rank propositional formula - exercise

Question

Let $r$ be the rank function of a propositional formula, show that $r(\phi)<r(\psi)$ if $\phi$ is a proper subformula of $\psi$.

I don't know how to prove it.

Answer 1

See the definition of rank function in Dirk van Dalen, Logic and Structure (5th ed - 2013), page 12.

For a non-atomic formula $\varphi$, $rank(\varphi)$ is at least $+1$ with respect to its "immediate" subformulas.

Thus, if $\psi$ is a proper subformula of $\varphi$ :

$rank(\varphi) \ge rank(\psi) +1 > rank(\psi)$.