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1934). Functionality in combinatory logic, Proceedings of the National Academy of Science USA 20, pp. 584–590. B. (1969). Modified basic functionality in combinatory logic, Dialectica 23, pp. 83–92. B. and R. Feys (1958). Combinatory Logic, Vol. I, North-Holland, Amsterdam. R. P. Seldin (1972). Combinatory Logic, Vol. II, North-Holland, Amsterdam. -Y. (1972). Interpr´etation fonctionelle et ´elimination des coupures dans l’arithm´etique d’ordre sup´erieur, Dissertation, Universit´e Paris VII. R.

4. Theorem. If M → →βδ N and N is in βδ-nf, then M → → βδ N. Proof. 22). 5. Example. One of the first versions of a δ-rule is in Church (1941). Here X is the set of all closed normal forms and for M, N ∈ X we have δC M N → true, if M ≡ N ; δC M N → false, if M ≡ N . Another possible set of δ-rules is for the Booleans. 6. Example. The following constants are selected in C. true , false , not , and , ite (for if then else). The following δ-rules are introduced. x). It follows that ite true x y → → x, ite false x y → → y.

B. (1994). Typability and type-checking in the second-order λ-calculus are equivalent and undecidable, Proceedings of the 9th Annual Symposium on Logic in Computer Science, Paris, France, IEEE Computer Society Press, pp. 176–185.