Causal rank-raising map on well-order codes
= Causal rank-raising map on well-order codes
There is a coordinatewise causal map $S:\omega^\omega\to\omega^\omega$ that recodes a relation after adjoining a new least element. If $x\in\mathrm{WF}$, then $S(x)\in\mathrm{WF}$ and $\lVert S(x)\rVert=\lVert x\rVert+1$; the tagged coding also ensures $S(x)\ne x$ when $x\notin\mathrm{WF}$. Because the first $n+1$ output coordinates depend only on the first $n+1$ input coordinates, Player II can produce $S(x)$ online.