@history()
The development of LiE was initiated in Jan 1988 by the computer algebra
group at CWI, Amsterdam. Members of this group, and their contributions,
have been the following: Arjeh M. COHEN (the idea, many mathematical
functions, the first manual, project leader), Ron SOMMELING (first version,
mathematical programs, trouble shooting), Bert LISSER (the kernel and the
interpreter), Bart de SMIT (mathematical programs), Bert RUITENBURG (tests,
texts, examples), Marc van LEEUWEN (math programs, rewriting and new
functions for version 2.0, manual for version 2.0, adaptation and source
documentation for version 2.1).
@version()
Version 2.1
Available publicly from URL http://www.cwi.nl/~maavl/LiE
Runs on UNIX platforms and with some limitations on all ANSI C platforms.
Tested ports exist to SGI, SUN and Atari ST
@size(vec)array
size(v).   Returns the size of a vector.
@degree(pol)polynomials
degree(p).   Returns the degree of a polynomial.
degree(p) = max { degree(p[i]) | 1 <= i <= length(p) }
degree(p[i]) = sum (j=1..n_vars(p)) expon(p,i)[j]
Example		degree(X[1,2]+3X[3,4]) = 7
@expon(pol,int)polynomials
expon(p,i).   Returns the exponent vector of the i-th term of p.
The terms are sorted first (sorting criterion can be set by `on ...').
Example		expon(X[1,2]+3X[3,4],2) = [3,4]
@coef(pol,int)polynomials
coef(p,i).   Returns the coefficient of the i-th term of p.
The terms are sorted first (sorting criterion can be set by `on ...').
Example		coef(X[1,2]+3X(3,4),2) = 3.
@n_rows(mat)array
n_rows(m).   Returns the number of rows of m.
@n_cols(mat)array
n_cols(m).   Returns the number of columns of m.
@n_vars(pol)polynomials
n_vars(p).   Returns the number of indeterminates of p.
@length(pol)polynomials
length(p).   Returns the number of non-zero terms of p.
@*(vec,vec)
v*w   Returns standard inner product `sum (i=1..size(v)) v[i]*w[i]'.
@*(mat,mat)
Matrix multiplication.
@*(int,pol)
n*p   Multiplies each term of p by n.
@*(grp,grp)
Direct product of groups.
@*(mat,vec)
Applies the matrix from the left to the (transposed) vector.
@*(mat)
Transposes the matrix.
@.make(int,int)
make(f,size) with f:int->int.   Makes the vector [f(1),f(2),..,f(size)].
@.make(int,int,int)
make(f,r,c) with f:int,int->int.   Makes a r*c-matrix with i,j-entry f(i,j).
@.make(int,vec)
make(f,v) with f:int->int.   Makes the vector [f(v[1]),f(v[2]),..,f(v[n])]
   where n=size(v).
@.make(int,vec,vec)
make(f,v,w) with f:int,int->int and vectors v,w. Makes the vector
   [f(v[1],w[1]),f(v[2],w[2]),..,f(v[n],w[n])] where n=size(v)=size(w).
@.make(int,mat)
make(f,m) with f:int->int and matrix m.   Makes a r*c-matrix with
   i,j-entry f(m[i,j]), where r=n_rows(m), c=n_cols(m).
@.make(int,mat,mat)
make(f,m,n) with f:int,int->int and matrix m,n.   Makes a r*c-matrix
   with i,j-entry f(m[i,j],n[i,j]), where r=n_rows(m), c=n_cols(m).
@operators()lieshell
arithmetic: + - * / %
logical: && || 
comparison: == != <= >= < >
@function()lieshell

<function definition)::=
   <identifier> ( <type> <var> , <var> , ... ; <type> ... ) = <series>
|  <identifier> ( <type> <var> , <var> , ... ; <type> ... ) { <series> }

Examples	f(int x,y;vec v)=x*v+y
		inc(int n) { print(n); n+1 }
@type()lieshell
'int'= integer,'bin'=big integer, 'vec'= vector, 'mat'= matrix,
'grp' = group, 'tex'=text, 'vid'=void and 'pol' = polynomial.
@integer()lieshell
Entries of vectors, matrices and polynomial exponents are limited in
magnitude to 2^31-1=2147483647. Other intgers are of unlimited magnitude.
@vector()array

<vector expression>::= [ <integer expression> , ..., <integer expression> ]

Example		[1,-2,3,5] 
@matrix()array

<matrix expression>::= [ <vector expression> , ..., <vector expression> ]

Example		[[1,2],[-3,5],[2,2345]] 
@group()group
Simple Lie groups: A1, A2 ..., B2, B3, ..., C2, C3, C4, D3, D4, ...,
                   E6, E7, E8, F4, G2.
Semisimple groups: Concatenation of simple liegroups.
Reductive groups:  Semisiple group possily followed by central torus `Tn'.

