refine notes

diff --git a/topics/week2_lambda_advanced.mdwn b/topics/week2_lambda_advanced.mdwn
index 1372e43..dbc62e7 100644 (file)
--- a/topics/week2_lambda_advanced.mdwn
+++ b/topics/week2_lambda_advanced.mdwn
@@ -1,4 +1,4 @@
-## Fine points concerning the lambda calculus ##
+## Advanced notes on the Lambda Calculus ##

 
 Hankin uses the symbol
 <code><big><big>&rarr;</big></big></code> for one-step contraction,
 
 Hankin uses the symbol
 <code><big><big>&rarr;</big></big></code> for one-step contraction,


@@ -7,10 +7,7 @@ zero-or-more step reduction. Hindley and Seldin use
 <code><big><big><big>&#8883;</big></big></big><sub>1</sub></code> and
 <code><big><big><big>&#8883;</big></big></big></code>.
 
 <code><big><big><big>&#8883;</big></big></big><sub>1</sub></code> and
 <code><big><big><big>&#8883;</big></big></big></code>.
 

-As we said in the main notes, 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 write that like
-this:
+As we said in the main notes, 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 write that like this:

 
     M <~~> N
 
 
     M <~~> N
 


@@ -31,11 +28,11 @@ when discussing the metatheory. Instead we'll use `<~~>` as we said
 above. When we want to introduce a stipulative definition, we'll write
 it out longhand, as in:
 
 above. When we want to introduce a stipulative definition, we'll write
 it out longhand, as in:
 

-> T is defined to be `(M N)`.
+> `T` is defined to be `(M N)`.

 
 or:
 
 
 or:
 

-> Let T be `(M N)`.
+> Let `T` be `(M N)`.

 
 We'll regard the following two expressions:
 
 
 We'll regard the following two expressions:
 


@@ -92,7 +89,7 @@ what (c) reduces to. So if we took (b) to reduce to `\y. y y`, we'd wrongly be
 counting (b) to be equivalent to (c), instead of (a).
 
 To reduce (b), then, we need to be careful to that no free variables in what
 counting (b) to be equivalent to (c), instead of (a).
 
 To reduce (b), then, we need to be careful to that no free variables in what

-we're substituting in get captured by binding &lambda;s that they shouldn't be
+we're substituting in get "captured" by binding &lambda;s that they shouldn't be

 captured by.
 
 In practical terms, you'd just replace (b) with (a) and do the unproblematic substitution into (a).
 captured by.
 
 In practical terms, you'd just replace (b) with (a) and do the unproblematic substitution into (a).


@@ -147,7 +144,7 @@ furthermore lack any notion of a bound position.
 
 ## Review: syntactic equality, reduction, convertibility ##
 
 
 ## Review: syntactic equality, reduction, convertibility ##
 

-Define N to be `(\x. x y) z`. Then N and `(\x. x y) z` are syntactically equal,
+Define `N` to be `(\x. x y) z`. Then `N` and `(\x. x y) z` are syntactically equal,

 and we're counting them as syntactically equal to `(\z. z y) z` as well. We'll express
 all these claims in our metalanguage as:
 
 and we're counting them as syntactically equal to `(\z. z y) z` as well. We'll express
 all these claims in our metalanguage as:
 


@@ -158,16 +155,16 @@ This:
 
     N ~~> z y
 
 
     N ~~> z y
 

-means that N beta-reduces to `z y`. This:
+means that `N` beta-reduces to `z y`. This:

 
     M <~~> N
 
 
     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 something 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
 
 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.
+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 <code>&vdash;</code>) 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.
 
 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.