si)/\rh)/\((la/\ka)/\mu)))


Method #2:  Recursive grouping
------------------------------

This parenthesization was first proposed by FL.  It is described by an
algorithm given by Andrew Salmon.

1. From left to right, group by three as many subexpressions as
possible.  Repeat until no more grouping occurs.

2. If there are two subexpressions, group them.

3. Done.

Example:
1 2 3 4 5 6 7 8 9 A B
(1 2 3) (4 5 6) (7 8 9) A B          Rule 1
( (1 2 3) (4 5 6) (7 8 9) ) A B      Rule 1
( ( (1 2 3) (4 5 6) (7 8 9) ) A B)   Rule 1

Example:
1 2 3 4
(1 2 3) 4      Rule 1
((1 2 3) 4)    Rule 2


For 2 through 12 conjuncts, we would have:

(ph/\ps)
(ph/\ps/\ch)
((ph/\ps/\ch)/\th)
((ph/\ps/\ch)/\th/\ta)
((ph/\ps/\ch)/\(th/\ta/\et))
((ph/\ps/\ch)/\(th/\ta/\et)/\ze)
((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si))
((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))
(((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))/\la)
(((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))/\la/\ka)
(((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))/\(la/\ka/\mu))



Method #3:  Symmetry
--------------------

This method has 3 rules:

  1. Minimum nesting.

  2. Minimum parentheses.

  3. Symmetrical parenthesization.

For n = 2 through 12 conjunctions, it appears that the following
parenthesizations are the only ones that satisfy these rules:

                                                      nesting sum
                         (ph/\ps)                            2
                       (ph/\ps/\ch)                          3
                    (ph/\(ps/\ch)/\th)                       6
                  (ph/\(ps/\ch/\th)/\ta)                     8
               ((ph/\ps/\ch)/\(th/\ta/\et))                 12
             ((ph/\ps/\ch)/\th/\(ta/\et/\ze))               13
          ((ph/\ps/\ch)/\(th/\ta)/\(et/\ze/\si))            16
        ((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))          18
     ((ph/\ps/\ch)/\(th/\(ta/\et)/\ze)/\(si/\rh/\la))       22
   ((ph/\ps/\ch)/\(th/\(ta/\et/\ze)/\si)/\(rh/\la/\ka))     25
((ph/\ps/\ch)/\((th/\ta/\et)/\(ze/\si/\rh))/\(la/\ka/\mu))  30


An algorithm for method #3:
---------------------------

Here is an algorithm.  I don't know if there is a better one.

The groupings for 0 through 5 conjuncts must be memorized, although they
have justifications that aren't too hard.  For >= 6 conjuncts, there is
a recursive algorithm starting from 0 through 5.

(The 0 and 1 cases aren't really conjuncts but help define the
algorithm.  Alternately, we could start with 2 conjuncts and start the
recursion with n >= 8, but then cases 6 and 7 need to be memorized.)

0 through 5 conjuncts:                      Justification:

          n = 0             [null]          only 1 symmetric possibility
          n = 1               ph            only 1 symmetric possibility
          n = 2            (ph/\ps)         only 1 symmetric possibility
          n = 3          (ph/\ps/\ch)       only 1 symmetric possibility

          n = 4       (ph/\(ps/\ch)/\th)    the other symmetric possibility
                                            ((ph/\ps)/\(ch/\th) is longer

          n = 5     (ph/\(ps/\ch/\th)/\ta)  the other symmetric possibility
                                            ((ph/\ps)/\ch/\(th/\ta)) is longer

For n >= 6 conjuncts, conjoin (./\./\.) and (./\./\.) around n-6.

Another way to look at it:  start the with the case (n mod 6) from
above, then successively wrap in (./\./\.)...(./\./\.) until n conjuncts
are achieved.

                   ((ph/\ps/\ch)  /\  (th/\ta/\et))
                 ((ph/\ps/\ch)/\  th  /\(ta/\et/\ze))
              ((ph/\ps/\ch)/\  (th/\ta)  /\(et/\ze/\si))
            ((ph/\ps/\ch)/\  (th/\ta/\et)  /\(ze/\si/\rh))
         ((ph/\ps/\ch)/\  (th/\(ta/\et)/\ze)  /\(si/\rh/\la))
       ((ph/\ps/\ch)/\  (th/\(ta/\et/\ze)/\si)  /\(rh/\la/\ka))
  ((ph/\ps/\ch)/\  ((th/\ta/\et)  /\  (ze/\si/\rh))  /\(la/\ka/\mu))


Notes for method #3:
--------------------

1. I conjecture that the algorithm above always results in minimum
   nesting given the symmetry requirement, but I don't have a proof.

2. Unfortunately, in some cases a grouping with minimal nesting does
   not have symmetry.  For example,

        ((ph/\ps/\ch)/\(th/\ta/\et))

   has nesting sum 12, whereas the non-symmetrical

        ((ph/\ps)/\ch/\(th/\ta/\et))

   has nesting sum 11.

3. Minimum nesting + symmetry by themselves don't imply minimum
   parentheses.  For example, the following groupings for 6 conjuncts
   each have minimum nesting sum of 12, but only the first has
   minimum parentheses:

                 ((ph/\ps/\ch)/\(th/\ta/\et))
                 ((ph/\ps)/\(ch/\th)/\(ta/\et))

4. Minimum parentheses + symmetry by themselves do not necessarily imply
   minimum nesting.  For example, for 11 conjuncts, the nesting sum
   from the above algorithm is 25:

      ((ph/\ps/\ch)/\(th/\(ta/\et/\ze)/\si)/\(rh/\la/\ka))  25

   But there exist 3 other symmetrical groupings with same parentheses
   but more nesting:

      (((ph/\ps/\ch)/\th/\ta)/\et/\(ze/\si/\(rh/\la/\ka)))  27
      ((ph/\(ps/\ch/\th)/\ta)/\et/\(ze/\(si/\rh/\la)/\ka))  27
      ((ph/\ps/\(ch/\th/\ta))/\et/\((ze/\si/\rh)/\la/\ka))  27

5. I don't know if (say for some larger n) there exist other symmetric
   patterns with both minimum nesting and minimum parentheses.  If so, then
   the algorithm would become the definition of the grouping, not just the 3
   rules.

(End of method #3 discussion)


Method #4
---------
                                                         nesting sum
(ph/\ps)                                                     2
(ph/\ps/\ch)                                                 3
((ph/\ps)/\ch/\th)                                           6
((ph/\ps/\ch)/\th/\ta)                                       8
((ph/\ps/\ch)/\(th/\ta)/\et)                                11
((ph/\ps/\ch)/\(th/\ta/\et)/\ze)                            13
((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si))                      16
((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh))                  18
(((ph/\ps)/\ch/\th)/\(ta/\et/\ze)/\(si/\rh/\la))            22
(((ph/\ps/\ch)/\th/\ta)/\(et/\ze/\si)/\(rh/\la/\ka))        25
(((ph/\ps/\ch)/\(th/\ta)/\et)/\(ze/\si/\rh)/\(la/\ka/\mu))  30

Algorithm for method #4
------------------------

From grouping n to grouping n+1:

1. Let m = max nesting depth of grouping n.

2. If there is a wff at depth m-1 that is not a triple conjunction, let m = m-1

3. Find first wff (call it w) at depth m that is not a triple conjunction.

4. If w is a wff variable, change it to a conjunction of two wff variables.
   If w is a conjunction of two wff variables, change it to a conjunction of
   three wff variables.

Examples of algorithm for method #4
-----------------------------------

Example: start with  ((ph/\ps/\ch)/\th/\ta).
1. m = max nesting depth = 2.
2. Non-triple-conjunct th is at depth m-1 = 1, so m = m-1 = 1.
3. w = th
2. Change w to (th/\ta).  Result is ((ph/\ps/\ch)/\(th/\ta)/\et).

Example: start with  ((ph/\ps/\ch)/\(th/\ta)/\et).
1. m = max nesting depth = 2.
2. Non-triple-conjunct (th/\ta) is at depth m-1 = 1, so m = m-1 = 1.
3. w = (th/\ta)
2. Change w to (th/\ta/\et).  Result is ((ph/\ps/\ch)/\(th/\ta/\et)/\ze).

Example: start with  ((ph/\ps/\ch)/\(th/\ta/\et)/\(ze/\si/\rh)).
1. m = max nesting depth = 2.
2. There is no non-triple conjunct at depth m-1 = 1, so m stays at 2.
3. w = ph
2. Change w to (ph/\ps).  Result is
   (((ph/\ps)/\ch/\th)/\(ta/\et/\ze)/\(si/\rh/\la)).

Example: start with  (ph/\ps).
1. m = max nesting depth = 1.
2. Non-triple-conjunct (ph/\ps) is at depth m-1 = 0, so m = m-1 = 0.
3. w = (ph/\ps)
2. Change w to (ph/\ps/\ch).  Result is (ph/\ps/\ch).

Example: start with  (ph/\ps/\ch).
1. m = max nesting depth = 1.
2. There is no non-triple conjunct at depth m-1 = 0, so m stays at 1.
3. w = ph
2. Change w to (ph/\ps).  Result is ((ph/\ps)/\ch/\th).

(End of method #4 discussion)


Examples
--------

(The examples use method #3 above and are analogous for the others)

As an example, to swap the 1st and 3rd conjuncts in a 4-conjunct antecedent,
we could have, in place of several ancomxxs forms, just:

    4com13.1 $e ( ( ph /\ ( ps /\ ch ) /\ th ) -> ta ) $.
    4com13   $p ( ( ch /\ ( ps /\ ph ) /\ th ) -> ta ) $=

To add an antecedent in the 3rd position to a 4-conjunct antecendent,
in place of the numerous adantlrl, adantlrr, etc. we would have just:

    4adant3.1 $e ( ( ph /\ ps /\ ch ) -> th ) $.
    4adant3   $p ( ( ( ph /\ ( ps /\ ta ) /\ ch ) -> th ) $=

There would be only one "import" statement for each conjunct count,
rather than e.g. imp41 through imp45 (plus others for the 3an cases) for
4 conjuncts:

    imp4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
    imp4   $p |- ( ( ph /\ ( ps /\ ch ) /\ th ) -> ta ) $=

Some things may become problematic.  For example, suppose we have
( ( ta /\ et ) -> ps ) that we would normally inject into an antecedent
with a sylan-like theorem.  We would need multiple cases for various
conjunct counts in both hypotheses, to ensure that the antecedent of the
result stays in canonical form.  I don't know how big a problem this
will be.

    syl32an1.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    syl32an1.2 $e |- ( ( ta /\ et ) -> ph ) $.
    syl32an1   $p |- (  ta /\ ( et /\ ps ) /\ ch ) -> th ) $=


Additional comments
-------------------

It is interesting that (with methods 2 and 3 above) df-3an usually
halves the number of parentheses in the canonical forms:

                 conjuncts:  2 3 4 5  6  7  8  9 10 11 12
                 ----------  ----------------------------
      parens w/ df-an only:  2 4 6 8 10 12 14 16 18 20 22
  parens w/ df-an + df-3an:  2 2 4 4  6  6  8  8 10 10 12

This is an argument in favor of keeping df-3an rather than having the
simpler syntax with df-an only.

If a theorem and its proof uses only conjuncts in canonical form, it
might be relatively straightforward to retrofit a possible future
df-4an (or even revert to just df-an) just by changing the canonical
forms and the utility theorems handling them.

Jonathan Ben-Naim has df-4an (called df-bnj17) in his mathbox.  I would
want to ponder whether its benefits outweigh its drawbacks before
moving it into the main set.mm.  My initial objection was the large
number of additional utility theorems that would be needed for
transformations back and forth to df-an and df-3an.  That might become
less important if we start using canonical parenthesizations.  If we add
df-4an, I think the table above would be:

                          conjuncts:  2 3 4 5  6  7  8  9 10 11 12
                          ----------  ----------------------------
               parens w/ df-an only:  2 4 6 8 10 12 14 16 18 20 22
           parens w/ df-an + df-3an:  2 2 4 4  6  6  8  8 10 10 12
  parens w/ df-an + df-3an + df-4an:  2 2 2 4  4  4  6  6  6  8  8

so the parenthesis savings, while nice, have less impact than with
adding just df-3an.  We could also go to the extreme and add
df-5an, ..., df-12an so we always have just 2 parentheses, but (among
other things) the run time of the current "improve all" algorithm in
MM-PA would grow exponentially; already the default search limit times
out when there are too many nested df-3ans.  It is possible to improve
that algorithm but it would take some work.

(End of 18-Aug-2011 canonical conjunctions proposal)
-------------------------------------------------------------------------


(22-Jun-2009) Gerard Lang's proof that ax-groth implies ax-pow was
apparently unrecognized.*  The page
http://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theory
(which calls ax-groth "Tarski's axiom") mentions only that
"Tarski's axiom also implies the axioms of Infinity and Choice."
Perhaps someone should update that page. :)
(In the interest of objectivity I do not personally edit
Wikipedia references to Metamath.)

* Later (24-Jun-2009):  Gerard pointed out that this was mentioned by
Bob Solovay here:

 http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html

I updated the Wikipedia entry to cite that.


(11-Dec-2008) I resynchronized Jeff Hoffman's nicod.mm to the recent
label changes and made it an official part of set.mm.  In connection
with this, the NAND connective (Sheffer stroke) was added to set.mm.

Jeff's original nicod.mm can be found at

  http://groups.google.com/group/metamath


(18-Nov-2008) Some stragglers I missed in yesterday's change were
updated, so if you downloaded yesterday's set.mm, you should refresh it.

   Old       New

   cdavalt   cdaval
   cdaval    cdavali
   unbnnt    unbnn3
   frsuct    frsuc
   fr0t      fr0g
   rdg0t     rdg0g
   ssonunit  ssonuni
   ssonuni   ssonunii
   eqtr3t    eqtr3
   eqtr2t    eqtr2
   eqtrt     eqtr
   3eqtr4r   3eqtr4ri
   3eqtr3r   3eqtr3ri
   3eqtr2r   3eqtr2ri
   3eqtr4r   3eqtr4ri
   3eqtr3r   3eqtr3ri
   3bitr3r   3bitr3ri
   3bitr4r   3bitr4ri
   3bitr2r   3bitr2ri
   biimpr    biimpri
   biimp     biimpi
   impbi     impbii

(17-Nov-2008) Many labels in set.mm were changed to conform to the
following convention:  an inference version of a theorem is now always
suffixed with "i", whereas the closed theorem version has no suffix.
For example, "subcli" and "subcl" are the new names for the old "subcl"
and "subclt" respectively.

Also, the inference versions of the various transitive laws now have an
"i" suffix, such as "eqtr3i" (for the old "eqtr3"), "bitr4i" (for the
old "bitr4"), etc.  This will make them consistent with the "i"/"d"
convention for inferences and deductions ("eqtr3i"/"eqtr3d", etc.)

There were 1548 labels involved in this change.  They are documented at
the top of the set.mm file.

For people used to the old labels, it will take some practice to get
used to this change, but I think it will be better in the long term
since the database now conforms to a single standard.

If anyone is using set.mm for something not in their sandboxes, you can
contact me for a script that updates the labels with these changes.


(14-Nov-2008) The following frequently-used theorems were renamed for
better consistency and to avoid confusion with negative numbers (thanks
to Stefan Allan for the suggestion):

    Old       New

    nega      notnot2
    negai     notnotri
    negb      notnot1
    negbi     notnoti
    negbii    notbii
    negbid    notbid
    pm4.13    notnot
    pm4.11    notbi


(8-Sep-2008) Although at first glance expimpd and expdimp seem rather
specialized, they actually amazingly shorten over 80 proofs, so with
them the net size of set.mm is reduced.


(1-Sep-2008) I am phasing in "A e. Fin" in place of the current
"E. x e. om { A } ~~ x" to express "A is finite".  The latter idiom is
now used frequently enough so that the net size of set.mm will hopefully
be reduced as a result.


(31-Aug-2008) I did a number of revisions to the Unicode font characters
so that all symbols now display in the Opera browser as well as Firefox.


(22-May-2008) Yesterday's derivation of axiom ax-4 from the others
required new versions of axioms ax-5 and ax-6.  The old versions were
renamed ax-5o and ax-6o.  Theorems ax5o and ax6o derive axioms
ax-5o/ax-6o from the new ax-5/ax-6; theorems ax5 and ax6 show the
reverse derivations.

The organization of the axioms in set.mm has been changed.  The new
complete set of non-redundant axioms is now introduced in a single place
in set.mm in a new section called "Predicate calculus axiomatization".
(Before, they were scattered throughout, introduced as they were
needed.)  We immediately derive ax-4 and the old ax-5 and old ax-6 (now
called ax-5o and ax-6o) as theorems ax4, ax5o, and ax6o.

The next section in set.mm, "Predicate calculus without distinct
variables", has the original gentle derivations from ax-4, ax-5o, and
ax-6o, and eventually the equality theorems not needing ax-17.  The idea
here is that as long as an inexperienced reader accepts ax-4 a priori,
there is no need to go through the advanced, $d-using proof of ax4.
This also provides us with more meaningful "proved from axioms" lists
for the section without distinct variables, without mention of the ax-17
etc. used to prove ax4.

We finally introduce the "normal" use of ax-17 in the section "Predicate
calculus with distinct variables" with essentially the same organization
as before.

The reason for proving ax4 at the beginning, and not after say the old
place where ax-17 used to be, is to conform to the following convention,
mentioned in the comment of ax4:

  Note:  All predicate calculus axioms introduced from this point forward
  are redundant.  Immediately before their introduction, we prove them
  from earlier axioms to demonstrate their redundancy.  Specifically,
  redundant axioms ~ ax-4 , ~ ax-5o , ~ ax-6o , ~ ax-9o , ~ ax-10o ,
  ~ ax-11o , ~ ax-15 , and ~ ax-16 are proved by Theorems ~ ax4 , ~ ax5o ,
  ~ ax6o , ~ ax9o , ~ ax10o , ~ ax11o , ~ ax15 , and ~ ax16 .

so that the proof of theorem ax4 can't have an accidental circular
reference to axiom ax-4 (which would be possible if we put the ax4 proof
later in the development).

The Metamath Proof Explorer Home Page has been updated with the new set
of non-redundant (as far as we know) predicate calculus axioms that
eliminates axiom ax-4.

With ax-4 omitted from the official list of non-redundant axioms, we no
longer have the former "pure" predicate calculus subsystem, that used to
be ax-4/ax-5o/ax-6o, as part of the non-redundant list.  Therefore it no
longer makes sense to subdivide the axioms into separate groups on the
MPE Home Page, and I combined them into one big table.  I moved the
description of the "pure" predicate calculus subsystem to the last entry
of the subsystem table
https://us.metamath.org/mpeuni/mmset.html#subsys

-----

On another matter, the user sandboxes have been moved to the end of
set.mm as suggested by O'Cat.  Unfortunately, this means the software
thinks they are in the "Hilbert Space Explorer" section during the web
page generation.  This will be a minor cosmetic inconvenience until I
address this.


(21-May-2008) With slightly modified ax-5 and ax-6, ax-4 becomes
redundant as shown by theorem ax4.  The ax-5 and ax-6 modifications have
the same total length as the old ones, renamed to ax-5o and ax-6o.


(17-May-2008) Axiom ax-10 was shortened.  The previous version was
renamed ax-10o.  Theorem ax10o shows that the previous version can be
derived from the new ax-10.  The Metamath Proof Explorer Home Page has
been updated to use the shortened axiom.


(14-May-2008) I am hoping that the supremum df-spw for weak orderings
will end up being easier to use in general than df-sup, because it
doesn't need a separate hypothesis to show that the supremum existence
condition is met.  Instead, the supremum exists iff the supw value
belongs to the relation's field.  If this turns out to be useful, I may
rethink the definition of df-sup as well.


(12-May-2008) The following frequently-used labels have been changed
to be slightly less cryptic and more consistent:

          old       new

12-May-08 a4w1      a4eiv
12-May-08 a4w       a4imev
12-May-08 a4c1      a4imed
12-May-08 a4c       a4ime
12-May-08 a4b1      a4v
12-May-08 a4b       a4imv
12-May-08 a4at      a4imt
12-May-08 a4a       a4im

For the new labels, "a4" means related to ax-4, "im" means the implicit
substitution hypothesis needs to be satisfied in only one direction, "i"
means inference, "e" means existential quantifier version, "v" means
distinct variables eliminate a bound-variable hypothesis, "d" means
deduction, and "t" means closed theorem.


(6-May-2008) The definitions of +oo and -oo (df-pnf and df-mnf) were
changed so that the Axiom of Regularity is not required for their
justification.  Instead, we use Cantor's theorem, as shown in
pnfnre, mnfnre, and pnfnemnf.

A standard version of the Axiom of Infinity, ax-inf2, has been added to
set.mm.  It is derived from our version as theorem axinf2, using
ax-inf and ax-reg.  I broke out ax-inf2 as a separate axiom so that
we can more easily identify "normal" uses of Regularity.  Before, this
was hard to do because any reference to omex would automatically
include Regularity as one of the axioms used.


(21-Apr-2008) Paul Chapman has replaced the real log with the more
general complex log.  The earlier real log theorems by Steve Rodriguez
have been revised to use the new definition.  Steve's original theorems
can temporarily be found under the same name suffixed with "OLD", using
the token "logOLD" rather than "log".


(10-Mar-2008) The complex number axioms use a different naming
convention than their corresponding theorems, e.g. we have axaddrcl
rather than readdclt, sometimes causing confusion for people entering
proofs.  Therefore, I added aliases for their names using 1-step proofs,
as follows:

     Axiom       Alias

     axaddcl     addclt
     axaddrcl    readdclt
     axmulcl     mulclt
     axmulrcl    remulclt
     axaddcom    addcomt
     axmulcom    mulcomt
     axaddass    addasst
     axmulass    mulasst
     axdistr     adddit
     ax0id       addid1t
     ax1id       mulid1t
     axlttrn     lttrt
     axmulgt0    mulgt0t


(6-Mar-2008) pm3.26, pm3.27, and pm3.28 were erroneously given with
logical OR expanded into negation and implication.  pm3.26OLD, pm3.27OLD,
and pm3.28OLD, which will eventually be deleted, are the erroneous
versions of these.  This error also found its way into pmproofs.txt
https://us.metamath.org/mmsolitaire/pmproofs.txt which has also
been corrected.  Going through my backups, I found that this error dates
back to pre-Metamath in the early 90's when I converted my manually
typed list of 193 Principia Mathematica theorems into the condensed
detachment notation of pmproofs.txt.  Fortunately, this has no effect on
the pmproofs.txt proof itself.  I checked against the original typed
list, and only these 3 theorems had the mistake.


(11-Feb-2008) Theorems whose description begins with "Lemma for" have
their math symbols suppressed in the Statement (Theorem) List in order
to reduce the bulk of the list for faster web page loading.  Sometimes,
though, it is useful to have the lemma displayed.  As an informal
standard, I will change "Lemma for" to "- Lemma for" when we want the
lemma displayed.  The first one is fsumcllem, requested by Paul Chapman,
since it will be used for multiple purposes and may make sense to
someday call a "theorem".  If there are lemmas you would like to have
displayed, let me know.

(3-Feb-2008) topbas provides a simpler definition of a basis when we
know its topology in advance.  It is interesting that the very complex
expansion of "( B e. Bases /\ ( topGen ` B ) = J )" simplifies to
"A. x e. J E. y ( y (_ B /\ x = U. y )" when J is known.  Proving it was
trickier than I thought it would be, although the final proof is
relatively short.

I updated the Description of istopg to explain why the variable name "J"
is used for topologies.


(16-Jan-2008) ax-12 is the longest predicate calculus axiom, and an open
problem is whether it can be shortened or even proved from the others.
After 15 years of on-and-off work on this problem with no success,
today's a12study finally gives us a first hint, showing that it is
possible to represent ax-12 with two shorter formulas.  While the
shortening of the starting formulas is modest, and of course their
combined length is much longer than ax-12, the result is still
significant:  before, it wasn't clear whether ax-12 had some intrinsic
property preventing it from being "broken up" into smaller pieces.

It is curious that the hypotheses of a12study have similar forms.  I
don't know how they might be related.  Note that by detaching ax-9 from
the second one, they can also be written:

    a12study.1 $e |- ( -. A. z z = y ->
                             ( A. z ( z = x -> z = y ) -> x = y ) ) $.
    a12study.2 $e |- ( -. A. z -. z = y ->
                             ( A. z ( z = x -> -. z = y ) -> -. x = y ) $.


(12-Jan-2008) cnnvg is designed to match hypotheses of the form
"$e |- G = ( +v ` U )" such as in nvass.  When nvass is applied to the
vector space of complex numbers, cnnvba and cnnvg will change X to CC
and G to + with no other manipulations, immediately producing the
standard associative law for addition of complex numbers (after
"U e. CVec" is detached with cnnv).  This method will allow us to make
efficient use of complex number theorems, such as when working with
linear functionals that map to complex numbers.

cnnvdemo shows how this is done.  While U is substituted with
"<. <. + , x. >. , abs >." in cnnvdemo, we keep the U separate in
cnnv, cnnvg, and cnnvba to allow simplifying the display of proof steps
(and reducing the web page size) in lemmas for long proofs, to avoid
having to repeat "<.  <. + , x. >. , abs >." over and over.

Analogous cnnv* theorems will be added for other vector space functions.


(21-Dec-2007) cofunex2g has a somewhat longer proof than might be
expected because A and B are not required to be relations but may be any
classes whatsoever.  In particular, B may be any proper class.

The recent hlxxx theorems are meant to complete the list of "(future)"
theorems referenced in the comment of ax-hilex.  These theorems will
allow us to eliminate the Hilbert Space Explorer axioms in special cases
(i.e. for concrete Hilbert spaces like CC), in order to use the Hilbert
Space Explorer theorems as part of a ZFC-only theory.


(17-Nov-2007) df-pm (with value theorem pmvalg) introduces the notion of
partial functions.  Although partial functions are ubiquitous in the
theory of operators in functional analysis, there seems to be no symbol
in the literature for them.  The closest I've seen is an occasional
"F : dom F --> B" in place of of the total function "F : A --> B",
with dom F subset A implicit.  But to do operator theory in set.mm, not
having a formal notation for for partial functions would make the theory
of operators clumsy to work with.

There are two ways to do this.  One way would be to define an analog of
df-f:

     df-fp $a |- ( F : A -|-> B <-> ( Fun F /\ F (_ ( X X. Y ) ) $.

or equivalently (by funssxp)

     df-fp $a |- ( F : A -|-> B <-> ( F : dom F --> B /\ dom F (_ A ) ) $.

Here, the standard mapping arrow with a vertical bar in the middle is
used by the Z language to denote a partial function, and it is
the only published symbol for it I've seen, although the Z language
isn't really "textbook mathematics."  I like this symbol because of its
similarity to the familiar "F : A --> B" of df-f, and I was very
tempted to use it.  The drawback is that it defines a new syntactical
structure, not just a new symbol, so we would need a whole
mini-development of equality theorems, bound variable hypothesis
builders, etc. as we do with df-f.

Such a new structure is unavoidable when the arguments could be proper
classes, as in the case of many uses of df-f.  But in the case of the
intended uses of partial functions, the domain and range will always be
sets (at least I've never seen a requirement for them in set theory
where proper classes sometimes arise).  This means that we can define a
constant symbol for an operation similar to df-map, making all of the
theorems relating to operations immediately available.

With that in mind, I chose "^pm" as a generalization of "^m" of df-map.
I am not thrilled with it because it doesn't seem intuitive or
suggestive of its meaning, but I didn't have any better ideas.  I am
open to suggestions for a better symbol to use in place of "^pm", and in
the meantime I'll continue to use "^pm" for lack of a better
alternative.


(15-Nov-2007) Baire's Category Theorem bcth was unexpectedly hard to
prove.  A big problem is that initially I didn't know that acdc5
(Dependent Choice) would be required to prove the existence of g.  The
textbook proof simply says we conclude the existence of g "by
induction," which certainly stretches the meaning of that word.


(2-Nov-2007) 0.999... is now proved, so the volunteer request of
30-Sep-2007 below is no longer applicable, although I appreciate
the attempts of individuals such as rpenner on the physorg.com forum.

The proof was more involved than I thought it would be, requiring new
theorems serzmulc1, isummulc1a, and geoisum1.

For the proof of 0.999... itself, quantifiers were avoided except for
the implicitly quantified summation variable k.  Hopefully this will
make it possible for more non-mathematicians to follow the proof.


(22-Oct-2007) Note that pm3.26bd, pm3.27bd were renamed pm3.26bi,
pm3.27bi.


(12-Oct-2007) Some of the kmlem* proofs were shortened by restating
the lemmas and using yesterday's eldifsn.


(30-Sep-2007) 0.999...=1 has been debated for many years on Usenet and
elsewhere on the Internet.

Example:  http://forum.physorg.com/index.php?showtopic=13177 from March
2007 with 267(!) pages of discussion still on-going as of today.
Includes a poll where 41% of people disbelieve 0.999...=1.  There is
even a brief reference to Metamath somewhere in the mess.

Example:
http://groups.google.com/group/sci.math/browse_frm/thread/3186915e0766f1ca
from May 2007, whose last post was September 26.

Does someone wish to volunteer to prove

  $( The repeating decimal 0.999... equals 1. $)
  0.999... $p |- sum_k e. NN ( 9 / ( 10 ^ k ) ) = 1 $= ? $.

to put an end to it once and for all?  (Wishful thinking of course.)  At
least you'll make a name for yourself. :)  Theorem geoisum may be useful
for the proof.


(28-Sep-2007) The symbol for floor was changed from "floor" to
"|_" (L-shaped left bracket) at the suggestion of Paul Chapman.


(21-Sep-2007) iccf was moved out of FL's sandbox to make it "official".
It was also renamed from the earlier "icof".

Compared to the old bl2iooOLD, the bl2ioo proof is shorter because it
incorporates Paul Chapman's recent absdifltt.


(17-Sep-2007) Perhaps a reader will volunteer to create Metamath proofs
for one or more of the following.  I hope I have stated them correctly.
They should be fun puzzles, and in the unlikely event that two people
submit the same one, the shortest proof will win.  :)  The tricks
provided by these theorems may simplify the use of theorem cnco and
relatives, because they have no dummy variables to deal with, unlike
class builder representations.  If no one responds, I'll prove them
myself eventually when I have time.

For fpar, note that each operand of i^i is not a function by itself -
the intersection cuts them down so that the final set of ordered pairs
is single-valued.  This should make it interesting to prove. :)

I think the other two are relatively straightforward, involving mainly
expansions of the definitions.  It may be possible to use a special case
of fpar for the proof of opr2f, using fconstg, but I'm not sure it would
help.  In order of increasing difficulty, I would guess fsplit, opr2f,
and fpar.  If anyone finds a simpler expression for the left-hand side
of the equality, let me know.

So, paste the below at the end of your set.mm and fire up mmj2...

(Note added 2/4/09:  fpar has been added.)
  ${
    $d x y z A $. $( etc. $)
    $( Merge two functions in parallel.  Use as the second argument of a
       composition with a (2-place) operation to build compound operations
       such as ` z = ( ( sqr ` x ) + ( abs ` y ) ) ` . $)
    fpar $p |- ( ( F Fn A /\ G Fn B ) ->
         ( ( `' 1st o. ( F o. 1st ) ) i^i ( `' 2nd o. ( G o. 2nd ) ) ) =
         { <. <. x , y >. , z >. |
            ( ( x e. A /\ y e. B ) /\ z = <. ( F ` x ) , ( G ` y ) >. ) } ) $=
?$.
  $}

(Note added 2/4/09:  I will be completing fsplit soon.)
  ${
    $d x y $.
    $( A function that can be used to feed a common value to both operands
       of an operation.  Use as the second argument of a composition with
       the function of ~ fpar in order to build compound functions such
       as ` y = ( ( sqr ` x ) + ( abs ` x ) ) ` . $)
    fsplit $p |- `' ( 1st |` I ) = { <. x , y >. | y = <. x , x >. } $=
?$.
  $}

(Note added 12/16/08:  opr2f is no longer needed; this will become curry2.)
  ${
    $d x y A $. $( etc. $)
    $( Turn an operation with a constant second operand into a function of the
       first operand only, such as ` y = ( x + 5 ) ` . $)
    opr2f $p |- ( ( F Fn ( A X. B ) /\ C e. B ) ->
        ( F o. `' ( 1st |` ( V X. { C } ) ) ) =
         { <. x , y >. | ( x e. A /\ y = ( x F C ) ) } ) $=
?$.
  $}



(13-Sep-2007) The astute reader will notice that df-ims was changed to a
more compact version (compare df-imsOLD).  imsval3 replaces imsvalOLD for
use in reproving the related *OLD theorems, although imsval3 may be
phased out with shorter direct proofs from the new imsval.

A clever technique was used in Paul Chapman's reret (of 8-Sep-2007) to
eliminate a hypothesis by using the if() function directly, without
invoking dedth.


(8-Sep-2007) hlcom is part of an eventual derivation of the Hilbert
Space Explorer axioms using ZFC only.  A small change in the Hilbert
Space Explorer axiomatization will then allow us to convert all theorems
to pure ZFC theorems, with no changes to the theorems themselves,
whenever we are dealing with a fixed Hilbert space (such as complex
numbers).  This axiomatization change is described in the comment of
ax-hilex https://us.metamath.org/mpegif/ax-hilex.html .

I probably will not actually make this change in axiomatization but will
only describe it.  It is very simple to do for anyone interested.  I
still think it is useful to have the axioms separated out - it makes the
Hilbert Space Explorer Home Page easier to describe and it also allows
us to see what axioms are used to prove specific theorems.

The Hilbert Space Explorer Home Page
https://us.metamath.org/mpegif/mmhil.html was updated to mention
this alternate approach (the first 3 paragraphs of "The Axioms"
section).


