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recursion in ordinal.txt
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recursion in ordinal.txt
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1 <= k < omega in all of the followings
MC=Main Conjecture, EF = Erickson's formula
assigning ordinal 1 to fusible number 0
(MC) exc(omega^alpha k + beta) = exc(omega^{[alpha]_{exc(alpha)+k-1}} + beta)
- (beta = omega^{[alpha]_{exc(alpha)+k-1} + 1} ? 1 : 0)
(EF) exc(omega^alpha k + beta) = exc(omega^{[alpha]_exc(alpha)} + beta)
- (beta = omega^{[alpha]_exc(alpha) + 1} ? 1 : 0)
for alpha,beta limits, beta<omega^alpha
(MC) exc(omega^{alpha+1} k + beta) = exc(omega^alpha (k+1) + beta)
(EF) exc(omega^{alpha+1} k + beta) = exc(omega^alpha 2 + beta)
for beta < omega^{alpha+1} limit
(MC) exc(omega^alpha k) = d(omega^alpha k) - d(alpha) + exc(alpha) - (k>1 ? 1 : 0)
(EF) exc(omega^alpha k) = d(omega^alpha) - d(alpha) + exc(alpha)
for alpha limit
(MC) exc(omega^alpha k) = d(omega^alpha k) - d(alpha) + (k>1 ? 1-k : 2)
(EF) exc(omega^alpha k) = d(omega^alpha) - d(alpha) + (k>1 ? 0 : 2)
for alpha successor.
d(k) = k
(MC) d(omega^alpha k + beta) = 1 + d(omega^{[alpha]_{exc(alpha)+k-1}} + beta)
(EF) d(omega^alpha k + beta) = k + d(omega^{[alpha]_exc(alpha)} + beta)
for alpha limit, beta < omega^alpha
(MC) d(omega^{alpha+1} k + beta) = 1 + d(omega^alpha (k+1) + beta)
(EF) d(omega^{alpha+1} k + beta) = k + d(omega^alpha 2 + beta)
for beta < omega^alpha