User:Vel!/BEAF

An array is a function \(A : \Omega \mapsto \Omega\), where the number of outputs greater than 1 is finite. Let \(b = A(0)\), \(p = A(1)\), \(\pi = \min\{\alpha > 1: A(\alpha) > 1\}\), \(\kappa = \pi - 1\), and finally \(\lambda = \max\{\alpha > 0: A(\alpha) \in \text{Lim}\}\).

Define the prime block \(\Pi(\alpha)\):


 * \(\Pi(0) = \{\}\)
 * \(\Pi(\alpha + 1) = \{\alpha\} \cup \Pi(\alpha)\)
 * \(\Pi(\alpha) = \Pi(\alpha[p])\) if \(\alpha\) is a limit ordinal

Define the passengers as \(S = \Pi(\pi) \backslash \{\pi, \kappa\}\).


 * 1) Base rule. If \(\pi\) does not exist, \(v(A) = b^p\).
 * 2) Prime rule. If \(p = 1\), \(v(A) = p\).
 * 3) Limit rule. If \(\lambda\) exists:
 * 4) * Define \(A_n'\) as \(A\) with \(A_n'(\lambda) := A(\lambda)[n]\).
 * 5) * \(v(A)[n] = v(A_n')\) and \(v(A) = \sup\{n < \omega : v(A)[n]\}\).
 * 6) Catastrophic rule. Otherwise, define \(A'\) as \(A\) with the following modifications:
 * 7) * Define \(B\) as \(A\), but with \(B(1) := p - 1\).
 * 8) * If \(\kappa\) exists, \(A'(\kappa) := v(B)\).
 * 9) * \(A'(\pi) := A(\pi) - 1\).
 * 10) * \(A'(\sigma) := b\) for \(\sigma \in S\).
 * 11) * \(v(A) = v(A')\).