(7-Sep-2007) The new cnmet (with Met) that will replace cnms (with
MetSp) also replaces the distance function "{ <. <. x , y >. , z >. | (
( x e. CC /\ y e. CC ) /\ z = ( abs ` ( x - y ) ) ) }" with
"( abs o. - )", which I think is nicer.  A more compact version of cnmet
could read simply "( abs o. - ) e. Met", but the D is separated out to
integrate more smoothly with other theorems.  It also makes the proof a
little easier to read.

By the way the "Base" extractor (df-ba) for normed metric spaces is
capitalized because, once it is fixed for a particular vector space U,
it is not a function, unlike e.g. the "norm" extractor (df-nm).  This is
usually our convention when there is no literature standard.  Another
example is the set closed subsets "Clsd" (df-clsd) vs. the closure "cls"
(df-cls).


(4-Sep-2007) The following major changes have been made to set.mm.

1. The token Met (metric space) has been changed to MetSp.  A new token
called Met is defined as the class of all metrics (df-met), and a metric
space (df-ms) is defined as the pair of a base set and metric.  To
extract the base set X from a metric D, we will usually use "dom dom D".

Note that this is consistent with what we now do for topologies (df-top
and df-topsp), with "U. J" for the base set of topology J.  It is also
consistent with groups, which are defined using only the group operation.

The advantages of the new convention is that proofs will be often be
shorter, and theorems will be shorter to state, e.g.

  OLD:
  msf.1 $e |- X = ( 1st ` M ) $.
  msf.2 $e |- D = ( 2nd ` M ) $.
  mscl $p |- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR ) $=

  NEW:
  metf.1 $e |- X = dom dom D $.
  metcl $p |- ( ( D e. Met /\ A e. X /\ B e. X ) -> ( A D B ) e. RR ) $=

2. Eventually, the theorems involving the old MetSp will be phased out
and replaced with equivalent theorems involving the new Met.  Note that
in topology, the TopSp definition has had little real value since
everything can be done more easily with Top, and the same should be
true with metric spaces.

3. The definitions making use of the old MetSp have been replaced with
ones using Met. The old definitions have been renamed *OLD, e.g. df-bl
vs. df-blOLD.  You can see the changed ones with 'show statement
df-*OLD'.

4. All theorems making use of a df-*OLD will eventually have their
labels suffixed with OLD, in the next few days.  Some of this has
already happened.  They will eventually be replaced with non-OLD
versions.

5. Based on a suggestion of Frederic Line (see the 16-Apr-2007 comment
in http://planetx.cc.vt.edu/AsteroidMeta/set.mm_discussion_replacement ),
the cryptic "( 1st ` ( 2nd ` U ) )" etc. will go away in normed
vector spaces (including pre-Hilbert spaces, Banach spaces, and Hilbert
spaces).  Instead, we will phase in the use of the named components
df-va, df-sm, df-nm and df-ba to make the theorems more readable as well
as shorter to state.  In addition, the theorems will become independent
of the details of the ordered pairs in the vector space definition.
E.g. nvge0 will be changed from

  ${
    nvge0OLD.1 $e |- W = ( 1st ` U ) $.
    nvge0OLD.2 $e |- G = ( 1st ` W ) $.
    nvge0OLD.3 $e |- N = ( 2nd ` U ) $.
    nvge0OLD.4 $e |- X = ran G $.
    $( The norm of a normed complex vector space is nonnegative. $)
    nvge0OLD $p |- ( ( U e. NrmCVec /\ A e. X ) -> 0 <_ ( N ` A ) ) $=...
  $}

to the new

  ${
    nvge0.1 $e |- X = ( Base ` U ) $.
    nvge0.2 $e |- N = ( norm ` U ) $.
    $( The norm of a normed complex vector space is nonnegative. $)
    nvge0 $p |- ( ( U e. NrmCVec /\ A e. X ) -> 0 <_ ( N ` A ) ) $=...
  $}


Again, the original versions will be renamed to *OLD.  Some of them
already have, and this renaming should be completed in a few days.

(In the future, I may extended this use of named components to metric
spaces, etc.  For now I am limiting it to normed vector spaces, which in
a way is a "final" application of topologies, metric spaces, groups, and
non-normed vector spaces.)

Over the next few days, the labels in the current set.mm will unstable,
with frequent changes, starting at df-ms, and individual label changes
there will not be documented in the "Recent label changes" at the top of
set.mm.  The labels _before_ df-ms are stable, and any changes will be
documented in "Recent label changes" as usual.  If you are working with
set.mm, it will be safe (and preferred) to use the latest version
provided you are using things above df-ms.

The last version of set.mm before these changes are available in
us.metamath.org/downloads/metamath.zip, for a week or so.


(3-Sep-2007) Tomorrow there will be a major change in the notational
conventions for metric and vector spaces.  Today's version of set.mm is
the last version prior to this change.  If you are working from the
current set.mm, you may want to archive today's version for reference,
to compare against the new version if needed.


(22-Aug-2007) Interestingly, hbxfr shortens 40 proofs and "pays" for
itself several times over in terms of set.mm size reduction.


(2-Aug-2007) I wouldn't have guessed a priori that proving addition is
continuous (plcn) would be so tedious.  Part of the problem might be
that we have defined continuity in the very general context of
topologies, but in the long run this should pay off.  I didn't use the
epsilon-delta method, but instead obtained a slightly shorter proof (I
think) by using the already available climadd together with cnmet4.
This is exactly the method used by Gleason, although his one sentence to
that effect expands to a very long proof.


(10-Jun-2007) The symbol "Cls" was changed to "Clsd".  See the
discussion at http://planetx.cc.vt.edu/AsteroidMeta/closed_and_closure


(24-May-2007) axnegex and axrecex are now no longer used by any proof,
and were renamed to axnegexOLD and axrecexOLD for eventual deletion.
The axiom list at
https://us.metamath.org/mpegif/mmcomplex.html#axioms was updated.

A note on theorem names like msqgt0:  a theorem name such as "msqgt0"
with "msq" (m=multiplication) means "A x. A", while a name such as
"sqgt0" with just "sq" means "A ^ 2".  Since we are working directly
with the axioms, we use A x. A rather than A ^ 2 because exponentiation
is developed much later.


(23-May-2007) Eric Schmidt has solved the long-standing open problem
(first posted to Usenet on Apr. 25, 1997) of whether any of the ax*ex
axioms for complex numbers are redundant.  Here are his proofs:

For axnegex:

  One thing to notice is that both 0re and 1re depend on axnegex for their
  proofs, potentially a problem if we will need to invoke these
  statements.  However, the proof of 0re only incidentally uses axnegex,
  mainly because it relies on 1re.  Instead, we note that the existence of
  any complex number implies by axcnre the existence of a real number,
  from which 0 in R follows from the (by now) usual inverse argument.  [So
  (R, +) is a group.]

  To prove axnegex, given a complex number a + bi, we would like to find
  the additive inverse as (-a) + (-b)i.  However, proving that this is an
  additive inverse requires us to know that 0i = 0, which itself depends
  on axnegex.  We can get by with a weaker statement, namely that xi is
  real for some real x. For there exist x, y in R such that 0 = y + xi, or
  xi = -y.

  Having such an x, we know there exists c in R such that b + c = x. Then
  a + bi + ci is real, and hence has an additive inverse d. Then ci + d is
  an additive inverse of a + bi, which proves axnegex.

  We can then prove 1 in R using the current Metamath proof, in case we
  will need it.

For axrecex:

  For axrecex, (a + bi) * (a - bi)/(a^2 + b^2) = 1 ought now to be
  provable without any hoops to jump through.  The two main points are
  proving (a + bi) * (a - bi) = a^2 + b^2 and that a^2 + b^2 != 0 if
  a + bi != 0 (from which, using the now provable 0i = 0, we readily
  obtain a != 0 or b != 0).

I formalized his axnegex proof, which was posted yesterday as negext.
The axrecex proof will need a reorganization of set.mm so that some of
the ordering theorems come before the reciprocal/division theorems, so
it may take a couple of days to formalize.  These kinds of proofs tend
to be somewhat long, because we can't make use of future theorems that
depend on the axioms we are trying to prove.  Eventually axnegex and
axrecex will be eliminated from the official set of complex number
axioms at https://us.metamath.org/mpegif/mmcomplex.html, reducing
the number of axioms from 27 to 25.


(19-May-2007) As you can see from its "referenced by" list, 3expia ends
up shortening 40 proofs, which was a suprise to me, and shrinks the size
of set.mm accordingly.


(18-May-2007) Paul Chapman's relatively sophisticated bcxmas was done
entirely with mmj2.  He writes, "using mmj2, I don't have to remember
the names of theorems.  What I do with steps like this is try something
and see if mmj2 finds a theorem which fits.  When I don't, I usually add
another step (or very occasionally try a different sequence).  For more
complex steps I tend to search set.mm for text fragments I expect to
find in theorems which I think might fit the problem, eg 'A + B ) e.'."


(17-May-2007) ssimaexg and subtop were taken from FL's "sandbox" and
made official, with slightly shorter proofs.  The originals were renamed
ssimaexbOLD, topsublem1OLD, topsublem2OLD, and topsubOLD, and will be
deleted eventually.


(30-Apr-2007) The definitions of +v, etc. of 26-Apr-2007 have been
retired and replaced with new ones.  See df-va and the statements
following it.


(26-Apr-2007) It seems the new symbols +v, etc., described in the
23-Apr-2007 note below, are not a good idea after all.  It quadruples
the proof size of ncvgcl (compared to ncvgclOLD), and in general is
going to lead to longer proofs, especially for theorems brought over
from more general theories (like ncvgcl is, from vcgcl).  I have several
other ideas I'm considering but need to think them over carefully.  In
the meantime, I'll probably continue to develop new theorems with the
"W = ( 1st ` U )" etc. hypotheses, for retrofitting later.


(23-Apr-2007) The symbols +v, .s, 0v, -v, norm, and .i were taken from
the Hilbert Space Explorer for use by new definitions df-va, df-sm,
df-0v, df-vs, df-nm, and df-ip.  This will allow us to use the less
cryptic "( +v ` U )" for vector addition in a normed complex vector
space U (and Banach and Hilbert spaces), instead of the old
"( 1st ` ( 1st ` U ) )".  This was brought up by fl and discussed in the
16-Apr-2007 entries at
http://planetx.cc.vt.edu/AsteroidMeta/set.mm_discussion_replacement .
The new definitions will also provide more "generic" theorems in case we
decide to change the ordered pair structure of df-ncv, etc.

The new definitions df-va and df-ba serve the purpose of fl's proposed
df-ahf and df-hilf in http://planetx.cc.vt.edu/AsteroidMeta/fl's_sandbox .

The symbols in the Hilbert Space Explorer have been replaced with
+h, .h, 0h, -h, .ih, and normh.


(18-Apr-2007) The old Hilbert Space Explorer axioms ax-hvaddcl and
ax-hvmulcl will be replaced by ax-hfvadd and ax-hfvmul so that the
operations can be used with our group, vector space, and metric space
theorems.


(12-Apr-2007) Eric Schmidt discovered that the old ax1re, 1 e. RR,
can be weakened to ax1cn, 1 e. CC.  I updated the mmcomplex.html
page accordingly.


(27-Mar-2007) Maybe this is REALLY REALLY the end of shortening
grothprim.  At least we broke through the 200 symbol barrier.

axgroth3 was used to shorten the previous grothprim.  Unfortunately,
that one (grothprim-8OLD) is now obsolete, so I'll probably delete
axgroth3.


(23-Mar-2007) grothprim was shortened a little more by exploiting the
Axiom of Choice (via fodom and fodomb).  As for shortening grothprim
further, this may REALLY be the end of what I am capable of doing.


(21-Mar-2007) Paul Chapman revised the proof of 0nn0 (compare 0nn0OLD)
to use olci, which he feels is more natural than the old one's use of
olc, "which seems to make a complicated wff out of a simple one."


(20-Mar-2007) Unlike df-f, dff2 avoids direct or indirect references to
df-id, df-rel, df-dm, df-rn, df-co, df-cnv, df-fun, and df-fn (all of
which are used when df-f is expanded to primitives) but is still almost
as short as df-f.  I was surprised at how long and difficult the proof
was, given the vast number of theorems about functions that we already
have.  Perhaps a shorter proof is possible that I'm not seeing.


(17-Mar-2007) df-hl was changed to an equivalent one that is slightly
easier to use.  Compare the old one, df-hlOLD.


(15-Mar-2007) dfid2 is the only theorem that makes use of the fact that
x and y don't have to be distinct in df-opab.  I doubt that dfid2 will
be used for anything, but I thought it was interesting to demonstrate
this.


(12-Mar-2007) This may be it for grothprim for a while.  I have stared
at this thing for a long time and can't see any way to shorten it
further.  If anyone has any ideas let me know.


(7-Mar-2007) impbid1 and impbid2 occupy 570 bytes in set.mm but reduce
other proofs by 1557 bytes, with a 987 byte net size reduction of
set.mm.


(5-Mar-2007) In spite of its apparent simplicity, abexex is quite
powerful and makes essential use of the Axiom of Replacement (and is
probably equivalent to it, not sure).  Chaining abexex can let us prove
the existence of such things as { x | E. y E. z E. w...} that arise from
non-trivial class builders (e.g. other than just the subsets of cross
products) corresponding to ordered pair abstraction classes, etc. and
which can be quite difficult to prove directly.


(4-Mar-2007) I found shorter proofs for elnei, neips, ssnei2, innei, and
neissex.  The previous proofs are in elneiOLD, neipsOLD, ssnei2OLD,
inneiOLD, and neissexOLD (which will be deleted in a few days).


(1-Mar-2007) The contributions by Frederic Line are new versions
provided by him, using the new definition df-nei (see the notes of
15-Feb-2007 below).  Compare the *OLD ones starting at df-neiOLD.  Most
have been renamed, as well, and description for each *OLD version gives
the corresponding new name.


(20-Feb-2007) I have incorporated new sections at the end of the set
theory part of set.mm (before the Hilbert space part), called
"sandboxes," that will hold user contributions that are too specialized
for the "official" set.mm or that I haven't yet reviewed for official
inclusion.  Here are the notes in set.mm about these sections.  And, to
prevent any future misunderstandings, some dire warnings.  :)

  "Sandboxes" are user-contributed sections that are not officially part
  of set.mm.  They are included in the set.mm file in order to ensure that
  they are kept synchronized with label, definition, and theorem changes
  in set.mm.  Eventually they may be broken out as separate modules,
  particularly in conjunction with future Ghilbert translations.

  Notes:

  1. I (N. Megill) have not necessarily reviewed definitions for soundness
     or agreement with the literature.
  2. Over time I may decide to make certain definitions and theorems
     "official," in which case they will be moved to the appropriate section
     of set.mm and author acknowledgments added to their descriptions.
  3. I may rename statement labels and constants at any time.
  4. I may revise definitions, theorems, proofs, and statement descriptions at
     any time.
  5. I may add or delete theorems and/or definitions at any time.
  6. I may decide to delete part or all of a sandbox at any time, if I feel
     it will not ultimately be useful or for any other reason.

  If you want to preserve your original contribution, keep your own copy
  of it along with the version of set.mm that works with it.  Do not depend
  on set.mm as its permanent archive.

  Syntax guideline:  if at all possible, please use only 0-ary class constants
  for new definitions, to make soundness checking easier.

  By making a contribution, you agree to release it into the public domain,
  according to the statement at the beginning of this file.

Today I added sandboxes for Fred Line and Steve Rodriguez.  The contents
of their sandboxes appear in the Theorem List, at the end of the "Metamath
Proof Explorer" part.


(15-Feb-2007) The old definition of neighborhood was somewhat awkward to
work in some situations.  In particular, "the set of all neighborhoods
of a point," which occurs when working with limit points, needed a class
abstraction.  So I have revised the definition of neighborhood to be a
function that maps each subset to all of its neighborhoods, rather than
a binary relation.  This also fits more consistently with some other
definitions, I think.

The neighborhood theorems will be revised so that

  N e. ( ( nei ` J ) ` S )

is used instead of

  N ( nei ` J ) S

to mean "N is a neighborhood of subset S".  Even though this seems
longer, I believe it will make certain future theorems more natural and
even have shorter proofs in some cases.  For example, "the set of all
neighborhoods of S" just becomes

 ( ( nei ` J ) ` S )

instead of

 { x | x ( nei ` J ) S }

so that working with a dummy variable becomes unnecessary.  (We could
also use

  ( ( `' ( nei ` J ) ) " { S } )

to avoid a dummy variable with the old definition, but I don't think
many people would enjoy deciphering that!)

The old neighborhood is called "neiOLD", with its theorems renamed to
*OLD, as in df-neiOLD, etc.  These will be deleted once the conversion
is complete.


(5-Feb-2007) df-10 was added to the database, and the comments under
df-2 were revised.  Since we don't have an explicit decimal
representation of numbers, df-10 will allow more reasonable
representations as powers of 10 than just having the digits defined.
E.g.  (omitting parentheses):

    old: 456 = 4*(9+1)^2 + 5*(9+1) + 6
    new: 456 = 4*10^2 + 5*10 + 6

Previously, I avoided defining 10 since a presumed future decimal
representation might have juxtaposed 1 and 0. But such a representation
seems far off and low priority at this time, so an explicit definition
of 10 will be helpful in the interim.

A sample theorem 7p3e10 was added to "test" the new definition;
additional simple theorems for the number 10 will be added shortly.


(5-Feb-2007) (cont.)  A new version of ax-11 was added.  The original
ax-11 was renamed ax-11o, and all uses of it were replaced with
references to the new theorem ax11o (proved from the new ax-11).  A new
axiomatization was placed on the mmset.html page, and a new table was
added that summarizes what is known about various possible subsystems.

Theorem ax11a (mentioned yesterday and earlier) was renamed ax11.


(2-Feb-2007) ax11a2 proves that ax11a can replace ax-11.  I have been
wondering off and on for over 10 years whether this is the case, so I am
pleased to see it proved.  This answers the open question of 22-Jan-2007
below:  "I don't know if ax-11 can be recovered from it (that would be
nice)..."  This now means we can replace ax-11 with the shorter
equivalent

      ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )

which I am taking under consideration.  However, it would be nicer if
ax-11 could be proved from ax11a without relying on ax-16 and ax-17, so
that the "predicate calculus without distinct variables" portion ax-1
through ax-15 (+ ax-mp + ax-gen) would have the same metalogical power
of proof.

Even if we can't prove ax-11 from ax11a without ax-16 and ax-17, the
axiom set ax-1 through ax-15 would still be logically complete in the
sense described at
https://us.metamath.org/mpegif/mmzfcnd.html#distinctors .  The
deficiency would be that more theorems would have dummy variables in
their distinctor antecedents, in particular the old ax-11 proved as a
theorem.  However, in a way this is only of cosmetic importance, since
no matter how many axioms without distinct variables we have, Andreka's
theorem tells us there will always be some theorems with dummy variables
in their antecedents.

Now, if we could just simplify the long and ugly ax-12...  I have
attempted that off and on also, trying to find a shorter axiom that
captures its "essence" in the presence of the others, but without
success.  (I don't care that much about ax-15, since it is redundant in
the presence of ax-17, as theorem ax15 shows.)  The basic statement it
makes is an atomic case of ax-17 using distinctors, just like ax-15, and
that basic statement should be provable in the same way as theorem ax15
if we have the right support theorems.  The problem so far is that those
support theorems seem to need ax-12 in a different role.

ax11a2 also shows that if we wish we can "weaken" ax11a2 by making
$d x y and $d x ph if we wish, and still have completeness.  Some people
might prefer this as part of an alternate axiomatization that tries to
reduce double binding in the axioms by having all set variables
distinct.


(1-Feb-2007) Interestingly, 3anidm23 will shorten 13 proofs, and
adding will result in a net decrease in the size of set.mm.


(31-Jan-2007) To prove that ipval has the inner product property
( C x. ( A ( ip ` U ) B ) ) = ( ( C S A ) ( ip ` U ) B ), i.e.
C.<A,B> = <C.A,B> in standard notation, for all complex C (in the
presence of the parallelogram law) is nonelementary:  it involves an
induction to show it holds for C e. NN, then we extend it to QQ, then to
RR using continuity and the fact that QQ is dense in RR (qbtwnre), then
to CC.  I think this was proved by Jordan and von Neumann in 1935.  The
difficulty of the proof may be why most (all?) books define a Hilbert
space as not just a special normed space but as having a "new" operation
of inner product, from which a norm is derived.

I had some misgivings because of the difficulty of the proof, but I
think it will pay off:  our definition has the nice property that
CHil (_ CBan (_ CNrmVec which the standard definition doesn't.  This
will allow these spaces to share theorems trivially, which isn't the
case with the "standard" textbook definition.  (Analogous to this is our
NN (_ ZZ (_ QQ (_ RR (_ CC.  The standard textbook definition of CC as
ordered pairs from RR doesn't have this property formally.)

We will needed some additional theory about continuous functions for the
proof, but that should be useful for other things as well.  Anyway, it
will be some time before all the inner product properties are proved.

I may add pre-Hilbert spaces CPreHil, which is CNrmVec in which the
parallelogram law holds.  Then we would also have
CHil (_ CPreHil (_ CNrmVec.  However, CBan and CPreHil are not
comparable as subclasses (one is complete; the other has the
parallelogram law).  CHil would have the trivial definition
CHil = ( CBan i^i CPreHil ).


(24-Jan-2007) I finally was able to prove a single theorem ax11inda that
covers all cases of the quantification induction step simultaneously.
It has no restrictions on z and needs no "tricks" to use it (so there
no associated uncertainty that some special case hasn't been
overlooked).  This makes all the other versions obsolete, which have
been renamed to *OLD.  Part of the problem before is that I didn't even
know what it should end up looking like, much less how to prove it.
While the previous two evenings of effort were thus wasted, perhaps
subconsciously they helped lead me towards this final solution.

Although it is unnecessary now, I reproved yesterday's ax11demo (whose
old proof is called ax11demoOLD) to show how simple its proof becomes
with the new ax11inda.

This completes, therefore, all the basis and induction steps needed to
derive any wff-variable-free instance of ax-11 without relying on ax-11,
thus showing that ax-11 is not logically independent of the other axioms
(even though it is metalogically independent).


(23-Jan-2007) I was unhappy with yesterday's ax11inda (now ax11indaOLD)
because it was deceptively difficult to use for actual examples I tried,
and it wasn't clear to me that it could handle all possible cases
theoretically (e.g. it wasn't clear that I could derive today's ax11demo
with it).  The new ax11inda is simple to use, but it only works when z
and y are distinct.  I added the more powerful ax11inda2 that can be used
otherwise.  I think ax11inda2 can cover all possible cases, although I'm
still working on a convincing argument for that.

ax11inda2 is still not as easy to use - the variable renaming to
eliminate the 2nd hypothesis can be very tricky.  I added ax11demo to
show how to use it.

ax11inda3 is really a lemma for ax11inda, but I thought it was
interesting in its own right because it has no distinct variable
restrictions at all, and I made it a separate theorem for now.  I might
rename it to ax11indalem, though.


(22-Jan-2007) The following email excerpt describes the new theorems
related to ax-11.

  Hi Raph,

  > 4. How important is ax-11?
  >
  > Clearly, all theorems of PA can be proved using your axioms, but it's
  > quite possible that ax-11 makes the statement of certain theorems more
  > general in a useful way, and thus the resulting proof files would be
  > shorter and clearer. I'm particularly interested in the quantitative
  > question: how _much_ shorter? This is more a question for Norm than
  > for Rob, but in any case it's entirely plausible that the only real
  > way to answer it would be to try to prove a corpus of nontrivial
  > theorems both ways.

  I don't know if I have the answer you seek, but I'll recap what I know
  about ax-11:

    ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )
    https://us.metamath.org/mpegif/ax-11.html

  (You may already know some of this.)  Before Juha proved its
  _metalogical_ independence, I spent some time in the other direction,
  trying to prove it from the others.  My main result was proving, without
  ax-11, the "distinct variable elimination theorem" dvelimf2 (which
  pleased me at the time):

    https://us.metamath.org/mpegif/dvelimf2.html

  that provides a method for converting "$d x y" to the antecedent
  "-. A. x x = y ->" in some cases.  This theorem can be used to derive,
  without ax-11, certain instances of ax-11.  Theorem ax11el shows an
  example of the use of dvelimf2 for this.

  In the remark under ax-11, I say:

    Interestingly, if the wff expression substituted for ph contains no wff
    variables, the resulting statement can be proved without invoking this
    axiom.  This means that even though this axiom is metalogically
    independent from the others, it is not logically independent.  See
    ax11el for a simple example of how this can be done.  The general case
    can be shown by induction on formula length.

  Yesterday I added the theorems needed to make this remark rigorous.  For
  the basis, we have for atomic formulas with equality and membership
  predicates:

    https://us.metamath.org/mpegif/ax11eq.html
    https://us.metamath.org/mpegif/ax11el.html

  (These were tedious to prove.  ax11el is the general case that replaces
  older, more restricted demo example also called ax11el, now obsolete and
  temporarily renamed ax11elOLD.)  As a bonus, we also have the
  special-case basis for any wff in which x is not free:

    https://us.metamath.org/mpegif/ax11f.html

  For the induction steps, we have for negation, implication, and
  quantification

    https://us.metamath.org/mpegif/ax11indn.html
    https://us.metamath.org/mpegif/ax11indi.html
    https://us.metamath.org/mpegif/ax11inda.html

  respectively.  I wanted the last one to be prettier (without the implied
  substitution and dummy variable) but wasn't successful in proving it
  that way; nonetheless it is hopefully apparent how it would be used for
  the induction.

  The "distinctor" antecedent of ax-11 can be eliminated if we
  assume x and y are distinct:

    ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )  where $d x y
    https://us.metamath.org/mpegif/ax11v.html

  I didn't try to recover ax-11 from this, but my guess is that we can.

  We can also eliminate the "distinctor" antecedent like this:

    ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
    https://us.metamath.org/mpegif/ax11a.html

  which has no distinct variable restriction.  This is a curious
  theorem; I don't know if ax-11 can be recovered from it (that would
  be nice) or if it can be proved without relying on ax-11.

  Norm


(20-Jan-2007) enrefg, sbthlem10, and sbth have been re-proved so that
the Axiom of Replacement is no longer needed.


(18-Jan-2007) The replacements for the clim* and climcvg* families are
complete.  In a few days, the old theorems will be made obsolete,
with their replacements indicated in the following list, which will
be added to the "Recent Label Changes" section of set.mm.

Date      Old       New         Notes

18-Jan-07 climcvgc1 ---         obsolete; use clmi1
18-Jan-07 climcvg1  ---         obsolete; use clmi2
18-Jan-07 clim1     ---         obsolete; use clm2
18-Jan-07 clim1a    ---         obsolete; use clm3
18-Jan-07 clim2a    ---         obsolete; use clm2
18-Jan-07 clim2     ---         obsolete; use clm4
18-Jan-07 climcvg2  ---         obsolete; use clmi2
18-Jan-07 climcvg2z ---         obsolete; use clmi2
18-Jan-07 climcvgc2z ---        obsolete; use clmi1
18-Jan-07 climcvg2zb ---        obsolete; use clmi2
18-Jan-07 clim2az   ---         obsolete; use clm3
18-Jan-07 clim3az   ---         obsolete; use clm3
18-Jan-07 clim3a    ---         obsolete; use clm3
18-Jan-07 clim3     ---         obsolete; use clm4
18-Jan-07 clim3b    ---         obsolete; use clm2
18-Jan-07 climcvg3  ---         obsolete; use clmi2
18-Jan-07 climcvg3z ---         obsolete; use clmi2
18-Jan-07 clim4a    ---         obsolete; use clm3
18-Jan-07 clim4     ---         obsolete; use clm4
18-Jan-07 climcvg4  ---         obsolete; use clmi2
18-Jan-07 climcvgc4z ---        obsolete; use clmi1
18-Jan-07 climcvg4z ---         obsolete; use clmi2
18-Jan-07 clim0cvg4z ---        obsolete; use clm0i
18-Jan-07 climcvgc5z ---        obsolete; use clmi1
18-Jan-07 climcvg5z ---         obsolete; use clmi2
18-Jan-07 clim0cvg5z ---        obsolete; use clm0i
18-Jan-07 climnn0   ---         obsolete; use clm4
18-Jan-07 climnn    ---         obsolete; use clm4
18-Jan-07 clim0nn   ---         obsolete; use clm0
18-Jan-07 climcvgnn ---         obsolete; use clmi2
18-Jan-07 climcvgnn0 ---        obsolete; use clmi2
18-Jan-07 clim0cvgnn0 ---       obsolete; use clm0i
18-Jan-07 climcvg2nn0 ---       obsolete; use clmi2
18-Jan-07 clim0cvg2nn0 ---      obsolete; use clm0i
18-Jan-07 climnn0le ---         obsolete; use clm4le
18-Jan-07 clim0nn0le ---        obsolete; use clm4le and clm0


(14-Jan-2007) The purpose of resiexg is to allow us to re-prove
(eventually) the Schroeder-Berstein theorem sbth without invoking the
Axiom of Replacement.


(11-Jan-2007) Right now there is a confusing mess of about 3 dozen
theorems in the clim* and climcvg* families.  It appears that these can
all be replaced by around 7 theorems that cover all possible cases, and
clm1 is the first in this new family.  These should allow us to get rid
of the old ones, which will probably happen soon.


(8-Dec-2006) In the comment of 17-Nov-2006 below, I mentioned
"ra4sbcgfOLD used some clever tricks to convert the hypothesis of
ra4sbcfOLD to an antecedent."  Since ra4sbcgfOLD will soon be deleted, I
extracted the "trick" into a neat stand-alone theorem, dedhb.  I
shortened the proof of ra4sbcfOLD with it to show how dedhb is used.


(6-Dec-2006) I put a detailed comment about the hypotheses in imsmslem
because it uses them all in one place.  I am making note of it here for
future reference.  I've been roughly trying to keep the variable names
consistent.  There are a few changes from one theory to the next, e.g.
the group theory unit U is changed to Z (zero) in normed vector space
because it seems more natural.

Even though all these hypotheses are getting cumbersome to drag around,
that is what happens when the implicit assumptions of analysis books are
made explicit.  Fortunately, many of them tend to disappear in final
applications, such as imsms or ccims.

While it would be theoretically nicer to allow general division rings
for the scalar product of vector spaces, I think that restricting it to
CC is a reasonable compromise from a practical point of view, since
otherwise we'd need up to 5 additional hypotheses to specify the
division ring components.  In any case, most proofs would be essentially
the same if we need that generality in the future, so much of the hard
work would already be done.  There may even be an additional advantage
to doing it with CC first:  the CC proofs would tell us the minimal
number of ring theorems we would need for the more general development,
so that we could get there more quickly.

Steve Rodriguez sent in his ncvcn of 4-Dec-2006 at a fortuitous time,
because it provided the special case needed for the weak deduction
theorem dedth used in the imsms proof.


(4-Dec-2006) vcm shows that we can equivalently define the inverse of
the underlying group in a complex vector space as either the group
inverse or minus 1 times a vector.  This shows that the requirement of
an underlying Abelian group is not necessary; it could be instead an
Abelian monoid (which generalizes an Abelian group by omitting the
requirement for inverse elements), although I didn't see any mention of
that in the literature.  In any case, for future theorems I am thinking
of using mostly minus 1 times a vector in order to be compatible with
the Hilbert Space Explorer, which does not postulate a negative vector
as part of its axioms, since it can be derived from the scalar product
in the same way as vcm does.  We can use vcm to obtain the other
approach.


(1-Dec-2006) dvdemo1 and dvdemo2 are discussed at:
http://planetx.cc.vt.edu/AsteroidMeta/U2ProofVerificationEngine


(17-Nov-2006) ra4sbc eliminates the hypothesis of ra4sbcf, making the
latter obsolete (and it will be deleted eventually).  It will also make
ra4sbcgf - renamed to ra4sbcgfOLD - obsolete, since its first antecedent
is now redundant.  (Kind of sad, because ra4sbcgfOLD used some clever
tricks to convert the hypothesis of ra4sbcfOLD to an antecedent; looking
at it again, I don't know if I could ever figure it out again.  Oh
well.) ra4sbc will also eliminate the distinct variable restriction x,A
in ra4sbca and ra4csbela (the preveious versions of which have been
renamed to ra4sbcaOLD and ra4csbelaOLD).


(15-Nov-2006) The redundant Separation, Empty Set, and Pairing axioms of
ZF set theory were separated out so that their uses can be identified
more easily.  After each one is derived, it is duplicated as a new
axiom:

  Immediately after axsep ($p), we introduce ax-sep ($a)
  Immediately after axnul ($p), we introduce ax-nul ($a)
  Immediately after axpr ($p), we introduce ax-pr ($a)

To go back to the "old way" that minimizes the number of axioms, we
would just delete each $a and replace all references to it with the $p
immediately above it.  Thus we can easily go back and forth between two
approaches, as our preference dictates:  a minimal ZF axiomatization or
a more traditional one that includes the redundant axioms.


