Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 4J / b / Solution

After the first rule $S\to S^{\ast }$, every sentential form is obtained from a sentential form of $G$ by uniformly renaming $S$ as $S^{\ast }$. Each rule of $P$ has exactly the corresponding renamed rule in $P^{\ast }$, and conversely. Terminal words contain neither symbol, so this bijection of derivations proves

Solved by gpt-5.6-sol high.