[Inquiry] Re: Theme One Program -- Commentary -- Axil

[Jon Awbrey]

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TOP.  Commentary Note 52

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Axil.

Axil is a transformation of ideas that is used to recognize
a couple of special-purpose configurations in cactus graphs.
It may be useful to work through the computation of Axil in
a step by step, case by case fashion.

Step 1.  The functional value of Axil (this) = nil unless
         Respect (this) is non-nil, as illustrated below:

            ^ ante (this) =/= nil
             \
              o---o
              |   | arch (this) = blank
              o---o
              ^
         this | = respect (this)
              @

Step 2.  Test the value of Equity (fore (this), next (this)).
         The following figures illustrate the possible cases:

Case 1.  this = fore (this) = next (this).

            ^
          o--\--------o
          |   o---o   |
          o-->|   |---o
              o---o
              ^
         this | = equity (fore (this), next (this))
              @

Case 2.  this =/= fore (this) = next (this).

            ^
          o--\----------------o
          |   o---o   o---o   |
          o-->|   |-->| x |---o
              o---o   o---o
              ^
         this | = equity (fore (this), next (this))
              @

Case 3.  equity (fore (this), next (this)) = nil.

            ^
          o--\------------------------------o
          |   o---o   o---o         o---o   |
          o-->|   |-->| x |-->···-->| y |---o
              o---o   o---o         o---o
              ^
         this |
              @

Step 3.  In Case 3, that is if Equity (fore (this), next (this)) = nil,
         then Axil (this) = this.  In Cases 1 and 2, Axil (this) is set
         equal to Suspect (stem (this)), as shown in the figures below:

Case 1.  suspect (stem (this)) = nil,
         therefore axil (this) = nil.

Case 2.  The only way to obtain a non-nil result
         is with the following type of structure:

               ^             o-------o
             o--\------------|---o   |
             |   o---o   o---o   |   |
             o-->|   |-->| ) |---o   |
                 o---o   o---o       |
                 ^                   |
                 |      o------------o
                 |     /
            this |    /
                 |   /
                 |  /
         o-------|-/---------------o
         |   o---o<        o---o   |
         o-->| ( |-->...-->| x |---o
             o---o         o---o
             ^
             | axil (this)
             @

Case 3.  equity (fore (this), next (this)) = nil.

            ^
          o--\------------------------------o
          |   o---o   o---o         o---o   |
          o-->|   |-->| x |-->···-->| y |---o
              o---o   o---o         o---o
              ^
         this | = axil (this)
              @

function axil (this: idea): idea;
var here: idea;
begin
 here := nil;
 if respect (this) <> nil then
  begin
   here := covert (equity (fore (this), next (this)), this);
   if here = nil then here := suspect (stem (this))
  end;
 axil := here
end;

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