theory Solutions+ −
imports Base "Recipes/Timing"+ −
begin+ −
+ −
chapter {* Solutions to Most Exercises\label{ch:solutions} *}+ −
+ −
text {* \solution{fun:revsum} *}+ −
+ −
ML{*fun rev_sum((p as Const(@{const_name plus},_)) $ t $ u) = p $ u $ rev_sum t+ −
| rev_sum t = t *}+ −
+ −
text {* \solution{fun:revsum typed} *}+ −
+ −
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_Data.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 ``\isacommand{sorry}'', that is the method @{ML SkipProof.cheat_tac}+ −
+ −
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] + −
"writeln (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 @{ML "cheat_tac" in SkipProof}.+ −
*}+ −
+ −
ML{*local+ −
fun mk_tac tac = + −
timing_wrapper (EVERY1 [tac, K (SkipProof.cheat_tac @{theory})])+ −
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+ −