Lambda terms definable as combinators
It si well known that for each ^-term there is a corresponding combinatory term formed using the combinators K and S instead of the ^-operator. Similarly for every combinatory term there is a ^-term. For weaker sets of combinators such as B, C and K or B, B', I and W we show how such a correspondence for 'translation; can be formulated and we determine in the case of several such sets of combinators the sets of ^ -terms that can be translated using them.
As combinators can represent Hilbert-style proofs of theorems of implicational logic and ^-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.