X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek2_lambda_advanced.mdwn;h=d84d5ee7a14f6a4a12cd3da0c7eba9a08c569e5c;hp=dbc62e76de6bbced7d0f1889ffd7bf40b3910f6a;hb=c70ea8653fd391ba094c8a64419f3532af2076ce;hpb=52b49c352b3d56d1a929694f22b0b94577e28ee3 diff --git a/topics/week2_lambda_advanced.mdwn b/topics/week2_lambda_advanced.mdwn index dbc62e76..d84d5ee7 100644 --- a/topics/week2_lambda_advanced.mdwn +++ b/topics/week2_lambda_advanced.mdwn @@ -107,10 +107,10 @@ Another way to think of it is to identify expressions not with particular alphabetic sequences, but rather with *classes* of alphabetic sequences, which stand to each other in the way that (a) and (b) do. That's the way we'll talk. We say that (a) and (b) are just typographically different notations for a -*single* lambda formula. As we'll say, the lambda formula written with (a) and -the lambda formula written with (b) are literally syntactically identical. +*single* lambda term. As we'll say, the lambda term written with (a) and +the lambda term written with (b) are literally syntactically identical. -A third way to think is to identify the lambda formula not with classes of +A third way to think is to identify the lambda term not with classes of alphabetic sequences, but rather with abstract structures that we might draw like this: @@ -159,12 +159,12 @@ means that `N` beta-reduces to `z y`. This: M <~~> N -means that `M` and `N` are beta-convertible, that is, that there's something they both reduce to in zero or more steps. +means that `M` and `N` are beta-convertible, that is, that there's some common term they both reduce to in zero or more steps. The symbols `~~>` and `<~~>` aren't part of what we're calling "the Lambda Calculus". In our mouths, they're just part of our metatheory for talking about it. In the uses of the Lambda Calculus as a formal proof theory, one or the other of these -symbols (or some notational variant of them) is added to the object language. But only in outermost contexts. It's like the "sequent" symbol (written `=>` or `⊢`) in [Gentzen-style proof systems](https://en.wikipedia.org/wiki/Sequent_calculus) for logic. You can't embed the `~~>` or `<~~>` symbol inside lambda terms. +symbols (or some notational variant of them) *is* added to the object language. But only in outermost contexts. It's like the "sequent" symbol (written `=>` or `⊢`) in [Gentzen-style proof systems](https://en.wikipedia.org/wiki/Sequent_calculus) for logic. You can't embed the `~~>` or `<~~>` symbol inside lambda terms. See Hankin Sections 2.2 and 2.4 for the proof theory using `<~~>` (which he writes as `=`). He discusses the proof theory using `~~>` in his Chapter 3.