Add note about β-reduction for type variables.

diff --git a/Chapters/Normalization.tex b/Chapters/Normalization.tex
index 55e00577e393dc0c3933ecb708ad4c394d95ca5c..cdd57bb9c52da6a0d885ae2cadf02d114f76f548 100644 (file)
--- a/Chapters/Normalization.tex
+++ b/Chapters/Normalization.tex
@@ -812,6 +812,11 @@
         sure that most lambda abstractions will eventually be reducable by
         β-reduction.
 
+        Note that β-reduction also works on type lambda abstractions and type
+        applications as well. This means the substitution below also works on
+        type variables, in the case that the binder is a type variable and teh
+        expression applied to is a type.
+
         \starttrans
         (λx.E) M
         -----------------
@@ -829,6 +834,17 @@
 
         \transexample{beta}{β-reduction}{from}{to}
 
+        \startbuffer[from]
+        (λt.λa::t. a) @Int
+        \stopbuffer
+
+        \startbuffer[to]
+        (λa::Int. a)
+        \stopbuffer
+
+        \transexample{beta-type}{β-reduction for type abstractions}{from}{to}
+
+
       \subsubsection{Empty let removal}
         This transformation is simple: It removes recursive lets that have no bindings
         (which usually occurs when unused let binding removal removes the last