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