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==Example==

RCGs are capable of recognizing the non-linear index language <math>\{ www : w \in \{a,b\}^{*} \}</math> as follows:

Letting x, y, and z be variable symbols:
<math display="block">
  \begin{aligned}
    S(xyz) &\to A(x, y, z) \\
    A(ax, ay, az) &\to A(x, y, z) \\
    A(bx, by, bz) &\to A(x, y, z) \\
    A(\epsilon, \epsilon, \epsilon) &\to \epsilon
  \end{aligned}
</math>
The proof for ''{{not a typo|abbabbabb}}'' is then

<math>S(abbabbabb) \Rightarrow A(abb, abb, abb) \Rightarrow A(bb, bb, bb) \Rightarrow A(b, b, b) \Rightarrow A(\epsilon, \epsilon, \epsilon) \Rightarrow \epsilon</math>

Or, using the more correct dotted notation for ranges:

<math>S(\bullet{}abbabbabb\bullet{}) \Rightarrow A(\bullet{}abb\bullet{}abbabb, abb\bullet{}abb\bullet{}abb, abbabb\bullet{}abb\bullet{}) \Rightarrow A(a\bullet{}bb\bullet{}abbabb, abba\bullet{}bb\bullet{}abb, abbabba\bullet{}bb\bullet{})</math> <math>\Rightarrow A(ab\bullet{}b\bullet{}abbabb, abbab\bullet{}b\bullet{}abb, abbabbab\bullet{}b\bullet{}) \Rightarrow A(\epsilon, \epsilon, \epsilon) \Rightarrow \epsilon</math>

For a string of <math>3n</math> letters, there are <math>\binom{3n+2}{2} = \frac{(3n+2)(3n+1)}{2}</math> different instantiations of that first clause, but only the one which makes <math>x,y,z</math> all <math>n</math> letters each allows the derivation to reach <math>\epsilon</math>.