(9-Nov-2006) An interesting curiosity:  I updated the longest path in
the "2+2 trivia" section on the Metamath Proof Explorer home page, and
the longest path changed dramatically.  The path length increased from
132 to 137 - an occasional increase is to be expected, as over time new
theorems (common subproofs) are found that shorten multiple proofs.  The
curious thing is that in the old path, not a single theorem of predicate
calculus was in the list:  it jumped over predicate calculus completely
with the path: eqeq1 (set theory) <- bibi1d (prop. calc.).  However, the
new path has 22 theorems of predicate calculus, mostly uniqueness and
substitution stuff.  This was caused by the change of 12-Sep-2006 (see
notes for that date below) that provided a different path for proving
0ex.  Here is the old path for comparison:

  The maximum path length is 132.  A longest path is:  2p2e4 <- 2cn <-
  2re <- readdcl <- axaddrcl <- addresr <- 0idsr <- addsrpr <- enrer <-
  addcanpr <- ltapr <- ltaprlem <- ltexpri <- ltexprlem7 <- ltaddpr <-
  addclpr <- addclprlem2 <- addclprlem1 <- ltrpq <- recclpq <- recidpq <-
  recmulpq <- mulcompq <- dmmulpq <- mulclpq <- mulpipq <- enqer <-
  mulasspi <- nnmass <- omass <- odi <- om00el <- om00 <- omword1 <-
  omwordi <- omword <- omord2 <- omordi <- oaword1 <- oaword <- oacan <-
  oaord <- oaordi <- oalim <- rdglim2a <- rdglim2 <- rdglimt <- rdglim <-
  rdgfnon <- tfr1 <- tfrlem13 <- tfrlem12 <- tfrlem11 <- tfrlem9 <-
  tfrlem7 <- tfrlem5 <- tfrlem2 <- tfrlem1 <- tfis2 <- tfis2f <- tfis <-
  tfi <- onsst <- ordsson <- ordeleqon <- onprc <- ordon <- ordtri3or <-
  ordsseleq <- ordelssne <- tz7.7 <- tz7.5 <- wefrc <- epfrc <- epel <-
  epelc <- brab <- brabg <- opelopabg <- opabid <- opex <- prex <- zfpair
  <- 0inp0 <- 0nep0 <- snnz <- snid <- snidb <- snidg <- elsncg <- dfsn2
  <- unidm <- uneqri <- elun <- elab2g <- elabg <- elabgf <- vtoclgf <-
  hbeleq <- hbel <- hbeq <- hblem <- eleq1 <- eqeq2 <- eqeq1 <- bibi1d <-
  bibi2d <- imbi1d <- imbi2d <- pm5.74d <- pm5.74 <- anim12d <- prth <-
  imp4b <- imp4a <- impexp <- imbi1i <- impbi <- bi3 <- expi <- expt <-
  pm3.2im <- con2d <- con2 <- nega <- pm2.18 <- pm2.43i <- pm2.43 <-
  pm2.27 <- id <- mpd <- a2i <- ax-1


(8-Nov-2006) The fact that dtru (and thus ax-16) can be proved without
using ax-16 came as something of a surprise.  Still open is whether
ax-16 can be derived from ax-1 through ax-15 and ax-17.

(Later...)  Well, it turns out ax-16 can be derived from ax-1 through
ax-15 and ax-17!  That is a complete surprise.  The "secret" lies in
aev, which is a nice little theorem in itself.  I've updated the
mmset.html page - it's not very often that a new result is found about
the axiom system.  Perhaps I'll still leave in the dtruALT proof since
it is an interesting exercise in predicate logic without the luxury of
definitions, although I might delete it since it is not very important
anymore.


(4-Nov-2006) To simplify the notation - which is still quite awkward - I
decided specialize vector spaces to complex fields, instead of defining
vector spaces on arbitrary division rings, since that is what I expect
we will use most frequently.  If we need to generalize later, most
proofs should be nearly the same.


(3-Nov-2006) isgrp and grplidinv replace the older versions, but use the
hypothesis "X = ran G" instead of "X = dom dom G".  This allows us to
eliminate the 5 theorems with the "X = dom dom G" hypothesis, and all
theorems with that hypothesis have now been deleted from the database.


(31-Oct-2006) All group theory theorems (except the first 5 leading up
to grprn) were re-proved with "X = ran G" instead of "X = dom dom G" as
the hypothesis.


(29-Oct-2006) Steve Rodriguez provided a shorter proof (by 190 bytes in
the compressed proof size and by 39377 bytes in the HTML page size) for
efnn0valtlem (the lemma for his efnn0valt).


(26-Oct-2006) See
http://planetx.cc.vt.edu/AsteroidMeta/Translation_Systems for discussion
related to isarep1 and isarep2.


(25-Oct-2006) Most books (at least the ones I looked at) that define
a group with only left identity/inverse elements appear to implicitly
assume uniqueness when they derive the right identity/inverse elements,
but you need the right identity/inverse elements to prove uniqueness.
This makes our proof, which involves careful quantifier manipulations to
circumvent circular reasoning, much more complicated than the ones in
textbooks.  I don't know of a simpler way to do it.


(22-Oct-2006) I remind the reader of the entry from (17-May-2006) below
called "Dirac bra-ket notation deciphered."

kbass6t completes the associative law series kbass1t-kbass6t.  I moved
them to one place in the database for easier comparison:
https://us.metamath.org/mpegif/mmtheorems80.html#kbass1t


(19-Oct-2006) The mmnotes.txt entry of (4-Sep-2006) describes the
general philosophy I have settled on for structures like metric spaces,
which seems to be working out well:

    hyp.1 $e |- X = ( 1st ` M ) $.
    hyp.2 $e |- D = ( 2nd ` M ) $.
    xxx   $p |- (metric space theorem involving M, X, D) $=...

For topologies, the "pure" approach analogous to metric spaces would be
to work with topological spaces df-topsp, which defines topological
structures as ordered pairs S = <. X , J >..  We would then have e.g.
(hypothetical example not in the database):

    1openA.1 $e |- X = ( 1st ` S ) $.
    1openA.2 $e |- J = ( 2nd ` S ) $.
    1openA   $p |- ( S e. TopSp -> X e. J ) $=...

However, I am treating topological spaces in a different way because it
is easy to recover the underlying set from the topology on it (just take
the union).  So theorems can be shortened as follows, still separating
the topology from the underlying set in the theorem itself:

    1open.1 $e |- X = U. J $.
    1open   $p |- ( J e. Top -> X e. J ) $=...

This last is the standard I am adopting for the special case of
topologies.  It saves a little bit of space in set.mm.  Switching to the
"pure" approach in the hypothetical 1openA would be trivial if we ever
wanted to do that for aesthetic consistency or whatever.

I bring this up because yesterday's grpass shows that we are taking a
similar approach for group theory, where the underlying set can be
recovered from the domain of the group operation:  X = dom dom Grp.
Again, it would be trivial to convert all theorems to the "pure"
approach if for some reason we wanted to do that in the future.


(1-Oct-2006) Note the parsing of ac9s.  The infinite Cartesian product
X_ x e. A ... takes a class (B in this case) and produces another class
(X_ x e. A B).  Restricted quantification A. x e. A ..., on the other hand,
takes a wff (B =/= (/)) and produces another wff (A. x e. A B =/= (/)).

    ac9s $p |- ( A. x e. A B =/= (/) <-> X_ x e. A B =/= (/) )
                           <------->     <--------->
                             wff            class
                 <----------------->     <----------------->
                       wff                   wff

If we were to use additional parenthesis (which are unnecessary for
unambiguous parsing), ac9s would read:

    ac9s $p |- ( A. x e. A ( B =/= (/) ) <-> ( X_ x e. A B ) =/= (/) )

So far in the database, the following definitions with "restricted"
bound variables take a class and produce a class:

    df-iun       U_ x e. A B
    df-iin     |^|_ x e. A B
    df-ixp       X_ x e. A B
    df-sum     sum_ x e. A B

If we wanted, we could define these surrounded by parentheses to
eliminate any possible confusion.  No proofs would have to be changed,
only the theorem statements.  However, there are already too many
parentheses in a lot of theorems.  Since the parenthesis-free notation
for these is unambiguous, I thought it would be best in the long run.
It's just a matter of getting used to it, and if in doubt one can always
consult the syntax hints or use "show proof ... /all".

A different example of this kind of possible confusion is sbcel1g:

    ( [ A / x ] B e. C <-> [_ A / x ]_ B e. C )
                <---->     <----------->
                 wff          class
      <-------------->     <---------------->
           wff                 wff

which is never ambiguous because of the different brackets:  [ A / x ]
takes a wff as an argument, and [_ A / x ]_ takes a class as an
argument.


(29-Sep-2006) eluniima allows us to reduce alephfp from 72 steps to 62
steps.  Compare the older version still at
https://us.metamath.org/mpegif/alephfp.html .  (I revisited alephfp
after the discussion on http://planetx.cc.vt.edu/AsteroidMeta/metamath ).
eluniima is interesting because there aren't any restrictions on A,
which can be completely unrelated to the domain of F.

rankxplim and rankxpsuc provide the answer to part of Exercise 14 of
Kunen, which asks the reader to "compute" the rank of a cross product.
(Some of the other ones can almost be "computed" - you take the previous
answer and add 1 - but it is a stretch to call this proof a
"computation".)  This is a very difficult and rather unfriendly problem
to give as a "homework exercise" - at least the answer should have been
provided as a clue to work out the proof, which is already hard enough
(especially since the answer has two parts, or three if we count the
empty cross product).  I wasted a lot of time on it, because I had to
prove something that I had no clue of what it would be.  I wonder how many
people have actually worked this out:  no one in sci.logic seemed able
to answer it.
http://groups.google.com/group/sci.logic/browse_frm/thread/41fad0ba18a9dce1

df-ixp is new.  I'm somewhat torn about the bold X - a capital Pi
is used in many books, but as the comment says I'd prefer to reserve
that for products of numbers.  I'm open to comments on the notation.


(28-Sep-2006) I decided to restore the ancient (12-year-old) proof of
pwpw0 for "historical" reasons (see discussion at
http://planetx.cc.vt.edu/AsteroidMeta/metamath ).  It has actually been
modernized slightly, to remove the requirement that the empty set exist.
This eliminates the need for the Axiom of Replacement, from which
empty set existence is derived.  The original can be seen at
https://de.metamath.org/metamath/set.mm .

rankuni improves rankuniOLD of 17-Sep by eliminating the unnecessary
hypothesis A e. V.  Although this will shorten future proofs, I
don't know know if such shortening will end up "paying" for the extra 16
steps of overhead needed to eliminate A e. V.  But at least rankuni will
be easier to use than rankuniOLD, having one less condition to satisfy.


(17-Sep-2006) foprab2 is a new version (of foprab2OLD) that no longer
requires the "C e. V" hypothesis.  The new proof, using the 1st and 2nd
functions, is very different from that of foprab2OLD and the other
*oprab* theorems.


(16-Sep-2006) Steve Rodriguez says about his efnn0valtlem/efnn0valt
proof, "It's not short, but it took far less time than I expected, and
the result seemed so obvious that I felt nagged to prove it somehow."


(15-Sep-2006) opntop is an important theorem, because it connects metric
spaces to our earlier work on topology, by showing that a metric space
is a special case of a topology.  This lets us apply the theorems we
have already developed for topologies to metric spaces.  (It took some
work to get there; many of the theorems in the last few days where
towards the goal of proving opntop.)

The members of a topology J are called its "open sets" in textbooks, and
this theorem provides a motivation for that term.  (We do not have a
separate definition for the open sets of a topology, since to say that A
is an open set of topology J we just say "A e. J".)


(12-Sep-2006) A number of proofs (some not shown in the Most Recent
list) were modified to better separate the various uses of the Axiom of
Replacement, and in particular to show where the Axiom of Extensionality
is needed.  The old zfaus was renamed to zfauscl, and the current zfaus
is new.

Some proofs in the axrep1 through axrep5 sequence were modified to
remove uses of Extensionality, so that zfaus now uses only Replacement
for its derivation.  The empty set existence zfnul now uses only zfaus
(and thus only Replacement) for its derivation.  The new zfnuleu then
shows how Extensionality leads to uniqueness (via the very useful bm1.1,
which uses only Extensionality for its derivation).  Finally, 0ex was
changed (with a slightly longer proof) so that it is now derived
directly from zfnuleu, to illustrate the path:

  ax-rep -> zfaus -> zfnul -> zfnuleu -> 0ex
                                 ^
                                 |
                               ax-ext

Some books try to postpone or avoid the Replacement Axiom when possible,
using only the weaker Separation (a.k.a.  Subset, a.k.a.  Aussonderung).
This can now be done in our database, if we wish, by changing zfaus and
zfpair from theorems to axioms.  (See the new last paragraph in the
ax-rep description.)


(7-Sep-2006) The set.mm database was reorganized so that the ZFC axioms
are introduced more or less as required, as you can see in the new Table
of Contents https://us.metamath.org/mpegif/mmtheorems.html#mmtc .
This lets you see what it is possible to prove by omitting certain
axioms.  For example, we prove almost all of elementary set theory (that
covered by Venn diagrams, etc.) using only the Axiom of Extensionality,
i.e. without any of the existence axioms.  And quite a bit is proved
without Infinity - for example, Peano's postulates, finite recursion,
and the Schroeder-Bernstein theorem (all of which are proved assuming
Infinity in many or most textbooks).


(4-Sep-2006) I will be changing the way that the theorems about metric
spaces are expressed to address some inconveniences.

Consider the following two ways of expressing "the distance function of
a metric space is symmetric".  In the present database, we use both
methods for various theorems.  (These examples, though, are hypothetical,
except that mssym1v1 = mssymt).

 (1) mssymv1.1 $e |- D e. V $.
     mssymv1 $p |- ( ( <. X , D >. e. Met /\ A e. X /\ B e. X ) ->
                   ( A D B ) = ( B D A ) ) $=

 (2) mssymv2 $p |- ( ( M e. Met /\ A e. ( 1st ` M ) /\ B e. ( 1st ` M ) ) ->
                   ( A ( 2nd ` M ) B ) = ( B ( 2nd ` M ) A ) ) $=

The first way, mssymv1, shows the base set and the distance function
explicitly with the helpful letters X and D.

But often we want to say things about the metric space as a whole, not
just its components.  The second way, mssymv2, accomplishes that goal,
at the expense of readability:  it is less reader-friendly and more
verbose to say ( 2nd ` M ) rather than just D.

Although it is possible to convert from one to the other, it can be
awkward, especially converting from mssymv1 to mssymv2.  So practically
speaking, we will end up creating two versions of the same theorem,
neither of which is ideal.

A solution to this is provided by the following third version:

 (3) mssymv3.1 $e |- X = ( 1st ` M ) $.
     mssymv3.2 $e |- D = ( 2nd ` M ) $.
     mssymv3 $p |- ( ( M e. Met /\ A e. X /\ B e. X ) ->
                   ( A D B ) = ( B D A ) ) $=

The conclusion is simpler than either of the first two versions and
clearly indicates the intended meaning of an object with letters M, X,
and D.  Although the hypotheses are more complex, in the database they
will typically be resused by several theorems.

To obtain mssymv1 from mssymv3 is trivial:  we replace M with
<. X , D >., then we use op1st and op2nd to eliminate the hypotheses.

To obtain mssymv2 from mssymv3 is trivial:  we use eqid to eliminate
the hypotheses.

So, with this approach, we should never need to prove mssymv1 and
mssymv2 separately, since the conversion to either one in a proof is
immediate.

My plan is to convert everything to this approach and make most of the
existing theorems obsolete.  msf is the first one using this approach,
and it will replace the existing msft.  As always, comments are
welcome.


(3-Sep-2006) Although ax16b is utterly trivial, its purpose is simply to
support the statement made in the 7th paragraph of
https://us.metamath.org/mpegif/mmzfcnd.html


(29-Aug-2006) The value of the ball function is a two-place function,
i.e. it takes in two arguments, a point and a radius, and returns a set
of points.  I define it as an "operation" in order to make use of the
large collection of operation theorems, and also to avoid introducing a
new syntactical form.  However, we have two choices for expressing "The
value of a ball of radius R around a point P".  Note that M is a metric
space, X is the underlying space of a metric space, and D is a distance
function.

  Operation value notation:

    ( P ( ball ` M ) R )
    ( P ( ball ` <. X , D >. ) R )      when M = <. X , D >.

  Function value notation:

    ( ( ball ` M ) ` <. P , R >. )
    ( ( ball ` <. X , D >. )  ` <. P , R >. )     when M = <. X , D >.

The former is shorter and will result in shorter proofs in general, but
I'm not sure that using infix notation like we would for an operation
like "+" is the most natural or familiar.  There is no standard
notation in the literature, which uses English and also does not make
the metric space explicit.  I am open to comments or suggestions.

Since ( ball ` <. X , D >. ) acts like an operation value, we also have
a third choice and could say, equivalently,

    ( P ( X ball D ) R )      when M = <. X , D >.

for an even shorter notation, although I'm not sure how odd it looks.
But maybe I should use it for efficiency.

Note that I am using <. X , D >. e. Met instead of X Met D (via df-br),
even though the latter results in shorter proofs (and is shorter to
state).  It seems that Met feels more like a collection of structures
that happen to be ordered pairs of objects, than it does a relation,
even though those concepts are technically identical.


(28-Aug-2006) Two of the standard axioms for a metric space, that the
distance function is nonnegative and that the distance function is
reflexive, are redundant, so they have been taken out of the definition
df-ms to simplify it.  (We will prove the redundant axioms later as
theorems.)

The first part of the ismsg proof (through step 18) is used to get rid
of the antecedent X e. V that occurs in step 43.  (If either side of the
ismsg biconditional is true, it will imply X e. V, making it redundant
as an antecendent.)


(26-Aug-2006) Some small items related to yesterday's unctb (with a new
version today) were cleaned up:

  1. The unused hypothesis B e. V was removed from unictb.
  2. unpr was renamed to unipr for naming consistency.
  3. The hypotheses A e. V and B e. V eliminated from unctb, with the help
     of the new theorem uniprg.


(22-Aug-2006) csbopeq1a will help make the '1st' and '2nd' function
stuff worthwhile; it lets us avoid the existential quantifiers that are
used in e.g. copsexg.  It is often easier to work with direct
computations rather than having to mess around with quantifiers.  I was
surprised that there are no distinct variable, class existence, or any
other restrictions on csbopeq1a.  There is no restriction on what B may
contain, which could be any random usage of x and y, not just ordered
pairs of them; but what we are doing is the logical equivalent of
substitution for ordered pairs in B as if it actually contained them.


(21-Aug-2006) I made some subtle changes to the little colored numbers.
Although this may seem like a trivial topic, and probably is, the
problems of obtaining a spectrum of colors with uniform brightness and
maximum distinguishability vs. hue changes aren't as easy as they might
first appear.  Perhaps someone has done it before, but all of the
spectrum mappings I could find have brightness that varies with color
and aren't suitable for fonts, in particular because yellow is hard to
read.  They also do not change color (as perceived by the eye) at a
uniform rate as you go through the spectrum, such as the color changes
crowded into the transition from red to green.  This is OK if you want
an accurate representation of color vs. wavelength, but our goal is to
be able to distinguish different colors visually in an optimal way.

Anyway I thought it was an interesting problem, so I thought I'd say
something about it.  Even though it pales in importance compared to
today's announcement of the evidence that dark matter exists. :)

The new "rainbow" color scheme now has the following properties:

1. All colors now have 50% grayscale levels.  Specifically, all colors
now have an L (level) value of 53 in the L*a*b color model (see
http://en.wikipedia.org/wiki/Lab_color).  This means if you convert a
screenshot to grayscale, the colored numbers will all have the same
shade of gray.  Yellow becomes brown at L=53, with an RGB value of
131-131-0.

2. Within the constraint of the 50% grayscale level, each color has
maximum saturation.  This is not as simple as it seems:  the RGB color
for the brightest pure blue, 0-0-255, has an L value of only 30 (because
the eye is not as sensitive to blue), so we have to use the 57%
saturated 110-110-255 to get L=53.  Green, on the other hand, just
requires 0-148-0 for L=53, which is 100% saturated.

3. The hues are not equally spaced numerically, but according to how the
eye is able to distinguish them.  I determined this empirically by
comparing the distinguishability of adjacent hues on an LCD monitor,
using a program I wrote for that purpose.  For example, the eye can
distinguish more hues between red and green than between green and blue.
This was a problem with the old colors, which seemed to have too many
undistinguishable blue-greens.  Now, as experimentally determined, the
transition from green to blue represents only 21% of the color values.

You can see the new color spectrum at the top of a theorem list page
such as https://us.metamath.org/mpegif/mmtheorems.html.

The spectrum position to RGB conversion is done by the function
spectrumToRGB in mmwtex.c of Metamath version 0.07.21 (20-Aug-2006).


(16-Aug-2006) abfii5 is the last in the series, and just chains together
the others to obtain the final connection between verion used by subbas
and the version of the left-hand side that does not involve
equinumerosity.  This can allow us to express subbas in more elementary
terms, if we wish.


(14-Aug-2006) abfii4 is an interesting brainteaser:  we show the that

  |^| { x | ( A (_ x /\ A. y ( ( y (_ x /\ -. y = (/) /\
                                   E. z e. om y ~~ z ) -> |^| y e. x ) ) }

is equal to

  |^| { x | ( A (_ x /\ A. y ( ( y (_ A /\ -. y = (/) /\
                                   E. z e. om y ~~ z ) -> |^| y e. x ) ) }

The second differs from the first by only one symbol:  the "x" is
changed to "A".  At first, it superficially looked like this would be an
elementary theorem involving set intersection.  But the proof turned out
quite difficult, not only the proof itself but the fact that it needs
some subtheorems that are somewhat difficult or nonintuitive in
themselves (fiint, intab via abfii2, abfii3, abexssex [based on the
Axiom of Replacement], intmin3), involving the theories of ordinals,
equinumerosity, and finite sets.  abfii2 and abfii3 are each used twice,
for different purposes, in different parts of the proof.  I found this
quite a challenge to prove and would be most interested if anyone sees a
more direct way of proving this.  (Note that "E. z e. om y ~~ z" just
means "y is finite".  A simpler way of stating this is "y ~< om", but
that requires the Axiom of Infinity, which I wanted - and was able - to
avoid for this proof.)

abfii4 arose while looking at equivalent ways to express the collection
of finite intersections of a set, which determines a basis for a
topology (see theorem subbas).  Textbooks never seem to mention the
exact formal ways of saying this, but just say, informally, "the
collection of finite intersections."


(11-Aug-2006) With today's 3bitr4rd, we now have the complete set of all
8 possibilities for each of the 3-chained-biconditional/equality series
3bitr*, 3bitr*d, 3eqtr*, and 3eqtr*d.


(9-Aug-2006) If you are wondering why it is "topGen" and not "TopGen", a
standard I've been loosely following is to use lowercase for the first
letter of classes that are ordinarily used as functions, such as sin,
cos, rank, etc.  Of course you will see many exceptions, mainly because
I try to match the literature when the literature gives something a
name.


(7-Aug-2006) intab is one of the oddest "elementary" set theory theorems
I've seen.  It took a while to convince myself it was true before I
started to work out the proof, and the proof ended up rather long and
difficult for this kind of theorem.  Yet the only set theory axiom used
is Extensionality.  Perhaps I just have a mental block - if anyone sees
a simpler proof let me know.  I haven't seen this theorem in a book;
it arose as part of something else I'm working on.

Two new definitions were added to Hilbert space, df-0op and df-iop.
The expressions ( H~ X. 0H ) and ( I |` H~ ), which are equivalent to
them, have been used frequently enough to justify these definitions.


(4-Aug-2006) 3eqtrr, 3eqtr2, 3eqtr2r complete the 8 possibilities for 3
chained equalities, supplementing the existing 3eqtr, 3eqtr3, 3eqtr3r,
3eqtr4, 3eqtr4r.  I was disappointed that the 3 new ones reduce the size
of only 33 (compressed) proofs, taking away a total of 185 characters
from them, whereas the new theorems by themselves increase the database
size by 602 characters.  So, the database will grow by a net 417
characters, and the new ones don't "pay" for themselves.  Nonetheless,
O'Cat convinced me that they should be added anyway for completeness.
He wrote:

  I would vote for adding them even though the net change is PLUS 400 some
  bytes.

  It just makes unification via utilities like mmj2 much easier -- input
  the formula and let the program find a matching assertion.

  Esp. now that you've done the work to analyze them, it is illogical not
  to make the change:  eventually enough theorems will be added to save
  the added bytes.  And within 10 years when we have bigger, faster
  memories people would think it quaint to be so stingy with theorems that
  serve a valid purpose.

  We're going to have PC's with a minimum of 1GB RAM next year as
  standard, and that will just grow as the new nanotech advances.

So, I guess I should also complete the 3eqtr*d, 3bitr*, and 3bitr*d
families at some point.


(2-Aug-2006) fiint replaces fiintOLD.


(24-Jul-2006) Note that the description of today's sucxpdom says it has
"a proof using AC", and you can see in the list of axioms below the
proof that ax-ac was used.  Exercise:  How does ax-ac end up getting
used?

Metamath has no direct command to show you this, but it is usually easy
to find by examining the outputs of the following two commands:

1. show trace_back sucxpdom

  ... numth2 fodom fodomg fnrndomg ccrd($a) df-card($a) cardval cardon cardid
  cardne carden carddomi carddom cardsdom domtri entri entri2 sucdom unxpdomlem
  unxpdom
  ^^^^^^^

2. show usage ax-ac / recursive

  ...imadomg fnrndomg unidom unidomg uniimadom uniimadomf iundom cardval cardon
  cardid oncard cardne carden cardeq0 cardsn carddomi carddom cardsdom domtri
  entri entri2 entri3 sucdom unxpdomlem unxpdom unxpdom2 sucxpdom...
                                        ^^^^^^^

The theorem we want is the last theorem in the first list that appears
in the second list.  In this case, it is unxpdom.  And, indeed, we see
that unxpdom appears in proof step 43 of sucxpdom, and we can check that
unxpdom uses ax-ac for its proof.

Of course, there may be other theorems used by the sucxpdom proof that
require ax-ac, and the only way to determine that is to see if any of
the other theorems used in the sucxpdom proof appear in the second list.
But for a quick indication of what ax-ac is needed for, the method above
can be useful.


(21-Jul-2006) oev2, etc. are part of Mr. O'Cat's cleanup project for
some of the remaining *OLDs.


(20-Jul-2006) istps5 is important because it reassures us that our
definitions of Top and TopSp match exactly the standard textbook
version, even though the latter is more complex when the words are
converted to symbols.  I don't think istps5 will have an actual
practical use, though, because of the simpler theorems we have
available.


(18-Jul-2006) It is awkward to eliminate the "J e. V" hypothesis from
istop and isbasis/isbasis2 when it is redundant (as in most uses) - it
requires working with a dummy variable then elevating it to a class
variable with vtoclga - so I'm replacing these with "g" versions that
have "J e. V" as an antecedent (actually "J e. A" as customary, to allow
easy use of ibi when we need only the forward implication).  I think the
non-g versions will be used so rarely that it's not worth keeping them,
so they will be deleted, and it is trivial to use instead the "g"
version + ax-mp.  [Update:  istop, isbasis, and isbasis2 have now been
deleted.]

I also modified Stefan's 0opn, uniopn to use an antecedent instead
of hypothesis.

(17-Jul-2006) It is interesting that Munkres' definition of "a basis for
a topology" can be shortened considerably.  Compare
https://us.metamath.org/mpegif/isbasis3g.html (Munkres' version)
with https://us.metamath.org/mpegif/isbasisg.html (abbreviated
version).  Munkres' English-language definition is (p. 78):

  "Definition.  If X is a set, a _basis_ for a topology on X is a
  collection _B_ of subsets of X (called _basis elements_) such that

    (1) For each x e. X there is at least one basis element B containing x.

    (2) If x belongs to the intersection of two basis elements B1 and B2,
    then there is a basis element B3 containing x such that B3 (_ B1 i^i B2."

-------

The symbols and statement labels for topology were changed
in order to conform more closely to the terminology in Munkres, who
calls a member of our (old) Open a "topology" and calls a member of our
(old) Top a "topological space".

  Old symbol      New symbol
  ----------      ----------
  Top             TopSp
  Open            Top

  Old label       New label
  ---------       ---------
  ctop            ctps
  cope            ctop
  df-top          df-topsp
  df-open         df-top
  dfopen2         dftop2
  optop           eltopsp
  elopen1         istop
  elopen2         uniopnt
  op-empty        0opnt
  empty-op        sn0top
  op-indis        indistop

-------

If J is a topology on a set X, then X is equal to the union of J.  For
this reason, the first member of a topological space <X,J> is redundant.
I am doubtful that the topological space definition will offer any real
benefit and am considering deleting it.  If anyone knows of a good
reason to keep it, let me know.


(12-Jul-2006) I added definitions for topology (df-open and df-top),
and added some theorems that Stefan Allan proved back in Feb. and
March.  Note that "e. Open" and "e. Top" are equivalent ways of saying
something is a topology, which will be shown by the (in progress)
theorem "( <. U. A , A >. e. Top <-> A e. Open )", and "e. Open" is
often simpler.


(7-Jul-2006) Over 100 *OLD's were removed from the database,
and 83 remain, 27 of which are in the Hilbert space section.  After
subtracting the *OLD's, there are 6053 theorems in the non-Hilbert-space
part and 1181 for Hilbert space, a total of 7243.

This is the first time I have become aware that the Metamath Proof
Explorer has officially passed the 6000 theorem "milestone", which
happened 53 (green) theorems ago.  The 6000th theorem was mt4d, added on
18-Jun-2006.


(6-Jul-2006) With the updated projlem31, all uses of ~~>OLD have been
eliminated.  Soon I will remove the corresponding *OLD theorems from
the database.

(5-Jul-2006) With ege2le3, we have finally completed the
non-Hilbert-space conversion of ~~>OLD based theorems, so that no
theorem in the non-Hilbert-space section depends on ~~>OLD anymore.  We
can't delete their *OLD theorems yet, because ~~>OLD is still used in a
few places in the Hilbert space section, but that should happen soon.
(If you type "show usage cliOLD/recursive" you will see, before the
Hilbert space section, the *OLDs that will be deleted.  I will delete
them when the Hilbert space stuff is finished being converted to ~~>.)

By the way, we can also now delete all uses of sum1oo ("show usage
csuOLD /recursive"), which I will do soon.


(4-Jul-2006) Currently there are separate derivations of 2 <_ e,
e <_ 3, and 2 <_ e /\ e <_ 3.  Eventually, I'll probably delete the
first two, but for now my immediate goal is to replace the *OLDs.


(1-Jul-2006) The proof of class2seteq was shortened.  Compare the
previous version, class2seteqOLD.


(30-Jun-2006) Continuing with the *OLD cleanup project, the future
ereALT (to replace the existing ereALTOLD) will be an alternate
derivation of ere.  The proof of ereALTOLD specifically focuses on the
number e and has a much simpler overall derivation than ere, which is a
special case of the general exponential function derivation.  The lemmas
for ereALTOLD are mostly reused to derive the bounds on e (ege2OLD,
ele3OLD, ege2le3OLD).  Since ereALTOLD ends up being a natural byproduct
of those theorems, I have so far kept it even though it is redundant, in
order to illustrate a simpler alternate way to prove e is real.

By the way, if you are wondering how the mmrecent page can "predict" a
future theorem, the web page generation program simply puts "(future)"
after the comment markup for any label not found in the database (and
also issues a warning to the user running the program).  The mmrecent
page is refreshed with the "write recent" command in the metamath
program.


(21-Jun-2006) The conversion of cvgcmp3ce to cvgcmp3cet is one of our
most complex uses to date of the Weak Deduction Theorem, involving
techniques that convert quantified hypotheses to antecedents.  The
conversion is performed in stages with 2 lemmas.  (Eventually I want to
look at proving the theorem form directly, with the hope of reducing the
overall proof size.  But for now my main goal is to replace the *OLDs.)


(20-Jun-2006) As mentioned below, most of the recent convergence stuff
like cvgcmp3ce is slowly replacing the *OLDs.  My goal is to be able to
delete a big batch of *OLDs, used to develop the exponential function
df-ef, within 2 weeks.  We can't get rid of them until the last piece in
the chain of proofs is completed.


(6-Jun-2006) caucvg will replace caucvgOLD as part of the *OLD
elimination project.  BTW Prof. Wachsmuth doesn't know where this proof
is from; he may have originated it:  "I wrote many of the proofs years
ago and I don't remember a good reference for this particular one"
(email, 6-Jun).  It is a nice proof to formalize because it is more
elementary than most, in particular avoiding lim sups.  (We have
df-limsup defined, but it needs to be developed.  As you can see,
df-limsup is not so trivial.  One of its advantages is that it is
defined - i.e. is an extended real - for all sequences, no matter how
ill-behaved.)


(5-Jun-2006) kbass2t is the 2nd bra-ket associative law mentioned in
the notes of 17-May-2006.


(4-Jun-2006) The description of ax-un was made clearer based on a
suggestion from Mel O'Cat.


(2-Jun-2006) unierr is our first theorem related to qubits and quantum
computing.  In a quantum computer, an algorithm is implemented by
applying a sequence of unitary operations to the computer's qubit
register.  A finite number of gates selected from a fixed, finite set
cannot implement an arbitrary unitary operation exactly, since the set
of unitary operations is continuous.  However, there is a small set of
quantum gates (unitary operations) that is "universal," analogous to the
universal set AND and NOT for classical computers, in the sense that any
unitary operation may be approximated to arbitrary accuracy by a quantum
circuit involving only those gates.  Theorem unierr tells us,
importantly, that the worst-case errors involved with such
approximations are only additive.  This means that small errors won't
"blow up" and destroy the result in the way that, say, a tiny
perturbation can cause completely unpredictable behavior in weather
prediction (the "butterfly effect").


(1-Jun-2006) Someone complained about not being able to understand infpn
(now called infpn2), so I decided to make the "official" infpn use only
elementary notation.


(27-May-2006) The label of the ubiquitous 'exp' (export) was changed to
'ex' to prevent it from matching the math token 'exp' (exponential
function), in anticipation of the 24-Jun Metamath language spec change.
Normally I don't mention label changes here - they're documented at the
top of set.mm - but ex, used 724 times, is the 5th most frequently used
theorem, so this change will almost certainly impact any project using
set.mm as a base.  Although ex is of course not new, I re-dated it to
show up in today's "Most Recent Proofs."


