User blog:KurohaKafka/Computability of Bashicu Matrix

The gist
Iff we have a matrix such as

\(G\frown B_0\frown B_1\frown\cdots\frown B_n\)

which β-reduced from \(G\frown B\frown N\), then it is holds that

\(B_k\frown N_k<_rB\frown N\)

\(N_k\) denotes a first sequence of \(B_{k+1}\).

\(<_r\) is well-founded and defined for every bad part.

For hydra tree's property, that is sufficient condition of computability.

The specific proof
This proof depends on Bashicu's definition.

It is supposed that \(G\frown B\) is computable. We organize compute process for every bad part and prove inductively. We divide whole proof as follows;
 * The bad part is well-founded by \(<_r\).
 * The tree is well-founded.