Some properties of conversion

Our purpose is to establish the properties of conversion which are expressed in Theorems 1 and 2 below. We shall consider first conversion defined by Church's Rules I, II, IIIt and shall then extend our results to several other kinds of conversion.: 1. Conversion defined by Church's Rules I, II, III. In our study of conversion we are particularly interested in the effects of Rules II and III and consider that applications of Rule I, though often necessary to prevent confusion of free and bound variables, do not essentially change the structure of a formula. Hence we shall omit mention of applications of Rule I whenever it seems that no essential ambiguity will result. Thus when we speak of replacing { Xx. MI (N) ? by S xMI it shall be understood that any applications of I are made which are needed to make this substitution an application of II. Also we may write bound variables as unchanged throughout discussions even though tacit applications of I in the discussion may have changed them. A conversion in which III is not used and II is used exactly once will be called a reduction. If II is not used and III is used exactly once, the conversion will be called an expansion. "A imr B," read "A is immediately reducible to B," shall mean that it is possible to go from A to B by a single reduction. "A red B," read "A is reducible to B," shall mean that it is possible to go from A to B by one or more reductions. "A conv-I B," read "A conv B by applications of I only," shall mean just that (including the case of a zero number of applications). "A conv-I-II B," read "A conv B by applications of I