(25-May-2006) I think csbnest1g is interesting because the first
substitution is able to "break through" into the inside of the second
one in spite of the "clashing x's".  Compare csbnestg, where x and y
must be distinct.  I found it a little tricky to prove, and I wasn't
even sure if it was true at first.


(18-May-2006) kbasst proves one of the associative laws mentioned
yesterday:

  Dirac:  ( |A> <B| ) |C> = |A> ( <B| |C> )
  Metamath: ( ( A ketbra B ) ` C ) = ( ( ( bra ` B ) ` C ) .s A )


(17-May-2006)   Dirac bra-ket notation deciphered

Most quantum physics textbooks give a rather unsatisfactory, if not
misleading, description of the Dirac bra-ket notation.  Many books will
just say that <A|B> is defined as the inner product of A and B, or even
say that it _is_ the inner product, then go off and give mechanical
rules for formally manipulating its "components" <A| and |B>.  For
physicists, "formally" means "mechanically but without rigorous
justification."

If the dimensions are finite, there is no problem.  A finite-dimensional
Hilbert space is essentially linear algebra, and it is possible to prove
that any vector and linear operator can be represented by an n-tuple
(column vector) and matrix respectively.  In finite dimensions, we just
define |A> to be a column vector and <A| its conjugate transpose, a row
vector.  Viewed this way, the various combinations such as <A|B> (the
inner product) and |A><B| (a matrix) make sense in finite dimensions.

But what is the "transpose" of a vector in infinite-dimensional Hilbert
space?  To answer this, some of the more mathematically "rigorous" books
say that |B> is a member of Hilbert space (a vector) and call <A| a
member of a "dual space" of functionals, which defined such that the
value of functional <A| evaluated at vector |B> equals the inner product
of A and B.  While this solves some of the problems, mysteries remain,
such as, what is the "outer product" |A><B|?  They don't say (it is very
rare to find a QM book with "outer product" in the index), and they
evade the problem by implicitly assuming an (unproven) associative law
that allows them to manipulate their way out of having to explain it.
But the in-between stages of manipulation can contain undefined objects,
and while the mechanical rules "work," there isn't an obvious way to
formalize them.  In particular, this associative law cannot be proved
from Hilbert space axioms from the partial definitions given in most
books.

Maybe it's just the way my brain works, but I find it difficult to
become comfortable with a concept that isn't precisely defined, even if
it "works."  An advantage (as well as a frustration) of Metamath is
that it forces you to resolve any such issues if you want to make
progress.

Fortunately, I found a book by Eduard Prugovecki (pronounced
Proo-go-vetch-kee), _Quantum Mechanics in Hilbert Space_, that (on p.
376, 1981 ed.; p. 370, 1971 ed.) presents a defining equation for
|A><B|, which you can see as today's theorem kbvalvalt, from which we
can deduce a direct definition (df-kb) that works in both finite- and
infinite-dimensional Hilbert space.  With this final clue, I was able to
map precise set-theoretical expressions uniquely to and from all
possible Dirac bra and ket combinations.  The mapping is somewhat
complicated, but it is complete and well-defined.  In order to
accomplish this, I introduced two new definitions.  The "bra" function
takes in a vector and outputs a functional.  The "ketbra" operation
takes two vectors and outputs an operator.  Their definitions are given
by df-bra and df-kb.

  Added 18-May-06:  I noticed that the Wikipedia bra-ket article, added
  last year, also defines outer product (in the same way as Prugovecki).

Note that a "functional" is any function from H~ (Hilbert space) to CC
(complex numbers), and an "operator" is any function from H~ to H~.

The rule for combining bras and kets is that a bra may be placed after a
ket and vice-versa, but two bras and two kets may not be juxtaposed.  In
other words, bras and kets must alternate.  Juxtaposition can be thought
of roughly, but not exactly, as a kind of "product."  Juxtaposition is
associative:

  ( |A> <B| ) |C> = |A> ( <B| |C> )
  ( <A| |B> ) <C| = <A| ( |B> <C| )

(Added 24-May-2006) There is a technicality in our development.
Mathematicians, and set.mm, define inner product such that
( C x. ( A .i B ) ) = ( ( C .s A ) .i B ).  Physicists define it such
that ( C x. ( A .i B ) ) = ( A .i ( C .s B ) ) where ".s" is the scalar
product of a number and a vector and ".i" is the inner product of two
vectors.  See the description for ax-his3.  I used to think this was
arbitrary, with physicists having a slightly less natural definition for
no good reason.  (I have never seen a book explain why.)  It turns out
that the physicist definition is necessary for the bra-ket notation to
work!  Specifically, the second associative law above fails with the
mathematicians' definition.  Since set.mm adopts the mathematicians'
definition in order to be compatible with math books, we therefore will
consider the bra-ket <A|B> to be defined as ( B .i A ).  So now we have
the "best of both worlds" and can choose either <A|B> (which physicists
consider synonymous with inner product) or (A .i B) = <B|A>, to match
whatever text we're working with.

There are actually 4 kinds of objects that result from different bra and
ket juxtapositions:  complex numbers, vectors, functionals, and operators.
This is why juxtaposition is not "exactly" a product, because its
meaning depends on the kind of objects that the things being juxtaposed
represent.  The starting operations on vectors are as follows:

                                                             Finite dim.
   Operation       Notation  Metamath        Value           analogy

   ket             |A>       A               vector          column vector
   bra             <A|       ( bra ` A )     functional      row vector
   inner product   <A|B>     ( B .i A )      complex number  complex number
   outer product   |A><B|    ( A ketbra B )  operator        matrix

The inner product <A|B> can also be expressed as ( ( bra ` A ) ` B )
- as today's theorem bravalvalt shows - and this will be needed to use
the table below (in line 5).  (Lines 3 and 4 in the above table are
redundant, since they are special cases of lines 5 and 4 below; line 4
above is computed ( A ketbra ( `' bra ` ( bra ` B ) ) ) = ( A ketbra B ).)

We will represent the four kinds of possible results in the "Value"
column above as v, f, c, and o respectively.  After accounting for the
restrictions on juxtaposing bras and kets (e.g., we can never have an
inner product followed by a ket), exactly the following cases can occur
when two Dirac subexpressions T and U are juxtaposed to produce a new
Dirac expression TU:

  T  U  TU  Metamath operation           Description

  c  c   c  ( T x. U )                   Complex number multiplication
  c  f   f  ( T .fn U )                  Scalar product with a functional
  v  c   v  ( U .s T )                   Scalar product (note T & U swap)
  v  f*  o  ( T ketbra ( `' bra ` U ) )  Outer product (with converse bra)
  f  v   c  ( T ` U )                    Inner product (bra value)
  f  o   f  ( T o. U )                   Functional composed with operator
  o  v   v  ( T ` U )                    Value of an operator
  o  o   o  ( T o. U )                   Composition of two operators

  *Note:  In line 4, U must be a continuous linear functional (which will
  happen automatically if U results from a string of kets and bras).
  This is needed by the Riesz theorem riesz2t, which allows the
  inverse bra to work.  The other lines have no restriction.

  See df-hfmul for ".fn", df-co for "o.", and df-cnv for "`'".

  Line 5 can be stated equivalently:

    f* v   c  ( U .i ( `' bra ` T ) )      Inner product (with converse bra)

So, the "operation" of juxtaposition of two Dirac subexpressions can
actually be any one of 8 different operations!  This is why we can't
easily express the Dirac notation directly in Metamath, since a class
symbol for an operation is supposed to represent only one object.

Supplementary note:  Physics textbooks will often have equations with an
operator sandwiched between a bra and a ket.  Its juxtaposition with a
bra or ket also now becomes easy to formalize:  match an entry from the
table above where the operator corresponds to an "o" input.

As an example of the use of the above table, consider the associative
laws above and their set-theoretical (Metamath) translations, which
we will eventually prove as theorems in the database.

  Dirac:  ( |A> <B| ) |C> = |A> ( <B| |C> )
  Metamath: ( ( A ketbra B ) ` C ) = ( ( ( bra ` B ) ` C ) .s A )

  Dirac:  ( <A| |B> ) <C| = <A| ( |B> <C| )
  Metamath:  ( ( ( bra ` A ) ` B ) .fn ( bra ` C ) ) =
                        ( ( bra ` A ) o. ( B ketbra C ) )

Note that ( A ketbra B ) above is really ( A ketbra ( `' bra ` ( bra ` B ) ) )
- see table line 4 - but we will adopt the convention of canceling
converse-bra bra since ( `' bra ` ( bra ` B ) ) = B.  In some cases
we can't cancel:

  Dirac:  ( | A >. <. B | ) ( | C >. <. D | ) =
          | A >. ( <. B | ( | C >. <. D | ) )

  Metamath:  ( ( A ketbra B ) o. ( C ketbra D ) ) =
             ( A ketbra ( `' bra ` ( ( bra ` B ) o. ( C ketbra D ) ) ) ) )

There you have it, a complete formalization of Dirac notation in
infinite dimensional Hilbert space!  I've never seen this published
before.

For an intuitive feel for the table above, it can be useful to compare
the finite dimensional case using vectors and matrices.  Suppose A and
B are column vectors of complex numbers

     [a_1]      [b_1]
     [a_2]      [b_2]

Then |B> is the same as B, and <A| is the row vector [a_1* a_2*] (where
* means complex conjugate).

Then all the 8 Dirac operations can be justified with vector and matrix
multiplications.  For example, <A|B> becomes the inner product
a_1* x. b_1 + a_2* x. b_2, and |B><A| becomes the 2x2 matrix

     [b_1 x. a_1*   b_1 x. a_2*]
     [b_2 x. a_1*   b_2 x. a_2*]

(It should be mentioned that Dirac notation can also be rigorously
modeled by an algebra known as "rigged Hilbert space," which can be
useful for certain theoretical purposes.  However, rigged Hilbert space
is rather abstract and somewhat removed from the Hilbert space we are
working with.)


(11-May-06) The Description for today's sumeqfv
https://us.metamath.org/mpegif/sumeqfv.html mentions that A
represents A(k) for those used to the standard way is would be written
in a textbook, meaning that the class substituted for A would normally
have a free variable k in it.  How do we know that A(k) is intended?  It
can be inferred from the fact that A and k do not appear together in a
distinct variable group, i.e. there is no $d A k constraint.

In principle, all theorems could be converted to the A(k) textbook
notation above, and explicit distinct variable groups could be
eliminated, if we make the following assumptions:

1. Implicitly assume all set variables are distinct (like textbooks do).

2. Implicitly assume that a set variable does _not_ occur in an
expression represented by a wff or class variable unless it is present
in the variable's argument list.  E.g., j may _not_ occur in a class
expression substituted for A(k); y _may_ occur (although it doesn't have
to) in a wff expression substituted for ph(x,y,z); etc.  This is also
what textbooks usually assume.

With the above assumptions, we could define, for most theorems (with
an exception explained below), an unambiguous mapping from Metamath's
notation (with explicit $d's and implicit free-variable arguments) to
textbook notation (with implicit $d's and explicit free-variable
arguments) and vice-versa.  Possibly a future display method could do
such a translation automatically, so that that notation becomes more
familiar to mathematically experienced readers.

So why don't we just adopt the textbook-style notation as the underlying
standard that Metamath is based on?  The answer is that the proof
checker would be much more complicated.  Also, see the 2nd paragraph on
https://us.metamath.org/mpegif/df-sb.html for a problem related to
using this notation to represent substitution, although this could be
avoided (but with associated algorithmic complexity) if we were strict
about assumptions (1) and (2) above.

Of course, a more complex proof checker is no big deal for a computer,
once the program is written.  In fact most proof checkers other than
Metamath do use the explicit-free-variable approach.  So why doesn't
Metamath just use a more complex algorithm?  First, the simpler the
algorithm, the more confidence you can have in its robustness.  But more
importantly, implicit in a complex algorithm is the associated
difficulty in learning how the algorithm works.  I believe that it is
important to understand how the algorithm works if you "really" want to
understand how each proof step was arrived at and not just to accept it
on faith.  In beginning logic courses, the complex rules for free
variables and proper substitution take a significant effort to learn,
whereas Metamath's substitution rule can presumably be learned in a few
minutes (although I have no data on this).  For me, it was a
philosophical goal to make the math as transparent as possible by using
the simplest possible algorithm, and the $d method accomplished that (to
my mind).  Or even better, in principle (although awkward in practice),
the simplest algorithm would use only axioms that avoid $d's entirely:
https://us.metamath.org/mpegif/mmzfcnd.html

A problem with the explicit-free-variable approach is that it cannot
represent set variables that don't have to be distinct, such as in
x = y -> y = x and A. x A. y ph <-> A. y A. x ph.  To accomodate this,
we could have a notation representing the opposite of a $d that says two
variables are not necessarily distinct.  Alternately, we could adopt the
approach of restating the axiom system so that set variables are always
distinct, as described on
http://planetx.cc.vt.edu/AsteroidMeta//mmj2Feedback (search for "Here is
an idea").  However, in set theory and beyond, situations where set
variables are not required to be distinct are not very common.


(10-May-06) A reader comments at http://www.jaredwoodard.com/blog/?p=5
that he wishes I'd spend less time on Hilbert space and more on cleaning
up *OLDs.

The cleaning up of *OLDs has actually been happening "behind the scenes"
even though people may not notice it.  Almost 200 *OLDs have been
eliminated since January (there were 380 then; now there are 185).
Yesterday's geoser and expcnv will eliminate their corresponding *OLDs,
for example.  Mel O'Cat is also working on a list of 27 of the *OLDs.

I realize hardly anyone cares about the Hilbert space stuff, and regular
visitors are probably bored to tears seeing the dreary purple theorems
day after day.  :)  (I do try not to go too long without a pretty green
one now and then.)  My long term goal is to build a foundation that will
let me explore rigorously some new ideas I have for Hilbert lattice
equations that may lead to writing a new paper.  I also want to build up
a foundation for theorems related to quantum computing.  In a few days
hopefully we will have a theorem related to error growth in qubits
(quantum gates).


(9-May-06) Compare geoser to the old one, geoserOLD, that it replaces.
Wikipedia entry:  http://en.wikipedia.org/wiki/Geometric_series


(5-May-06) The Riesz representation theorem is used to justify the
existence and uniqueness of the adjoint of an operator.  In particular,
the rigorous justification of Dirac bra-ket notation in quantum
mechanics is dependent on this theorem.  See also Wikipedia:
http://en.wikipedia.org/wiki/Riesz_representation_theorem


(13-Apr-06) One thing to watch out for in the literature is how the
author defines "operator".  I put some notes at
https://us.metamath.org/mpegif/df-lnop.html on the various
definitions:  for some they are arbitrary mappings from H to H, for
others they are linear, for still others they are linear and bounded.
In set.mm, "operator" means an arbitrary mapping.


Today's goal is:  a linear operator is continuous iff it is bounded.
This will be called "lncnbd" when it is completed later today. lnopcon
provides the basic proof of this:  it is not a trivial proof, 220 steps
long, and to me it is non-intuitive.  Many authors forget about the case
of the trivial Hilbert space, where sometimes a result holds and other
times not.  lnopcon does hold, but we have to prove the trivial case
separately, and in step 219 we combine the nontrivial case of step 194
with trivial case of step 218.


(12-Apr-06) The astute observer may have noticed that the dates on
the "Recent Additions to the Metamath Proof Explorer" now have 4-digit
years, e.g. 12-Apr-2006 instead of 12-Apr-06.  Version 0.07.15 of
the metamath program implements 4-digit date stamps, and set.mm has
been updated with 4-digit years.  The program still recognizes 2-digit
years (for the 'write recent' command) but assumes they fall between
1993 and 2092.


(10-Apr-06) The reason the xrub proof is long is that it involves 9
cases:  3 cases for B (real, -oo, and +oo), and for each B, 3 cases for
x. (We handle some of them simultaneously in the proof.)  This theorem
means we only have to "scan" over the reals, rather than all of the
extended reals, in order to assert that B is an upper bound for set A.
This is true even if A contains non-reals or if B is non-real (+oo
or -oo).

When we quantify over all extended reals, often we have to consider the
3 cases of real, -oo, +oo separately.  The advantage of this theorem is
that we don't have to handle the last two cases anymore, so some proofs
will become significantly shorter as a result.


(6-Apr-06) The proof of unictb is very different from Enderton's, which
I found somewhat awkward to formalize.  Instead, it is effectively a
special case of Takeuti/Zaring's much more general uniimadom.

It is also interesting how simple it is to go from the indexed union to
the regular union version of a theorem, whereas it can be rather
difficult in the other direction.  For example, iunctb to unictb is
trivial through the use of uniiun.  But for the finite version of this
theorem, compare the difficulty of going from the regular union version,
unifi to iunfi, requiring the not-so-trivial fodomfi, which was
proved for this purpose.

The conversion of unifi to iunfi involved substituting z for x and
{ y | E. x e. A y = B } for A in unifi, using dfiun2 on the consequent,
and doing stuff to get the antecedents right.  The "doing stuff"
ends up being not so simple.

Perhaps if I had to do it over, it might have been simpler to prove
iunfi first, then trivially obtain unifi via uniiun, although I'm not
really sure.

Both iunctb and iunfi are intended ultimately to be used by a Metamath
development of topology, which Stefan Allan has started to look at.


(1-Apr-06) Today's avril1 https://us.metamath.org/mpegif/avril1.html
is a repeat of last year's, except for a small change in the
description.  But I bring it up again in order to reply to last year's
skeptics.

Unlike what some people have thought, there is nothing fake about this
theorem or its proof!  Yes, it does resemble an April Fool's prank, but
the mathematics behind it are perfectly rigorous and sound, as you can
verify for yourself if you wish.  It is very much a valid theorem of ZFC
set theory, even if some might debate its relative importance in the
overall scheme of things.  The only thing fake is that Prof. Lirpa uses a
pseudonym, since he or she wishes to remain anonymous.

Tarski really did prove that x=x in his 1965 paper.  While it is
possible he wasn't the first to do so, he did not attribute the theorem
to anyone else.

The theorem

  -. ( A P~ RR ( i ` 1 ) /\ F (/) ( 0 x. 1 ) )

importantly tells us we cannot prove, for example,

  ( A P~ RR ( i ` 1 ) /\ F (/) ( 0 x. 1 ) )

if ZFC is consistent.  If we utter the latter statement, that will
indeed be a hilarious joke (assuming ZFC is consistent) for anyone who
enjoys irony and contradiction!  But anyone who could prove the latter
statement would achieve instant notoriety by upsetting the very
foundation that essentially all of mathematics is built on, causing it
to collapse like a house of cards, into a pile of (Cantor's) dust that
would blow away in the wind.  That assumes, of course, that the paradox
is not hushed by the established mathematical community, whose very
livelihoods would be at stake.  In that case, the discoverer might
achieve wealth instead of fame.

So, in effect the theorem, being preceded by the "not" sign, really
tells us:  "I am _not_ an April Fool's joke."  Thus we are reminded of
the Liar Paradox, "this sentence is not true," but with an important
difference:  paradoxically, avril1 is not a paradox.

For those whose Latin is rusty, "quidquid germanus dictum sit, altum
viditur" means "anything in German sounds profound."  Just as logicians
have done with Latin ("modus ponens" and so on), set theorists have
chosen German as their primary obfuscating language.  For example, set
theory texts lend great importance and mystery to the otherwise trivial
subset axiom by calling it "Aussonderung."  This helps keep the number
of set theorists at a comfortable level by scaring away all but a few
newcomers, just enough to replace those retiring.

To derive avril1, we have used an interdisciplinary approach that
combines concepts that are ordinarily considered to be unrelated.  We
have also used various definitions outside of their normal domains.
This is called "thinking outside of the box."  For example, the
imaginary constant i is certainly not a function.  But the definition of
a function value, df-fv, allows us to substitute any legal class
expression for its class variable F, and i is a legal class expression.
Therefore ( i ` 1 ) is also a legal class expression, and in fact it can
be shown to be equal the empty set, which is the value of "meaningless"
instances of df-fv, as shown for example by theorem ndmfv.

https://us.metamath.org/mpegif/df-fv.html
https://us.metamath.org/mpegif/ndmfv.html

Now that the technique has been revealed, I hope that next year someone
else will make a contribution.  You have a year to work on it.


(28-Mar-06) sspr eliminates the hypotheses of the older version, which
has been renamed to ssprOLD.


(27-Mar-06) As of today's version of set.mm, 183 out of the 315 theorems
with names ending "OLD" were removed, so there are only 132 *OLDs left.
This has made set.mm about 300KB smaller.  (The 132 remaining can't just
be deleted, since they currently are referenced by other proofs, which
will have to be revised to eliminate their references.  Mel O'Cat has
started working on some of them.)


(20-Mar-06) Stefan has done the "impossible," which is find an even
shorter proof of id.  (See the note of 18-Oct-05 below.)  His new proof
strictly meets the criteria I use for accepting shorter proofs
(described at the top of the set.mm file).  He writes, "Too bad you
don't get a special prize for shortening this one!"  I agree; any
suggestions?

About a1d, he writes, "[a1d] is not a shorter proof in compressed
format, and is in fact the same size as the old one.  However it has
fewer steps if expanded out into axioms, so you might want to include it
in set.mm anyway."


(24-Feb-06) Stefan's sylcom proof has 1 fewer character in set.mm than
the previous, and 9 fewer characters in its HTML file.  I think we may
be approaching the theoretical limit.  :)


(22-Feb-06) The proof of efcj uses some obsolete theorems with the old
convergence ~~>OLD, but I don't have the updated ones ready yet and just
wanted to get efcj out of the way since we will need it for more
trignometry.  Eventually the proof of efcj will be updated.  Note that
"obsolete" doesn't mean "unsound"; the proof is perfectly rigorous.  The
purpose of the new notation is to make proofs more efficient (shorter)
once everything is updated with it.

(17-Feb-06) efadd was completed a little sooner than I thought.  Here
are some statistics:  the set.mm proof (efadd + 28 lemmas) takes 47KB
(out of 4.5MB for set.mm).  The HTML pages take 4.7MB (out of 372MB
total for mpegif).

(13-Feb-06) Over the next couple of weeks, we will be proving what has
turned out to be a difficult theorem - the sum of exponents law for the
exponential function with complex arguments, i.e. e^(a+b)=e^a.e^b.  Even
though textbook proofs can seem relatively short, the ones I've seen
gloss over many of the tedious details.  After several false starts I
came up with a proof using explicit partial sums of product series and
explicit comparisons for the factorial growth (we will use the recent
fsum0diag and faclbnd4 for this).  The whole proof will have around
30 lemmas.

(12-Feb-06) nonbool demonstrates that the Hilbert lattice is
non-Boolean.  This proves that quantum logic is not classical.  Of
course this is well known, but I've only seen it stated without proof,
so I came up with a formal demonstration.  It seems the dimension
must be at least 2 to demonstrate non-Boolean behavior.

Note that we have already shown that it is orthomodular (in pjoml4),
but Boolean is a special case of orthomodular, so that in itself doesn't
demonstrate that quantum logic is not classical.

(11-Feb-06) Even though the climmul proof is long, I'm not unhappy about
it, since Gleason dedicates over 2 pages to the proof (although some of
it is an informal discussion of how one goes about coming up with such a
proof).

While our proof roughly follows Gleason, our trick of constructing a new
positive number less than both A and 1, given a positive number A, is
not in Gleason - he uses the infimum of A and 1 (bottom of page 170),
which would be more awkward (for us at least) to deal with.  This trick
is proved by recrecltt and is used in climmullem5.

(9-Feb-06) I found a slightly shorter equivalent for ax-groth expanded to
primitives.  The idea was to use fun11 at step 42, so that the old steps
42-60 become 42-48.  But the result was a little disappointing.  I had
higher hopes for the idea but it only ended up removing one binary
connective.  At least the proof is 59 instead of 71 steps.  (The old one
has been kept temporarily as grothprimOLD.)  Probably the biggest problem
is the repeated use of grothlem (4 times) to expand binary relations.
I wonder if there is a shorter way to effectively express this concept.

(8-Feb-06) dummylink was added for a project to interface O'Cat's mmj2
Proof Assistant GUI with the metamath program's Proof Assistant, but
I've discovered that it can be quite handy on its own as suggested by
its description.  For more background see "Combining PA GUI and CLI - an
interim solution?" at the bottom of web page
http://planetx.cc.vt.edu/AsteroidMeta/mmj2ProofAssistantFeedback

(Downloaders - the Metamath download containing this proof will be in
tomorrow's download.  In general, the Most Recent Proofs usually take
about a day to propagate to the downloads.)

(4-Feb-06) More shorter proofs by O'Cat - pm2.43d, pm2.43a.

rcla4cv ended up shortening 26 proofs by combining rcla4v and com12.
The result was a net reduction in the database size, even after
accounting for the new space taken by rcla4cv.

(3-Feb-06) Mel O'Cat found shorter proofs for sylcom, syl5d, and syl6d
while having fun with his new toy, the Proof Assistant GUI.

Note:  the new proofs of of syl5d and syl6d have the same number of
logical steps, but proofs are shorter if we include the unseen
wff-building steps.  Out of curiosity I restored the original syl5d
proof, since it had already been shortened by Josh Purinton, and called
it syl5OLDOLD.  Here are the complete proofs for the syl5d versions:

syl5d:  14 steps

  wph wps wta wch wth wph wta wch wi wps syl5d.2 a1d syl5d.1 syldd $.

syl5dOLD:  16 steps

  wph wps wch wth wi wta wth wi syl5d.1 wph wta wch wth syl5d.2 imim1d
  syld $.

syl5dOLDOLD:  19 steps

  wph wta wps wth wph wta wch wps wth wi syl5d.2 wph wps wch wth syl5d.1
  com23 syld com23 $.


(30-Jan-06) Today we start a brand new proof of the binomial theorem
that will be much shorter than the old one.  It should also be much more
readable.  This is what the new one will look like (A e. CC, B e. CC):

( N e. NN0 -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N )
         ( ( N C. k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )

Compare to the old, binomOLD:

( ( N e. NN0 /\ A. k e. ( 0 ... N ) ( F ` k ) =
               ( ( N C. k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ) ) ) ) ->
                   ( ( A + B ) ^ N ) = ( <. 0 , N >. sumOLD F ) )

(24-Jan-06) (Compare note of 22-Oct-05.)  Per the request of Mel O'Cat,
I eliminated all connective overloading in set.mm by making weq, wel,
and wsb "theorems" so that he can use set.mm with his GUI Proof
Assistant.  This involved moving set theory's wceq, wcel, and wsbc
up into the predicate calculus section, which is somewhat confusing,
so I added extensive commenting to explain it hopefully.

Note that the web page "proofs" of weq, wel, and wsb have only one step:
this is because they are syntax proofs, and all syntax building steps
are suppressed by the web-page generation algorithm, which doesn't
distinguish weq, etc. from "real" theorems.  I'm not yet sure if it's
worth changing the algorithm for this special case.  To see the actual
steps, in the Metamath program type "show proof weq /all".

(20-Jan-06) supxrcl shows the usefulness of the extended reals:  the
supremum of any subset always exists.  Compare the non-extended real
version suprcl, which has a complicated antecedent that must be
satisfied.

(19-Jan-06) A new set theory axiom, ax-groth, was added to the database.
This axiom is used by Mizar http://mizar.org to do category theory (that
ZFC alone cannot do), and I think it is appropriate to add it to set.mm.
One of the negative aspects of this axiom (aesthetically speaking) is
that it is "non-elementary" and very ugly when expanded to primitives,
unlike the ZFC axioms.  I worked out grothprim because I was curious to
see what it actually looks like.  I don't think grothprim will actually
be used for anything since it is impractical; instead, ax-groth would be
the starting point.  However, grothprim can provide a benchmark for
anyone seeking a shorter version.  There may be a theoretical reason why
it can't be as short as say ax-ac, but I don't think anyone knows what
the shortest possible equivalent is.

mmset.html has also been updated to include ax-groth below the ZFC
axioms.

I wrote to Jeff Hankins:

  I added ax-groth partly in response to your email on Mycielski's ST set
  theory,* although it's been on my backburner for a while.  In my mind,
  ax-groth "completes" set theory for all practical purposes.  (The Mizar
  people, who use this axiom, also think so.)  Unlike the controversial
  assertions of ST, ax-groth is relatively mild and uncontroversial - I
  don't know of any debate over it, unlike the debate on the Continuum
  Hypothesis.  I am pretty sure that Mycielski's Axiom SC implies ax-groth
  from his comments, although I haven't worked out a proof.  So
  ZFC+ax-groth is most likely a subset of ST.

* http://www.ams.org/notices/200602/fea-mycielski.pdf - free AMS
sign-up required to view article

(12-Jan-06) The exponential function definition df-ef is new.
Yesterday's version of df-ef was reproved as theorem dfefOLD that
will eventually be deleted after the old infinite summation notation is
phased out.

Compare the new exponential function value formula, efvalt, with the old
one, efvaltOLD.  Don't you agree that it is much easier to read?  This
kind of thing makes me believe that the effort to introduce the new
summation notation was worthwhile.  :)  In addition, will have a much
nicer version of the binomial theorem (whose old version has already
been renamed to binomOLD), with a much shorter proof - stay tuned!

(11-Jan-06) isumvalOLDnew links the old and new definitions of infinite
sum, allowing us to temporarily reuse theorems in the old notation until
they are phased out.  See comments below of 1-Nov-05, 2-Nov-05,
20-Dec-05, and 21-Dec-05 regarding the new finite/infinite sum notation
df-sum.

The present definition of the exponential function, df-ef, makes use of
the obsolete infinite sum notation. dfefOLDnew will replace df-ef in the
next few days and become its new official definition.  The old df-ef
will become a (temporary) theorem that will be used to support the old
infinite sum notation until it is phased out.

gch-kn was updated with new hyperlinks.

(10-Jan-06) Regarding the 9-Jan-06 item in "Also new", the primary
reason I added the "/except" switch to "minimize_with" was to exclude
3syl.  While 3syl may shorten the uncompressed normal proof, it often
makes compressed proofs grow longer.  This happens when the intermediate
result of two chained syl's is used more than once.  When the
intermediate result disappears due to 3syl, two formerly common
subproofs have to be repeated separately in the compressed proof - part
of the compression is achieved by not repeating common subproofs.  So,
typically I exclude 3syl then minimize with it separately to see if the
compressed proof shortens or lengthens.  Maybe I'll add an option to
also check the compressed proof length instead of or in addition to the
normal proof length, but the "/exclude" was easier to program, and
curiously 3syl is the only problematic theorem I'm aware of.

(9-Jan-06) climOLDnew is a temporary theorem that links the old and new
limit relations, ~~>OLD and ~~>.  This will let us "jump ahead" and work
on exponentiation, etc. with the new notation before cleaning up all the
~~>OLD stuff (which I will still clean up eventually).  The linkage is
needed to avoid any gaps in the released set.mm.  (The metamath command
"verify proof *" should always pass for the daily releases of set.mm,
ensuring absolute correctness of its contents - even the stuff that's
obsolete.)

The main difference between ~~>OLD and ~~> is that ~~>OLD has the rigid
constraint that the sequence F be a function from NN to CC, whereas ~~>
allows F to be any set with lots of irrelevant garbage in it as long as
it eventually has function values in CC beyond some arbitrary point.
This can make ~~> much more flexible and easier to use.

The uppercase "OLD" in climOLDnew means the theorem will go away; for my
cleanup I will be phasing out and deleting all theorems matching *OLD*.
Currently there are 380 *OLD* theorems due to be phased out.  They can
be enumerated by counting the lines of output of the metamath command
"show label *OLD*/linear".

(6-Jan-06) syl*el* are all possible combinations of syl5,syl6 analogs
for membership and equality.  I added them to shorten many proofs, since
these patterns occur frequently.  (Since 1999 I've added the syl*br*
versions of these for binary relations and found them useful, so I
decided to add the syl*el* versions.)  There is a curious asymmetry in
which ones ended up being useful:  syl5eqel got used over 30 times,
whereas syl5eleq isn't used at all so far.  I don't know if this is
because I wrote the original proofs with certain patterns subconsciously
repeated, or if there is something more fundamental.

(3-Jan-06) r19.21aiva adds 319 bytes to the database, but it reduces the
size of about 50 (compressed) proofs by 765 bytes total, for a net
reduction in database size of 466 bytes.

(21-Dec-05) All theorems that involved df-fsum have been updated to use
the dual-purpose (finite and infinite) df-sum instead.  So, we now have:

       Definition               Token                Symbol
  Yesterday   Today       Yesterday Today     Yesterday   Today
  df-sum      df-sum      sum_NEW   sum_      \Sigma_NEW  \Sigma
  df-fsum     df-fsum     sum_      sum_OLD   \Sigma      \Sigma_OLD
  df-fsumOLD  df-fsumOLD  sumOLD    sumOLD    \Sigma_OLD  \Sigma_OLDOLD

The names with "OLD" are now kind of oddly inconsistent, but everything
with "OLD" in it (whether label or token) will eventually be deleted so
it doesn't really matter.

(20-Dec-05) The new finite sum stuff looks like it will be very useful,
and we will need an infinite sum version to replace the current df-isum.
Rather than repeat the whole development with new equality, bound
variable, etc. utility theorems, I decided to combine the two
definitions.  The new combined definition is called df-sum, which is
basically the union of two definitions.  The index range (finite or
infinite) determines whether the sum is finite or infinite.  See the
comments in df-sum.  We need about half a dozen utility theorems.  Then,
after changing the "sigmaNEW" to "sigma", we can "plug in" the new
definition and re-use the theorems we have already developed for finite
sums without further modification.

fzneuzt is the basic theorem that lets us distinguish the finite
half vs. the infinite half of df-sum.

(14-Dec-05) fsum1sNEW (to be renamed fsum1s) exploits class2set to
eliminate the hypothesis of fsum1slem, so that we require only existence
of A(M) as a member of some arbitrary class B, rather than requiring
that it be a complex number (as yesterday's fsum1s requires).  This will
shorten future proofs by allowing us to apply fsum1sNEW directly when A
is a real, rational, etc.  I had almost forgotten about class2set, which
I think is a neat trick.  Yesterday's fsum1s will be renamed to
fsum1sOLD and eventually deleted.

