Proof via induction for small-step semantics
سؤال
I'm doing a course in Computer Programming Languages and I'm trying to prove the following (roughly following Pierce's Types and Programming Languages book):
if $t \rightarrow^* t'$ then $if\; t\; then \; t2 \; else \; t3 \rightarrow^* if\; t'\; then \; t2 \; else \; t3$
I'm a little confused about where to start; as far as I know, I'm supposed to define a base case, then prove it via induction.
I've started my proof by assuming these base cases:
$P(\frac{}{true \rightarrow^* true})$ and $P(\frac{}{false \rightarrow^* false})$.
I'm stuck at this point, and I don't really how know to proceed.
edit: I've added the syntax in an effort to clear things up.
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