Problem 909

[Solarion4131]

I'm having trouble understanding how the L-equations can always be simplified. What if you reach a point where none of the three rules works? Presumably it shouldn't be possible by the wording of 'the final result does not depend on the order of the transformations', but I'm having this issue when evaluating $S(S)(S(S))(S(Z))(A)(0)$. Can someone explain why the answer to this is 6?

[neverforget]

Problem 909 (View Problem)

For this question, seems like 0 is considered a natural number.

I'm having trouble understanding how the L-equations can always be simplified. What if you reach a point where none of the three rules works? Presumably it shouldn't be possible by the wording of 'the final result does not depend on the order of the transformations', but I'm having this issue when evaluating $S(S)(S(S))(S(Z))(A)(0)$. Can someone explain why the answer to this is 6?

For the relevant L-expressions that the question asks about, the only point where none of the rules work is when the L-expression has been transformed down to a single number. For evaluating $S(S)(S(S))(S(Z))(A)(0)$ you (must) start by using the third rule, taking $u=S$, $v=S(S)$, and $w=S(Z)$. Make sure to apply the rules carefully as it is easy to make mistakes, especially if doing so by hand.