.Open Names
. Idea
Documentation gives names to things. This may be its most important 
function.

Algebra and natural languages have two different paradigms for creating 
names[Gr??, SIGPLAN].  Thus "x" is clearly an algebraic name, and "cats" an
English name. Programming languages do not  include a list of valid English
words because they predate the existence of spelling checkers and because 
of the divine right of programmers to confuse their colleagues.

MATHS allows both to be used in formulae. A `natural` MATHS name is an 
unabbreviated phrase in a natural language with spaces replaced by an 
underline symbol. It will be constructed from words that are in a 
dictionary of that language.  However to keep formula and proofs 
manageable, short algebraic `variables` or `symbolic names` are needed.

. Style Guide
Use algebraic or symbolic names for variables bound 
inside a definition or formula(local).  Use natural identifiers when they 
are declared in a piece of documentation and shared across two or more 
formulae (public).  Avoid short global identifiers unless you are
working with complex formula and proofs or the identifier is
already in use.

. Example
.Net
 Set::Sets. 
 op::infix(Set).
 for all a,b,c:Set( a op (b op c) = (a op (b op c)).
.Close.Net
In the above `Set`, `op` are public and `a`,`b`, and `c` are local.
The words "Sets", "infix" are defined in the STANDARD MATHS package.
Words like "for" and "all" are part of the MATHS language.

. Introduction to Syntax
A local or private name is always introduced by a private/local declaration
or definition with the following syntax:
	variable ":" set_expression
	Name_of_Type variable
	variable ":=" expression
These forms are used inside expressions and inside summaries of
extended pieces of documentation.

A public name, or parameter needs to be easy to retrieve or spot
by eye.  They are always introduced (defined or declared) by using two 
colons:
	name ":" ":" Set_expression.
	name ":" ":" "=" expression.
	name ":" ":" set_expression "=" expression.
	name ":" ":" "=" "following".
	name ":" ":" set_expression "=" "following".

The above forms are used inside documents and should be interspersed with
comments.  The keyword "following" above means that the following closed 
piece of document forms the expression.

. Development of Syntax of names
name::= symbolic_name | natural_identifier,

A MATHS environment needs to be able to keep an index of names that are 
current.

index :: @name, the index is a set of names, partitioned into
 |- index>=={bound_names, declared_names, defined_names, used_identifier},

A name that is not in the index is free:
  free_names::=name~index.

variable::=  symbolic_name | variable_structure,

variable_structure::= R(variable) | P(variable).
.Box
.See http://www/dick/maths/math_64_Meta_Macros.html
|- For X:@#lexeme,
	R(X) ::="(" #(X comma) identifier "=>" X #( comma identifier "=>" X) 
")",
	P(X) ::= "(" X #( comma X ) ")".
.Close.Box

symbolic_name::= (mathematical_symbol #(subscript|superscript)) & 
(bound_symbolic_name | non_bound_symbolic_name),

At each point in a piece of documentation there exists a set of bound 
symbolic names (the current index). Bindings, definitions and declarations
add to this set. Variables are removed at the end of their  scope. At the 
start no names are bound.

A binding always creates a new variable or name - even when the symbol or 
identifier is already bound.  The old meaning is recovered at the end of 
the new symbols scope. Variables are uniquely identified by their name and
the place where they were bound.  Compare
.See ./notn_12_Expressions.html#Free Variables
for more examples.


mathematical_symbol::= letter | symbol | back_slashed_symbol | (identifier
& mathematical),

A mathematical symbol can be a character in another alphabet represented, 
in ASCII,  by a word - typically prefixed by a backslash. Notice that in 
keeping with mathematical tradition single character identifiers are used 
for local variables. Identifiers like 'sin', 'cos', 'abs', 'sinh', 'asin',
'sqrt', 'pr', 'id'... etc can also be found in mathematical dictionaries, 
and can be used as mathematical symbols. Significantly they have not been 
typeset in italics as variables however. Thus there is an exception to the
rule that names that are words must be in a standard dictionary.  A (short)
glossary of mathematical words can therefore be included as part of a 
document - and a tool should be made that scans a document and extracts 
names that are defined but are not in the standard dictionary.

subscript::= digit # digit | "[" expression"]",
subscripts should be type set below the line.

superscript::= prime.

To make reuse easier, natural names whose scope is larger than a single 
formula.  This is especially true for ideas that someone else may want to 
remember.. Some f these can be given internal algebraic abbreviations are 
used inside a piece of documentation.

natural_identifier::= word #(underline word)  & (declared_identifier | 
defined_identifier | unknown_identifier)

The words should exist in a standard a dictionary of the user's and 
client's natural language. The use of underline to link words is 
incompatible with \TeX  subscripting. A preprocessor that outputs \TeX must
therefore escape underlines within natural identifiers. It will also have 
to insert formatting of sub- and super-scripts.

Strings like "I" and "a" are ambiguous - they are both in the English 
dictionary and are algebraic variables.  Within a formal piece of 
documentation they are taken to be algebraic. Within informal commentary 
they are taken to be words.  However, inside a comment, a reverse quotes 
should enclose a formula.  Identifiers of defined terms may be
preceded by a dollar sign to indicate a cross-reference or hypertext
link.
Note.  This is not really necessary outside comments if suitable tools
are available for interpreting formulae.

. Dangerous Bend
Notice that it is easy to assume that different names
refer to different (unequal) object.  In MATHS it may be
necessary to add some explicit assumptions that names are different:
.As_is 		|- treedle_dee <> tweedle_dum.
.Close Names