Example		A3B6G2T5
@text()lieshell
<text>::= "Any string, without carriage returns, enclosed in double quotes"
"This is an example."
@polynomial()polynomials
An object of type `pol' is a polynomial. It is a sum of terms of the form
`n X v', where n is an integer `coefficient' and v a vector `exponent'.
Each term of the polynomial a vector of the same length as exponent, which
length is called the number `n_vars' of indeterminates of the polynomial.
A polynomial can be interpreted by choosing that many indeterminates, and using
the first entry of each exponent as the power of the first indeterminate, etc. 
For example, the polynomial  `X[2,0,7]+3X[0,2,0]' can be interpeted as

			 2  0  7        0  2  0 	 2  7	 2
			x  y  z  + 3 * x  y  z      ( = x  z  + 3 y) .
@special()lieshell
$ = last printed value ,
\ = command will be continued at next line ,
# = the characters typed in until the next '#' will be considered 
    as comment.
? = request for information about the string typed after '?'.
: = the string typed after ':' is command to the shell.
'?' and ':' have to be the first characters of the command.
@help()
LiE has some features of a pocket calculator, but all numbers are integers.
Formulas can be entered in the usual way like `(((2+3)*2-1)*2^(2-1)-2/3)'.
Moreover, you can assign values to variables (a=2) and define functions
(square(int i)=i*i). There exist objects of type integer, vector, matrix and
group. The if-then-else-fi, while-do-od and for-do-od control structures are
also available. Information about functions, variables and what more appears
on the screen upon entering a question mark followed by the name of a query.
Commands (like this ?feature) are executed when a <Return> is entered,
unless LiE finds that you have an incomplete expression (e.g., an unclosed
parenthesis), which can be forced by typing '\' as last character.

Other useful help available: ?index, ?functions, learn index.
@assignment()lieshell

<assignment>::= <identifier> = <expression>
	      | loc <identifier> = <expression>
	      | <identifier> += <expression>
	      | <selection> = <expression>
	      | <selection> += <expression>

Examples	my_var=4711,	v=[2,4]
		loc count=0
		count += 1
		v[5]=3,		m[3,-4]= 3*count
		m[3,4]+=7
@+(tex,tex)
String concatenation
@+(tex,bin)
a+b   appends to textual representation of b to string a
@*(tex,int)
a*b   repeats string a b times. 
@if()lieshell

<conditional clause>::= if <expression> then <series> else <series> fi
		      | if <expression> then <series> fi

@for()lieshell

<loop clause>::= for <ident>=<expression> to <expression> do <series> od
	       | for <ident>=<expression> downto <expression> do <series> od
	       | for <ident> in <expression> do <series> od
	       | for <ident> row <expression> do <series> od

Examples	for i=19 to 68 do print(i^3) od
		for j=47 downto 11 do print(j); f(j) od
		sum=0; v=[3,3,2,8]; for x in v do sum+=x od; sum
		for v row id(4) do print(v) od
@commands()lieshell
Special commands: quit, edit, read, write, monfil, on, off, maxobjects,
   maxnodes, listvars, listfuns, listops, help, ?, learn
Type ? followed by command keyword for more information. 
@on()lieshell
on		shows parameter settings
on runtime	shows runtime after each executed command.
on gc		puts garbage collector on.
on monitor	maintains logfile.
on prompt	shows prompt.
on bigints      computation with integers of unlimited size.(default)

on lprint	lineair print.
on + lex	increasing lexicographic ordening (polynomials)
on - lex	decreasing lexicographic ordening
on + degree	increasing degree ordening
on - degree	decreasing degree ordening
on + height	increasing height ordening 
on - height	decreasing height ordening

These last 2 ordenings are connected to the default group (see setdefault).
@off()lieshell
clears the modes set by 'on'; see there for more details.
@quit()lieshell
exit
@edit()lieshell
'edit' edits the most recently read or edited file (default: initfile), and
executes the commands in that file after leaving the editor.

'edit "filename"'   edits the file "filename". 
@null(int,int)array
null(n,m)   A zero matrix of n rows and m columns.
@null(int)array
null(n)   A zero vector of size n.
@all_one(int,int)array
all_one(n,m)   A matrix of n rows and m columns, with all entries 1.
@all_one(int)array
all_one(n)    A vector of size n, with all entries.
@poly_null(int,int)polynomials
poly_null(n)   A zero polynomial with n indeterminates.
@poly_one(int,int)polynomials
poly_one(n)   A unit polynomial with n indeterminates.
@id(int)array
id(n)   The n by n identity matrix
@break()lieshell

