X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek2_lambda_intro.mdwn;h=6bbb53991b855a73c57de344a56d24c0756ea83b;hp=091e1234dbaef36e5cd163b03bacad2eac19dbc1;hb=a4d2693effe839524592f4427465ff8d97625302;hpb=b6e2ba40d18b2c1efb84e764d0e07d7f97d57319

diff --git a/topics/week2_lambda_intro.mdwn b/topics/week2_lambda_intro.mdwn
index 091e1234..6bbb5399 100644
--- a/topics/week2_lambda_intro.mdwn
+++ b/topics/week2_lambda_intro.mdwn
@@ -198,9 +198,7 @@ and:
     z (\x (x y))
 
 
-**Dot notation** Dot means "assume a left paren here, and the matching right
-paren as far to the right as possible without creating unbalanced
-parentheses". So:
+**Dot notation** Dot means "Insert a left parenthesis here, and the matching right parenthesis as far to the right as possible without creating unbalanced parentheses---so long as doing so would enclose an application or abstract not already wrapped in parentheses." Thus:
 
     \x (\y (x y))
 
@@ -228,7 +226,15 @@ abbreviates:
 
     ((\x (\y (x y))) x)
 
-We didn't have to insert any parentheses around the inner body of `\y. (x y)` because they were already there.
+The outermost parentheses were added because we have an application. `(\x. \y.
+...)` became `(\x (\y. ...))` because of the rule for dots. We didn't
+insert any parentheses around the inner body of `\y. (x y)` because they were
+already there. That is, in expressions of the form `\y. (...)`, the dot abbreviates
+nothing. It's harmless to write such a dot, though, and it can be conceptually
+helpful especially in light of the next convention.
+
+Similarly, we permit `\x. x`, which is shorthand for `\x x`, not for `\x (x)`, which
+our syntax forbids. (The [[lambda evaluator|/code/lambda_evaluator]] however tolerates such expressions.)
 
 
 **Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
@@ -247,7 +253,7 @@ function?  One popular answer is that a function can be represented by
 a set of ordered pairs.  This set is called the **graph** of the
 function.  If the ordered pair `(a, b)` is a member of the graph of `f`,
 that means that `f` maps the argument `a` to the value `b`.  In
-symbols, `f: a` ↦ `b`, or `f (a) == b`.  
+symbols, `f: a` ↦ `b`, or `f (a) == b`.
 
 In order to count as a *function* (rather
 than as merely a more general *relation*), we require that the graph not contain two
@@ -283,7 +289,7 @@ lambda calculus, note that duplicating, reordering, and deleting
 elements is all that it takes to simulate the behavior of a general
 word processing program.  That means that if we had a big enough
 lambda term, it could take a representation of *Emma* as input and
-produce *Hamlet* as a result. 
+produce *Hamlet* as a result.
 
 Some of these functions are so useful that we'll give them special
 names.  In particular, we'll call the identity function `(\x x)`
@@ -312,7 +318,7 @@ are distinct lambda expressions:
 
     (\z z)
 
-yet when applied to any argument M, all of these will always return M. So they
+yet when applied to any argument `M`, all of these will always return `M`. So they
 have the same extension. It's also true, though you may not yet be in a
 position to see, that no other function can differentiate between them when
 they're supplied as an argument to it. However, these expressions are all
@@ -427,12 +433,12 @@ reasons sketched above.
 ## The analogy with `let` ##
 
 In our basic functional programming language, we used `let`
-expressions to assign values to variables.  For instance, 
+expressions to assign values to variables.  For instance,
 
     let x match 2
-    in (x, x) 
+    in (x, x)
 
-evaluates to the ordered pair (2, 2).  It may be helpful to think of
+evaluates to the ordered pair `(2, 2)`.  It may be helpful to think of
 a redex in the lambda calculus as a particular sort of `let`
 construction.
 
@@ -440,7 +446,7 @@ construction.
 
 is analogous to
 
-    let x match ARG 
+    let x match ARG
     in BODY
 
 This analogy should be treated with caution.  For one thing, our `letrec`