Lambda Terms Definable as Combinators

It is well known that for each l-term there is a corresponding combinatory term formed using the combinators K and S instead of the I-operator. Similarly for every combinatory term there is a i.-term. For weaker sets of combinators such as B, C and K or B, B’, I and W we show how such a correspondence or “translation” can be formulated and we determine in the case of several such sets of combinators the sets of I-terms that can be translated using them. As combinators can represent Hilbert-style proofs of theorems of implicational logic and 2.terms natural deduction style proofs, this work allows us to formulate natural deduction systems equivalent to the Hilbert-style systems for various implicational logics and can form a basis for proof generating programs for these logics.