the maths rules for the BKO scheme

Here are the maths rules behind BKO:

1) <x||y> == 0 if x != y.  
2) <x||y> == 1 if x == y.
3) <!x||y> == 1 if x != y. (NB: the ! acts as a not. cf, the -v switch for grep) 
4) <!x||y> == 0 if x == y.
5) <x: *||y: z> == 0 if x != y.
6) <x: *||y: z> == 1 if x == y, for any z. 
7) applying bra's is linear. <x|(|a> + |b> + |c>) == <x||a> + <x||b> + <x||c>
8) if a coeff is not given, then it is 1. eg, <x| == <x|1 and 1|x> == |x>
9) bra's and ket's commute with the coefficients. eg, <x|7 == 7 <x| and 13|x> == |x>13  
10) in contrast to QM, in BKO operators are right associative only.
<a|(op|b>) is valid and is identical to <a|op|b>
(<a|op)|b> is invalid, and undefined.
11) again, in contrast to QM, <a|op|b> != <b|op|a>^* (a consequence of (10) really)
12) applying projections is linear. |x><x|(|a> + |b> + |c>) == |x><x||a> + |x><x||b> + |x><x||c>
13) kets in superpositions commute. |a> + |b> == |b> + |a>
14) kets in sequences do not commute. |a> . |b> != |b> . |a>
Though maybe in the sequence version of simm, this would be useful:
|a> . |b> = c |b> . c |a>, where usually c is < 1. (yeah, it "bugs out" if you swap it back again, but in practice should be fine)
another example: 
  |c> . |a> . |b> = c |a> . c |c> . |b>
                  = c |a> . c |b> . c^2 |c>
15) operators (in general) do not commute. <b|op2 op1|a> != <b|op1 op2|a>
16) if a coeff in a superposition is zero, we can drop it from the superposition without changing the meaning of that superposition. 
17) we can arbitrarily add kets to a superposition if they have coeff zero without changing the meaning of that superposition.
18) |> is the identity element for superpositions. sp + |> == |> + sp == sp.
19) the + sign in superpositions is literal. ie, kets add.
|a> + |a> + |a> = 3|a>
|a> + |b> + |c> + 6|b> = |a> + 7|b> + |c>
20) <x|op-sequence|y> is always a scalar/float
21) |x><x|op-sequence|y> is always a ket or a superposition