X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week4.mdwn;h=58d6bc38204bd3bcbf4afabe6d9c4306b82735de;hp=077884cc0df70cfb962f2b03ffad70f121ca1f32;hb=HEAD;hpb=1bac07d66d424599f9a06d374cde4e2b904847a2 diff --git a/week4.mdwn b/week4.mdwn deleted file mode 100644 index 077884cc..00000000 --- a/week4.mdwn +++ /dev/null @@ -1,37 +0,0 @@ -[[!toc]] - -#Q: How do you know that every term in the untyped lambda calculus has a fixed point?# - -A: That's easy: let `T` be an arbitrary term in the lambda calculus. If -`T` has a fixed point, then there exists some `X` such that `X <~~> -TX` (that's what it means to *have* a fixed point). - -
-let L = \x. T (x x) in
-let X = L L in
-X ≡ L L ≡ (\x. T (x x)) L ~~> T (L L) ≡ T X
-
-
-Please slow down and make sure that you understand what justified each
-of the equalities in the last line.
-
-#Q: How do you know that for any term `T`, `Y T` is a fixed point of `T`?#
-
-A: Note that in the proof given in the previous answer, we chose `T`
-and then set `X = L L = (\x. T (x x)) (\x. T (x x))`. If we abstract over
-`T`, we get the Y combinator, `\T. (\x. T (x x)) (\x. T (x x))`. No matter
-what argument `T` we feed `Y`, it returns some `X` that is a fixed point
-of `T`, by the reasoning in the previous answer.
-
-#Q: So if every term has a fixed point, even `Y` has fixed point.#
-
-A: Right:
-
-let Y = \T. (\x. T (x x)) (\x. T (x x)) in
-Y Y ≡ \T. (\x. T (x x)) (\x. T (x x)) Y
-~~> (\x. Y (x x)) (\x. Y (x x))
-~~> Y ((\x. Y (x x)) (\x. Y (x x)))
-~~> Y (Y ((\x. Y (x x)) (\x. Y (x x))))
-~~> Y (Y (Y (...(Y (Y Y))...)))
-
-