Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 76 / 3 / ii / Solution

Fix a closed $T$ and an arbitrary closed target $M$. If $S$ is an easy lambda term, the extension by $S=\lambda x.M$, with fresh $x$, is consistent. In that extensionThe theory obtained by adding only $ST=M$ is a subtheory of this consistent extension, so it is consistent too. Thus $ST$ is easy for every closed $T$. The printed unrestricted wording requires this qualification because easiness was defined only for closed terms: if $T$ is open, $ST$ need not be closed. With easiness extended to equations between open terms, the same argument works for arbitrary $T$.

If an easy closed $S$ were a solvable lambda term, some finite sequence of closed arguments would give $S{P}_1\cdots P_k{\equiv }_{\beta }I$. Repeated application of the result above would make this term easy. Easiness is invariant under beta equivalence, so $I$ would be easy, contrary to the preceding calculation. Hence every easy term is unsolvable.