manip trees: tweaks

Signed-off-by: Jim Pryor <profjim@jimpryor.net>

diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index a05fa25..69b5131 100644 (file)
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -280,7 +280,7 @@ interesting functions for the first argument of `treemonadizer`:
 We could simulate the tree state example too, but it would require
 generalizing the type of the continuation monad to
 
-       type ('a -> 'b -> 'k) continuation = ('a -> 'b) -> 'k;;
+       type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;;
 
 The binary tree monad
 ---------------------
@@ -297,29 +297,29 @@ monad, the binary tree monad:
 
 For once, let's check the Monad laws.  The left identity law is easy:
 
-    Left identity: bind (unit a) f = bind (Leaf a) f = fa
+    Left identity: bind (unit a) f = bind (Leaf a) f = f a
 
 To check the other two laws, we need to make the following
 observation: it is easy to prove based on `tree_bind` by a simple
 induction on the structure of the first argument that the tree
 resulting from `bind u f` is a tree with the same strucure as `u`,
-except that each leaf `a` has been replaced with `fa`:
+except that each leaf `a` has been replaced with `f a`:
 
 \tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
 
                        .                         .
                      __|__                     __|__
                      |   |                     |   |
-                     a1  .                    fa1  .
+                     a1  .                   f a1  .
                         _|__                     __|__
                         |  |                     |   |
-                        .  a5                    .  fa5
+                        .  a5                    .  f a5
           bind         _|__       f   =        __|__
                        |  |                    |   |
-                       .  a4                   .  fa4
+                       .  a4                   .  f a4
                      __|__                   __|___
                      |   |                   |    |
-                     a2  a3                 fa2  fa3
+                     a2  a3                f a2  f a3
 
 Given this equivalence, the right identity law
 
@@ -344,18 +344,18 @@ proceed.  Let `f a = Node (Leaf a, Leaf a)`.  Then
                  .               .            |       |
        bind    __|__   f  =    __|_    =      .       .
                |   |           |   |        __|__   __|__
-               a1  a2         fa1 fa2       |   |   |   |
+               a1  a2        f a1 f a2      |   |   |   |
                                             a1  a1  a1  a1
 
 Now when we bind this tree to `g`, we get
 
-                  .
-              ____|____
-              |       |
-              .       .
-            __|__   __|__
-            |   |   |   |
-           ga1 ga1 ga1 ga1
+                   .
+              _____|______
+              |          |
+              .          .
+            __|__      __|__
+            |   |      |   |
+          g a1 g a1  g a1 g a1
 
 At this point, it should be easy to convince yourself that
 using the recipe on the right hand side of the associative law will