By Alonzo Church

The description for this e-book, The Calculi of Lambda Conversion. (AM-6), could be forthcoming.

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By the hypothesis of induction, applied to ' \ and '\• the result of a special terminating sequence of contractions is unique to within possible applications of Rule I. Therefore the result of any terminating sequence of contractions, µ, is unique to within possible applications of Rule I. VI. xN)lf), then: ( 1) B1 is B. "l. N)lf) of A, and the number m, such that, starting with Bm, at most l1ru consecutive Bic's occur for which it is not true that Bk red Bk+ 1 , If A 1mr B (1) is obvious. To prove ( 2), let ( (i'.

1 or 2, this f'ollows f'rom 1 4 I. For larger values of' n, we prove it by induction. Let L2 be a f'ormula such that t 2 1 conv A1 , and let L1 be a f'ormula such that L 11 conv A2 , L_1 2 conv A', ••• , L1ftl conv ~ (where /ti represents n-1 ). Also let (; be a f'ormula such that (;1 conv L1 and C2 conv L 2 • Then a f'ormula L having the required property is: ~i 14 III. r il (Pi>. If' A1 , A2 , ••• ,An, F1 , F2 , ••• , Fm contain no f'ree variables, a f'onnula E can be f'ound which represents an enumeration of the least set of' f'ormulas which contains A, , A2 , ••• , An and is closed under each of' the operations of' f'orming FcrXY from the f'ormulas X,Y ~ = 1, 2, ••• , n ) , in the sense that every f'onnula of this set is convertible into one of' the fonnulas in the infinite sequence £1 I £21 •••I and every f'onnula in this infinite sequence is convertible into one of the f'onnulas of the set.

Man. OXIV. OXV'. 0"7NI. OXV'II. OXV'III. OXIX. OXX. F-y 1- F(f3{f3{f3I)h>· F ((3((3((3/)}-y) I- F-y. FI I- F ((3//). F ((3 I/) I- FI. F {-y((3(3((3(3(3}}(3) I- F((3((3(3)(3). F ((3((3(3)(3) I- F(-y((3(3((3(3(3))(3), F ("'1((3(3((3(3(3))-y) F {(3{(3-y){(3f3f3)) I- F((3{f3-y)(f3(3f3)}. I- F{-y{f3f3(f3f3f3))-y). OXXI. 1)((3(3(3)). OXXII. 1), OXXIII. F (-yf3I) I- F(f3((3I)/), OXXIV. OXXV'. OXXV'I. OXXVII. OXXVIII. OXXIX. OXXX. OXXXI. OXXXII. F(f3(f3/)I) F ((3(3-y) I- I- F(-y(31). F{f3(f3(f3-y)-y)(f3f3)).