--- a/CookBook/Solutions.thy	Thu Mar 12 18:39:10 2009 +0000
+++ b/CookBook/Solutions.thy	Fri Mar 13 01:15:55 2009 +0100
@@ -144,21 +144,21 @@
 text {* \solution{ex:addconversion} *}
 
 text {* 
-  To measure the difference, we will create mechanically some terms involving 
-  additions and then set up a goal to be simplified. To prove the remaining 
-  goal we use the ``lemma'':
+  We use the timing function @{ML timing_wrapper} from Recipe~\ref{rec:timing}.
+  To measure any difference between the simproc and conversion, we will create 
+  mechanically terms involving additions and then set up a goal to be 
+  simplified. We have to be careful to set up the goal so that
+  other parts of the simplifier do not interfere. For this we set up an
+  unprovable goal which, after simplification, we are going to ``prove'' with
+  the help of the lemma:
 *}
 
 lemma cheat: "A" sorry
 
 text {*
-  The reason is that it allows us to set up an unprovable goal where we can
-  eliminate all interferences from other parts of the simplifier and
-  then prove the goal using @{thm [source] cheat}. We also assume
-  the timing function @{ML timing_wrapper} from Recipe~\ref{rec:timing}.
-
-  First we define a function that returns a complete binary tree whose 
-  leaves are numbers and the nodes are additions.
+  For constructing test cases, we first define a function that returns a 
+  complete binary tree whose leaves are numbers and the nodes are 
+  additions.
 *}
 
 ML{*fun term_tree n =
@@ -175,14 +175,14 @@
 end*}
 
 text {*
-  For example
+  This function generates for example
 
   @{ML_response_fake [display,gray] 
   "warning (Syntax.string_of_term @{context} (term_tree 2))" 
   "(1 + 2) + (3 + 4)"} 
 
-  The next function generates a goal of the form @{text "P \<dots>"} with a term 
-  filled in.
+  The next function constructs a goal of the form @{text "P \<dots>"} with a term 
+  produced by @{ML term_tree} filled in.
 *}
 
 ML{*fun goal n = HOLogic.mk_Trueprop (@{term "P::nat\<Rightarrow> bool"} $ (term_tree n))*}
@@ -190,7 +190,7 @@
 text {*
   Note that the goal needs to be wrapped in a @{term "Trueprop"}. Next we define
   two tactics, @{text "c_tac"} and @{text "s_tac"}, for the conversion and simproc,
-  respectively. The tactics first apply the conversion (respectively simproc) and 
+  respectively. The idea is to first apply the conversion (respectively simproc) and 
   then prove the remaining goal using the lemma @{thm [source] cheat}.
 *}
 
@@ -198,26 +198,25 @@
   fun mk_tac tac = timing_wrapper (EVERY1 [tac, rtac @{thm cheat}])
 in
 val c_tac = mk_tac add_tac
-val s_tac = mk_tac 
-             (simp_tac (HOL_basic_ss addsimprocs [@{simproc add_sp}]))
+val s_tac = mk_tac (simp_tac (HOL_basic_ss addsimprocs [@{simproc add_sp}]))
 end*}
 
 text {*
-  This is all we need to let them run against each other.
+  This is all we need to let the conversion run against the simproc.
 *}
 
-ML{*val _ = Goal.prove @{context} [] [] (goal 8) (K c_tac);
+ML{*val _ = Goal.prove @{context} [] [] (goal 8) (K c_tac)
 val _ = Goal.prove @{context} [] [] (goal 8) (K s_tac)*}
 
 text {*
-  As you can see, both versions perform relatively the same with perhaps some
-  advantages for the simproc. That means the simplifier, while much more
-  complicated than conversions, is quite good for tasks it is designed for. It
-  usually does not make sense to implement general-purpose rewriting using
+  If you do the exercise, you can see that both ways of simplifying additions
+  perform relatively the same with perhaps some advantages for the
+  simproc. That means the simplifier, even if much more complicated than
+  conversions, is quite efficient for tasks it is designed for. It usually does not
+  make sense to implement general-purpose rewriting using
   conversions. Conversions only have clear advantages in special situations:
   for example if you need to have control over innermost or outermost
-  rewriting; another situation is when rewriting rules are prone to
-  non-termination.
+  rewriting, or when rewriting rules are lead to non-termination.
 *}
 
 end
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