By [edited by] Haskell B. Curry, J. Roger Hindley, Jonathan P. Seldin.
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Extra info for Combinatory logic. / Volume II
REMARK 2. x(xa))(AyS). Then x 2 P(Pa), and P(Pa) and Pa are both residuals of S according to the definitions of 8 4B. Property (vii) can be proved thus: If S is one of the R i or does not overlap with any R i , the result follows by (iv). Suppose that L = L' u L" where the members of L' are parts of S and those of L do not overlap with S. Let S be the a-redex aU,U,. U,, and let the result of substituting U,,. , U , for x,, . , x , in A be V. Let a complete reduction relative to L' carry U , into U i ; then it will carry S into S' E a U ; .
The BmI of 4 5F is an example of I,. Let X cnv Y be defined (in this context) to mean that, for some m, X and Y are both 0,combinations such that, for new variables x l , . ,x,, x x l . . x , = o Y xl . . x , . Then the theorems of 5 5F all remain valid if BmI is replaced by I , and cnv is defined as just stated. 5. ~'Thus if the primitive combinators are I, K, S, a canonical redex is of one of the following forms Ix, KUlx, SUiUZX, in which U1, U 2 do not contain x. A canonical reduction is a reduction in which only canonical weak redexes are contracted.
G. 8 C2. 11. This is the only case considered in 8 SC1. In a letter dated March 6, 1960 J. R. Wallace called attention to the fact that under the definition given in 9 5C1 only a proper combinator corresponds to anything at all. 12. Cf. [EVR], especially footnote 19. There ‘regular combinator’ is defined by the specifications: (i) The combinators B, C, I, K, S, W, are regular. (ii) If X i s regular so is BX. (iii) If X and Y are regular so is X . e. B X Y ) . Among regular combinators we can specify an equality as in [EVR].
Combinatory logic. / Volume II by [edited by] Haskell B. Curry, J. Roger Hindley, Jonathan P. Seldin.