<statement>::= break
	     | break <expression>

Exits the inner most do- or while-loop, returning the expression if present.
@matvec(vec,int)array
matvec(v,n)   Returns a matrix with n rows and size(v)/n columns. The
   vector v is split into subvectors, which form the successive rows of m.
@vecmat(mat)array
vecmat(m)   Returns a vector, which is the concatenation of the rows of m.
@error()lieshell
error(text) prints text and terminates the command being executed.
@print()lieshell
print(a)   (any type a) prints a.
@void()lieshell 
void(a) (any type a)   Returns void (useful to match type of other expr.)
@n_comp(grp)group
n_comp(g)   The number of simple components of g.
@+(vec,int)
v+n   Extens v by the entry n.
Example		[1,2]+3=[1,2,3] 
@+(int,vec)
n+v   Prefix v by the entry n.
Example: 1+[2,3]=[1,2,3] 
@+(mat,vec)
m+v   The matrix m with additional row v at the end.
Example: [[1,2],[3,4]]+[5,6]=[[1,2],[3,4],[5,6]] 
@-(vec,int)
v-i   Remove the i-th entry v[i] from v.
Example: [1,2,3]-2 = [1,3]
@files()lieshell
1 In the Lie shell it is possible to execute an external text file
  with Lie shell commands. In order to execute the file `prog'
  one has to issue:
	read prog
2 It is possible to edit an external file in the Lie shell by entering
  the command
	edit prog
  The Lie shell chooses the editor defined in shell variable $EDITOR
  (only of UNIX). This variable has to be defined and exported by the
  shell from which Lie was invoked.
  After the editing the updated file `prog' will be executed. 
  When a filename is omitted after the edit command, the most recently
  read or edited file will be taken.
3 It is possible to write the user defined functions on an external
  file `dest' by entering the command:
	write dest
4 If there is a file named `initfile' in the same directory as Lie,
  this file will first be read by entering Lie. The initfile must
  be a text file containing Lie shell commands. It is intended to
  contain your own data and functions.
5 It is possible to monitor a session. This will happen after issuing
  the command
	on monitor .
  All The Lie shell commands and their results will be written in the file
  named 'monfil'. It can be turned off by entering
        off monitor
  It is possible to specify an other file for monitor output by entering 
	monfil otherfile.
@write()lieshell
It is possible to write the user defined functions on an external
file `dest' by entering the command:
        write dest
It is possible to write/append a value of a single variable 
or a definition of a single function on a file by 
entering 
	?name>dest		(write)
or
	?name>>dest		(append)
@read()lieshell
In the Lie shell it is possible to execute an external text file with Lie
shell commands. In order to execute the file `prog' one has to issue:
        read prog
@monitor()lieshell
All commands and results, as they appear on your screen, will be stored
in the file 'monfil' after the command:
        on monitor
It can be turned off with 'off monitor'. The file name used can be altered by
setting 
	monfil filename
This can be used to prevent overwriting the previous monitor file.
@save()lieshell
See write or files.
@save_mat(mat,tex)lieshell
save_mat(m,f) Saves matrix m in file f in Lie intern format.
@get_mat(tex)lieshell
get_mat(f) Returns a matrix which is stored in file f in Lie intern format.
@save_string(tex,tex)lieshell
save_string(s,f) Saves string s in file f in Lie intern format.
@get_string(tex)lieshell
get_string(f) Returns a string which is stored in file f in Lie intern format.
@^(vec,vec)
v^w   Concatenates v and w
@^(pol,pol)
p^q   extends all exponents of p with n_vars(q) zeros,
      prefixes all exponents of q with n_vars(p) zeros; returns their product
@%(int,int)
n%m = n modulo m
@%(vec,int)
v%n =v modulo n entry-wise
@%(mat,int)=
m%n =m modulo n entry-wise
@/(int,int)
n/m = the division of n by m (and rounded off towards 0 to an integer).
@/(vec,int)
v/n = v/n entry-wise
@/(mat,int)
m/n = m/n entry-wise
@^(mat,mat)
a^b   vertical concatenation of a and b
@^(int,int)
a^n   a to the n-th power
@^(mat,int)
a^n   a to the n-th power
@!(int)
!b   Boolean negation: if b==0 then 1 else 0 fi
@int()silence
Type indication for integer
@bin()silence
Type indication for big integer
@vec()silence
Type indication for vector
@mat()silence
Type indication for matrix
@grp()silence
Type indication for group
@tex()silence
Type indication for text
@vid()silence
Type indication for void
@factor(int)int
factor(n).   Prints a tentative factorization of n into prime factors.
   (Returns void, i.e., nothing.) Only prime factors smaller than 2^15 are
   found.
@unique(mat)array
unique(m). Returns a canonical form for a matrix representing a set of
   vectors: it sorts the matrix and removes any multiple rows.
   Algorithm: heap sort.
@setdefault()group
setdefault g.   Set default Lie group equal to g.  This Lie group is used in
other functions if the optional argument g is omitted.  Without an argument
the command setdefault prints the default group.
@sort(mat)array
sort(m).  Returns the matrix obtained from `m' by sorting its rows into the
   order selected by the user (defualt is lexicographically decreasing).
   Algorithm: quicksort.
@sort(vec)array
sort(v) Returns the vector obtained from `v' by sorting its entries into
   decreasing order. Algorithm: quicksort.
@finish()