I'm not sure if fsum1s2 will be useful, but it lets us show off an
application of the interesting fz1sbct.

(13-Dec-05) fsum1slem shows an example of a use for the new
substitution-for-a-class notation.  Compare it to the implicit
substitition version fsum1.

fsum1s turns the hypothesis A e. V of fsum1slem into an antecedent.
Since A is quantified, we have to work a little harder than usual to
accomplish this.

(4-Dec-05) See http://planetx.cc.vt.edu/AsteroidMeta//metamathMathQuestions
for comments on equsb3.

(30-Nov-05) csbnestg is the same as csbnestglem except that it has fewer
distinct variable restrictions.  Its proof provides another example of a
way to get rid of them; the key is using a dummy variable that is
eliminated with csbcog.  I think it is a neat theorem and was pleasantly
surprised that so few distinct variable restrictions were needed.  The
antecedents just say that A and B are sets and are easily eliminated in
most uses.  By having antecedents instead of A e. V, B e. V hypotheses,
we can make more general use of the theorem when A and B sethood is
conditioned on something else; hence the "g" after "csbnestg".

(23-Nov-05) In all versions of set.mm from 18-Nov-05 to 22-Nov-05 inclusive,
the line

  htmldef "QQ" as "<IMG SRC='bbieq.gif' WIDTH=13 HEIGHT=19 ALT='QQ' ALIGN=TOP>";

should be

  htmldef "QQ" as "<IMG SRC='bbq.gif' WIDTH=13 HEIGHT=19 ALT='QQ' ALIGN=TOP>";

Thanks to Jeff Hankins for pointing this out.


(18-Nov-05) sbccom is the same as sbccomlem except that it has fewer
distinct variable restrictions.  Its proof shows an example of how to
get rid of them when they are not needed.

-------

I made around 80 changes to the bixxx series names to be
consistent with earlier bixxx -> xxbix changes in prop. calc.  E.g.
biraldv was changed to ralbidv.

  r - restricted
  al - for all
  bi - biconditional
  d - deduction
  v - $d instead of bound-variable hypothesis

Also, bi(r)abxx were changed to (r)abbieqxx e.g. biabi was changed to
abbieqi.

  ab - class abstract (class builder)
  bi - hypothesis is biconditional
  eq - conclusion is equality
  i - inference

As usual, all changes are listed at the top of set.mm, and as instructed
there can be used to create a script to update databases using set.mm as
their base.  As always, better naming suggestions are welcome.

(17-Nov-05) abidhb is a very neat trick, I think!  It will let us do
things that the Weak Deduction Theorem by itself can't handle.  For its
first use, we create a "deduction" form of the bound-variable hypothesis
builder for function values, hbfvd - this is actually a closed form that
allows _all_ hypotheses to be eliminated, since 'ph' is arbitrary and
can be replaced with a conjunct of the hypotheses.  And the only thing
hbfvd needs is the "inference" version hbfv in step 5!  Before I thought
of abidhb, hbfvd was going to require a long chain of hbxxd's (hbimd,
hbabd,...) that would build up to the function value definition.  I was
actually starting to get depressed about the amount of work that would
have been needed.  But as they say, laziness is the mother of invention.
Now, we can just add hbxxd's as needed, starting from the hbxx
"inference" versions!

(15-Nov-05) Note that fsumeq2 requires a $d for A and k, whereas fsumeq1
doesn't.  On the other hand, we have analogously iuneq1 and iuneq2,
neither of which require the bound variable to be distinct!  I spent a
lot of time trying to get rid of it for fsumeq2 by changing the
definition df-fsum, but it always seemed that if I got rid of it in
fsumeq2 it would show up in fsumeq1.  So I don't know whether it is
theoretically possible to get rid of it.  In the current version of the
fsumeq2 proof, the $d is needed to satisfy resopab in steps 9 and 10.

Getting rid of $d A k in fsumeq2 would be advantageous if I add an
"explicit substitution" form of induction like (for ordinals) Raph
Levien's findes, where the hypothesis findes.2 has the substituted
variable free in the expression to be substituted.  So, if anyone can
solve this, let me know!

(14-Nov-05) Today we introduce a new definition, df-csbc, the proper
substitution of a class variable for a set into another class variable.
We use underlined brackets to prevent ambiguity with the wff version,
otherwise [ x / y ] A R B could mean either x e. { y | A R B } for the
df-sbc wff version or <. [ x / y ] A , B >. e. R for the df-csbc class
version.  So instead we use [_ x / y ]_ A for the class version.  One
reason I chose the underline is that it is easy to do in Unicode and
LaTeX, but if you have another idea for the notation let me know.  See
notes of 5-Nov-05 for other notes on the definition.

(13-Nov-05) I decided to make the new finite summation notation df-fsum
official.  The old has been renamed to df-fsumOLD.  I am uncertain about
whether to keep the old (under a different name yet to be determined) or
delete it eventually.  There are 61 theorems using it (21 of which are
the binomial theorem binom) which I hope to eventually re-prove with the
new notation.

(5-Nov-05) Regarding sbabex:  The notation "[ y / x ] ph" means "the
proper substitution of y for x in phi".  We do not have a separate
notation for the class version of this, so until such time (if it
becomes common enough to warrant a special notation), the idiom
"{ z | [ y / x ] z e. A }" means "the proper substitution of y for x in
class variable A".  In other words we turn the class into a wff - the
predicate "is in A" - then do the proper substitution, and finally turn
the result back into a class by collecting all sets with this predicate.
I think that's a neat trick, and it will become the definition if we
introduce a notation for it.  Note that the argument of "[ y / x ]" is
"z e. A", which is a wff.

(2-Nov-05) I have about a dozen theorems in progress with the current
'df-fsum' notation, that I might as well finish before switching to the
new notation.  These will provide reference proofs that will make the
corresponding versions in the new notation easier to prove, but they
will eventually be deleted (assuming I adopt the new notation, whose
definition I'm still fine tuning.)

Not all theorems will be shorter with the new notation, which is one
reason for my indecision.  For example:

Old: (fsumserz)

|- F e. V  =>  |- ( N e. ( ZZ> ` M ) ->
       ( <. M , N >. sum F ) = ( ( <. M , + >. seq F ) ` N ) )

New: (fsumserzNEW)

|- F e. V  =>  |- ( N e. ( ZZ> ` M ) ->
        sumNEW k e. ( M ... N ) ( F ` k ) = ( ( <. M , + >. seq F ) ` N ) )


(1-Nov-05) The proof of the binomial theorem painfully illustrates that
the current notation for summations is very awkward to work with,
particularly with nested summations.

A new definition I'm experimenting with is df-fsumNEW, which, unlike
df-fsum (which is a class constant with no arguments), has a dummy
variable k and two other arguments.  To indicate the sum A^1 + A^2 +...+
A^N, we can write

  sumNEW k e. ( 1 ... N ) ( A ^ k )

instead of the present

  ( <. 1 , N >. sum { <. k , y >. | ( k e. ZZ /\ y = ( A ^ k ) ) } )

(where usually the class builder is stated as a hypothesis like
F = { <. k , y >. | ( k e. ZZ /\ y = ( A ^ k ) ) } to keep the
web page proof size down).  Nested sums are even more awkward, as
the hypothesis "G =" in the binomial lemmas shows.

With the new notation, the binomial theorem would become:

  ( N e. NN0 -> ( ( A + B ) ^ N ) = sumNEW k e. ( 0 ... N )
          ( ( N C. k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ) ) ) )

The price we pay is that 'sumNEW' is not just a set-theoretical class
constant like 'sum', but instead a symbol with arguments and a bound
variable, analogous to indexed union df-iun.  In particular, its
soundness verification, while still simple, is not as "trivial" as with
new class constants.  There is nothing wrong with this in principle,
but it is contrary to my simplicity goal of introducing only new class
constants for new definitions, thus keeping the number of "primitive"
syntactical structures to a bare minimum.  But in this case I think
practicality will win out.  The proofs should be more elegant with
'sumNEW' (later to be changed to 'sum' if I decide to keep it),
and I also think it is more readable.

Of course, soundness justification will not be an issue with the
eventual Ghilbert version.

To further elaborate on my simplicity preference (for which df-fsumNEW
will be an exception), below I reproduce my response to an email Josh
Purinton wrote me (on Oct. 18) regarding the notation for fzvalt.

  > Consider using square brackets for 'compatibility' with the
  > distinction between a closed and open interval.

My response:

  I understand what you are getting at, but there is a slight problem.
  df-fz is just the class symbol "..." which is used as an operation,
  and the parentheses are just the normal parentheses that surround an
  operation value.  Thus "( 1 ... 3 )" means "( ... ` <. 1 , 3 > )".

  I could define a new syntactical structure or pattern "[ A ... B ]" but
  then I couldn't use all the equality, hb*, etc. theorems we have for
  operation values.  After basic set theory development, which is more or
  less finished, I've been trying to introduce only new class constant symbols
  (with exceptions for a few very common things like the unary minus "-u";
  actually that is the only exception so far).  In addition to allowing us
  to reuse general-purpose theorems, the soundness justification is
  trivial for a new constant class symbol, which is what I like most about
  that approach.

  Also, "( m ... n )" is really more of a discrete, unordered list
  than an a continuous closed interval.

  I will probably never be completely happy with "..." in particular
  because it is nonstandard and unfamiliar, but on the other hand it has
  turned out to be very useful for theorems involving finite sums.  But I
  didn't consider it so important that it justifies its own new
  syntactical pattern.  It is so rare in the literature (if it ever
  occurs) that I was pleased to stumble across Gleason's partial version
  of the notion.

  For the four real intervals (x,y), (x,y], [x,y), [x,y] I haven't decided
  what to do yet.  It would be preferable to have them be just operation
  in the form of new class constant operation symbols, but I haven't
  thought of any good notation to accomodate them in this form.  We could
  have e.g.  "(.,.]" or "(]" so we'd have "( A (.,.] B )" or "( (.,.] ` <.
  A , B >. )" or "( A (] B )" etc. but these are odd-looking.  What I will
  end up doing is very open at this point.  Maybe it's time to start using
  words like "closed", "rclosed", "lclosed", "open", etc. in some way?

  Right now we have only the two workhorses "( F ` A )" and "( A F B )"
  for general function/operation values.  Analogously we have "A e. R" and
  "A R B" for predicates/binary relations.  They are the only general
  patterns the reader has to be familiar for virtually all new
  definitions.  In theory these are all that we need, although certain
  notations become very awkward (e.g. extending them to more arguments via
  ordered pairs, and the real intervals you have brought up).

Note that right now, we are using the "workhorse" ( A F B ) for
virtually all of the new definitions of sums, sequences, shifts,
sequential integer sets, etc.  I like it because there is only one
underlying notation, i.e. operation value, that you have to be aware of.
But I think the present df-fsum stretches the limit of what is
practical and "user-friendly".

--------------------

(24-Oct-05) Today we introduce the superior limit limsup, which will
be one of our principal uses of the extended reals.


(22-Oct-05) It appears I mispoke yesterday when I said "The new syntax
allows LALR parsing," and I changed it to "The new syntax moves us
closer to LALR parsability."  From Peter Backes:

  It makes it more LALR than before, but not completely. ;)

  What remains is

    1) set = set (trivial, since redundant)      [i.e. weq vs. wceq]
    2) set e. set (trivial, since redundant)     [i.e. wel vs. wcel]
    3) [ set / set ] (trivial, since redundant)  [i.e. wsb vs. wsbc]
    4) { <. set , set >. | wff } (we already discussed it and agreed it
    was not easy to solve)                       [i.e. copab]
    5) { <. <. set , set >. , set >. | wff } (ditto)   [i.e. copab2]

These are all easy to fix by brute force (eliminating weq, wel, and wsb,
and changing "{", "}" to "{.", "}." in copab and copab2) but I don't
want to be too hasty and am looking into whether there are "nicer" ways
to do this first.


(21-Oct-05) A big change (involving about 121 theorems) was put into the
database today:  the indexed union (ciun, df-iun) and indexed
intersection symbols (ciin, df-iin) are now underlined to distinguish
them from ordinary union (cuni, df-uni) and intersection (cint, df-int).
Although the old syntax was unambiguous, it did not allow for LALR
parsing of the syntax constructions in set.mm, and the proof that it was
unambiguous was tricky.  The new syntax moves us closer to LALR
parsability.  Hopefully it improves readability somewhat as well by
using a distinguished symbol.  Thanks to Peter Backes for suggesting
this change.

Originally I considered "..." under the symbol to vaguely suggest
"something goes here," i.e. the index range in the 2-dimensional
notation, but in the end I picked the underline for its simplicity (and
Peter prefered it over the dots).  Of course I am open to suggestion and
can still change it.  In the distant future, there may be
2-dimensional typesetting to display Metamath notation (probably
programmed by someone other than me), but for now it is an interesting
challenge to figure out the "most readable" 1-dimensional representation
of textbook notation, where linear symbol strings map 1-1 to the ASCII
database tokens.

iuniin is the same as before but has an expanded comment, and also
illustrates the new notation.

(18-Oct-05) Today we show a shorter proof of the venerable theorem id.
Compare the previous version at https://de.metamath.org/mpegif/id.html .

fzvalt is the same as before but has an expanded comment.

(15-Oct-05) Definition df-le was changed to include the extended reals,
and df-le -> xrlenltt -> lenltt connects the new version to the existing
theorems about 'less than or equal to' on standard reals.

(14-Oct-05) The set of extended reals RR*, which includes +oo and -oo,
was added, with new definitions df-xr, df-pinf, df-minf, and df-ltxr.
The old < symbol was changed to <_RR, the new df-ltxr symbol was called
<, and the ordering axioms were reproved with the new < symbol (and they
remain the same, since in RR, < and <_RR are the same by ltxrlt.  This
allows us to use all remaining theorems about RR in the database
unchanged, since they are all restricted to elements of RR.  The
theorems proved today are the minimum necessary to retrofit the database
in this way.  I was pleasantly surprised at how easy it was to add in
the extended reals.

Unlike textbooks, that just say +oo and -oo are "new" distinguished
elements without saying what they are, we must state concretely what
they are in order to use them.  So I picked CC for +oo and { CC }
for -oo.  Many other choices are possible too.  The important thing
is not what elements are chosen for them, but how the new < ordering
is defined.

Unlike some analysis books, Gleason finds it unnecessary to extend the
arithmetic operations (only the ordering) for +oo and -oo, so I will be
avoiding that too unless a clear advantage becomes apparent.  E.g. some
books define +oo + A = +oo, A / +oo = 0, etc. but for us that is now
undefined.

(6-Oct-05) modal-b is analogous to the Brouwer modal logic axiom if we
map "forall x" to box ("necessarily") and "exists x" to diamond
("possibly").  See http://plato.stanford.edu/entries/logic-modal/ and
also http://www.cc.utah.edu/~nahaj/logic/structures/systems/s5.html

In fact, our axioms ax-4, ax-5, and ax-6 (plus ax-1/2/3, modus ponens,
and generalization) are *exactly* equivalent to modal logic S5 under
this mapping!  This was not intended when I first came up with the
axioms.  Our axioms have a different form because I arrived at them
independently when I didn't know what modal logic was, but they (or
Scott Fenton's ax46/ax-5) are provably equivalent to S5 and can be used
(under the mapping) as alternate axioms for S5.  Conversely, all the
theorems of S5 will automatically map to theorems of our "pure"
predicate calculus.

Axiom ax-7 has no modal logic analog, since it has two variables.
However, if we restrict x and y to be distinct, it might be possible to
make an analogy between it and the Barcan Formula BF (see the
plato.stanford.edu page), particularly because the BF converse is also
true (http://plato.stanford.edu/entries/actualism/ltrueCBF.html) as it
also is for ax-7.

(4-Oct-05) ser1f0 - The difficulty of proving this "obvious" fact was
surprising.

(30-Sep-05) df-isum is a new definition for the value of an infinite sum
that will eventually replace df-sumOLD.  Its advantage is that the sum
can start at any index N instead of the fixed index 1.  isumvalt is the
first use of the new definition.

The notation of df-isum perhaps isn't as nice as df-sumOLD, but I don't
know how else to do it since we are using a linear notation with no
subscripts.  (The infinity superscript is not a separate symbol but part
of the fixed infinite summation symbol, represented as "sumoo" in the
database.)

(27-Sep-05) The obsolete definitions df-seq0OLD and df-climOLDOLD, along
with all theorems dependent on them, have finally been purged from the
set.mm database, lightening it a bit.

dfseq0 is nice.  Perhaps I'll interchange df-seq0 and dfseq0 some day.

(19-Sep-05) Scott Fenton found a shorter proof of ax46.

(18-Sep-05) It is becoming apparent that the recently introduced new
version of df-clim (now called df-climOLDOLD), although very useful
because of its arbitrary starting point, has some limitations:  since
there is no built-in requirement that the limit be a complex number or
that the sequence have complex values, additional conditions would have
to be stated for proving convergence that will make a lot of theorems
awkward.  Therefore I changed the definition to today's df-clim, called
the old ~~> as ~~>OLDOLD, and we will reprove most or all of the old
theorems then delete them.  The new definition df-clim is more complex
but it should be worth it in the end.

(There is still df-climOLD, which is severely limited to sequences
starting at 1, that must eventually be replaced with df-clim.  This will
be a longer-term project, since df-clim is directly or indirectly
affects around 500 theorems.  df-climOLDOLD, with its short-lived
existence, only affects around 20 theorems.)

(14-Sep-05) elfz4b shows the converse of elfz4t also holds - a nice
surprise.  Maybe I'll use it instead of elfz4t.

(11-Sep-05) Today we introduce an new definition, df-uz.  The idiom "set
of all integers greater than M" keeps recurring, so I decided to add a
notation for it to shorten proofs, even though it is nonstandard.  "ZZ
subscript >=" is a function that maps an integer to the set of all
integers greater than or equal to it.  I think I chose a good notation
for it that should be easy to remember; if you don't think so let me
know.

(8-Sep-05) fsumserz is an important theorem that expresses a finite sum
as a partial sum of a sequence builder.  In fact, it shows that the
finite sum notation is redundant, although we'll keep it because it
slightly more compact and seems like a more natural notation.

(7-Sep-05) A small change was made to df-fz to restrict its domain to
ZZ X ZZ, requiring a new version of fzvalt.  All other theorems remain
compatible, but the change allows us to state the useful elfz7t, where
(provided N is a set) we can deduce that M and N are in ZZ simply from
the fact that ( M ...  N ) has a member.  This will allow us to simplify
proofs by not requiring M e. ZZ and N e. ZZ as additional hypotheses.
(The fact that N must be a set is an artifact of our operation value
definition.  I'm currently pondering changing the operation value
definition so that N would not be required to be a set in elfz7t, but
that would be a huge change throughout the db - perhaps in the long term
future.)

seq0seqz and seq1seqz are yesterday's promised special cases of "seq".

(6-Sep-05) The old symbol "seq" for a 1-based infinite sequence builder
has been changed to "seq1" (df-seq1) for consistency with the 0-based
version "seq0".  The symbol "seq" (df-seqz) has been (re)defined to be
an infinite sequence builder with an arbitrary starting index, and we
will show, today or tomorrow, that "seq0" and "seq1" are special cases
of it.

(26-Aug-05) I didn't like the notation for finite sums so I decided to
do it all over again.  Everything in the last few days has been renamed
to *OLD.  These will be reproved with the new notation and the *OLDs
deleted.  (Also, I extended the definition so the value is zero if the
lower limit is greater than the upper limit, like some books do.)

So, instead of the (to me) awkward "F Sigma <M,N>" for

                ---   N
                \          F_i
                /
                ---  i = M

we now can state this as "Sigma`<<M,N>,F>", which seems more natural.
By df-opr this is equivalent to "<M,N> Sigma F", which results in
shorter proofs, so that's what I'll use for most proofs.  But that's
just a technicality that has nothing to do with the definition; anyone
can trivially reprove them with the "Sigma`<<M,N>,F>" notation if they
want.

(20-Aug-05) Many new or revised definitions today:

df-shft replaces df-shftOLD and df-shftOLDOLD - I extended it to all
    complex numbers, not just integers, for more flexible long-term use.
df-clim replaces df-climOLD - Convergence is now independent of the
    domain (NN, NN0, ZZ) of the underlying sequence - much nicer!
    In fact the input function can be garbage at the beginning, as
    long as there exists an integer somewhere beyond which it behaves.
df-seq0 replaces df-seq0OLD and df-seq0OLDOLD in order to use the new
    df-shft
df-fsum is the definition of a finite series summation.  I have mixed
    feelings about the notation (see fsumvalt comment), and comments are
    welcome.
df-plf is the addition of two functions.  I made it so it can apply to
    complex functions in general, not just sequences.  For sequences,
    we'll restrict the function sum to NN, etc. to strip out meaningless
    values.
df-muf multiplies a constant times a function, again for complex functions
    in general.

Slowly the obsolete *OLD versions will be replaced and eventually
deleted.  Yesterday's shftcan1t, etc. are already obsolete!

The lesson learned from the multiple versions of df-shft was that it
seems more useful to keep the definitions simple and as general as
possible.  Individual theorems can impose the necessary restrictions as
needed, rather than having the restrictions "hard-coded" into the
definition.  For example, df-clim is now dramatically more useful by
not restricting the domain of the underlying sequence to NN.

-------

I am thinking about a general 'seq' recursive sequence generator with an
arbitrary starting point.  The present 'seq' would be renamed to 'seq1'.
What would be nice would be:

    ( + seq0 F ) = ( + ( seq ` 0 ) F )
    ( + seq1 F ) = ( + ( seq ` 1 ) F )  etc.

Unfortunately seq0 and seq1 are proper classes and can't be produced as
function values, but restricting them to be sets would limit their
usefulness.  On the other hand, defining seq so that

    ( + seq0 F ) = ( < + , 0 > seq F )

or

    ( + seq0 F ) = ( + seq < F , 0 > )

or

    ( + seq0 F ) = ( < + , F > seq 0 )

etc. can be made to work without a restriction but none of the 12
possibilites seem natural to me.  What do you think?

(17-Aug-05) cvgratlem1,2,3 will replace the longer old versions (renamed
*OLD) in a re-proof of cvgrat that I have planned.

(5-Aug-05) The old definitions of the shift and seq0 operations have
been SCRAPPED.  They have been renamed df-shftOLD and df-seq0OLD (to be
deleted eventually), and replaced by new ones df-shft and df-seq0.  All
of the old theorems are obsolete and have been renamed *OLD.  The old
symbols have been changed to prevent accidental re-use.

The new definitions will provide simpler and more general theorems.  For
example, seq01 and seq0p1 are now the exact analogs of seq1 and seqp1 -
compare seq01OLD and seq0p1OLD, which required an annoying functionality
hypothesis.

(31-Jul-05) Per a suggestion from Scott Fenton, I renamed the following
theorems:

  Old       New

  syl34d    imim12d
  syl4d     imim2d
  syl3d     imim1d
  syl34     imim112i
  syl4      imim2i
  syl3      imim1i
  syl2      imim2
  syl1      imim1

(27-Jul-05) I was finally able to find a shorter proof of uzind.
Veteran visitors to this site will recall the 3/4 megabyte proof
described on 18-Jun-04 in mmnotes2004.txt, then called zind, and
currently renamed to uzindOLD.

(11-Jul-05) Back to the drawing board...  I decided to change binary
coefficient df-bc so that it is now defined (as 0) outside of its
"standard" domain of 0 <_ k <_ n, as is often done in the literature.
With the old definition, I can now see that many proofs using it would
have been very awkward.  Accordingly, several proofs were changed to
accomodate the new definition (not shown on the 'most recent' page - I
usually do not re-date modified proofs) and today's new ones were added.

(6-Jul-05) peano2re, although it is trivial and seems silly, shortens a
dozen proofs and reduces the net size of the set.mm database file.

(5-Jul-05) peano5nn is a simplification of the previous version.  df-n
was also simplified.

(28-Jun-05) pm4.83 finally completes the entire collection of the 193
propositional calculus theorems in Principia Mathematica.  This had been
done before for the Metamath Solitaire applet in
https://us.metamath.org/mmsolitaire/pmproofs.txt - but the set.mm
proofs are hierarchically structured to be short, indeed as short as I
(or Roy Longton for some of them) could find.

An ordered index of these can be found on the xref file
https://us.metamath.org/mpegif/mmbiblio.html in the
[WhiteheadRussell] entries.

(26-Jun-05) Yesterday's reuunixfr probably ranks among the most cryptic
in the database.  :)  Today's reuunineg shows an application of it that
is much easier to understand, with most of the confusing hypotheses of
reuunixfr eliminated.

(21-Jun-05) rabxfr lets us conclude things like the following:
(z e. RR -> (z e. {x e. RR | x < 1} <-> -z e. {y e. RR | -y < 1})).
The first two hypothesis just specify that y mustn't be free in B and C
(a less strict requirement than distinct variable groups y,B and y,C).

pm4.42 is Roy Longton's first Metamath proof.

(20-Jun-05) shftnnfn and shftnnval show the example of NN shifted to NN0
described in df-shft.  Hopefully these two theorems make it clear, in
a simple and intuitive way, what the 'shift' operation does.

(19-Jun-05) df-shft is a new definition; see its comment for an
explanation of the sequence shift operation.  In general I dislike
introducing a made-up explicit notation for a concept that exists in the
literature only implicitly in informal proofs, and I try to avoid it
when possible, because the notation will be completely unfamiliar even
to mathematicians.  But in the case of df-shft, after careful
consideration I felt the benefits will outweigh this disadvantage.  Once
the initial complexity is overcome with some basic lemmas, it is a
relatively simple concept to understand intuitively.

(18-Jun-05) We will start using j,k,m,n for integer set variables and
J,K,M,N for integer class variables.  I hope this will improve
readability a little.  Over time I will retrofit old theorems.  This
will be a major change involving hundreds of theorems, so if you have
comments on this let me know.

In the retrofitted proof of bcvalt, you can see the effect of this
change.

(17-Jun-05) imret provides us with a simpler way to define the imaginary
part, compared to df-im.  I may do that eventually.

(11-Jun-05) I finally caved in and revised df-exp so that 0^0=1 (as
can be seen with expnn00) instead of being undefined.  I have decided
that otherwise, some future things I have in mind are just going to be
too awkward.

Raph Levien came up with the original df-exp, where 0^0=1.  But he's a
computer scientist.  From a more purist perspective, I felt it was an
"inelegant patch," as it has been called, and I changed his definition
to exclude it.  For the past year we've trodded along merrily without
it.  But I'm starting to see that 0^0=1 will lead to simpler proofs and
statements of theorems in many cases.  So now we have 0^0=1 again.

Gleason's book defines 0^0=1 and uses it extensive, e.g. in the
definition of the exponential function (where we avoided it by breaking
out the 0th term outside of the infinite sum).

For a further discussion of this see:
http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/

Another reason I wanted to avoid defining 0^0 is that years ago on
Usenet, and probably still, there were endless arguments about it.  I
wanted to distance Metamath from that.  :)

(10-Jun-05) The choose function df-bc was added.  The literature
uses math italic capital C - but that conflicts with our purple C for
classes (when printed on a black-and-white printer).  So I decided to
use a Roman C.

bcvalt is somewhat awkward to prove because of its "Pascal triangle"
restricted domain instead of the full NN0 X. NN0.  Thus we have to use
oprabvali instead of the more efficient oprabval2.

(4-Jun-05) As far as I know, inf5 has never been published.  I think it
is neat.

pm2.13 seems like a rather silly variant of excluded middle (exmid).
What can I say - I'm just implementing what's in the book.

(2-Jun-05) efclt makes use of (in efcltlem1) the very important and
useful ratio test for convergence, cvgrat of 28-May-05, to show (in
efcltlem3) the convergence of the exponential function.  This in turn
lets us show that the value of the exponential function is a complex
number.  This will open a lot of doors with what we can do with the
exponential function.  Note that all of the confusing (or at least
unconventional) limit, seq, and infinite sum stuff have disappeared,
having served their purpose, and we're back into familiar territory.

Interestingly, the bounds we found earlier for Euler's constant e, in
ege2le3, didn't require all this work.  That is because e is a special
case of the exponential function that is much easier to work with.

(27-May-05) sercj tells us the the complex conjugate of each term in an
infinite series is the sum of the complex conjugates of the underlying
sequence.  We prove it by induction.  Recall that (+ seq F)`A means

                ---   A
                \         F_i
                /
                ---  i = 1

Theorem minusex just says that the negative of any class whatsoever
(even a proper class) is a set.  While this is not very meaningful when
the argument is not a complex number, it saves the effort of proving
that the argument is a complex number, making it trivial, for example,
to eliminate the hypothesis "A e. V" of yesterday's cvgcmp3cet.

(26-May-05) cvgcmp3cet is a pretty massive application of the Weak
Deduction Theorem https://us.metamath.org/mpegif/mmdeduction.html that
converts 8 hypotheses into antecedents.  A number of tricks were
employed to make the proof sizes manageable.  I didn't bother with the
final hypothesis, "A e. V", because it's trivial to eliminate with
vtoclg if needed (you don't need the Weak Deduction Theorem for that)
and in most cases A will exist anyway.

(25-May-05) The theorems expgt1t and oewordri have little to do with
each other.  There is an isomorphism between finite ordinal
exponentiation and exponentiation of the natural number subset of reals,
that could be exploited in principle, but they are independently
developed in our database.  A common root for both can be traced back to
ordinal multiplication (which is a starting point for the construction
of the reals), but from that point they diverge.  And when ordinals get
into the transfinite, the behavior of exponentiation becomes bizarrely
different, as we will see when (hopefully) I eventually get to it.

(24-May-05) Two of the hypotheses of cvgcmp3ce, cvgcmp3ce.4 and
cvgcmp3ce.7, are theoretically unnecessary.  However, eliminating them
is tedious (it involves dinkering around in the hidden regions of F and
G prior to index B; these regions were purposely left undefined to make
the theorem more general) and for most practical purposes unnecessary,
so I decided to leave the theorem "less than perfect," so to speak, at
least for now.

We could also, with only a few more steps (by changing y to a dummy
variable z and using cbvexv and mpbi at the end) eliminate the
requirement that x and y be distinct variables.  I may do this if it
ever becomes useful to do so.  In that case, the distinct variable group
"x,y,G" would split into "x,G" and "y,G".

The new rcla4 series swaps the antecedents.  I think this makes their
use more natural in a lot of cases.  However, I'm wondering if this was
a mistake:  rcla4v was used in around 90 theorems, and it took several
hours just to convert a couple dozen of the easiest ones.  In maybe 75%
of those cases the proof size was reduced, but in others it was
increased, and the hoped for net "payback" in terms of reduced database
size hardly seems worth it, if there will be a net reduction at all.
The old rcla4* versions were renamed with an "OLD" suffix, and I'll be
eliminating them over time (on dreary days when I'm feeling otherwise
uninspired).

By the way here is an informal breakdown of the cryptic name "rcla42v":

  'r' - uses restricted quantification (vs. "cla4*")
  'cl' - deals with substitution with a class variable
  'a4' - an analog to the specialization axiom ax-4 (and Axiom 4 in
         many systems of predicate calculus, which is stdpc4 in our
         database)
  '2' - deals with two quantifiers
  'v' - distinct variables eliminate the hypothesis that occurs in rcla4

(21-May-05) eqtr2t and eqtr3t were added because they shortened 16
proofs, with the net effect of reducing the total database size.

(20-May-05) odi is essentially the same proof as the 2/3 smaller nndi
for natural numbers, except that it uses transfinite induction instead
of finite induction.  So we have to prove not only the 0 and successor
cases but also the limit ordinal case.  But the limit ordinal case was a
monstrosity to prove, taking up 2/3 of the proof from steps 59 through
257.  Eventually I'll shorten nndi as a special case of odi.

(16-May-05) drex2 is part of a cleanup of some old lemmas.  The notable
feature of this theorem and others like it is that x, y, and z don't
have to be distinct from each other for the theorem to hold (normally, z
can't occur in the antecedent, as in for example biexdv).  The
"Distinctor Reduction Theorem" provides a way to trim off unnecessary
antecedents of the form (not)(forall x)(x=y), called "distinctors," in a
system of predicate calculus with no distinct variable restrictions at
all (which makes automated proof verification trivial, like for
propositional calculus).  (That system is the same as ours minus ax-16
and ax-17.  The paper shows that it is complete except for antecedents
of the form (not)(forall x)(x=y).  To translate its theorems to standard
predicate calculus, these antecedents are discarded and replaced with
restrictions of the form "x and y must be distinct variables.")

We can also translate distinctors to distinct variable pairs in the
logic itself (after ax-16 and ax-17 are added) by detaching them with
dtru.

The reverse can be done (distinct variable pairs to distinctors) by
using dvelim.  This comes in handy when a distinct variable restriction
is unnecessary, e.g. x and y in ralcom2; we convert the distinct variable
pair to a distinctor with dvelim then eliminate the distinctor with the
algorithm of the Distinctor Reduction Theorem.

(13-May-05) Thank goodness caucvg is out of the way...  The lemmas just
seemed to grow bigger and bigger as I scrambled to complete it and is
quite a mess towards the end.  When the proof author said "this
direction is much harder" he/she is not joking.  There is often much
hidden detail you end up discovering, that isn't apparent at first, when
you try to formalize a proof.  (For example, the very first stumbling
block was how to formalize "the set of numbers less than all values of F
except for finitely many".  Certainly "finitely" isn't to be taken
literally, i.e. strictly less equinumerous than omega, unless we want an
incredibly complex proof.)

It looks like I should eventually introduce an abbreviation for Cauchy
sequences, like I do for Hilbert space.  Then these proofs can be redone
with a somewhat simplified notation.  (That's easy to do, once you have
the proof.)

