A solution to Curry and Hindley’s problem on combinatory strong reduction

It has often been remarked that the metatheory of strong reduction $$\succ$$, the combinatory analogue of βη-reduction $${\twoheadrightarrow_{\beta\eta}}$$ in λ-calculus, is rather complicated. In particular, although the confluence of $$\succ$$ is an easy consequence of $${\twoheadrightarrow_{\beta\eta}}$$ being confluent, no direct proof of this fact is known. Curry and Hindley’s problem, dating back to 1958, asks for a self-contained proof of the confluence of $$\succ$$ , one which makes no detour through λ-calculus. We answer positively to this question, by extending and exploiting the technique of transitivity elimination for ‘analytic’ combinatory proof systems, which has been introduced in previous papers of ours. Indeed, a very short proof of the confluence of $$\succ$$ immediately follows from the main result of the present paper, namely that a certain analytic proof system Ge[$$\mathbb {C}$$] , which is equivalent to the standard proof system CLext of Combinatory Logic with extensionality, admits effective transitivity elimination. In turn, the proof of transitivity elimination—which, by the way, we are able to provide not only for Ge[$$\mathbb {C}$$] but also, in full generality, for arbitrary analytic combinatory systems with extensionality—employs purely proof-theoretical techniques, and is entirely contained within the theory of combinators.