X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek7_introducing_monads.mdwn;h=1dbd5101872f6245b30b145ae4718680900d75a6;hp=7acc77cfad44a56d1f9b2cadb4414f1631ae5fc4;hb=435a68843444ebdaa28d6f8ac83bf78f750eeae2;hpb=8be622c594f003684e8348973c5e85911161457b

diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn
index 7acc77cf..1dbd5101 100644
--- a/topics/week7_introducing_monads.mdwn
+++ b/topics/week7_introducing_monads.mdwn
@@ -152,8 +152,8 @@ The name "bind" is not well chosen from our perspective, but this is too deeply
 
 To take a trivial (but, as we will see, still useful) example,
 consider the Identity box type: `α`. So if `α` is type `bool`,
-then a boxed `α` is ... a `bool`. In terms of the box analogy, the
-Identity box type is a completely invisible box. With the following
+then a boxed `α` is ... a `bool`. That is, <code><u>α</u> = α</code>.
+In terms of the box analogy, the Identity box type is a completely invisible box. With the following
 definitions:
 
     mid ≡ \p. p
@@ -161,24 +161,24 @@ definitions:
 
 Identity is a monad.  Here is a demonstration that the laws hold:
 
-    mcomp mid k == (\fgx.f(gx)) (\p.p) k
-                ~~> \x.(\p.p)(kx)
-                ~~> \x.kx
-                ~~> k
-    mcomp k mid == (\fgx.f(gx)) k (\p.p)
-                ~~> \x.k((\p.p)x)
-                ~~> \x.kx
-                ~~> k
-    mcomp (mcomp j k) l == mcomp ((\fgx.f(gx)) j k) l
-                       ~~> mcomp (\x.j(kx)) l
-                        == (\fgx.f(gx)) (\x.j(kx)) l
-                       ~~> \x.(\x.j(kx))(lx)
-                       ~~> \x.j(k(lx))
-    mcomp j (mcomp k l) == mcomp j ((\fgx.f(gx)) k l)
-                       ~~> mcomp j (\x.k(lx))
-                        == (\fgx.f(gx)) j (\x.k(lx))
-                       ~~> \x.j((\x.k(lx)) x)
-                       ~~> \x.j(k(lx))
+    mcomp mid k ≡ (\fgx.f(gx)) (\p.p) k
+              ~~> \x.(\p.p)(kx)
+              ~~> \x.kx
+              ~~> k
+    mcomp k mid ≡ (\fgx.f(gx)) k (\p.p)
+              ~~> \x.k((\p.p)x)
+              ~~> \x.kx
+              ~~> k
+    mcomp (mcomp j k) l ≡ mcomp ((\fgx.f(gx)) j k) l
+                      ~~> mcomp (\x.j(kx)) l
+                        ≡ (\fgx.f(gx)) (\x.j(kx)) l
+                      ~~> \x.(\x.j(kx))(lx)
+                      ~~> \x.j(k(lx))
+    mcomp j (mcomp k l) ≡ mcomp j ((\fgx.f(gx)) k l)
+                      ~~> mcomp j (\x.k(lx))
+                        ≡ (\fgx.f(gx)) j (\x.k(lx))
+                      ~~> \x.j((\x.k(lx)) x)
+                      ~~> \x.j(k(lx))
 
 The Identity Monad is favored by mimes.
 
@@ -393,5 +393,5 @@ theory of accommodation, and a theory of the situations in which
 material within the sentence can satisfy presuppositions for other
 material that otherwise would trigger a presupposition violation; but,
 not surprisingly, these refinements will require some more
-sophisticated techniques than the super-simple Option monad.)
+sophisticated techniques than the super-simple Option/Maybe monad.)