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Sun Oct 2 14:35:00 PDT 2011

# SuperStrings

Superstrings are a genralization of normal strings - all the same axioms but on a larger class of objects.

A superstring (unlike a normal string of symbols like "abc") can hhave holes in it -- some subscripts are missing:

1. (1+>"a" | 3+>"c") is a super string.

2. For Sets A, finite A, super(A)::= Nat<>->A
3. For A, elements(A)::=Nat><A.
4. For x: super(A), end(x)::Nat= if(x={}, 0, max(pre(x)) ).
5. For x,y:super(A), x!y::@(Nat,A)= x | y(_+end(x)).
6. (above)|-MONOID( super(A), (!), {}).
7. |()- (monoid) #A==>super(A).
8. |()- super(A) generated_by elements(A).
9. (above)|-Unique factorisation!

10. For all x:super(A), x={} or for one e:element(A), y:super(A), x=e!y.

So define ... first(x)....rest(x)...last(x)...,

As always we have

11. |-For P,Q:@super(X), x!P, P!x, P!Q are defined and in @super(A).

We have a typical induction schema:

12. SUPER_INDUCTION::=
Net{
1. P::@super(A).
2. {} in P.
3. For all e:elements(A), a!P ==>P

4. (above)|-P=super(A).

}=::SUPER_INDUCTION.

13. |-SUPER_INDUCTION.

We can reduce a superstring to just the elements in a subset of the natural numbers:

14. For N:@Nat, x:super(A), N!x::=N;x.
15. (above)|-N!x={ i+>a || i:N and (i+>a) in x }.

. . . . . . . . . ( end of section SuperStrings) <<Contents | End>>

# HyperStrings

Replace Nat by Positive Real,
1. hyper(A)::= { x: Real<>->A || one lub(x) and one glb(x) },
2. end(x)::=lub(pre(x)), ...

. . . . . . . . . ( end of section HyperStrings) <<Contents | End>>

# histories

1. histories(A)::= { x: Real<>->bag(A) || one lub(x) and one glb(x) },
2. end(x)::=lub(pre(x)), ...
3. For x,y:histories(A), x!y::@(Real,bag(A))= { (t,e) || t in pre(x)|pre(y) and e=x(t)+y(t) }.

. . . . . . . . . ( end of section histories) <<Contents | End>>

# Super-general

A is any set. T is any set S is a subset of @T such that each set t in S has a special maximum value + is an associative operation that preserves maximum values the sum of two T's is greater than the max of them S conatins all the finite subsets of T. supergenstring=S<>->A.

. . . . . . . . . ( end of section Super-general) <<Contents | End>>

# Notes on MATHS Notation

Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

# Notes on the Underlying Logic of MATHS

The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.

For a more rigorous description of the standard notations see

1. STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html

# Glossary

2. above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
3. given::reason="I've been told that...", used to describe a problem.
4. given::variable="I'll be given a value or object like this...", used to describe a problem.
5. goal::theorem="The result I'm trying to prove right now".
6. goal::variable="The value or object I'm trying to find or construct".
7. let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
8. hyp::reason="I assumed this in my last Let/Case/Po/...".
9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.