--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ProgTutorial/Solutions.thy	Thu Mar 19 13:28:16 2009 +0100
@@ -0,0 +1,218 @@
+theory Solutions
+imports Base "Recipes/Timing"
+begin
+
+chapter {* Solutions to Most Exercises\label{ch:solutions} *}
+
+text {* \solution{fun:revsum} *}
+
+ML{*fun rev_sum t =
+let
+  fun dest_sum (Const (@{const_name plus}, _) $ u $ u') = u' :: dest_sum u
+    | dest_sum u = [u]
+in
+   foldl1 (HOLogic.mk_binop @{const_name plus}) (dest_sum t)
+end *}
+
+text {* \solution{fun:makesum} *}
+
+ML{*fun make_sum t1 t2 =
+      HOLogic.mk_nat (HOLogic.dest_nat t1 + HOLogic.dest_nat t2) *}
+
+text {* \solution{ex:scancmts} *}
+
+ML{*val any = Scan.one (Symbol.not_eof)
+
+val scan_cmt =
+let
+  val begin_cmt = Scan.this_string "(*" 
+  val end_cmt = Scan.this_string "*)"
+in
+  begin_cmt |-- Scan.repeat (Scan.unless end_cmt any) --| end_cmt 
+  >> (enclose "(**" "**)" o implode)
+end
+
+val parser = Scan.repeat (scan_cmt || any)
+
+val scan_all =
+      Scan.finite Symbol.stopper parser >> implode #> fst *}
+
+text {*
+  By using @{text "#> fst"} in the last line, the function 
+  @{ML scan_all} retruns a string, instead of the pair a parser would
+  normally return. For example:
+
+  @{ML_response [display,gray]
+"let
+  val input1 = (explode \"foo bar\")
+  val input2 = (explode \"foo (*test*) bar (*test*)\")
+in
+  (scan_all input1, scan_all input2)
+end"
+"(\"foo bar\", \"foo (**test**) bar (**test**)\")"}
+*}
+
+text {* \solution{ex:addsimproc} *}
+
+ML{*fun dest_sum term =
+  case term of 
+    (@{term "(op +):: nat \<Rightarrow> nat \<Rightarrow> nat"} $ t1 $ t2) =>
+        (snd (HOLogic.dest_number t1), snd (HOLogic.dest_number t2))
+  | _ => raise TERM ("dest_sum", [term])
+
+fun get_sum_thm ctxt t (n1, n2) =  
+let 
+  val sum = HOLogic.mk_number @{typ "nat"} (n1 + n2)
+  val goal = Logic.mk_equals (t, sum)
+in
+  Goal.prove ctxt [] [] goal (K (arith_tac ctxt 1))
+end
+
+fun add_sp_aux ss t =
+let 
+  val ctxt = Simplifier.the_context ss
+  val t' = term_of t
+in
+  SOME (get_sum_thm ctxt t' (dest_sum t'))
+  handle TERM _ => NONE
+end*}
+
+text {* The setup for the simproc is *}
+
+simproc_setup %gray add_sp ("t1 + t2") = {* K add_sp_aux *}
+ 
+text {* and a test case is the lemma *}
+
+lemma "P (Suc (99 + 1)) ((0 + 0)::nat) (Suc (3 + 3 + 3)) (4 + 1)"
+  apply(tactic {* simp_tac (HOL_basic_ss addsimprocs [@{simproc add_sp}]) 1 *})
+txt {* 
+  where the simproc produces the goal state
+  
+  \begin{minipage}{\textwidth}
+  @{subgoals [display]}
+  \end{minipage}\bigskip
+*}(*<*)oops(*>*)
+
+text {* \solution{ex:addconversion} *}
+
+text {*
+  The following code assumes the function @{ML dest_sum} from the previous
+  exercise.
+*}
+
+ML{*fun add_simple_conv ctxt ctrm =
+let
+  val trm =  Thm.term_of ctrm
+in 
+  get_sum_thm ctxt trm (dest_sum trm)
+end
+
+fun add_conv ctxt ctrm =
+  (case Thm.term_of ctrm of
+     @{term "(op +)::nat \<Rightarrow> nat \<Rightarrow> nat"} $ _ $ _ => 
+         (Conv.binop_conv (add_conv ctxt)
+          then_conv (Conv.try_conv (add_simple_conv ctxt))) ctrm         
+    | _ $ _ => Conv.combination_conv 
+                 (add_conv ctxt) (add_conv ctxt) ctrm
+    | Abs _ => Conv.abs_conv (fn (_, ctxt) => add_conv ctxt) ctxt ctrm
+    | _ => Conv.all_conv ctrm)
+
+fun add_tac ctxt = CSUBGOAL (fn (goal, i) =>
+  CONVERSION
+    (Conv.params_conv ~1 (fn ctxt =>
+       (Conv.prems_conv ~1 (add_conv ctxt) then_conv
+          Conv.concl_conv ~1 (add_conv ctxt))) ctxt) i)*}
+
+text {*
+  A test case for this conversion is as follows
+*}
+
+lemma "P (Suc (99 + 1)) ((0 + 0)::nat) (Suc (3 + 3 + 3)) (4 + 1)"
+  apply(tactic {* add_tac @{context} 1 *})?
+txt {* 
+  where it produces the goal state
+  
+  \begin{minipage}{\textwidth}
+  @{subgoals [display]}
+  \end{minipage}\bigskip
+*}(*<*)oops(*>*)
+
+text {* \solution{ex:addconversion} *}
+
+text {* 
+  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 construct an
+  unprovable goal which, after simplification, we are going to ``prove'' with
+  the help of the lemma:
+*}
+
+lemma cheat: "A" sorry
+
+text {*
+  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 =
+let
+  val count = ref 0; 
+
+  fun term_tree_aux n =
+    case n of
+      0 => (count := !count + 1; HOLogic.mk_number @{typ nat} (!count))
+    | _ => Const (@{const_name "plus"}, @{typ "nat\<Rightarrow>nat\<Rightarrow>nat"}) 
+             $ (term_tree_aux (n - 1)) $ (term_tree_aux (n - 1))
+in
+  term_tree_aux n
+end*}
+
+text {*
+  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 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))*}
+
+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 idea is to first apply the conversion (respectively simproc) and 
+  then prove the remaining goal using the @{thm [source] cheat}-lemma.
+*}
+
+ML{*local
+  fun mk_tac tac = timing_wrapper (EVERY1 [tac, rtac @{thm cheat}])
+in
+val c_tac = mk_tac (add_tac @{context}) 
+val s_tac = mk_tac (simp_tac (HOL_basic_ss addsimprocs [@{simproc add_sp}]))
+end*}
+
+text {*
+  This is all we need to let the conversion run against the simproc:
+*}
+
+ML{*val _ = Goal.prove @{context} [] [] (goal 8) (K c_tac)
+val _ = Goal.prove @{context} [] [] (goal 8) (K s_tac)*}
+
+text {*
+  If you do the exercise, you can see that both ways of simplifying additions
+  perform relatively similar 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, or when rewriting rules are lead to non-termination.
+*}
+
+end
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