Induction on Well-Formed Formulas

Theorem
Let $\mathcal L$ be a formal language with a bottom-up grammar.

Let $\Phi$ be a proposition about the well-formed words of $\mathcal L$.

Suppose that $\Phi$ is true for all letters of $\mathcal L$.

Suppose further that every rule of formation preserves $\Phi$, i.e. when fed well-formed words satisfying $\Phi$, it yields new well-formed words satisfying $\Phi$.

Then all well-formed words of $\mathcal L$ satisfy $\Phi$.

Proof
By definition of bottom-up grammar, the well-formed words of $\mathcal L$ comprise:


 * letters of $\mathcal L$;
 * expressions resulting from rules of formation.

Either case is dealt with by the assumptions on $\Phi$.

Hence the result, from Proof by Cases.