(12-May-05) For the caucvg proof, I am formalizing the proof found at
http://pirate.shu.edu/projects/reals/numseq/proofs/cauconv.html . I
couldn't find this proof in a textbook (most of those proofs use "lim
sup" instead).  If someone has a textbook reference for this particular
proof, it will be appreciated.

cruOLD has been phased out.

(11-May-05) cru generalizes the old version (now called cruOLD until it
is phased out) to include the converse.

(10-May-05) relimasn is a version of imasng that doesn't require that A
be a set (in the case where R is a relation, which is most of the time).
When A is not a set, the theorem isn't really meaningful - both sides of
the equality become the empty set - but relimasn allows us to prove more
general theorems overall.

(9-May-05) This morning a correspondent wrote me:

> Do you know of a rigorous formulation of Wang's single axiom schema for
> first order identity theory? I saw one in Copi's 'Symbolic Logic [Fourth
> Edition]' page 280, but I don't quite follow his notation nor do I see how
> to precisely state the stipulations for the variables in a "phi and psi"
> style axiom schema. And I didn't see it in your proof explorer as a theorem.

I added sb10f and answered:

I have the 5th edition, and I think you mean P6 of system RS_1 on p.
328.  (The 5th ed. added a chapter on set theory, which moved all the
later pages up, probably by 48 pages or so.  Copi was noted for killing
the used-book market by releasing new editions every few years.)

In the way it is stated, this axiom apparently has 2 errors.  First,
there are no restrictions (that I could find) stated for Fx and Fy.  Let
us suppose Fz is x=z.  Then Fx is x=x and Fy is x=y.  Putting these into
P6, we get:

   A. y (Fy <-> E. x (x=y /\ Fx))
   A. y (x=y <-> E. x (x=y /\ x=x))
   A. y (x=y <-> E. x (x=y))
   A. y (x=y <-> true)
   A. y (x=y)

The last line is false in a universe with 2 or more elements, so the
system is inconsistent.

The correction is to add a proviso that x must not occur free in Fy.

The second mistake is that there is no requirement (that I could find)
that x and y must be be distinct variables.  But if they are not
distinct, an inconsistent system results.

With these corrections, the proofs of the usual axioms on p. 329 still
go through.

The "A. y" prefix is redundant in P6, since it can be added by R2
(generalization).  In a logic that allows an empty universe, the A. y
would be needed, but on p. 319 it is stated that RS_1 is intended to be
true in a nonempty universe (and the rest of the axioms won't work in an
empty universe).  So, it seems like R6 is an afterthought tacked onto
the system.  Even the notation Fx and Fy is different from the Q that
represents the substitution instance of P in earlier axioms e.g. P5
p. 294.

I added what I thought was a close approximation to P6 (without the
redundant quantifier) here:

  https://us.metamath.org/mpegif/sb10f.html

The hypothesis specifies that x must not occur free in phi, and x and y
must be distinct, as must necessarily be the case.

Three other variants that are similar to P6 are:

  https://us.metamath.org/mpegif/sb5.html
  https://us.metamath.org/mpegif/sb5rf.html
  https://us.metamath.org/mpegif/equsex.html ,

the last one implicitly substituting y for x in phi to result in psi.

By the way, even though we can express a logical equivalent to P6 in
Metamath, this does not mean that it becomes the sole axiom replacing
the other equality/substitution axioms.  (It is possible that one or
more of the others could become redundant, but I haven't thought about
it too much.)  The reason is that in RS_1, substitution is done at the
metalogical level, outside of the primitive system.  In Metamath, we do
this "metalogic" at the primitive level of the system itself, and we use
additional axioms involving equality to accomplish this.  In many ways
substitution and equality are closely related, and the standard
formalization "hides" this by moving substitution outside of the axioms.

You might want to re-read these that explain this in more detail:

  https://us.metamath.org/mpegif/mmset.html#axiomnote
  https://us.metamath.org/mpegif/mmset.html#traditional


(8-May-05) While Euclid's classic proof that there are infinitely many
primes is easy to understand intuitively, I found the proof used by
infpnlem1 and infpnlem2 simpler to formalize.  (Specifically, this proof
avoids the product of a hypothetical finite set of all primes, which I
found cumbersome to formalize.)

Here is the proof:

  For any number n, the smallest divisor (greater than 1) of n!+1 is a
  prime greater than n.  Hence there is no largest prime.

Or, in explicit detail:

  Suppose there are a finite number of primes.  Let p be the largest.  Let
  q be the smallest divisor (greater than 1) of p!+1.  (The set of
  divisors of p!+1 is nonempty, since p!+1 is one of them, so by the
  well-ordering principle there is a smallest, which we will call q.)  Then
  none of 2,3,4,...,p divides q since otherwise it would also divide
  p!+1, which is impossible.  (2,3,4,...,p all divide p!, so none divides
  p!+1.)  And none of p+1,...,q-1 divides q since otherwise it would also
  divide p!+1, and then q would not be the smallest divisor of p!+1.
  Therefore q is prime, and q > p, so p is not the largest prime.

(6-May-05) funcnvres2 is a tongue-twister, or perhaps a brain-twister...

I reproved cvgcmp as a special case of cvgcmp2.  However, I have
(temporarily?) left in the original proof and called it cvgcmpALT, as I
think it might be instructive.  Comparing cvgcmpALT to
cvgcmp2lem+cvgcmp2, i.e. 33 vs. 68+17=85 steps, you can see how much
extra work was needed just to ignore up to the Bth term in cvgcmp2.

(5-May-05) cvgcmp2c is useful because it allows the test function
to be much larger (via a large C) than the reference function, yet
still converge.

(4-May-05) divclt, divrect, redivclt are slight modifications of their
older versions, which have been renamed divcltOLD, divrectOLD,
redivcltOLD and which will disappear when they are phased out over time

(3-May-05) prodgt0t also works for A=0, not just A strictly greater than
0.  This makes the theorem more general - something I like to do when I
can - but requires more work.  In prodgt0 (from which prodgt0t is
derived with dedth2h) you can see the separate derivation that we need
for the A=0  case.

(2-May-05) reccl* and rereccl* shorten many proofs (by replacing explicit
division closure that was used before) - e.g. I shortened 18 proofs
with rereccl.  So even though these are trivial they were worth adding.

(1-May-05) cvgcmp2 will be used to build a general-purpose comparison
test for complex number sequences.  cvgcmp2 tests for convergence of a
nonnegative real infinite series (+ seq G) (which normally will be a
series of absolute values), which is compared to a known convergent
series (+ seq F).  This version of cvgcmp allows us to ignore the
initial segment up to the Bth term; this was a tricky thing to do.  To
achieve this I compare G to an auxilliary sequence H (see cvgcmp2lem)
instead of F; H adds the supremum of the initial segment of G to F, so
it is guaranteed to be bigger than G everywhere including the initial
segment.

Originally I planned to use climshift of 24-Apr and sertrunc of 27-Apr
to achieve this (the ignoring up to the Bth term); now it looks like
they are no longer needed.  Too bad; they were a lot of work.  Perhaps
I'll leave them in in case a use for them shows up in the future.

In cvgcmp2 we show the actual value it converges to (i.e. the sup)
rather than just existence.  This will allow us to use hypotheses
instead of antecedents, which will make some proofs smaller.  For our
final theorem we will eliminate the hypotheses with the Weak Deduction
Theorem dedth then produce a simpler-to-state existence version.

(24-Apr-05) 2eu8 is a neat little discovery that I doubt has ever been
published.  It is fun seeing what you can do with the E! quantifier.
Hardly anything about it exists in the literature, and apparently double
E! has never even been published correctly; see 2eu5.  Exercise:  Can
you change E!x E!y to E!y E!x in either side of 2eu8 by using 2eu7 as
suggested?  (Hint:  use ancom.)  Note that E!x E!y doesn't commute
generally, unlike Ex Ey; probably not too many people know that.
Another interesting thing about 2eu7 and 2eu8:  x and y don't have to be
distinct variables.

(23-Apr-05) climshift shows that we can ignore the initial segment of a
sequence when computing the limit.  This is intuitively obvious (since
it's only what happens towards infinity that counts) but is a little
tedious to prove.

(22-Apr-05) It is curious that max1 doesn't require B to be a real
number.

(21-Apr-05) In steps 1-19 of climre, you may wonder why we have extra
steps using vtoclga to switch from variable x to variable w, when in
fact variable x could have been used throughout up to step 19.  The
answer is that by using w's, we can reuse step 14 at step 43, without
having to repeat its work.  This is a little trick that shortens the
compressed proof and the web page.  (The uncompressed proof, on the
other hand, is lengthened because it does not reuse previous steps, but
normally set.mm is stored with compressed proofs.)

(20-Apr-05) For serabsdif, note that (+ seq F)`n - (+ seq F)`m is the
sum from m+1 to n of the terms of F, i.e.
F`(m+1) + F`(m+2) + ... + F`n.  So even though our notation for series
(+ seq F) is limited for notational simplicity to start at the fixed
lower index of 1, we can represent a more general lower limit using
this method.  (A more general notation for series may be introduced in
the future.)

(19-Apr-05) We're going to be using (+ seq F) a lot, so here's a
refresher, since this notation is not standard.  We are using a special
case of our general-purpose "seq" operation, and there seems to be no
standard notation for (+ seq F) in the literature other than the
informal "a_1 + a_2 + ...  + a_n" which is not suitable for formal math.
(Gleason uses "F-" in front of an infinite sum to indicate the partial
sum function underlying the infinite sum, but it is not standard.)  If
you are following these theorems it might be useful to keep the
following note in mind.  It is straightforward if you understand the
correspondence to the conventional notation.

(+ seq F) is the sequence of partial summations in an infinite
series.  E.g. for a sequence of squares:

  argument sequence   partial sum of series

   n         F`n     (+ seq F)`n
   1         1          1
   2         4          5
   3         9         14
   4        16         30
          ...

Of course this series diverges, so the infinite sum doesn't exist, but
all partial summations exist as shown above.  Today's theorem serft
expresses a very obvious fact:  if F is a function from the positive
integers to the complex numbers, then so is (+ seq F).

Another example:

   n          F`n     (+ seq F)`n
   1         0.5        0.5
   2         0.25       0.75
   3         0.125      0.875
   4         0.0625     0.9375
          ...

This series converges to 1, so the infinite sum sigma-1-oo`F exists and
equals 1 (see theorem geosum1).  By "converges" we mean the limit as
n->oo of (+ seq F) is 1; we express this as (+ seq F) ~~> 1 (see theorem
geolim).

(17-Apr-05) ege2le3 proves with absolute rigor that scientific
calculators display the value of e correctly to at least 1 decimal
place. :)  This is a nice "sanity check" of all the abstract stuff that
finally collapses to this result.

(13-Apr-05) It surprised me that it took so much work to prove that
e is a real number.

(5-Apr-05) equid2 shows how you'd prove x=x in a traditional system of
predicate calculus which has a9e as an axiom.  This proof is almost
identical to Tarski's.  Still, I think the no-dummy-variable equid
proof is neat and unexpected.

(1-Apr-05) The Usenet announcement of Poisson d'Avril's theorem is here:
http://groups.google.com/group/sci.logic/browse_frm/thread/7aa9265da2819705/ee8862dd6adb3fad#ee8862dd6adb3fad

(26-Mar-05) geosum1, which you may be familiar with from high-school
algebra, is the culmination of our initial development of infinite
series.  As you can see, it is much harder than the high-school version
if you want to prove it with absolute rigor!

geosum0 shows the sum from 0 to infinity instead of 1 to infinity.  It
provides a good illustration of why I chose to have our "fixed limit"
infinite series operator start at 1 instead of 0, so that we just add
the 0 case outside of the summation when we need a lower limit of 0. If
we started at 0, we would have to worry about the meaning of 0^0 in this
case.  A long time ago, when I introduced the ~~> symbol (mainly for use
with Hilbert space), I initially defined it for functions starting at 0
instead of 1. But I ran into so many special cases (divide-by-0 and so
on) that eventually I redefined it to start at 1. This greatly
simplified many proofs.  Who knows, this may be the real reason that
analysts start with 1 for the natural numbers (unlike set theorists, who
start at 0).  In fact Schechter's _Handbook of Analysis_ doesn't even
have a separate symbol for the set of nonnegative integers; instead he
uses "NN (natural numbers) union {0}" in the one case I could find where
he needed them.

(16-Mar-05) In sumvalt and sumval2t, the function Sigma_1^oo is one
symbol (object), not three symbols, even though it is shown as three as
a mnemonic device.  To make the proofs and notation simpler, the lower
limit is fixed at 1, and even with this constraint it should be useful
in a wide variety of applications.  I chose 1 instead of 0 as the lower
limit because the 0th term often involves messy special cases (e.g. to
avoid divide-by-0), and it's easy enough to add the 0th term outside the
summation when we want.  In addition, our limit relation ~~> is defined
for sequences that start at 1, making it compatible.  (Later we may
introduce a more general summation with variable upper and lower limits,
but that wouldn't buy us a whole lot now and would just make the
notation complex.)  The Sigma_1^oo function has a single argument, which
is an infinite sequence that starts at 1 (i.e. is a function from natural
numbers to complex numbers).  The value of Sigma_1^oo is the infinite
sum of all the numbers in the sequence, if that sum exists.  I thought
the the conventional summation sign would be neat reminder that this is
what the function does, even though it is used slightly differently from
the textbook summation (e.g. there are no dummy index variables; these
are implicit in the sequence that it takes for an argument).

The Sigma_1^oo object is, on the one hand, just a set like any other
set; more specifically it is a set of ordered pairs that gives us a
function of one variable.  But if you think about it, it contains
implicitly an amazingly complex "machine" inside.  It can take in any
infinite sequence whatsoever and produce the infinite sum, if the sum
exists.  If you peel away the layers of its definition, you'll find
df-rdg, our recursive function generator on ordinals.

(14-Mar-05) For sersuc, recall the notes of 7-Mar-05 below.  sersuc
tells us the value of the next partial sum in an infinite series.
Yesterday's ser1 tells us the value of the first partial sum i.e.
F ` 1 .

(9-Mar-05) ltsor arises as part of our complex number construction; it
is needed for the proof of a couple of our complex number axioms.  ltso,
on the other hand, is a result derived from the complex number axioms
after those axioms have been proved.  I got tired of having to clean up
accidental uses of ltsor in place of ltso (which destroys the
portability of our complex number construction), so I decided to prevent
it once and for all by adding the artificial right conjunct.  Clever,
eh?

(7-Mar-05) This is what the 'seq' operation in sercl is all about:
  ( + seq F ) ` 1  =  F ` 1
  ( + seq F ) ` 2  =  F ` 1 + F ` 2
  ( + seq F ) ` 3  =  F ` 1 + F ` 2 + F ` 3     etc.
In other words we are using the powerful 'seq' recursion operation to
create the terms of an infinite series from an arbitrary infinite
sequence F. We will encountering this device frequently as we get into
infinite series.  (With 'times' instead of '+', 'seq' will give us
infinite products.)  Earlier we used 'seq' to define exponentiation
df-exp and factorial df-fac.

By the way, note that in theorem sercl, the class A is normally
dependent on x. In other words, A should be thought of as A(x).  The way
you can tell is that A and x are not both in the same distinct variable
group, yet A is in the scope of x (i.e. inside the braces in the first
hypothesis and in the scope of the quantifier in the second hypothesis).
THIS IS AN IMPORTANT PRINCIPLE THAT CONFUSES SOME PEOPLE.  We also know
that A is _not_ dependent on y because A and y must be distinct.  (On
the other hand, the fact that B and x must be distinct is an irrelevant
artifact resulting from a sloppy proof.  I may improve the proof at some
point to get rid of that unnecessary restriction, unless it lengthens
the proof too much.  Sometimes I leave such artifacts in, especially in
lemmas used once, if it results in a shorter proof as it often does.)

As a background project I'm slowly filling in the remaining
propositional calculus theorems from Principia Mathematica.  I figure we
might as well have the complete collection eventually.

(5-Mar-05) By using brrelexi and the new climrel in the proof of clim,
clim's old "F in V" hypothesis was eliminated.  The new version of clim
also allowed us to prove new versions of the other clim* theorems with
this hypothesis eliminated.  Overall the database size decreased since
theorems referencing clim* no longer have to prove "F in V".

(4-Mar-05) oancom requires the Axiom of Infinity for its proof.
Virtually all other results on ordinal arithmetic, remarkably, do not
require the Axiom of Infinity (which starts being used with theorems
inf4 and after; inf4 currently has the colored number 3567).

(1-Mar-05) alephval3 - I finally figured out how to prove this
convoluted, self-referencing definition that at least two authors prefer.
Even though in some sense it may be "intuitive," it wasn't an easy thing
to prove formally.

(20-Feb-05) Well, I proved that ax0re is redundant as a complex number
axiom, the first redundancy that anyone has found in 8 years.  0re
replaces the old ax0re, and ax0re was removed from
https://us.metamath.org/mpegif/mmcomplex.html .  0cn is a completely
new proof of the old 0cn that doesn't depend on the old ax0re, and it
provides the key to eventually proving 0re.  The table row on the
mmcomplex.html page, after the grayed-out ax0re axiom label, shows the
connections between the important theorems for the proof.

(18-Feb-05) dfsb2 was one of the two choices I was considering a long
time ago when I finally chose df-sb somewhat arbitrarily.  I decided to
prove dfsb2 as a theorem, well, just to have it proved.  Although the
proof isn't long, it's actually somewhat involved all the way back to
axioms.  Axiom ax-11 seems essential:  ax-11 -> equs2 -> equs5 -> sb4 ->
dfsb2.

By the way it is an open problem whether ax-11 is redundant.  I
can prove from the others all cases of ax-11 where phi is replaced by
any expression without a wff variable e.g. (x=z -> -.w=y).
ddelimf2 was, I felt, a major step towards helping prove the
redundancy of ax-11, but I haven't made any progress since.

(17-Feb-05) I was somewhat surprised hbequid could be proved without
ax-9.  (I found the proof while toying with the open problem of item #16
at https://us.metamath.org/award2003.html .  That problem is still
unsolved, though.)

(16-Feb-05) dfrdg2 (which uses yesterday's eqif) allows us to introduce
a new definition df-rdg of the rec operation.  The new df-rdg now fits
on one line if you put your browser in full screen mode!  I rewrote the
df-rdg comment slightly.  df-rdg is still almost "impossible" to
understand, but I think the use of the "if" operation improves its
understandability a little.

I cleaned up the list of traditional predicate calculus with equality
axioms at https://us.metamath.org/mpegif/mmset.html#traditional and
added stdpc6 to match Mendelson's system exactly.

What is very strange is why stdpc6 is quantified.  It is possible this
may make it sound in empty domains to make it compatible with the system
on p. 148, but I couldn't find any indication of this.  Does anyone
know?  If you have the 3rd or earlier (pre-1997) edition of Mendelson's
book, could you check for me that Axiom 6 p. 95 (or the equivalent page)
is quantified?

I renamed sbequ2a from yesterday to become stdpc7 and enhanced its
description.

(15-Feb-05) eqif shows an example of what elimif is intended for.

stdpc7 is a faithful emulation of the 5th traditional axiom
at https://us.metamath.org/mpegif/mmset.html#traditional and
replaces sbequ2, which wasn't quite right for that purpose.

(13-Feb-05) 2eu6 is nice in that it, unlike 2eu4, only has one mention
of phi on the right-hand side.  This can be useful when phi is a long
expression, so we don't have to repeat it twice.  It also makes less
urgent the need for a rarely-used and nonstandard new definition for
double existential uniqueness such as E!!xy or something, which is
nonexistent in the literature (which incorrectly uses E!xE!y; see 2eu5).
As you can see 2eu6 was somewhat tedious to prove.

(9-Feb-05) axaddopr and axmulopr can't be proved directly from the
axioms for complex numbers shown at
https://us.metamath.org/mpegif/mmcomplex.html , so I had to dig back into
their construction.  Brings back memories - I haven't looked at it
in years.

The construction was not easy, with very tedious equivalence relations
(look at oprec and ecoprass for example) to jump from one level to the
next - positive integers, positive rationals, positive reals (using
Dedekind cuts), signed reals, and finally complex numbers.  Nonetheless
it appeared to be the simplest of several methods I looked at.  The
final axioms are cleanly broken out, so a different construction could
be "plugged in" trivially for anyone who prefers a different approach
(and is willing to go through the tedious construction).

axaddopr and axmulopr will be used for some infinite sequence stuff
because it will make some things slightly simpler.  However, I decided
not to replace the "official" axioms at
https://us.metamath.org/mpegif/mmcomplex.html with these because I think
the official ones look nicer, and in principle they are sufficient.


(8-Feb-05) In flval2t, "U{x in ZZ|...}" can be read:  "The unique
integer x such that..."  (provided there is exactly one such x, which in
this case there is).

(4-Feb-05) The two variations ordsssuc and ordsssuc2 show that either A
or B can be a general Ord class.  However, it fails if both of them are
Ord classes.  In other words at least one of them must exist as a set,
i.e. a member of On.  Of course both may be sets, as shown by the weaker
onsssuc.

ordunisuc2 is one of those neat counterintuitive results - you wonder
what one side of the equivalence could possibly have to do with the
other...

(3-Feb-05) An error made in some published math books for the layman is
the statement that there are aleph-one reals (or irrationals).  In fact,
there are 2^aleph-zero reals, which is equal to aleph-one only when we
assume the Continuum Hypothesis.  Without CH we can only state that
there are at least aleph-one reals (or irrationals), but there could be
more.

To prove aleph1irr we invoke the somewhat exotic infdif.

(31-Jan-05) I realized that onpwsuc could be generalized to all ordinal
classes, not just ordinal numbers, so the result is ordpwsuc.  I also
simplified the proof of onpwsuc so as to derive it trivially from
ordpwsuc.  By the way I mentioned onpwsuc on Usenet; search for
"onpwsuc" in groups.google.com.

(20-Jan-05) alephfp is one of several theorems where we show an explicit
expression, vs. textbooks, which typically show only existence.  To me,
an actual example is a lot more satisfying.  Two other examples in
set.mm are trcl and cardaleph.  The rec function is one of the main
tools that lets us construct explicit expressions.

(19-Jan-05) oev2 is the simplest expression for the "traditional"
definition I could find, using only the rec function and set theory
primitives (without the "if" function of oev).  The left-hand side of
the intersection is the "natural" definition that arises from
transfinite recursion, and the right-hand side suppresses 0^(limit
ordinal) to be 0 instead of 1, while keeping 0^0=1.

(12-Jan-05) Yesterday I suggested proofs from biluk and mpbi as an
exercise.  I looked them up in the _Polish Logic_ book and they are not
obvious at all.  pm4.2 can be proved in 13 steps, but the others are much
longer.  So if you're interested, get the book to avoid a lot of
frustration.

(11-Jan-05) The proof of biass was reduced to 35 steps (with the help of
or42 and xor).

biluk is an easy consequence of biass.  Note that biass, bicom, and
pm4.2 can be derived from biluk and mpbi and nothing else, although we
won't be doing it.  (Exercise for the reader...)

or42 is the 'or' version of an42, which "rotates" the right-most 3
conjuncts.  an42 and an42s are used surprisingly often (for such an
apparently obscure operation).  Other analogues are add42 and caopr42.

(7-Jan-05) tfinds3 can shorten certain proofs that used to use tfinds by
8 steps, which is the number of steps in the proof of tfinds3 from
tfinds.  By applying it to the 5 theorems in its referenced-by list, I
reduced the net size of the set.mm database file (vs. before I added
tfinds3).  In some cases I shortened them more:  compare (until the
mirror site gets refreshed)
https://us.metamath.org/mpegif/oacl.html (29 steps) vs.
https://us.metamath.org/mpegif/oacl.html (41 steps)

(5-Jan-05) Again, compare oesuc to omsuc to see the additional
complexity caused by the awkward traditional definition.
oelim (compared to omlim) isn't too bad, though.

(4-Jan-05) According to oe0m1, zero to the omega power is zero, not
one.  This is the requirement that complicates df-oexp.

(1-Jan-05) oev is our first theorem involving ordinal exponentiation.
It uses the somewhat awkward traditional definition.  Compare it to omv
for example; an alternate, but nontraditional, definition for
exponentiation would be exactly as simple.  I decided on the current
definition after a discussion here:
http://groups.google.com/group/sci.logic/browse_frm/thread/fac0ce315e8ea855
However it will be making a number of proofs, such as this one, more
complex.

(30-Dec-04) I think iunss2 is rather nice (even though its proof is
very short), and I might be able to use it to shorten oaass.

(25-Dec-04) oaass is a huge proof!  This has been on my to-do list for a
long time, so getting it done is a relief.  Before we only had nnaass,
which is limited to natural numbers and which I did a long time ago for
the real/complex number construction.  To build the oaass proof I
basically copied nnaass, then added the induction hypothesis for limit
ordinals (step 164).  But proving this hypothesis is 2/3 of the final
proof, or steps 44-164!  (And this doesn't count oalimcl, which is
really a lemma for oaass with no other use I'm aware of.)  For the other
steps, you may want to compare oaass to nnaass - they are identical
except for closure laws.  (Eventually I'll use oaass to shorten nnaass.)

By the way to understand why nnaass was needed for our complex number
construction, you can follow its path as it propagates forward through
the stages of construction of complex numbers (and the associative law
for complex number addition in particular), which you can see in the
"referenced by" lists:

nnaass -> addasspi -> addasspq -> addasspr -> addasssr -> axaddass

BTW Merry Christmas!

(23-Dec-04) sumdmd is Holland's theorem (finally).

(21-Dec-04) sbequ5 is just a cleanup of an earlier version.  By the way
the theorem eq5 used in its proof was the first "significant" theorem
(so I felt) that I proved from the unusual predicate calculus subsystem
ax-4 through ax-15, and it was the initial clue towards eventually
proving completeness for my "Finitely Axiomatized..." paper.  I remember
feeling very excited that it held, some dozen years ago.  I still think
its proof is strange and unintuitive.  Axiom ax-11 is not used for its
proof and was added later.  An open problem is whether ax-11 is
redundant; it can be proved as a "metamatheorem" (outside of Metamath)
from the others but I've never been able to prove it directly from the
others.

The Hilbert space stuff is more progress towards Holland's theorem.

(20-Dec-04) Q is strictly less equinumerous than R as promised.  (Now,
that wasn't hard, was it?)

(19-Dec-04) qbtwnre, that Q is dense in R, is an important result about
real numbers.  This is one of those counterintuitive results about
infinite sets:  Q is strictly less equinumerous than R (I guess I should
add a theorem for that - yes, I'll do it for tomorrow), but on the other
hand, in between any two reals, no matter how close they are, you'll
find a rational!  How can that be?  Think about it.  This used to bother
me when I was younger, and I suppose it still gives me an uneasy feeling
now and then.:)

(17-Dec-04) I finally received from the library Holland's 1969 paper
(listed in the Hilbert Space Explorer references) which turns out to
have a marvelous proof of subspace closure of the sums of dual modular
pairs.  I've been looking for such a proof for a long time.  Maeda's
version that I tried to use has a gap I haven't been able to figure out,
but Maeda died a couple of years ago so I can't ask him to clarify it
(and no one else seems to know).  The Hilbert space results for the last
couple of days are some basic facts we'll eventually draw on for
Holland's proof.

(15-Dec-04) Some simple but frequently-used theorems that let me shorten
a bunch of proofs.

(14-Dec-04) The proof of Prop. 8.8 in Takeuti&Zaring p. 59 has a typo:
the index on the big union should be "delta < gamma", not "delta <
beta".  (This brings to total typos I've found in the book to about 30.
In spite of this, it is by far the best book for the technical details
of many proofs that other books gloss over.  Too bad it is out of
print.)

I put a lot of work into oawordeulem because I was afraid it was going
to be huge.  The end result is that it is actually 1/3 smaller than the
finite version that it replaces, and I'm kind of proud of it.  (The
finite version was nnordexlem, which I deleted but you can still find on
some of the mirrors.)  The intersection of the temporary class S
corresponds to gamma in the T&K proof.

(13-Dec-04) These variations of modus ponens and syllogism are used to
shorten a number of proofs and will be handy to have available in the
future.

(12-Dec-04) In our database, ordinal arithmetic is not well developed.
The history is that I did just enough to construct real numbers, and
some operations (e.g. the associative law nnaass) are proved for
natural numbers only.  Over time I'll be extending these to the
transfinite.  I'm discovering that some textbook proofs have subtle
mistakes (or to be generous, they gloss over details) that only become
apparent when you try to work them out, making for a somewhat
frustrating task.  Perhaps I'll mention some of them here as they come
up.

Today's oawordri is the commuted version of oaword of 8-Dec.  Unlike the
natural number version it must be proved separately since ordinal
addition ceases to be commutative in the transfinite.  Note that the
commuted version of oaord of 7-Dec does not hold in the transfinite.
Also, the converse direction of oawordri does not hold in the
transfinite.

(11-Dec-04) In ordunidif, note that we're not removing a single element
but an entire subset of elements up to that element.  (Otherwise it
would be A\{B}, not A\B.)  For example:  if A is 5={0,1,2,3,4} and B is
4={0,1,2,3}, then A\B is {4}.  Then the union of A is 4, and the union
of A\B is also 4.  Notice that for finite (and other successor) ordinals,
the union operation subtracts 1.

(9-Dec-04) All the previous work leading to rebtwnz, involving heavy use
of the archimedean principle and well ordering of bounded integer sets,
finally pays off by letting us show the properties of the floor function
in flleltt.  It is interesting that so much work is needed for such a
"trivial" thing as the floor function.

(7-Dec-04) I have been wanting to prove oaordi for some time but the
limit case of the transfinite induction (steps 32-51) kept proving
illusive.  Finally I realized ssiun2s was the trick I needed, and we
don't even need the available "for all y less than x" antecedent (added
with a1d at step 51, and that proved to be a red herring in earlier
proof attempts).

(3-Dec-04) Look at how easy the definition of factorial df-fac becomes
using our new seq operation.  I'm very pleased.  (See
https://us.metamath.org/mpegif/mmset.html#function for why the
notation is (!`n) instead of n!.)

(2-Dec-04) I am torn whether to add a definition for the class of all
cardinal numbers.  Well, it actually has been added - df-cardn - but I
consider it provisional and so far I've avoided using it.  With the
definition, ondomcard could be simplified to {x e. On|...} e. Card.  But
df-cardn saves only a few symbols, and already we have so many
definitions that it's hard to remember them all.  I've never seen both
df-card and df-cardn in the same textbook, although textbooks have the
advantage of informal English like "is a cardinal number".

shsumval3 is nice, compared to the "official" subspace sum
value, shsumvalt.  I'm tempted to redefine df-shsum accordingly,
although the current definition does capture the "intent" of subspace
sum even though it's more complicated.

(1-Dec-04) zmax shows an application of reuxfr that we proved a
few days ago.  Note the use of reuhyp to eliminate one of reuxfr's
hypotheses.

I added a new "feature" for those people who check the most
recent proofs page every day.  You can go directly to the new theorems
via https://us.metamath.org/mpegif/mmrecent.html#table without
having to scroll down.

(30-Nov-04) It is surprisingly difficult to generalize the integer B in
uzwo2 with the real B in uzwo3, when on the surface this seems so
obvious.  Integers and reals don't mix naturally.

(19-Nov-04) euxfr, even though it is simple-looking, is a culmination of
a lot of results about existential uniqueness.  As you backtrack through
its proof you will find mopick, moexex, 2moswap, 2euswap, and euxfr2,
all of which I find interesting as somewhat non-obvious, but none of
which appear in the literature to my knowledge.  But euxfr is often used
in informal proofs implicitly; for example:  if we know that there is
exactly one x such that f(x)=4, then we can conclude there is exactly
one y such that f(3y+5)=4.  And vice-versa.  The formalization of this
argument is euxfr.  Who would have guessed that something so "obvious"
is apparently so hard to prove?  (I don't know how hard it would be to
prove euxfr directly, though.)

(17-Nov-04) Takeuti/Zaring have a rather long proof of ac6s2 that I
wasn't looking forward to formalizing.  Then it occurred to me that I
already have ac6s, which is proved rather simply from the AC variation
ac6 plus the Boundedness Axiom bnd2.  The final proof of ac6s2 is very
simple.

(11-Nov-04) uzwo generalizes nnwo, whose proof has been shortened to
become a special case of uzwo.

Interesting tidbit:  I did a quick count, and there are now 5177
theorems (including Hilbert space) in set.mm, and 1480 of them reference
969 bibliographic entries (theorems, exercises, etc.) on the
mmbiblio.html Bibliographic Cross Reference page.  (This is more
bibliographic entries than I would have guessed offhand.)

(10-Nov-04) mapunen, with its long, tortuous proof, completes (whew) the
"basic" equinumerosity theorems needed for the 3 operations (addition,
multiplication, exponentiation) of cardinal arithmetic.

(5-Nov-04) syli is a little "discovery" that shortens 14 proofs, more
than paying for itself by reducing the total size of the database.

(4-Nov-04) I think Takeuti/Zaring's definition of omega, dfom2, is
rather strange and unintuitive.  But this proof confirms it is correct.

(2-Nov-04) For infcntss, Takeuti/Zaring write after their Exercise 8 for
this:  "Hint:  Use AC".  Our antecedent expresses "is infinite" in a
different way that allow us to avoid AC.  (We also don't need the Axiom
of Infinity because the antecedent is false if omega doesn't exist.
This is a fortuitous quirk of our definition of dominance, and a
different definition of dominance might require Infinity for this
theorem.)

(1-Nov-04) qexpclt nicely illustrates the use of our general-purpose
exponentiation closure lemma expcllem.

