lambda calculus

;;;lambda calculus

;;;booleans
true = \x.\y.x
false = \x.\y.y
if = \v.\t.\f. v t f

;;;ex
if true M N = M
--->(\v.\t.\f. v t f) (\x.\y.x) M N
--->(\x.\y.x) M N
--->M

;;;Pairs
;fst(mkpair M N) = M
;snd(mkpair M N) = N

makpair = \x.\y.\f. f x y
fst =\p.p true
snd =\p.p false

;;;Church number
;;;a lambda function stand for natrual number
;;;encoded by a function of two arguments, f and x
;;;where the function applies f to x n tims (number n)

0 = \f.\x.x
1 = \f.\x.f x
2 = \f.\x.f (f x)
3 = \f.\x.f (f (f x))
...
n = \f.\x.f...ntimes...(f x)

add1 = \n.\f.\x f (n f x) ;apply f n+1 times

add = \m.\n. m add1 n ;m haapens to be a function that will take add1 and apply it m times

iszero = \n. n (\x.false) true

;;;减法,主要就是要减少一次f的调用
;;;把n的f全部替换为warp f,x替换成<true,x>
;;;其中这个true主要是用来表示是否为第一次执行f,如果是那么就是x,如果不是那么就继续保持f x
;;;很简单,带入展开就知道了
sub1 = \n.\f.\x. snd(n (warp f) <true,x>)
warp = \f.\p.<false , if (fst p) (snd p) (f (snd p))>

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