From: Jim Pryor Date: Wed, 15 Sep 2010 20:53:06 +0000 (-0400) Subject: week1 tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=f259b9c18ecee2715349fefba2de8209a896cd69 week1 tweaks Signed-off-by: Jim Pryor --- diff --git a/week1.mdwn b/week1.mdwn index 8124ffe6..729b14d1 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -167,7 +167,7 @@ For instance: > T is defined to be `(x (\x (\y (x (y z)))))` -The first occurrence of `x` in `T` is free. The `\x` we won't regard as being an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free. +The first occurrence of `x` in T is free. The `\x` we won't regard as being an occurrence of `x`. The next occurrence of `x` occurs within a form that begins with `\x`, so it is bound as well. The occurrence of `y` is bound; and the occurrence of `z` is free. Here's an example of beta-reduction: @@ -185,7 +185,7 @@ Different authors use different notations. Some authors use the term "contractio M ~~> N -We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol -> for one-step contraction, and the symbol ->> for zero-or-more step reduction. Hindley and Seldin use (triangle..sub1) and (triangle). +We'll mean that you can get from M to N by one or more reduction steps. Hankin uses the symbol → for one-step contraction, and the symbol →> for zero-or-more step reduction. Hindley and Seldin use |>1 and |>. When M and N are such that there's some P that M reduces to by zero or more steps, and that N also reduces to by zero or more steps, then we say that M and N are **beta-convertible**. We'll write that like this: @@ -193,7 +193,7 @@ When M and N are such that there's some P that M reduces to by zero or more step This is what plays the role of equality in the lambda calculus. Hankin uses the symbol `=` for this. So too do Hindley and Seldin. -In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol (three bars) for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing metatheory for the lambda calculus. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in: +In the metatheory, it's also sometimes useful to talk about formulas that are syntactically equivalent *before any reductions take place*. Hankin uses the symbol ≡ for this. So too do Hindley and Seldin. We'll use that too, and will avoid using `=` when discussing metatheory for the lambda calculus. Instead we'll use `<~~>` as we said above. When we want to introduce a stipulative definition, we'll write it out longhand, as in: > T is defined to be `(M N)`.