By Alonzo Church
The description for this booklet, The Calculi of Lambda Conversion. (AM-6), should be forthcoming.
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OII. OIII. orv. OV. OVI. OVII. OVIII. OIX, IA I- A. A I- IA, F (IA) I- FA. FA I- F(I A) • F (IAB) I- f(AB), F (AB) I- F(IAB). F (JABCD) I- F(AB(AOC)). F (AB(AOC)) I- F(JABCO). 1((3-y{(3((3((3-y) /((3((3((3(3(3))/)))). OX. 1((3-y((3(r,((3-y))((3((3((3(3(3))I)))) I- FJ, OXI. F(3 I- F((3((3((3/)}(3). OXII. F ((3((3((3/))(3) I- F(3. 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).
Nce of contractions on these residuals in Bk gives Bk+l by 7 XXV. Thus Bk red "ic+l unless the reduction of ~ into ~+l involves a contraction of a residual of ( (llxN'JW); but if we start with any particular Ak this can be the case only a finite number of successive times by 7 XXV. Hence (3) 1s proved, um being defined as rollows : Perrorm m successive reductions on A in all ;possible ways. This gives a r1n1te set of formulas (since, ror this purpose, we need not distinguish rormulas d1rfering only by applications or Rule I).
13 27 Dj •s occur for which it is not true that Oj red Oj+i. Since the sequence C imr C1 , C1 1mr C2 , • • • leads to B in k reductions, there is, by hypothesis of incl>lction, a number v C,k such that any sequence of reductions starting from c leads to a normal form (and thus terminates) a~ter at most v C,k reductions. f(x+l) ... f( v C,k) reductions an Aj is reached on which no reductions are possible. form. f A to. a formula C, such that there is a sequence of k reductions leading from C to a normal form of A, daterm1nes an upper bound, f(vC,k), which holds for all sequences of reductions starting from A.
The calculi of lambda-conversion by Alonzo Church
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