(* Author: Tobias Nipkow *)
theory Radix_Sort
imports
"HOL-Library.List_Lexorder"
"HOL-Library.Sublist"
"HOL-Library.Multiset"
begin
text \<open>The \<open>Radix_Sort\<close> locale provides a sorting function \<open>radix_sort\<close> that sorts
lists of lists. It is parameterized by a sorting function \<open>sort1 f\<close> that also sorts
lists of lists, but only w.r.t. the column selected by \<open>f\<close>.
Working with lists, \<open>f\<close> is instantiated with \<^term>\<open>\<lambda>xs. xs ! n\<close> to select the \<open>n\<close>-th element.
A more efficient implementation would sort lists of arrays because arrays support
constant time access to every element.\<close>
locale Radix_Sort =
fixes sort1 :: "('a list \ 'a::linorder) \ 'a list list \ 'a list list"
assumes sorted: "sorted (map f (sort1 f xss))"
assumes mset: "mset (sort1 f xss) = mset xss"
assumes stable: "stable_sort_key sort1"
begin
lemma set_sort1[simp]: "set(sort1 f xss) = set xss"
by (metis mset set_mset_mset)
abbreviation "sort_col i xss \ sort1 (\xs. xs ! i) xss"
abbreviation "sorted_col i xss \ sorted (map (\xs. xs ! i) xss)"
fun radix_sort :: "nat \ 'a list list \ 'a list list" where
"radix_sort 0 xss = xss" |
"radix_sort (Suc i) xss = radix_sort i (sort_col i xss)"
lemma mset_radix_sort: "mset (radix_sort i xss) = mset xss"
by(induction i arbitrary: xss) (auto simp: mset)
abbreviation "sorted_from i xss \ sorted (map (drop i) xss)"
definition "cols xss n = (\xs \ set xss. length xs = n)"
lemma cols_sort1: "cols xss n \ cols (sort1 f xss) n"
by(simp add: cols_def)
lemma sorted_from_Suc2:
"\ cols xss n; i < n;
sorted_col i xss;
\<And>x. sorted_from (i+1) [ys \<leftarrow> xss. ys!i = x] \<rbrakk>
\<Longrightarrow> sorted_from i xss"
proof(induction xss rule: induct_list012)
case 1 show ?case by simp
next
case 2 show ?case by simp
next
case (3 xs1 xs2 xss)
have lxs1: "length xs1 = n" and lxs2: "length xs2 = n"
using "3.prems"(1) by(auto simp: cols_def)
have *: "drop i xs1 \ drop i xs2"
proof -
have "drop i xs1 = xs1!i # drop (i+1) xs1"
using \<open>i < n\<close> by (simp add: Cons_nth_drop_Suc lxs1)
also have "\ \ xs2!i # drop (i+1) xs2"
using "3.prems"(3) "3.prems"(4)[of "xs2!i"] by(auto)
also have "\ = drop i xs2"
using \<open>i < n\<close> by (simp add: Cons_nth_drop_Suc lxs2)
finally show ?thesis .
qed
have "sorted_from i (xs2 # xss)"
proof(rule "3.IH"[OF _ "3.prems"(2)])
show "cols (xs2 # xss) n" using "3.prems"(1) by(simp add: cols_def)
show "sorted_col i (xs2 # xss)" using "3.prems"(3) by simp
show "\x. sorted_from (i+1) [ys\xs2 # xss . ys ! i = x]"
using "3.prems"(4)
sorted_antimono_suffix[OF map_mono_suffix[OF filter_mono_suffix[OF suffix_ConsI[OF suffix_order.order.refl]]]]
by fastforce
qed
with * show ?case by (auto)
qed
lemma sorted_from_radix_sort_step:
assumes "cols xss n" and "i < n" and "sorted_from (i+1) xss"
shows "sorted_from i (sort_col i xss)"
proof (rule sorted_from_Suc2[OF cols_sort1[OF assms(1)] assms(2)])
show "sorted_col i (sort_col i xss)" by(simp add: sorted)
fix x show "sorted_from (i+1) [ys \ sort_col i xss . ys ! i = x]"
proof -
from assms(3)
have "sorted_from (i+1) (filter (\ys. ys!i = x) xss)"
by(rule sorted_filter)
thus "sorted (map (drop (i+1)) (filter (\ys. ys!i = x) (sort_col i xss)))"
by (metis stable stable_sort_key_def)
qed
qed
lemma sorted_from_radix_sort:
"\ cols xss n; i \ n; sorted_from i xss \ \ sorted_from 0 (radix_sort i xss)"
proof(induction i arbitrary: xss)
case 0 thus ?case by simp
next
case (Suc i)
thus ?case by(simp add: sorted_from_radix_sort_step cols_sort1)
qed
corollary sorted_radix_sort: "cols xss n \ sorted (radix_sort n xss)"
apply(frule sorted_from_radix_sort[OF _ le_refl])
apply(auto simp add: cols_def sorted_iff_nth_mono)
done
end
end
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