# Theory Separation_Logic_Imperative_HOL.To_List_GA

```section ‹Generic Algorithm to Convert Sets to Lists›
theory To_List_GA
imports Imp_Set_Spec Imp_List_Spec Hash_Set_Impl Open_List
begin
text ‹This theory demonstrates how to develop a generic to-list
algorithm, and gives a sample instantiation for hash sets and open lists.
›

subsection ‹Algorithm›
partial_function (heap) to_list_ga_rec where [code]:
"to_list_ga_rec
it_has_next it_next
l_prepend
it l
=
do {
b ← it_has_next it;
if b then do {
(x,it) ← it_next it;
l ← l_prepend x l;
to_list_ga_rec it_has_next it_next
l_prepend it l
} else
return l
}
"

lemma to_list_ga_rec_rule:
assumes "imp_set_iterate is_set is_it it_init it_has_next it_next"
assumes "imp_list_prepend is_list l_prepend"
assumes FIN: "finite it"
shows "
< is_it s si it iti * is_list l li >
to_list_ga_rec it_has_next it_next l_prepend iti li
< λr. ∃⇩Al'. is_set s si
* is_list l' r
* ↑(set l' = set l ∪ it) >⇩t"
proof -
interpret imp_set_iterate is_set is_it it_init it_has_next it_next
+ imp_list_prepend is_list l_prepend
by fact+

from FIN show ?thesis
proof (induction arbitrary: l li iti rule: finite_psubset_induct)
case (psubset it)
show ?case
apply (subst to_list_ga_rec.simps)
apply (sep_auto heap: psubset.IH)
apply (rule ent_frame_fwd[OF quit_iteration])
apply frame_inference
apply solve_entails
done
qed
qed

definition "to_list_ga
it_init it_has_next it_next
l_empty l_prepend s
≡ do {
it ← it_init s;
l ← l_empty;
l ← to_list_ga_rec it_has_next it_next l_prepend it l;
return l
}"

lemma to_list_ga_rule:
assumes IT: "imp_set_iterate is_set is_it it_init it_has_next it_next"
assumes EM: "imp_list_empty is_list l_empty"
assumes PREP: "imp_list_prepend is_list l_prepend"
assumes FIN: "finite s"
shows "
<is_set s si>
to_list_ga it_init it_has_next it_next
l_empty l_prepend si
<λr. ∃⇩Al. is_set s si * is_list l r * true * ↑(set l = s)>"
proof -
interpret imp_list_empty is_list l_empty +
imp_set_iterate is_set is_it it_init it_has_next it_next
by fact+

note [sep_heap_rules] = to_list_ga_rec_rule[OF IT PREP]
show ?thesis
unfolding to_list_ga_def
by (sep_auto simp: FIN)
qed

subsection ‹Sample Instantiation for hash set and open list›
definition "hs_to_ol
≡ to_list_ga hs_it_init hs_it_has_next hs_it_next
os_empty os_prepend"

lemmas hs_to_ol_rule[sep_heap_rules] =
to_list_ga_rule[OF hs_iterate_impl os_empty_impl os_prepend_impl,
folded hs_to_ol_def]

export_code hs_to_ol checking SML_imp

end
```