Quantifier-free induction for lists

We investigate quantifier-free induction for Lisp-like lists constructed inductively from the empty list \( nil \) and the operation \({\textit{cons}}\), that adds an element to the front of a list. First we show that, for \(m \ge 1\), quantifier-free \(m\)-step induction does not simulate quantifier-free \((m + 1)\)-step induction. Secondly, we show that for all \(m \ge 1\), quantifier-free \(m\)-step induction does not prove the right cancellation property of the concatenation operation on lists defined by left-recursion.