I sense that the following simple argument is invalid, but cannot figure out why.
Base Case:No variable or constant is an initial segment of a term, since the only terms that begin with variables or constants have length 1, and so lack initial segments.
Inductive Case:Suppose s is an initial segment of a term, t, where s is $f t_1 t_2 \cdots t_n$ for terms $t_i$. Then t is su, i.e., the string, s, followed by the string, u, for some non-empty string u. But, then, t is not a term, since an n-ary function symbol followed by n terms, followed by a non-empty string is a not a term.
Hence, no term is an initial segment of any other.
I did not make use of the inductive hypothesis, that no ti in s is the initial segment of any term. This is what makes me think I cheated somehow. Assuming that I did, I wonder if someone could show me how to prove the conclusion by means of induction on the complexity of terms without assuming that the tis in t are not initial segments of any terms (i.e., by merely assuming that the tis in s are not).
Any feedback would be greatly appreciated! Thanks in advance.