--- a/CookBook/Solutions.thy Wed Mar 18 23:52:51 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,218 +0,0 @@
-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
\ No newline at end of file