(29-Oct-04) nominpos was written specifically to refute an incorrect
statement on Usenet:
http://groups.google.com/groups?selm=XY-dnRTa4ISmyhzcRVn-vw%40rcn.net

A somewhat significant change was put into the site today.  The old
object "ded", used for the weak deduction theorem, has been renamed to
"if", and it is now called the "conditional operator" instead of the
"deduction class".  This is more in keeping with its true function,
especially since it is being used more and more for
non-deduction-theorem things.  df-ded was renamed to df-if, along with
some other renaming.  The Deduction Theorem page was revised to reflect
this change.  You may want to look at the quick example at the beginning
of the Deduction Theorem page mmdeduction.html to see if it makes more
sense now.

(25-Oct-04) I forgot to mention that on 19-Oct I added onpwsuc
specifically to answer this Usenet question:
http://groups.google.com/groups?threadm=Ts6dnc3uTY-bLejcRVn-qQ%40rcn.net

(23-Oct-04) infdif was quite frustrating, particularly because Enderton
mentions it in passing almost as an obvious, trivial fact.  His one-line
proof gives no hint of the difficulties lurking underneath.  I could see
no shorter way to prove it.

(22-Oct-04) The new, elegant df-exp, that makes use of our new infinite
sequence builder df-seq, now completely replaces the old awkward one.

Decisions, decisions.  I don't like that the new df-exp requires the
artificial complex number axiom axmulex.  I might redefine it so that in
place of "( x. seq ( NN X. { x } ) )" it would use
"( ( x. |` ( CC X. CC ) ) seq ( NN X. { x } ) )", so that the sethood
of multiplication "x." becomes irrelevant.  But that complexifies
df-exp, and I'm also thinking about future applications of df-seq.
So I'm not sure what I'll do.

(20-Oct-04) By using the Axiom of Choice, qnnen has the shortest proof
I'm aware of.  The standard proof uses somewhat complex (from Metamath's
point of view) algebraic manipulations.  Of course most authors want to
avoid AC when possible, but I thought it interesting to show how it can
be shortened with AC.

(18-Oct-04) I finally completed what turned out to be a bigger project
that I expected, which is to prove that the reals are uncountable.
Today's proof ruc is the culmination of several days' work.  It actually
involves 39 lemmas, ruclem1 through ruclem39.

(I artificially backdated the lemmas ruclem1 through ruclem39 so they
won't clutter up the "most recent proofs" page.)

See https://us.metamath.org/mpegif/mmcomplex.html#uncountable for a
detailed description.

(17-Oct-04) I think df-seq is going to be extremely useful in the
future.  For a long time I've been stuggling with a good way to
represent finite and infinites sequences, series, products, etc. which
are universally prevalent in the study of limits, calculus, and pretty
much all of higher analysis.  Each one seemed to require its own messy
development - see our ugly df-exp for example, and that's just the tip
of the iceberg.  Well, df-seq seems to be the answer!  For example,
df-exp can be alternately defined so its value becomes (in set.mm
notation):

  ( A ^ B ) = ( ( x. seq ( NN X. { A } ) ) ` B )

where NN X. { A } is the constant function with value A. The seq
operation produces the sequence A, A x. A, (A x. A) x. A,... that we
evaluate at B. How much simpler could it get?  Eventually I'll redefine
df-exp accordingly.  df-seq will even be useful for showing that real
numbers can be represented by an infinite decimal expansion (a mundane
thing in everyday mathematics but quite difficult formally).  So expect
to see a lot more of the seq object!

Fortunately you don't have to be concerned with df-seq itself (with its
horrendously complex definition) but only with its consequences seq1 and
seqsuc, which should be enough to handle most work with sequences and
series.  In fact the only purpose of df-seq is to provide us with an
object that has those simple properties.

I defined seq to start a 1, not 0, to make it compatible with our
df-clim.  A long time ago I initially defined df-clim to start at 0
but ran into messy problems avoiding divide-by-zero when actually
computing limits, and 1 makes certain things a lot simpler.  This is
possibly one reason most analysts have the natural numbers start at 1.
Anyway, if the need for starting at 0 or some other integer arises, I
can redefine a more general df-seq, but for now I wanted to keep it as
simple as possible.

Internally, df-seq uses an ordered pair <counter, value> to increment a
counter that is used to look up the value of the input sequence, so in a
way it is almost like it has a little "computer program" inside.  Unlike
a computer program, though, the "counter" counts to infinity!  The
incredible power of this concept almost boggles the mind.

By the way, with regard to "computer program like" devices, it is
becoming clear that the df-ded object, now used almost exclusively for
the weak deduction theorem, is in fact a far more applicable kind of
general-purpose "if statement".  Raph Levien pointed this out to me, and
I've already used it as such in unxpdomlem.  ded(phi,A,B) means "if phi
is true, return A, otherwise return B" and is almost exactly analogous
to the C-language conditional operator "condition ? A : B".  It can be
useful for defining functions such as

            /
           |    1 / x,  if x =/= 0
  f(x) =  <
           |    0, otherwise
            \

and if you look at Takeuti/Zaring's proof of unxpdomlem you'll see that
this is exactly what I am doing, with nested ded's to represent 3
possible alternatives.  Maybe I should rename "ded" to be "sel" for
selector or even "if".  What do you think?

The "seq", "ded", and "rec" objects in set.mm apparently do not appear
in the literature, so there is no standard notation for them, which I
think is unfortunate.

(12-Oct-04) Prof. Nambiar communicated his paper to me earlier this
year.  gch-kn shows that the Generalized Continuum Hypothesis is
equivalent to a generalized version of his Axiom of Combinatorial Sets,
from which the result in his paper falls out as the special case of A=0.

(10-Oct-04) ssenen doesn't seem to appear in any book but is apparently
assumed implicitly in the proof in Lemma 6.2 of Jech's _Set Theory_ p.
43 (otherwise I can't see how that proof could work).  At first I
thought the proof of ssenen would be trivial - it seems intuitive enough
on the surface.  But I soon discovered it's a rather difficult thing to
prove, and I hope you like the proof I came up with.

(9-Oct-04) The theorem cdaval has a comment explaining how we do
cardinal arithmetic, which you may find confusing.  The only new
operation we actually introduce is +c (disjoint union), and we re-use
existing set operations for the others.  Even +c is not really an
operation from cardinal numbers to cardinal numbers, but just a general
set operation.  Here is the mapping:

  +c is used to represent cardinal number addition
  cross product is used to represent cardinal number multiplication
  set exponentiation is used to represent cardinal number exponentiation
  equinumerosity is used to represent cardinal number equality

With this mapping, we can easily obtain the "real" cardinal arithmetic
operations, in the sense of operations that take actual cardinal numbers
and produce new actual cardinal numbers.  To get the actual cardinal
number (which is the smallest ordinal equinumerous to it) corresponding
to any set, we use the "card" function.  A cardinal number is its own
cardinal number as shown by cardid and cardcard.

A cardinal number is nothing more than a measure of the "size" of a set.
For finite sets, it is easy to see how the set operations correspond.
For example, the cross product of a set with 3 elements and a set with
5 elements will result in a set with 15 ordered pairs, corresponding to
3 x 5 = 15.

Our approach is exactly the approach used by Mendelson, and it saves us
having to have a whole bunch of new operations and theorems on them that
would essentially just be window dressing disguising what we already
have.

(5-Oct-04) infxpidm is a deep and important theorem that is typically
one of the most difficult in elementary set theory textbooks, and some
books don't even get that far (or omit the proof).  The only other
formal proof of this I'm aware of is on the Isabelle prover.  About half
of the paper "Mechanizing set theory:  cardinal arithmetic and the axiom
of choice" http://www.cl.cam.ac.uk/users/lcp/papers/Sets/AC.pdf is
devoted to the Isabelle proof of infxpidm.  Of course that was in 1996,
so Metamath is about 8 years behind the times.:)

(2-Oct-04) Some important milestones today!

In our development there are two completely different sets, omega (a
subset of ordinal numbers) and N (a subset of complex numbers), that are
both called "natural numbers," undoubtedly causing a great deal of
confusion.  The former are the natural numbers of set theorists, the
latter the natural numbers of analysts, and both sets satisfy Peano's
axioms.  nnenom relates these two sets, that live in completely different
worlds, by showing that they are equinumerous.

xpomen is a milestone theorem showing the cross product of omega with
itself is countable, and will be used as a key stepping stone for the
multiplication of infinite cardinal numbers.

xpomen is derived directly from xpnnen with nnenom.  xpnnen in turn is
essentially dependent on Raph Levien's nn0opth.  It is pleasant to see
everything finally all fits together, with one thing building on
another.

(1-Oct-04) Right now, the proof of sucdom uses some heavy-duty AC
(entri2) and Infinity (omex, nnsdom) stuff for such a trivial theorem
(that is casually used, without mention of how it's proved, in
textbooks).  I could not for the life of me figure out how to prove it
without AC and Infinity.  Does anyone know?

(30-Sep-04) As unrelated as they may seem, these are all preliminaries
towards getting into transfinite cardinal arithmetic.  dedex was a
surprise:  I expected a somewhat involved proof, and behold! it is just
a special case of keepel.

(29-Sep-04) An interesting curiosity is that xp2cda is an actual
equality, not just an equinumerosity relation.

I added a note to yesterday's cdaval, explaining our approach to
cardinal arithmetic.

(28-Sep-04) Today, with cdacomen, we take the first baby step in what
promises to be a fascinating journey into the deeply profound world of
cardinal arithmetic, where we will uncover the mysterious properties the
objects in "Cantor's paradise" (transfinite cardinal numbers).  The new
definition added is df-cda.

Some books define cardinal arithmetic operations explicitly, whereas
other books use cardinality and equinumerosity applied to ordinary set
operations.  We will use Mendelson's hybrid approach, where cardinal sum
is defined as a kind of "disjoint union" operation, and where cross
product and set exponentiation serve the roles of product and powers.
So, equinumerosity will mean "equals", and dominance will mean "less
than or equal to".

(27-Sep-04) I changed the turnstile |- color from green to gray.  The
color had no purpose other than to make it stand out, and with so many
other colors now (especially with the new rainbow-colored little
numbers) it seems like a gratuitous and pointless use of color.  In the
old days I suppose it added a splash of color to an otherwise dreary
page, but now it just makes the page busy.  In addition, it is
inconsistent since color otherwise distinguishes variables from
constants (and |- is a constant).  I hope the gray is a sufficient
distinction to impress the reader that it is a meta symbol that somehow
transcends the "normal" math.  The indentation indicator next to it is
also gray, but I don't think it will cause confusion.

It seems another proof system called "DC Proof" is comparing itself to
Metamath.  I suppose I should take that as flattery.  :)
http://groups.google.com/groups?selm=OQh5d.2688%24KF.21330%40tor-nn1.netcom.ca

(24-Sep-04) Our proof of undom is much simpler (when formalized) than
Mendelson's, who defines a complicated one-to-one function from the
left-hand to right-hand side of the dominance relation in the
conclusion.  Basically we exploit unen, domen, and ssdom2g, avoiding any
use of functions at all.

(17-Sep-04) dmsnsnsn allows us to complete the "theory" of domains of
iterated singletons of the empty set 0.  Thus:

    dom 0 = 0
    dom {0} = 0
    dom {{0}} = 0
    dom {{{0}}} = {0}
    dom {{{{0}}}} = {{0}}
    dom {{{{{0}}}}} = {{{0}}   etc.

The first 3 are dm0, dmsn0, dmsnsn0.  By the way this has absolutely
no practical value whatsoever. :)

Historical trivia:  The singleton {0} reminded me of the theorem pwpw0,
which is considered too obvious even to mention as a separate theorem in
most books.  Suppes leaves its proof as an exercise.  Even when a book
gives a "proof" it goes something like "{0} has 2 possible subsets, 0
and {0}, so its power set is {0,{0}}."  In the early days of Metamath I
was completely baffled how to formalize this and spent several
frustrating days with this theorem.  I finally stumbled upon exintr,
which I don't think is in any book, as the key to its proof.  As a kind
of cynical joke, in its description I said (as many books do) that we
"compute" its power set, although the proof seems to shed little light
on what the algorithm might be. :)

(16-Sep-04) The key theorem for qaddclt is divadddivt at step 26, which
allows us to prove that the sum of two ratios of integers is a ratio of
integers.  The other stuff is the messy overhead, made more so by having
to avoid dividing by zero and having to work with N, Z, Q, and C in the
same proof.

(14-Sep-04) I like abrexex2.  Simple to state yet intrinsically quite
deep and powerful.  The Axiom of Replacement plays an important role in
its proof, and this theorem might be equivalent to it (although I'm not
sure).  Note that it is essentially derived from iunex, which in turn
derived from abrexex and the Axiom of Union.  On the other hand, abrexex
can be easily recovered from abrexex2 by replacing phi with y=B, since
the 2nd hypothesis {y|y=B} always exists (it is either a singleton, or
the empty set when B is a proper class).  Wow!  I wonder if this has
ever been published.

(13-Sep-04) I think uni0b would make a nice textbook exercise.  By the
way sssn can be used to show that A in uni0b is either the empty set or
the singleton of the empty set.  sssn is another nice little theorem that
seems obvious but whose proof is not quite as simple as one might think.

(12-Sep-04) zltle1 is one of those unfuriatingly long proofs of
something seemingly extremely trivial.  This is in spite of the fact we
already have nnlelt1 (with an insanely long proof that has so far defied
attempts to shorten it), on which zltle1 is ultimately based.

elab2 and elab2g replace the venerable vtoclab and vtoclabg,
generalizing them so as not to require that x and B be distinct
variables.  This is important in cases where B is a class variable in a
hypothesis, usually as part of an intermediate lemma, that will
eventually be eliminated with cleqid.  By removing this restriction I
was able to shorten a couple of additional proofs that previously used
elab.

(7-Sep-04) Our proof of Abian's "A most fundamental fixed point theorem"
(abianfp) is, as far as I know, the first credible validation of this
interesting and not-so-trivial theorem.  In his last years, Abian
espoused some rather unconventional theories on Usenet, causing him to
be considered a crank and not taken seriously.  The following Usenet
post appears to be the original one announcing this theorem:
http://groups.google.com/groups?selm=68v5ij%24bar%241%40news.iastate.edu&output=gplain
Abian claimed for this theorem, "The fundamental significance of the
Theorem lies in the fact that a great many fixed point theorems can be
reduced to the special cases of the Theorem."  After some effort I
located a post where Abian seems to have proved Tarski's classical Fixed
Point Theorem as a corollary to this one:
http://groups.google.com/groups?selm=6972ve%24t96%241%40news.iastate.edu&output=gplain
This seems correct (although I need to work out the details to be
completely convinced).  If so, I think this is significant, but
apparently no one on Usenet took it seriously nor indicated they
understood it.

iunconst, fnresdm, and abianfplem are prerequisites for abianfp.

(6-Sep-04) tfinds2 should produce shorter proofs than tfinds for some
problems (analogous to finds vs. finds2).  I derived it from tfindes to
show it can be done (i.e. going from explicit back to implicit
substitution), but it required the new (and somewhat unusual) theorems
sbralie (what a brain-teaser), sbcco2 (allows us to obtain
[ suc x / x ] phi where suc x and x share a free variable), and sbcie
added today.  An interesting exercise, and the technique should be
useful in the future.

(2-Sep-04) dedlem1 and dedlem2 are not new but revisions of older
versions.  The "ded" operation can be useful for more than just the weak
deduction theorem - as these lemmas show, it acts as a "selector" from
two classes depending on the truth or falsity of a wff.

(31-Aug-04) undm may be the shortest proof in the database.  ("V \ A"
means the "complement of A", and most authors introduce a symbol
abbreviating this expression.  We need it so rarely that I decided not
to define it separately, at least not yet.)

Mr. O'Cat provides us with a shorter proof for ja.

difun was used to shorten the proof of dif23 of Aug 27.

(30-Aug-04) Yesterday's cfub is used to simplify the proof of cf0.

(29-Aug-04) A neat thing about snsspr is that neither A nor B have to be
sets (which we accomplish with the use of snprc in its proof).  I was
actually surprised that I didn't have this in the database, and was able
to shorten a couple of proofs with it (see the "referenced by" list).

cfub is (for me) an important breakthrough that will let us finally get
some cofinality stuff proved efficiently.  I haven't seen this in a
textbook.

(28-Aug-04) fconstfv is tricky - the cases of A = empty and A =
non-empty must be proved separately.  The reason is the non-emptiness
requirement of r19.9rzv in step 23.  This is one of those little
surprises you don't think of until you actually try to prove it.  But
it's neat the theorem still holds when A is empty.  It's also neat the
theorem still works when B is a proper class, and it is a pleasant
coincidence that the theorem still works when generalized to these
cases.

By contrast, fconst2 (without its hypothesis) fails when B is a proper
class.  Specifically, if F is empty and A is non-empty, the left-hand
side will be false but the right-hand side will be true.  See snprc for
what happens to a singleton of a proper class, and see f00 for what
happens when a mapping has an empty codomain.

(26-Aug-04) In dminss, R"A is the image of R under A - it "maps" a
subset of the domain to a subset of the range.  Taking the converse
image of this, we go back and get something at least as big as the
original subset of the domain.  Example:  consider the constant
function; the converse image of any (non-empty) subset of the range
(i.e. the whole of the range, which is a singleton) will be the entire
domain.  And this of course is at least as big as the original subset of
the domain.  Suppes calls this "somewhat surprising" but doesn't say
why.

(21-Aug-04) I saw iuniin for the first time on the web page
http://en.wikipedia.org/wiki/Union_(set_theory) (go to bottom of the
page) and thought it was neat, so here is the proof.  Does anyone
know what book it comes from?

(20-Aug-04) A long time ago when I was first learning logic, I was so
enamored with dfor2 that I wrote it on an index card for my "curious
math facts" collection.  I can't remember if I read it somewhere or
stumbled across it on my own.

At one point I thought looinv was intuitionistic and sent Mr. O'Cat on a
wild goose chase trying to prove it from ax-1 and ax-2.  It turns out
it's as nonintuitionistic as it could possibly be, being equivalent in
fact to Peirce's axiom (theorem peirce).

(17-Aug-04) ac6s had me stumped for a long time.  It seemed reasonable
to me that there was no reason to require that B be a set in ac6, but
proving it was a different matter.  It now seems we need the very deep
and strong Boundedness Axiom (whose proof is the culmination of our
development of rank and the cumulative hierarchy of sets).  The proof of
ac6s is a nice example of an application of the Boundedness Axiom.

(16-Aug-04) Using a technique inspired by Mel O'Cat, which is just
deleting the whole proof then letting the metamath program try to prove
it using "improve all/depth n" for n=1,2,3,4 in succession, I found
shorter proofs for the following 43 theorems:

  pm2.21i pm2.21d pm2.18 pm2.65 orc jctil jctir imp42 imp44 imp45 adantl
  adantr adantld anidm anidms ancom imdistanri abai anabs1 anabs7 anabsi5
  anabsi8 anabss1 anabss3 anabss4 anabss5 anandis anandirs ibi bilimd iba
  ibar 3impdi 3impdir pm5.1 baibr biantr niabn ninba ax6 19.33 euorv
  exmoeu2

I tried this on the first 984 theorems in the database (everything up to
set theory).  ("improve all" was not designed to find proofs, but just
to fill out syntax constructions.)  Out of these, "improve all" was able
to find proofs for 237 of them!  134 of these were identical to the
existing proofs found by hand, 60 were longer, and the 43 above were
shorter.

(16-Aug-04) zorn2 is a simpler version of Zorn's Lemma than zorn, so I
made it the "official" Zorn's lemma on the Theorem Sampler of the
Metamath Proof Explorer Home Page.

(15-Aug-04) qlax* are the new theorems that directly correspond to the
Quantum Logic Explorer axioms.

Most of today's non-Hilbert-space theorems are just slight modifications
to older ones for better overall consistency.  Sometimes the older ones
are still there if you click the "Next" link, until I slowly weed them
out of the theorems that use them.

If you have any comments on the new "Theorem list" links let me know.
For example, is the upper right corner the best place to put it?  Do you
find it useful?

(14-Aug-04) Mr. O'Cat comes through for us again with loolin.

Well, well, it looks like _both_ directions of reluni hold!  Neat.  Only
the reverse direction is given in Takeuti/Zaring's exercise (and in
set.mm since 1994, in the older version of reluni, which was based on
the excercise).  I wonder if they knew that both directions hold.

(13-Aug-04) fh1 is (one-half of) the very famous Foulis-Holland theorem
of orthomodular lattices, which was proved and published independently
by Foulis and Holland at almost exactly the same time.  I'm not sure who
really came first but both are always credited.  This proof is identical
in structure to the Quantum Logic Explorer version
https://us.metamath.org/qlegif/fh1.html which you may wish to
compare it to, and in fact I "borrowed" it from there.  You can see why
the Quantum Logic Explorer is simpler to work with for these kinds of
things:  we don't need "member of CH" hypotheses, and we don't have to
keep proving operation closure over and over.  I put fh1 in the Hilbert
Space Explorer also because we will later need it for deeper Hilbert
space things that the Quantum Logic Explorer axioms can't do (e.g.
involving quantification).

A bit of trivia regarding this theorem:  I used this theorem to
win a $2.00 bet with physicist John Baez, who was unaware of it.  See
http://groups.google.com/groups?selm=MaVe6.4799%24JG.614310%40news.shore.net

(12-Aug-04) Compare the proofs of un00 and chj00 - they're very similar!

(8-Aug-04) The last half of my talk at the Argonne workshop
http://www-unix.mcs.anl.gov/~mccune/award-2004/ is related to theorem
mdsym (of 2-Aug-04).  (As you can see, work on this site continued from
my hotel room thanks to Linux and ssh...:)

(4-Aug-04) Most of today's theorems prove the "theorem" form of the
result rather than the "inference" form of the result, and most steps do
nothing more that manipulate antecedents with propositional calculus.
Their proof displays give us excellent examples of the usefulness of the
little colored numbers.

Look, for example, at div23t.  At first it seems to be a formidable,
hopelessly complicated mess.  But if you ignore the steps with
red/orange numbers, there are only 5 steps left that have blue numbers.
Now, if you look at only those steps and ignore the rest, you can pretty
much see "in your head" how to connect the equalities to obtain an
informal proof.  Doesn't it make more sense this way?

The other way to prove the "theorem" form is to use the weak deduction
theorm dedth and its variants.  But as the number of antecedents grows,
so does the size of the proof using the weak deduction theorem.
Depending on the proof, sometimes you'll get a shorter proof proving it
directly, like with today's.  Sometimes I would prove it both ways and
pick whichever way was shortest (in terms of size of the compressed
proof in set.mm).

(3-Aug-04) expcllem shortens the proof of reexpclt (closure of natural
number exponentiation of reals) and will later be reused for integers,
rationals, etc.

(2-Aug-04) mdsym (M-symmetry in Hilbert space) is the final goal of
the recent Hilbert space work.  I will be giving a talk on this
remarkable theorem Aug. 5-7 at the Automated Reasoning workshop
http://www.mcs.anl.gov/~mccune/award-2004/ and I can now claim that I
understand its proof in complete detail.

(1-Aug-04) halfnz is a curiously long proof for such a simple fact.
If anyone sees a way to shorten it let me know.  On the other hand
it may illustrate the implicit complexity underlying even trivial
arithmetic facts (even if we start with the axioms for arithmetic).

(31-Jul-04) 2cn is a long-overdue theorem that shortens the proof of the
venerable 2p2e4 (2+2=4) as well as a couple of dozen other theorems.

(28-Jul-04) oncardid and oncardon are used to reprove cardnn so that it
doesn't use the Axiom of Choice.  I didn't like that overkill for
something as basic as cardnn.  I also reproved cardom so it also doesn't
need the Axiom of Choice.

(26-Jul-04) dom2d will help us prove some equinumerosity dominance
theorems more easily, i.e. without directly exhibiting the 1-1 function
required by brdom.

(25-Jul-04) nthruz is the 4000th theorem added to the non-Hilbert-space
part of set.mm.  (Hilbert space adds another 600 or so.)  I'll have to
update the summaries that say the Metamath Proof Explorer has "over 3000
theorems" to say "over 4000 theorems", once the 4001st is added.  The
date stamp was added to the metamath program on 5-Aug-93, which is the
earliest date in set.mm, at which point there were 450 theorems covering
propositional calculus, predicate calculus, and the beginnings of set
theory (subsets, union, intersection, abstraction classes, unordered and
ordered pairs).

(24-Jul-04) nn0enom provides an important link between between two very
different theories - complex numbers (nonnegative integer subset) and
ordinal numbers (finite subset).  With it, we can exploit our many
complex number theorems to get some equinumerosity results that will
ultimately be used for cardinal arithmetic.  I guess it's kind of
indirect and ugly but it will save a lot of work.  (And formally it's
absolutely rigorous of course.)  Actually we'll need the reals anyway to
study the continuum so maybe it's not that ugly.

(23-Jul-04) bnd2, like bnd, is one of those "obvious" things that is
actually quite deep.  The proof (if you trace it back) ties together
key results about ranks, the Axiom of Regularity, and the Axiom of
Infinity.

It was quite a journey to get atom1d - whereas Beltrametti/Cassinelli
merely say "The one-dimensional subspaces of H are obviously the atoms
of C(H)."  I guess this is the difference between formal and informal
math.

(22-Jul-04) ranksn, rankuni, rankuniss are some interesting little facts
about rank.  Most textbooks seem to have them, so why not.  (But we
don't have a use for them yet.)

(21-Jul-04) I am torn about the notation "Mod" for modular pair.  Maeda
uses (A,B)M for our A Mod B, but a plain "M" just seems too unspecific
in a general set theory environment.  On the other hand "Mod" suggests
the arithmetic modulo, but I'll use "mod" (lower case) for that when we
start to need that.  But I still may change A Mod B to something else
after I think about it - it is nonstandard and I dislike nonstandard
notation in set.mm unless it is unavoidable.

(18-Jul-04 - 19-Jul-04) isfinite2 does not require the Axiom of Infinity
for its proof, whereas the stronger isfinite does require it.

(14-Jul-04) unbnn will eventually give us a proof that a set strictly
dominated by omega is finite, without invoking the axiom of infinity.
(This is the hard part of the proof.)  See Suppes Th. 42 p. 151 for more
information.  unbnn is also a useful general result in its own right; for
example it will become useful for proving certain cofinality results.

(12-Jul-04 - 13-Jul-04) Cardinal exponentiation, when we define it in
the future, will be the cardinality of set exponentiation. mapen and
xpmapen provide two fundamental theorems used to establish properties of
cardinal exponentiation.  (xpmapen was tough to formalize - ugh - I
wonder if there is an easier way to do it.)

(11-Jul-04) canth3 gives us an unlimited supply of ever-bigger infinite
cardinal numbers.

(10-Jul-04) cardnn gives us a very convenient property of finite
cardinal numbers:  the natural numbers and the finite cardinal numbers
are the same thing!  The alternate definition of cardinals given by
karden (using the Axiom of Regularity instead of the Axiom of Choice)
does not have this property, which is one of their disadvantages.  Once
I worked out the first few finite cardinals from karden (it is an
exercise in Enderton) and they become very messy and complicated very
quickly.

(9-Jul-04) sucprcreg is probably not useful for anything, but it was
fun.  (We already have sucprc that sometimes helps prove more "general"
theorems where A does not have to be a set.  And I like to avoid
theorems requiring the Axiom of Regularity whenever possible.)

(8-Jul-04) aleph1 is an important theorem piecing together some recent
results (alephnbtwn2, canth2, pw2en).  The cardinality of reals is
2^aleph0 (which hopefully we will prove here someday), and in the
absence of the Continuum Hypothesis this theorem is the best we can do -
the reals are at least as numerous as aleph-one, but possibly more so.

(7-Jul-04) I had been wanting to prove spansncv for a long time but was
unable to (it is an exercise in Kalmbach with no answer given).  Prof.
Eric Schechter (http://www.math.vanderbilt.edu/~schectex/) finally
provided me with the missing piece, which is the theorem spansnj.

(6-Jul-04) The proof of pw2en seems much longer than it should be.
I'll have to revisit it someday.

(1-Jul-04) ssext is interesting because A and B may be proper classes,
yet we need to compare only their (non-proper-class) subsets to
determine whether they are equal.

nssinpss is one of those neat little "it ought to be a theorem or
example or exercise in a book" things, but apparently it isn't.  (Any of
our theorems that omits a bibliographical reference doesn't appear in
any book that I'm aware of.  If you find one that does let me know!)

(30-Jun-04) elsuc vs. sucel - We have many theorems expanding membership
of a class in a defined object, such as elsuc, but hardly any expanding
membership of a defined object in a class.  I'm not sure why that is,
but the need just doesn't seem to arise that often.  Today I added two,
0el and sucel, and used them to shorten a couple of existing proofs.
These actually seem to be the only such theorems in the database - I
looked through set.mm for others, but I couldn't find any others of the
form "(defined object) in (class variable) <-> ..." whose right-hand
side didn't reference the defined object.  On the other hand there are
probably over 100 theorems showing membership _in_ defined objects, like
elsuc.  Strange.  By the way there is no "membership in" version of 0el
in our database but rather we use the simpler noel.

(29-Jun-04) I used to be in awe of the fact that Cantor's theorem failed
in NF.  With ncanth it now seems kind of obvious why it would fail,
because V is a set in NF.  (The actual proof in NF is somewhat different
since the axioms are different, but the idea is the same.)  Anyway
ncanth takes away some of the mystery, which is good, but unfortunately
it also means I'm not as awed as I used to be.  I guess a little mystery
can sometimes make things more appealing.

(28-Jun-04) xpsnen shows another use of our friend en2 of 25-Jun.  The
proof makes essential use of the interesting op1st that extracts the
first member of an order pair.

eqssd was used to shorten 11 proofs that previously used eqss, jca, and
sylibrd.  The number of bytes trimmed off of their compressed proofs
exceeded the number of bytes in eqss, its comment, and its proof, so the
database size SHRUNK when eqssd was ADDED!  I guess that's what you'd
call a positive ROI.  :)

spanun and spansn are two important theorems of Hilbert space.  spanun
allows us to work in set theory to take the union of two sets of
vectors, then take the span to get their subspace sum.  This can be a
very powerful simplification over working directly with subspace sum.
spansn equates two very different ways of expressing a one-dimensional
subspace.  (Familiar example:  if the Hilbert space is the ordinary
3-dimensional RxRxR of real numbers, a one-dimensional subspace is a
line that goes through the origin.  Any non-zero vector A on the line
essentially specifies the line.  To get the line we take the span of the
singleton of A.  Alternately, we take the orthocomplement of the
singleton of A, which becomes the plane (through the origin)
perpendicular to A.  Then the second orthocomplement recovers the line
from that plane.  By the way there are 4 kinds of objects in the Hilbert
lattice of R^3:  the point at the origin, lines, planes, and the
entirety of R^3.  These are such different things, like apples and
oranges, that it seems counterintuitive that they could all be "like"
each other in the sense of being the elements of a lattice.  But it is
often useful to think of the R^3 example when trying to understand
Hilbert lattice theorems intuitively.)

(27-Jun-04) Normally rank is defined in terms of R1 (the cumulative
hierarchy of sets) - see df-rank.  Monk takes the opposite approach - he
defines rank first, then defines R1.  Here we prove, with r1val2, Monk's
definition as a theorem of our approach.  R1 and rank are in a sense
equivalent concepts, although very different in their approach and uses,
analogous to say the time domain vs. frequency domain representations of
a signal in Fourier analysis:  either representation can be recovered
from the other.  And each representation can have advantages over the
other depending on the problem.

dfom2 is a traditional way of defining the set omega of natural numbers
(as a subset of the ordinal numbers, not the natural number subset of
the reals) that is used in many textbooks.  However, dfom2 requires the
Axiom of Infinity in order to be valid, whereas our df-om does not.  (If
you look under the proof of dfom2, you'll see that ax-inf is used, and
in fact it is needed to ensure that at least one set x in the class
abstraction exists.)  This means that in these books, all theorems about
natural numbers presuppose the the Axiom of Infinity.  The reason they
do this is that it makes their development easier.  Our philosophy is to
avoid the Axiom of Infinity when it is not necessary, so our development
starting with df-om is quite a bit more complex than in these books.
(Takeuti/Zaring's book takes our approach.  However they take a shortcut
by using the Axiom of Regularity, which is still undesireable.  By
carefully reworking their proofs, I was able to avoid both Infinity and
Regularity in our development of natural numbers.  For example, if you
look at say nnacom you'll see infinity is not used.  T/Z also give a
somewhat confusing definition of omega.  Our equivalent but simpler
definition df-om doesn't seem to occur in the literature, although it is
related to Bell/Machover's natural number predicate.)

(26-Jun-04) map1 is our first theorem using en2.  See how easy the proof
is - no messy one-to-one onto function stuff!  (There is a function that
comes from elmap, but that has to do with the definition of set
exponentiation, and we quickly get rid of it with fconst2.)

In Hilbert space I'm building up a collection of basic theorems that
will be needed to develop the theory of atoms.

(25-Jun-04) I think en2 is quite neat - it is not immediately obvious
(to me at least) that you can conclude equinumerosity from the
innocent-looking hypotheses.  But look at its simple, beautiful proof:
the function stuff magically appears then vanishes!  I don't think en2
is published anywhere.  It should let us arrive at some equinumerosity
proofs much more quickly.

(24-Jun-04) omsmo is one of those "intuitively obvious" or "trivial"
facts that textbooks never bother to prove.  Unfortunately for us we
must actually prove it to use it since Metamath will not let us overlook
such minor details. :)

(20-Jun-04) I haven't had any luck shortening zind.  I am tired of
working on it and will put it aside for a while.

