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 {* 

  An alternative solution using the function @{ML [index] mk_binop in HOLogic} is:

 *}

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