User blog comment:Ikosarakt1/Formal definition of tetrational arrays/@comment-5529393-20130329075222

I think in rule A2

AX+1 & b[p] = AX * X & b[p]

should be replaced by

XA+1 & b[p] = XA * X & b[p].

Also, you need a rule of the form

XA & b[p] = XA[p] & b[p]

in order to go up in exponent level.