(* Author: Florian Haftmann, TU Muenchen *)

 header {* A simple cookbook example how to eliminate choice in programs. *}

 theory Execute_Choice

 imports Main AssocList

 begin

 text {*

   A trivial example:

 *}

 definition valuesum :: "('a, 'b :: ab_group_add) mapping \<Rightarrow> 'b" where

   "valuesum m = (\<Sum>k \<in> Mapping.keys m. the (Mapping.lookup m k))"

 text {*

   Not that instead of defining @{term valuesum} with choice, we define it

   directly and derive a description involving choice afterwards:

 *}

 lemma valuesum_rec:

   assumes fin: "finite (dom (Mapping.lookup m))"

   shows "valuesum m = (if Mapping.is_empty m then 0 else

     let l = (SOME l. l \<in> Mapping.keys m) in the (Mapping.lookup m l) + valuesum (Mapping.delete l m))"

 proof (cases "Mapping.is_empty m")

   case True then show ?thesis by (simp add: is_empty_def keys_def valuesum_def)

 next

   case False

   then have l: "\<exists>l. l \<in> dom (Mapping.lookup m)" by (auto simp add: is_empty_def expand_fun_eq mem_def)

   then have "(let l = SOME l. l \<in> dom (Mapping.lookup m) in

      the (Mapping.lookup m l) + (\<Sum>k \<in> dom (Mapping.lookup m) - {l}. the (Mapping.lookup m k))) =

        (\<Sum>k \<in> dom (Mapping.lookup m). the (Mapping.lookup m k))"

   proof (rule someI2_ex)

     fix l

     note fin

     moreover assume "l \<in> dom (Mapping.lookup m)"

     moreover obtain A where "A = dom (Mapping.lookup m) - {l}" by simp

     ultimately have "dom (Mapping.lookup m) = insert l A" and "finite A" and "l \<notin> A" by auto

     then show "(let l = l

         in the (Mapping.lookup m l) + (\<Sum>k \<in> dom (Mapping.lookup m) - {l}. the (Mapping.lookup m k))) =

         (\<Sum>k \<in> dom (Mapping.lookup m). the (Mapping.lookup m k))"

       by simp

    qed

   then show ?thesis by (simp add: keys_def valuesum_def is_empty_def)

 qed

 text {*

   In the context of the else-branch we can show that the exact choice is

   irrelvant; in practice, finding this point where choice becomes irrelevant is the

   most difficult thing!

 *}

 lemma valuesum_choice:

   "finite (Mapping.keys M) \<Longrightarrow> x \<in> Mapping.keys M \<Longrightarrow> y \<in> Mapping.keys M \<Longrightarrow>

     the (Mapping.lookup M x) + valuesum (Mapping.delete x M) =

     the (Mapping.lookup M y) + valuesum (Mapping.delete y M)"

   by (simp add: valuesum_def keys_def setsum_diff)

 text {*

   Given @{text valuesum_rec} as initial description, we stepwise refine it to something executable;

   first, we formally insert the constructor @{term AList} and split the one equation into two,

   where the second one provides the necessary context:

 *}

 lemma valuesum_rec_AList:

   shows [code]: "valuesum (AList []) = 0"

   and "valuesum (AList (x # xs)) = (let l = (SOME l. l \<in> Mapping.keys (AList (x # xs))) in

     the (Mapping.lookup (AList (x # xs)) l) + valuesum (Mapping.delete l (AList (x # xs))))"

   by (simp_all add: valuesum_rec finite_dom_map_of is_empty_AList)

 text {*

   As a side effect the precondition disappears (but note this has nothing to do with choice!).

   The first equation deals with the uncritical empty case and can already be used for code generation.

   Using @{text valuesum_choice}, we are able to prove an executable version of @{term valuesum}:

 *}

 lemma valuesum_rec_exec [code]:

   "valuesum (AList (x # xs)) = (let l = fst (hd (x # xs)) in

     the (Mapping.lookup (AList (x # xs)) l) + valuesum (Mapping.delete l (AList (x # xs))))"

 proof -

   let ?M = "AList (x # xs)"

   let ?l1 = "(SOME l. l \<in> Mapping.keys ?M)"

   let ?l2 = "fst (hd (x # xs))"

   have "finite (Mapping.keys ?M)" by (simp add: keys_AList)

   moreover have "?l1 \<in> Mapping.keys ?M"

     by (rule someI) (auto simp add: keys_AList)

   moreover have "?l2 \<in> Mapping.keys ?M"

     by (simp add: keys_AList)

   ultimately have "the (Mapping.lookup ?M ?l1) + valuesum (Mapping.delete ?l1 ?M) =

     the (Mapping.lookup ?M ?l2) + valuesum (Mapping.delete ?l2 ?M)"

     by (rule valuesum_choice)

   then show ?thesis by (simp add: valuesum_rec_AList)

 qed

 text {*

   See how it works:

 *}

 value "valuesum (AList [(''abc'', (42::int)), (''def'', 1705)])"

 end