Quantifier-free induction for lists

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