Quantifier-free induction for lists

Abstract

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