I have been learning Predicate Logic recently and I get a following question:

Assumed $ \ F = \{m,\ f\}, P = \{S,\ B\}$ , in which $ m$ is constant, $ f$ is ternary function, $ g$ is binary function, prove $ g(d,\ f(g(d,\ d),\ d,\ d))$ as a **correct Term** in Predicate Logic.

I want to prove it by drawing a **parse tree** or **inductive reasoning**, but I don’t know the which one is mathematical rigor.

If not mind, could anyone tell me the mathematical rigor way to prove **the correctness of Term and Formula** in Predicate Logic and prove it?

Thanks in advance.