Today's theorems finish up the list of single-digit operations, where
the result is a single digit.  As you probably have noticed we don't yet
have a standard way of expressing multiple digit numbers like they are
normally written down.  I used to have a definition for it but it was
clumsy and I took it out.  The interesting thing is that in abstract
math there is rarely a need for numbers larger than say 4 (which is the
largest number I have used so far outside of the list that we completed
today).  Glancing through the proof of Fermat's Last Theorem - one of
the most difficult proofs ever - I don't think I saw a number larger
than 12.  So I'll probably postpone multiple-digit numbers until it is
really needed.  And even so we can represent them with expressions of
single-digit numbers; for example 42 can be expressed as 6 times 7.  Now
that is an interesing problem:  find the shortest representation of all
numbers from say 1 to 1000 in terms of operations on single digits
(addition, subtraction, multiplication, division, and exponentiation).
Any takers?

Raph will have multiple-digit numbers in Ghilbert.

(18-Jun-04) At first zind looked trivial.  After all, we already have
induction on natural numbers (ind at step 290), and this theorem is just
"shifting" it to start at a different number.  But as I got deeper into
it more and more unpleasant surprises kept popping up.  After hours of
tedious work I ended up with one of the biggest and ugliest proofs in
set.mm that I see no way to simplify, and there are no "neat" lemmas
that stand out (if it were broken up it would only be for the purpose of
breaking it up and would not make the proof more efficient).  It doesn't
make any sense why it should be so huge and it makes me depressed.  I'll
have to look at it again after I recover.:)

(17-Jun-04) See isomin for an explanation of Takeuti and Zaring's
initial segment idiom in isoini.  This theorem, though cryptic, is
copied symbol for symbol from T&Z.

oprprc1 is philosophically ugly but will allow us to "cheat" and shorten
a few proofs in the future, by not having to worry so much about whether
a class is proper or not.

(16-Jun-04) erthi is the "key" theorem involved in the proof of erdisj.

(14-Jun-04) zrevaddclt basically says, if B is an integer and A is a
complex number and A+B is an integer, then A is an integer.  I think
reverse closure laws are interesting twists on closure laws.

(13-Jun-04) isotr and isotrALT are two different proofs of the same
thing, because I wanted to see which approach is shorter.  Well, one is
shorter in set.mm and the other produces a shorter web page.  So which
do I pick?

f1owe uses the new isomorphism stuff to dramatically simplify its proof.
Compare the old version (that will eventually be deleted), f1oweOBS.

(12-Jun-04) The definitions df-un and df-in have been changed to
be more traditional.  What used to be df-un is now dfun2 and vice
versa; what used to be df-in is now dfin2 and vice versa.  The price
we pay is a dummy variable, but I finally decided that dfun2 and dfin2
are just too unconventional.

The proof of isowe was left as an exercise for the reader in Takeuti and
Zaring, and thankfully I was able to figure it out!  It will provide a
great reduction in size of the enormous proof for f1owe - stay tuned...

The new definition df-cv for covering allows us to have a much simpler
definition for atoms, and df-at is the new version of the defintion.
Theorems ch0psst, atcv0, and atelt are supporting theorems for the new
df-at.

(11-Jun-04) bitr2d, psseq2i, psseq2d shorten some existing proofs.
znegclt, zaddclt, zsubclt will be needed when we get into proofs about
integers.  cvbr, cvbr2 express the basic property of the new df-cv
(covers) definition.  There is much literature on the properties related
to covering in Hilbert space, and these properties are ultimately tied
to the superposition of states (Schroedinger's cat, etc.)

(10-Jun-04) funimass2 arose in a proof in a to-be-published logic book
I'm reviewing.  I thought it was very neat that the domain and range of
F are completely irrevelevant to the result, nor does F have to be
one-to-one or onto.  funimass1 is similar but depends on the range of F.

(9-Jun-04) elnnz1 could be used as a definition for natural numbers if
we started with integers. nn0subt is a kind of closure law for
subtraction of nonnegative integers.  It is kind of surprising that its
proof is so long, given that we already have the almost-identical
nnsubt, but I couldn't find a simpler proof.  An example of shintclt is
that the intersection of two planes (linear subspaces) is a line (a
linear subspace). spanclt shows that the span function "grows" a
arbitrary set of vectors into (the smallest) linear space containing
them.

(8-Jun-04) anadis, anandirs shorten a bunch of proofs previously using
anandi, anandir plus sylbir.  By the way the "s" suffix on anandis,
anandirs means "eliminates a syllogism".  ssel2 is a trivial variant of
ssel but it is needed so often that I decided to put it in - it makes a
bunch of proofs slightly shorter.  The proof of isofr was left as an
exercise for the reader in Takeuti and Zaring - I always worry about
these because figuring out the answers to exercises is often the
hardest part of doing these proofs, and if I can't do it the whole
project comes to a standstill because there can't be any missing
pieces the database.  hvm1negt, hvsubcan1t, hvsubcan2t - some
more elementary Hilbert space elementary results; by the way the suffix
"t" means "theorem form" (vs. "inference form" where the membership
requirements are hypotheses).

(7-Jun-04) If you think isomin is incomprehensibly cryptic, don't blame
me. :)  It is stated exactly as written in Takeuti and Zaring's book,
symbol for symbol.  I did add a note explaining the initial segment
idiom.

I was surprised there's never been a need for something as basic
as unss12 yet.  funimacnv tells us that if A is a subset of the
range, the image of the converse image of A is A - neat and in a
sense intuitive, but its proof took some thought...  elznn0
and elnn0z express some more number class relationships.

(3-Jun-04) df-span is the same as the span of a vector space you learned
about in linear algebra.  It does not require any special properties of
Hilbert space.

(2-Jun-04) onxpdisj may ultimately find use in extending the set of
complex numbers with +infinity, -infinity (for reals) that is useful in
analysis.  Basically, in our definition df-c (and most textbook
definitions) complex numbers are ordered pairs, so per onxpdisj any
ordinal provides a disjoint set that could extend C with artificial
things like +infinity.

(1-Jun-04) sqr9 extends the series sqr0, sqr1, sqr4.
r1pwcl is a new version of the May 30 version that now has Lim B
as an antecedent (instead of a hypothesis).

(31-May-04) h1dle simplifies the proof of hatomic and should be useful
for other things too.  Note:  "atom" has nothing to do with physical
atoms but is just a mathematical term describing a smallest non-zero
element of a lattice, where there is nothing smaller than it except the
zero element.  In the lattice of closed subspaces of Hilbert space, all
atoms are one-dimensional subspaces (I hope to show the proof of this
soon).  To get a one-dimensional subspace, we start with a singleton
containing any non-zero vector, then take the double orthogonal
complement to grow the singleton to a closed subspace (which in this
case would just be the set of all vectors that differ by a scalar
factor, i.e. a one-dimensional "line").  I haven't seen the idiom
_|_ _|_ {B} for a one-dimensional subspace in the literature, but it is
convenient.  (The antecedent of h1dle doesn't require that B be nonzero,
but that's because the theorem also works when B is zero - although
_|_ _|_ {B} is not a 1-dimensional subspace in that case but instead the
zero subspace.)

(30-May-04) Raph contributed 3 really nice results - rankr1a, r1pw,
and r1pwcl - on rank and the cumulative hierarchy of sets (R1).  I
haven't seen any of them in a textbook.

(28-May-04) exp0 and expp1 are the final goals of exponentiation of
complex numbers to nonnegative integer powers.  These effectively
provide the recursive definition found in most textbooks (which
typically do not go through the tedious work of justifying the
recursion).  We could have started with exp0 and expp1 instead of the
direct definition df-exp, but then we would have had to introduce expp1
as an axiom because it is self-referential.  Instead, we want to ensure
direct traceability back to ZF set theory axioms.  From this point
forward we will develop all further properties from exp0 and expp1 only,
never referring back to the development leading up to them (and never
referring directly to df-exp again).  The work we did to develop
recursion on nonnegative integers - nn0rzer and nn0rsuc - will be
reusuable in the future for other recursive definitions.

It turns out that nn0rone and nn0rfnnn0 are not needed.  In particular,
we can prove closure of exponentiation from exp0 and expp1 using
induction, eliminating the need for nn0rfnnn0.  However I'll leave them
in because they may useful in future applications.

(24-May-04) Raph Levien developed the definition of exponentiation of
complex numbers to nonnegative integer powers in Ghilbert and translated
it to Metamath.  (He has a Ghilbert to Metamath translator now.)  I am
adapting it for the official set.mm.  It will replace the current
"dummy" definition df-expOBS that only works for powers of 2.

(18-May-04) None of these function-related theorems appear in any
textbook I'm aware of (if there is no bibliographical reference), but
most will be needed later to prove some theorems (about cardinality
etc.) that do appear in textbooks.  I guess that once set theory
advances beyond a certain point, these kinds of theorems are expected to
be "obvious" to the reader and are not explicitly mentioned.  Some of
them might make nice homework exercises for a set theory class.  I
wonder if any teacher has ever borrowed from our collection for that
purpose.

(15-May-04) I've often thought Takeuti/Zaring's initial segment notation
in today's eliniseg, iniseg, dffr3 is a rather cryptic and convoluted
idiom just to avoid the dummy variable in {y|y<x}.  Maybe it's just me,
but even after seeing this for years I still think "the image of the
converse of the less-than relation under the singleton of x" is harder
to grasp intuitively than "the set of all y less than x".  Nonetheless
they use it frequently in their isomorphism stuff we'll be getting into,
so I yielded and will adopt their approach.

(13-May-04) In case you are wondering - the cryptic theorem names
usually aren't random but typically follow some kind of rough
convention.  For example, today's "cla42gv" has the following breakdown:

  "cl" - uses class variables instead of set variables
  "a4" - related to axiom ax-4
  "2" - the theorem is essentially cla4gv applied twice
  "g" - introduces the hypotheses of the non-"g" version as an antecedent
        ("g" indicates "a more general version")
  "v" - uses distinct variables instead of bound-variable hypotheses

The idiom "a4" above appears a lot, and they are all relatives of axiom
ax-4:

  MM> show labels *a4*
  The assertions that match are shown with statement number, label, and type.
  1971 a4i $p        1979 a4s $p        1983 a4sd $p       2402 a4a $p
  2407 a4c $p        2413 a4c1 $p       2533 sbea4 $p      2534 sbia4 $p
  2535 sbba4 $p      2643 a4b $p        2648 a4b1 $p       2653 a4w $p
  2659 a4w1 $p       3905 ra4 $p        3906 ra4e $p       3907 ra42 $p
  4642 cla4gf $p     4643 cla4egf $p    4649 cla4gv $p     4650 cla4egv $p
  4657 cla4e2gv $p   4658 cla42gv $p    4665 cla4v $p      4666 cla4ev $p
  4673 rcla4v $p     4674 rcla4ev $p    4686 rcla42v $p    4687 rcla42ev $p
  4696 cla4e2v $p    4891 a4sbc $p

A few people have commented that the theorem labels are obscure, and in
general I'm not thrilled with them either.  Every now and then I revise
a few for better uniformity.  But no one has been able to suggest a
better approach that still keeps the names short.  For example, I use
"sbth" instead of "SchroederBernsteinTheorem" because the latter would
be annoying to type while entering a proof, and lists like the above
would not be very compact.  In general, an "English-like"
descriptive label would be have to be very long to be meaningful - how
would you label cla42gv?  I wonder if a breakdown for each label like
the "cla42gv" example above would be useful enough to be worth the
effort of maintaining it.  Maybe this could be done with an annotated
label in each description, something like:

  cl+a4+2:4+g+v

where cl, a4, etc. are looked up in a list and 2:4 refers to the
4th meaning of "2" in that list.


(10-May-04 - 12-May-04) Some miscellaneous stuff and simple arithmetic
facts we'll need later on.  A bold attempt to venture beyond 2+2=4...

Someday of course we'll have to complete the elementary school addition
and multiplication tables.  (At least Raph Levien will do this for his
Ghilbert database, and has ideas for decimal numbers - I used to have
them but took them out because I thought the notation was ugly and I
wasn't happy with it.)  However it is interesting that "higher math"
rarely uses numbers beyond 2.  For example scanning the proof of Fermat's
last theorem http://math.stanford.edu/~lekheng/flt/wiles.pdf the largest
number I see for most of the proof is 4 (although the last paragraph
mentions some larger numbers, but this may be material beyond FLT, I'm
not sure).

(9-May-04) exists2 is another one from the existential uniqueness list I
made several years ago.  I think it is interesting because it shows a
way to "talk" about multiple objects without using set theory.

(7-May-04) rdglem1 was a final loose end that was cleaned up. zfreg* are
cleanups of older versions that didn't use restricted quantification.
Other stuff you don't see behind the scenes:  over 2 dozen theorems were
deleted.  These were obsolete versions have been completely eliminated
from all proofs using them, and there are now no *OBS theorems in
set.mm.

zfregs (the strong version) is curious because the Axiom of Infinity
is apparently needed for its proof (at step 5 of tz9.1 used in its
proof) yet there is no obvious component of Infinity in the result.  No
one, including some prominent mathematicians, has been able to tell me
why or even whether Infinity is _required_ for a proof of this theorem.
All textbook proofs I've seen just implicitly use Infinity (in the form
of a finite recursion) at this step without further comment.  In other
proofs I've been able to avoid the Axiom of Infinity (vs. textbook
proofs) by using transfinite recursion instead - interestingly
transfinite recursion does not require the Axiom of Infinity, whereas
finite recursion (the textbook version) does.  This is because the
latter takes a shortcut by just showing the existence of the necessary
function over omega, without showing what it looks like, whereas the
former (with far more difficulty) must state explicitly the required
function over ordinals because it is proper class and does not exist as
a set.  In set.mm we don't take these shortcuts, and in most cases we
use finite recursion (frfnom, frzer, frsuc) without invoking the Axiom
of Infinity.

(6-May-04) ackm completes the proof of Maes' axiom by establishing
the final link to our Axiom of Choice ax-ac.

(4-May-04) More work on the new transfinite recursion proof.  Done with
all the lemmas now.  By the way, when I revise existing proofs, I
usually suffix the old one with OBS (obsolete), so the old tfrlem1
becomes tfrlem1OBS.  The OBS version will stay until all references to
it are removed.  This way I can ensure that, at any point in time, the
set.mm database as a whole is consistent and complete, even while things
are being modified.

(3-May-04) More work on the new transfinite recursion proof.

(2-May-04) isoid, cflem, and cfval are simple theorems to try out the
new df-iso and df-cf definitions.  The proof of transfinite recursion
(tfrlem*) is being redone in order to be slightly shorter.

(1-May-04) fv2, fv3, tz6.12*, etc. are part of a cleanup of function
value stuff to use the more compact xFy (df-br) in place of <x,y> e. F.

(30-Apr-04) Finally, aceq5.

(28-Apr-04 - 29-Apr-04) We need aceq3 and aceq4 to get the final piece
of aceq5, and they will also fill in part of an AC equivalence series I
have in mind.  (The eventual goal here is to get a link from Mae's AC to
ours.)

(27-Apr-04) I figured I might as well complete the equality/subclass
series with eqsstrd, eqsstr3d, etc.  These are named for their
resemblance to bitr (simple chaining), bitr3 (useful for eliminating
definitions), and bitr4 (useful for introducing definitions).

limelon is cleaned up from an older version and is needed for some
future stuff related to df-cf.

(26-Apr-04) I cleaned up some function value stuff to improve the proofs
leading to the very useful abrexex, which is one of those simple-looking
theorems that hides a lot of power.

After noticing several times the "need" for syl5ss, syl5ssr, syl6ss,
syl6ssr, I finally added them in.  (These, like others in the syl5xx,
syl6xx "family", are named after their resemblance to syl5 and syl6.)  I
was able to shorten over 2 dozen proofs with them, so I was curious what
this actually meant in terms of set.mm file size.  It turns out that a
total of about 300 bytes were trimmed from the (compressed) shortened
proofs, vs. about 1000 additional bytes for the 4 new theorems.  So they
haven't quite "payed" for themselves yet.  Perhaps they will when set.mm
is 3 times as big...

(23-Apr-04 - 24-Apr-04) These are some function and function value
theorems that we will need later.  And a step closer to aceq5...

(17-Apr-04 - 22-Apr-04) I have a list of uniqueness and "most one"
theorems I worked out several years ago, and I decided to add some of
them to the database.  Here is an excerpt from an email I wrote.

  Let us consider ax-1 through ax-16 (plus ax-mp, ax-gen).  We will ignore
  ax-17 (which, external to set.mm, is a metatheorem derivable from the
  others).

  I mentioned earlier that ax-1 through ax-15 (the $d-free fragment)
  cannot prove all $d-free theorems, and indeed by Andreka's theorem a
  complete $d-free fragment is impossible.  For completeness we need to
  add ax-16, which states:

      A. x x = y -> ( phi -> A. x phi )  where  $d x y

  This is the axiom that, one way or another, ultimately allows us to
  eliminate dummy variables from a proof.  But it has a $d condition.

  There is a trivial theorem (exists1 in the current set.mm) that states:

      E! x x = x  <->  A. x x = y    where $d x y

  E! means "there exists exactly one."  With this we can replace the
  antecedent of ax-16 and restate it so that it does not have a $d
  condition.  E! effectively "hides" the distinct variable y otherwise
  needed for ax-16.

  Of course E! is a defined connective, and when we eliminate the
  definition the distinct variable returns.  I would be surprised if, by
  treating E! as a primitive connective and adding $d-free axioms for it,
  we could exploit its "hiding" feature to defeat Andreka's theorem.  But
  I don't have a proof that we couldn't defeat it.

  In any case I think E! is an interesting connective, and it seems poorly
  developed in the literature, usually mentioned briefly in passing and
  sometimes not (formally) at all.  Maybe it's considered too trivial.
  But its theorems are sometimes not that obvious, and in fact E! x E! y
  has been mistakenly used to mean "there exists exactly one x and exactly
  one y."  (Nowhere, to my knowledge, is the correct expression for this
  found, except in set.mm).

  I have sometimes wondered what a complete axiomatization of E! might
  look like under (standard, not set.mm's) predicate calculus without
  equality (that would allow one to prove all theorems mentioning E! but
  not mentioning =).  In set.mm there are many theorems of this sort such
  as

      E! x E. y phi  ->  E. y E! x phi

(15-Apr-04) I want to link Maes' AC all the way to our Axiom of Choice
ax-ac, but I'm missing some standard equivalents needed to do that, and
I'll be working on those as I can fit them in. aceq5lem1 is a start at
this.  When it is complete, aceq5 will prove:

    aceq5 $p |- ( A. x E. f ( f (_ x /\ f Fn dom x ) <->
           A. x ( ( A. z e. x -. z = (/) /\
           A. z e. x A. w e. x ( -. z = w -> ( z i^i w ) = (/) ) ) ->
          E. y A. z e. x E! v v e. ( z i^i y ) ) ) $=

(14-Apr-04) Here, finally, we present Maes shorter version of his
5-quantifier AC.  (The longer one has been deleted from the database,
since it's no longer "interesting.")  Not only does this version have
only 5 quantifiers (our ax-ac has 7 or 8 if put in prenex form), but if
we expand out the biconditional in our ax-ac, Maes' axiom is the same
length as ours, tying my claim that ax-ac is the shortest possible.
This result refutes a conjecture of Harvey Friedman that a 5-quantifier
AC was not possible (and he thanked me for verifying Maes' result).

(7-Apr-04) Maes discovered a SHORTER version of his 5-quantifier Axiom
of Choice.  That means the old one is obsolete, so we'll
scrap it and reprove some lemmas to derive the shorter one.

(6-Apr-04) exists1 is an interesting little observation.  If we treat E!
as a primitive connective, we can use exists1 to replace the antecedent
of ax-16.  Now, disregarding ax-17 (which is techically redundant), the
only axiom requiring distinct variables is ax-16, so exists1 makes that
requirement go away.  Of course we would need to add axioms for E!, and
I'm not sure what they would be.

(5-Apr-04) eldm2, elrn2 shorten several proofs by eliminating the need
for df-br.

(3-Apr-04) Some more r19.* stuff, while we're on a roll...

(2-Apr-04) I thought I'd need these for Maes' theorem - I didn't - but
they still might be handy in the future.

(1-Apr-04) As you can guess, I've been busy formalizing the proof of
Kurt Maes' theorem.  It's now done!  AC with only 5 quantifiers in
prenex normal form is a new record and has not been published yet.  The
Metamath proof now confirms with absolute certainty that there is not a
mistake in Kurt Maes' proof.
References:
http://www.cs.nyu.edu/pipermail/fom/2003-November/007653.html
http://www.cs.nyu.edu/pipermail/fom/2003-November/007690.html

(25-Mar-04) Miscellaneous stuff.

(25-Mar-04) 19.28v shortens some proofs that used to use 19.28.
uni0 revises an earlier version.  funfvima3 might be useful in the
future to help prove Axiom of Choice equivalents.

(24-Mar-04) inf5 - Now, is this neat, or what?

(23-Mar-04) unidif0 and yesterday's disj4 will find use in a neat
theorem tomorrow - check back!

(22-Mar-04) mprgbir, even though kind of specialized, found use in
shortening 6 proofs.  I haven't seen disj4 in books, but it is kind of
interesting - in particular it implies the right-hand side is
symmetrical when A and B are swapped.

(17-Mar-04 - 21-Mar-04) More odds and ends for future use.

(13-Mar-04 - 16-Mar-04) I thought dmuni is kind of interesting; I'd
never seen it before I glanced at it in Enderton so I put it in;
don't know if it has any use yet.  In addition to residm, resabs1,2,
we also have rescom (commutative) law proved earlier.  elimasn
is often used implicitly by Takeuti/Zaring to represent minimal
elements - I don't think it is intuitive but it does save a bound
variable the way they use it e.g. in their Def. 6.21 p. 30.

(12-Mar-04) hta is a nice theorem - it tells us the closest we can get
to Hilbert's epsilon in ZFC, and provides a way to eliminate it from
proofs.  For a discussion see http://ghilbert.org/choice.txt

(11-Mar-04) Theorem schemes can be represented equivalently with class
variables and wff variables in set theory.  scottexs and scott0s provide
excellent examples of converting theorems with class variables
(scottex, scott0) to theorem schemes with wff variables.

(10-Mar-04) A surprising result communicated by Gregory Bush.  If "B"
represents df-bi, then in D-notation the proof reads:
DDD3DD2D1D3D1B11B.  George wrote:  'It's strange for two reasons:
first, it's radically more direct than the listed proof (17 steps
instead of 859).  Second, it doesn't matter what axiom (or definition)
goes in the second-to-last place.  When I first found this, it was
actually coming up with df-or instead of ax-1 in that spot, even though
"or" never appears in the expression, which led me through hours of
debugging.'

(9-Mar-04) Strengthen some class substitution theorems.  elabs
(which used to be called sbc7) now does not have any distinct variable
restrictions.

(8-Mar-04) These will be needed to work with abstraction classes
substituted for class variables.

(7-Mar-04) heplem is something I'm working on with Raph Levien and needs
to be cleaned up at some point.  The second hypothesis is not very
intuitive.  Basically I took a shortcut and reused zornlem1 to get this
result fast, in order to prove that it could be done, but the result is
a very complicated Hilbert epsilon formula (although mathematically it's
perfectly sound).  Raph indicated he would post some stuff on Hilbert
epsilons soon at ghilbert.org.

(6-Mar-04) The old transfinite induction theorems were converted so that
they now use restricted quantification.

(5-Mar-04) find updates an older version of finite induction by using
restricted quantifiers.  dmeqi, dmeqd, rneqi, and rneqd shorten a bunch
of proofs by eliminating ax-mp or syl needed for these commonly used
variants of dmeq and rneq.  Trivial as they are, last night I shortened
32 proofs with them.

(4-Mar-04) r19.20i2 will help us in some future proofs. zfinf shortens
an older version by using restricted quantification.  The proof of omex
was shortened by using r19.20i2, the new version of zfinf, and the new
version of peano5 (proved a couple of days ago).  ralxp is kind of neat,
letting us convert from a single quantification over a cross product to
a double quantification; it should be useful for some future theorems
involving operation values.

(3-Mar-04) reuuni, reuunis clean up older versions.  dfchj3 is a simpler
definition of closed subspace join in terms of supremum; I might make
it the official definition some day.

(2-Mar-04) peano5 has been restated with restricted quantification and
reproved with a 25% shorter proof (although it is still somewhat long
and difficult).  Unlike many textbooks, we prove peano5 without invoking
either the Axiom of Infinity nor the Axiom of Replacement.  This makes
the proof longer than those textbook proofs.  However, I think it is
philosophically nicer to be able to work with natural numbers without
having to assume that an infinite set exists.

(1-Mar-04) tfis, tfis2f, tfis2 update older versions of these theorems
by using restricted quantifiers.  Their proofs are shorter as a result
(30% shorter for tfis), and they will also shorten proofs that use them
as they are phased in.

(29-Feb-04) tfi updates an older version of the same theorem by using a
restricted quantifier.

(28-Feb-04) r1val1 is one of several alternate definitions of R1 in the
literature.

(27-Feb-04) cardaleph not only shows us that there exists an aleph for
every transfinite cardinal, but it gives us an explicit expression for
that aleph!  The textbook theorems I've seen only show existence, but I
think it's nicer to have a closed expression for the actual thing.  The
proof is quite long and brings together a lot of the cardinal/aleph
results we've proved up to this point.  The idea behind the proof is to
show it separately for 0, successor, and limit alephs.  Miraculously,
each of the three cases evaluate to the exact same expression.  Then we
combine the 3 cases with jaod at the end, then eliminate the triple OR
with ordzsl.

(26-Feb-04) alephord2i finally completes all the aleph ordering stuff we
should need.  alephle may seem "obvious" - after all, 0 is certainly less
than aleph`0 (countable infinity), 1 is less than aleph`1 (first
uncountable infinity), and so on.  In fact you'd almost think that it
should be "less than" instead of "less than or equal to".  Well, think
again - it turns out that A=aleph`A when A is big enough!  I think
that's amazing, and hopefully we'll prove it eventually.

(25-Feb-04) hbeleq nicely shortens proofs when it can used in place of
hbeq - I don't know why I didn't think of it before!  I like this little
theorem.  I shortened 12 proofs with it as you can see in the
"referenced by" list.  Exercise:  Does hbeleq hold if y and A are not
distinct?

(24-Feb-04) These are some boring utility theorems that will come in
handy later and also shorten some existing proofs. fnopab2 has already
"paid" for itself (meaning it shortens enough proofs so that the net
effect of adding it to the database is to reduce the database size)
(look at its "referenced by" list).

(23-Feb-04) elxp2 "modernizes" elxp with restricted quantification.
I think elxp4 is really cool - no dummy variables - it exploits
op1sta and op2nda we proved on 17-Feb.

(22-Feb-04) r19.22dv2, r19.26m, and ralcom give us some tools to use
when two different quantifier bounds are involved.  This will shorten
some future and existing proofs.  eusn and euuni are revisions of
earlier versions of these.  Someday I might introduce an iota, but
so far it wouldn't be used much.  I think Raph uses iota a lot in
his Ghilbert HOL stuff.  There is a person who _may_ have volunteered
to write a gh2mm translator; if that happens we'll have a lot of
new stuff to put here.

(21-Feb-04) onuninsuc, orduninsuc, nnsuc restrict the existential
quantifier of older versions to strengthen them.

(20-Feb-04) nndiv may become the basis for a new defined relation x|y or
"x divides y" for number theory.  divcan2t and divcan3t are needed for
nndiv.  divcan2t and divcan3t actually were already proven as divcan2z
and divcan3z; we just use the weak deduction theorem to get the "t"
versions.  It would have been nicer to prove them directly but we would
need divmult and divasst, and we only have divmulz and divassz.
Maybe someday I'll prove divmult and divasst, then I can shorten the
proofs of divcan2t and divcan3t.  The weak deduction theorem produces
horrible-looking proofs but it is a quick and dirty way to get stronger
theorems.

(19-Feb-04) Added some elementary theorems for the ordinal number 2.

(18-Feb-04) disj is a version of disj1 with restricted quantification
that will shorten some proofs. wefrc, tz7.5 are new versions of older
theorems with restricted quantification and shorter proofs. alephgeom
shows all alephs are infinite; the converse arises because (with our
definition) a function's value outside its domain is the empty set per
ndmfv.

(17-Feb-04) r19.23aivv lets us shorten several proofs.  dffr2, frc,
dfepfr, and epfrc are new versions of older theorems that have been
shortened with restricted quantifiers and re-proved with shorter proofs.
op1sta is from Raph's HOL Ghilbert stuff, where he calls it fst, and
op2nda is a modified version of his snd.

(16-Feb-04) birex2i, r19.21ad, cleqrabi are simple but we'll use them
later.  fvres is a revision of an older version.  dmsn0 completes the
series:  dm0, dmsn0, dmsnsn0.  Curiously dom 0 = dom {0} = dom{{0}} = 0,
but dom {{{0}}} and beyond are not 0.  For example dom {{{0}}} = {0}.

(15-Feb-04) Enderton p.222 gives an interesting discussion of kard vs.
card.  One thing I don't like about kard (that Enderton doesn't mention)
is that kard A is not always equinumerous to A, a property that card has
by cardid.  For example, kard of 0, 1, and 2 are the singletons {0},
{1}, and {2}, which are all equinumerous to 1 by ensn1.  (kard 3 is not
{3} but a messy expression, and they get messier as n grows.)

(14-Feb-04) We prove kardex using Scott's trick.  This is a little
different than the proof suggested in Enderton.

(13-Feb-04) I've never seen alephsuc2 in a book, but it looks like a
nice way to define the aleph function, and it may be the shortest
possible.  It exploits yesterday's cardval2.

(12-Feb-04) alephord3 is yet another ordering theorem.  (In the
end we will need all of these.)  I think cardval2 is kind of nice; I
came across it in Suppes only after I started, and maybe one day I'll
change df-card.  Suppes is the only place I've seen it.  The equivalence
of cardval and cardval2 is a very long journey and uses the Axiom of
Choice.

(11-Feb-04) alephord2 finally gives us a real textbook theorem (and its
converse as well).  alephsucdom is a consequence of the ordering stuff;
actually one of the earlier lemmas, alephordlem1, is more convenient to
prove it, and alephnbtwn2 also gets used.

(10-Feb-04) alephord generalizes alephordi to include the converse.  I
decided not to combine alephord and alephordi because the latter has one
less antecedent, making it slightly easier to use.  It is also
referenced twice in the alephord proof, so combining would cause its
proof to be redundantly expanded twice.

(9-Feb-04) alephordi is a key theorem essential for further aleph
results.  The proof provides a nice example of applying transfinite
induction.  This version (using strict dominance) doesn't seem to appear
in textbooks but later we will derive the textbook version that uses
cardinal ordering.

(8-Feb-04) alephnbtwn2 is the equinumerosity version of alephnbtwn.

(7-Feb-04) sneqi,sneqd are trivial but convenient and will shorten some
proofs.  ordsucun will be needed later to compute the rank of an
unordered pair.

(6-Feb-04) fvreseq shortens an earlier version with wral
notation.  inv and unv are trivial and probably useless things to
do when feeling lazy...

(5-Feb-04) sbcel1 and sbcel2 complete class substitution into a
membership relation; equality is defined in terms of membership so now
we can do that too in principle.  I can't think of a use for dmco2 yet
but I thought it was neat and couldn't resist proving it.  It wasn't
immediately obvious (to me at least) but the proof is relatively simple.

(4-Feb-04) With sbci, sbcn, and sbcal we now have the basic tools to
move class substitution in and out of any wff, since all other wff
connectives are defined in terms of implication, negation, and
universal quantification.

(3-Feb-04) rabid is trivial but it can simplify some proofs
such as nnwos.  sbcn gets a basic class substitution property out of the
way for future use.

(2-Feb-04) hbf is needed for fopab2. fvopab2 is a more general form
of an earlier version that had separate hypotheses. fopab2 is a
generalization of an earlier version that only proved the forward
direction of the conclusion; it is expected to be a nice tool for some
future equinumerosity theorems.

(1-Feb-04) dmopabss is a simple result that previously was proved
explicitly in a couple of other proofs, and it simplifies those proofs.
dmopab2 is a powerful equivalence that is expected to be useful in the
future and has already been used to simplify some proofs; previously
only the forward direction was proved.

(31-Jan-04) Got a bunch of bound-variable hypothesis builders out of
the way.  Actually they all chain together to help prove fnopabg, which
replaces an earlier version with a more powerful bidirectional
conclusion. dmsn is kind of curious - it holds with <I>all</I>
of its class existence hypotheses eliminated, although the proof of this
is somewhat tedious; but it shortens proofs that use it, over an
earlier version that required that the classes exist.

(30-Jan-04) Eliminated the requirement that C be a set in elmap.
fnopabfv is now a bidirectional result, making it more powerful. dmsnsn0 is
a cute technical result that will be used in the future.  Curiously dom
0 = dom {0} = dom{{0}} = 0, but dom {{{0}}} and beyond are not 0.

(29-Jan-04) sb7 should make some logicians happy who don't like
the x both free and bound in df-sb.  However I still think df-sb is
neat because it doesn't have any distinct variable requirements.

(28-Jan-04) Self-explanatory.

(27-Jan-04) I think sbidm is nice but I don't know if it will ever
be useful.  I simplified Raph's proof of nfunv, but later he wrote about
my version, "Neat.  I still kinda like my proof, because it's portable
to second-order arithmetic with Cantor pairing (in which the universe is
a relation, the complete relation to be specific)."  (His Ghilbert HOL
database has his version.)

(26-Jan-04) cleq2tr is on the one hand trivial, on the other hand to
me it seemed non-intuitive at first.  As an exercise, try to convince
yourself of it informally.

(25-Jan-04) The very basic cleqtr had never been proved in the
database before, and suprisingly it seems to have little use.  Only one
proof was shortened by it.  alephnbtwn, on the other hand, is a pretty
deep and basic "must have" fact about alephs that will be of much use in
the future.

(24-Jan-04) opnz is an easy theorem I thought should be there although
I don't know if it will ever be useful.  iunrab gets a basic fact of
indexed unions out of the way for future use.


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