R is basicially a function from rationals to algorithms computing irrationals.

L_R is decidable by a master algorithm running every y of x#y, e.g. 3.14#<y_pi> runs y_pi(|3.14|), so y runs up to 2 decimals.

The master algorithm runs y(|x|) and then compares the output to x.

It halts correctly for every x#y unless you can provide a counter-example?