User blog comment:! cook the lettuce/Despacit's Number Competition/@comment-37485018-20191208215742

Definition of chains:

\(0\in\text{Chain}\)

For \(A_1,A_2\in\text{Chain}\), \([A_1,A_2]\in\text{Chain}\)

Define a function \(C:\) that maps chains to chains:

\(C(0)=0 \\ C([a_0,a_1])=a_0\)

\(a<b\) is a relation over chains, here is its comparison algorithm:
 * 1) If \(a=0\), then \(a<b\iff b\neq0\).
 * 2) Suppose \(a\neq0\). Therefore, \(a=[a_0,a_1]\).
 * 3) If \(b=0\), then \(a<b\) is false.
 * 4) Suppose \(b\neq0\). Therefore, \(b=[b_0,b_1]\).
 * 5) If \(a_0<b_0\), then \(a<b\) is true.
 * 6) If \(b_0<a_0\), then \(a<b\) is false.
 * 7) Suppose \(a_0=b_0\).
 * 8) If \(C(a_1)=C(b_1)\), then \(a<b\iff a_1<b_1\).
 * 9) Suppose \(C(a_1)\neq C(b_1)\).
 * 10) If \(C(a_1)<C(b_1)\), then \(a<b\iff a_1<[C(a_1),b_1]\).
 * 11) Otherwise, jump to line 3.
 * 12) Otherwise, if \(\lnot(b1\), \(C(y,h,k)\) has a zero count of at most \((y+k)!\)